Modelling Longevity Dynamics for Pensions and Annuity Business

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Author: Ermanno Pitacco | Michel Denuit | Steven Haberman | Annamaria Olivieri

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Modelling Longevity Dynamics for Pensions and Annuity Business

MATHEMATICS TEXTS FROM OXFORD UNIVERSITY PRESS David Acheson: From Calculus to Chaos: An introduction to dynamics Norman L. Biggs: Discrete Mathematics, second edition Bisseling: Parallel Scientific Computation Cameron: Introduction to Algebra A.W. Chatters and C.R. Hajarnavis: An Introductory Course in Commutative Algebra René Cori and Daniel Lascar: Mathematical Logic: A Course with Exercises, Part 1 René Cori and Daniel Lascar: Mathematical Logic: A Course with Exercises, Part 2 Davidson: Turbulence D’Inverno: Introducing Einstein’s Relativity Garthwaite, Jollife, and Jones: Statistical Inference Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction G.R. Grimmett and D.R. Stirzaker: Probability and Random Processes, third edition G.R. Grimmett and D.R. Stirzaker: One Thousand Exercises in Probability, second edition G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers John Heilbron: Geometry Civilized Hilborn: Chaos and Nonlinear Dynamics Raymond Hill: A First Course in Coding Theory D.W. Jordan and P.Smith: Non Linear Ordinary Differential Equations Richard Kaye and Robert Wilson: Linear Algebra J.K. Lindsey: Introduction to Applied Statistics: A modelling approach, second edition Mary Lunn: A First Course in Mechanics Jiˇrí Matouˇsek and Jaroslav Neˇsetˇril: Invitation to Discrete Mathematics Tristan Needham: Visual Complex Analysis John Ockendon, Sam Howison, : Applied Partial Differential Equations H.A. Priestley: Introduction to Complex Analysis, second edition H.A. Priestley: Introduction to Integration Roe: Elementary Geometry Ian Stewart and David Hall: The Foundations of Mathematics W.A. Sutherland: Introduction to Metric and Topological Spaces Dominic Welsh: Codes and Cryptography Robert A. Wilson: Graphs, Colourings and the Four Colour Theorem Adrian F. Tuck: Atmospheric Turbulence André Nies: Computability and Randomness Pitacco, Denuit, Haberman, and Olivieri: Modelling Longevity Dynamics for Pensions and Annuity Business

Modelling Longevity Dynamics for Pensions and Annuity Business Ermanno Pitacco University of Trieste (Italy)

Michel Denuit UCL, Louvain-la-Neuve (Belgium)

Steven Haberman City University, London (UK)

Annamaria Olivieri University of Parma (Italy)

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Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Ermanno Pitacco, Michel Denuit, Steven Haberman, and Annamaria Olivieri 2009 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2009 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by CPI Antony Rowe, Chippenham, Wiltshire ISBN 978–0–19–954727–2 10 9 8 7 6 5 4 3 2 1

Preface

Actuarial science effectively began with the bringing together of compound interest and life tables, some of which had been derived from observed mortality rates. One of the first categories of financial problems that early actuaries tackled was the calculation of annuity values. Thus, the subject matter of this book can be traced back to the beginnings of the discipline of actuarial science in the mid-17th century. At this time, states and cities often raised money for public purposes by the sale of life annuities to their citizens. One of the first to write on this subject was Jan de Witt, who was the Prime Minister of the States of Holland, and who demonstrated in 1671 how to calculate the value of annuities using a constant rate of interest and a hypothetical life table (that was piecewise linear). Another early investigation of the calculation method for annuity values is the seminal work of Edmund Halley in 1693 which uses population mortality rates. From an overview of this early history, we can identify two key developments that feature in this book. First, there was the recognition of the importance of using life tables that were not hypothetical and were not based on general population data but rather were based on observed mortality data from registers of annuitants. This came in the mid-18th century – through the work of Nicholas Struyck in 1740, William Kersseboom in 1742, and Antoine Deparcieux in 1746 (Haberman and Sibbett, 1995). In modern terminology, we would present this in the context of adverse selection among the holders of life annuities – the tendency for purchasers of life annuities to live longer than the general population. It was the work of John Finlaison (the UK’s first Government Actuary) in 1829, which cogently demonstrated the financial problems that could emerge from overlooking this fundamental phenomenon. During the first two decades of the 19th century, the British Government had been selling annuities in an attempt to reduce the National Debt. The annuities were priced using mortality rates from a contemporary population-based life table, which failed to allow for the adverse selection effect and hence the mortality rates were too high for a portfolio of annuitants. Finlaison uncovered this problem and showed that the annuities were being sold at a loss. He identified the problem in 1819 and then produced scientific evidence based on a painstaking analysis

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of a range of data sets that led to recommendations that were accepted by the British Government in 1829. Thus, subsequently, Government annuities were sold on a sound basis in line with Finlaison’s recommendations. It is noteworthy that issues connected with mortality, annuities and adverse selection are common features of western 19th century literature. As pointed out by Skwire (1997), the novels of Jane Austen are a particularly rich source of actuarial references. In the words of Fanny Dashwood in Sense and Sensibility, ‘people always live for ever when there is any annuity to be paid them; . . . An annuity is very serious business and there is no getting rid of it’. The second development related to the understanding that, in the presence of a downward secular trend in mortality rates, mortality tables for application to annuities should include some allowance for the expected future improvements in mortality rates in order to protect the seller of annuities against future loss. The first tables to make such an allowance were those produced in the United Kingdom based on insurance company data covering the period 1900–1920. Thus, we see that this book is closely related to fundamental practical problems that actuaries have been trying to address for some years. But the more immediate history of this book can be traced to the research that has been carried out by the four authors and to two Summer Schools on which the authors taught and which were organized by the Groupe Consultatif Actuariel Européen (i.e. Consultative Group of Actuarial Organizations in the European Union) in 2005 at the MIB School of Management of Trieste and in 2006 at the University of Parma. The presentations at the summer schools were centered on disseminating the results of this research work in a manner that was accessible to both practitioner and academic audiences. The book is specifically aimed at final year undergraduate students, MSc students, research students preparing for an MPhil or PhD degree and practising actuaries (as part of their continuing professional development). This book deals with some very important topics in the field of actuarial mathematics and life insurance techniques. These concern mortality improvements, the uncertainty in future mortality trends and their relevant impact on life annuities and pension plans. In particular, we consider the actuarial calculations concerning pensions and life annuities. As we have noted above, the insurance company (or the pension plan) must adopt an appropriate forecast of future mortality in order to avoid underestimation of the related liabilities. These concepts and models could be extended to apply to other living benefits, which are provided, for example, by long-term care insurance products and whole life sickness covers.

Preface

Considerable attention is currently being devoted in actuarial work to the management of life annuity portfolios, both from a theoretical and a practical point of view, because of the growing importance of annuity benefits paid by private pension schemes. In particular, the progressive shift in many countries from defined benefit to defined contribution pension schemes has increased the interest in life annuities with a guaranteed annual amount. This book aims at providing a comprehensive and detailed description of methods for projecting mortality, and an extensive introduction to some important issues concerning longevity risk in the area of life annuities and pension benefits. The following topics are dealt with: life annuities in the framework of post-retirement income strategies; the basic mortality model; recent mortality trends; general features of projection models; a discussion of stochastic projection models, with numerical illustrations; and measuring and managing longevity risk. Chapter 1 has an introductory role, and aims to present the basic structure of life annuity products. Moving from the simple model of the annuitycertain, typical features of life annuity products are presented. From an actuarial point of view, the presentation progressively shifts from the traditional deterministic models to the more modern stochastic models. With an appropriate stochastic approach, we are able to capture the riskiness inherent in a life annuity portfolio and in particular the risks that arise from random mortality. Cross-subsidy mechanisms which may operate in life annuity portfolios and pension plans are then described. Our presentation of the actuarial structure of life annuities focuses on a very simple annuity model, namely the immediate life annuity. So, problems arising in the so-called accumulation phase (as well as problems regarding the annuitization of the accumulated amount) are initially disregarded. The chapter then provides a comprehensive description of a number of life annuity models; actuarial aspects are briefly mentioned, in favour of some more practical issues with the objective, in particular, of paving the way for the subsequent formal presentation. Some elements of the basic mortality model underlying life insurance, life annuities, and pensions are introduced in Chapter 1, while presenting the structure of life annuities. In Chapter 2, the mortality model is described in more depth, by adopting a more structured presentation of the fundamental ideas. At the same time we introduce some new concepts. In particular, an age-continuous framework is defined, in order to provide some tools needed when dealing with mortality projection models. Indices summarizing the probability distribution of the lifetime are described, and parametric models (often called mortality ‘laws’ in the literature) are presented. Transforms of the survival function are briefly addressed. We also

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consider two further topics that are of great importance in the context of life annuities and mortality forecasts but which are less traditional as far as actuarial books are concerned. These are mortality at the very old ages (i.e. the problem of ‘closing’ the life table) and the concept of ‘frailty’ as a tool to represent heterogeneity in populations due to unobservable risk factors. Chapter 3 considers mortality trends during the past century. The wellknown background is that average human life span has roughly tripled over the course of human history. Compared to all of the previous centuries, the 20th century has been characterized by a huge increase in average longevity. As we demonstrate in several chapters, there is no evidence which shows that improvements in longevity are tending to slow down. This chapter aims to illustrate the observed decline in mortality over the 20th century, on the basis of Belgian mortality statistics, using several of the mortality indices that have been introduced in Chapters 1 and 2. We also illustrate the trends in mortality indices for insurance data from the Belgian insurance market, which have been provided by the Banking, Finance and Insurance Commission (in Brussels). We note the key point that emerges from actuarial history that, in order to protect an insurance company from mortality improvements, actuaries need to resort to life tables incorporating a forecast of the future trends of mortality rates (the so-called projected tables). The building of these projected life tables is the main topic of the next chapters. Chapter 4 aims at describing the various methods that have been proposed by actuaries and demographers for projecting mortality. Many of these have been used in an actuarial context, in particular for pricing and reserving in relation to life annuity products and pension products and plans, and in the demographic field, mainly for population projections. First, the idea of a ‘dynamic’ approach to mortality modelling is introduced. Then, projection methods are presented and our starting point is the extrapolation procedures which are still widely used in current actuarial practice. More complex methods follow, in particular those methods based on mortality laws, on model tables, and on relations between life tables. The Lee–Carter method, which has been recently proposed, and some relevant extensions are briefly introduced (while a more detailed discussion, together with various examples of its implementation, is presented in Chapters 5 and 6). The presentation is thematic rather than following a strict chronological order. In order to obtain an insight into the historical evolution of mortality forecasts, the reader can refer to the final section of this chapter, in which some landmarks in the history of dynamic mortality modelling are identified. There is a variety of statistical models used for mortality projection, ranging from the basic regression models, in which age and time are viewed

Preface

as continuous covariates, to sophisticated robust non-parametric models. In Chapter 5, we adopt the age-period framework and we first consider the Lee–Carter log-bilinear projection model. The key difference with the classical generalized linear regression model approach centers on the interpretation of time which in the log-bilinear approach is modelled as a factor and under the generalized linear regression approach is modelled as a known covariate. In addition to the Lee–Carter model, we also consider the alternative Cairns–Blake–Dowd mortality forecasting method. Compared with the Lee–Carter approach, the Cairns–Blake–Dowd model includes two time factors. This allows the model to capture the imperfect correlation in mortality rates at different ages from one year to the next. This approach can also be seen as a compromise between the generalized regression approach and the Lee–Carter views of mortality modelling, in that age enters the Cairns–Blake–Dowd model as a continuous covariate whereas the effect of calendar time is captured by two factors (time-varying intercept and slope parameters). These two approaches are applied to Belgian mortality statistics and the results are interpreted. In Chapter 6, our aim is to extend the mortality models described in Chapter 5 in order to incorporate cohort effects as well as age and period effects. The cohort effect is a prominent feature of mortality trends in several developed countries including the United Kingdom, the United States, Germany, and Japan. It relates to the favourable mortality experience that has been observed for those born during the decades between the two world wars. Given that this is a significant feature of past experience, it is necessary first to be able to model and then to forecast its impact on future mortality trends. First, we discuss the evidence for the cohort effect, with particular reference to the United Kingdom. The age–period–cohort version of the Lee–carter model is then introduced, along with a discussion of the error structure, model fitting, and forecasting. A detailed case study is then presented involving historic data from England and Wales. The cohort versions of the Cairns–Blake–Dowd and P-splines models are also presented and their principal features are reviewed. In Chapter 7, we deal with the mortality risks borne by an annuity provider, and in particular with the longevity risk which originates from the uncertain evolution of mortality at adult and old ages. First, we describe possible approaches to a stochastic representation of mortality, as is required when modelling longevity risk. Then, an analysis of the impact of longevity risk on the risk profile of the provider of immediate life annuities is developed. Taking a risk management perspective, possible solutions for risk mitigation are then examined. Risk transfers as well as capital requirements for the risk retained are discussed. As far as the latter are concerned, some

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rules which could be implemented within internal models are tested and a comparison is also developed with the requirement for longevity risk set by Solvency 2, in its current state of development. With regard to risk transfers, particular attention is devoted to capital market solutions, that is, to longevity bonds. The possible design of reinsurance arrangements is examined in connection with the hedging opportunities arising from some of these capital market solutions. The main issues concerning policy design and the pricing of longevity risk are sketched. The possible behaviour of the annuitant with respect to the planning of her/his retirement income, which should be carefully considered in order to choose an appropriate design of life annuity products, is also examined. Our approach to writing this book has been to allocate prime responsibility for each chapter to one or two authors and then for us all to provide comments and input. Thus, Chapters 1 and 4 were written by Ermanno Pitacco; Chapter 2 by Ermanno Pitacco and Annamaria Olivieri jointly; Chapters 3 and 5 by Michel Denuit; Chapter 6 by Steven Haberman; and Chapter 7 by Annamaria Olivieri. We would like to add that a book like this will never be the result of the inputs of just the authors. Thus, we each would like to acknowledge the support that we have received from a range of colleagues. First, we would each like to thank our respective institutions for the stimulating environment that has enabled us to complete this project. Michel Denuit would like to acknowledge the inputs by Natacha Brouhns and Antoine Delwarde, who both worked on the topic of this book as PhD students under his supervision at UCL. Andrew Cairns kindly provided detailed comments on an earlier version of Chapters 3 and 5, which led to significant improvements, in particular with regard to mortality projection models. Discussions and/or collaborations with many esteemed colleagues helped to clarify the analysis of mortality and its consequence for insurance risk management, including Enrico Biffis, Hélène Cossette, Claudia Czado, Pierre Devolder, Jan Dhaene, Paul Eilers, Esther Frostig, Anne-Cécile Goderniaux, Montserrat Guillen, Étienne Marceau, Christian Partrat, Christian Robert, Jeroen Vermunt, and Jean-François Walhin. Luc Kaiser, Actuary at the BFIC kindly supplied mortality data about the Belgian life insurance market. Particular thanks go to all the participants to the ‘Mortality’ task force of the Royal Society of Belgian Actuaries, directed by Philippe Delfosse. Interesting discussions with practising actuaries involved also helped to clarify some issues. In that respect, Michel Denuit would like to thank Pascal Schoenmaekers from Munich Re for stimulating exchanges. Michel Denuit would like to stress his beneficial involvement in the working party appointed by the Belgian federal government in order to

Preface

produce projected life tables for Belgium. Special thanks in this regard go to Micheline Lambrecht and Benoît Paul from FPB. Also, Michel Denuit has benefited from partnerships with (re)insurance companies, especially with Daria Khachakidze and Laure Olié from SCOR, and with Lucie Taleyson from AXA. The financial support of the Communauté française de Belgique under contract ‘Projet d Actions de Recherche Concertées’ ARC 04/09-320 and of Banque Nationale de Belgique under grant ‘Risk measures and Economic capital’ are gratefully acknowledged. Steven Haberman would like to express his deep gratitude to his longterm research collaborator, Arthur Renshaw, for his contributions to their joint work which has underpinned the ideas in Chapters 5 and 6 and for stimulating discussions about mortality trends. He would also like to thank his close colleague, Richard Verrall, for his contributions and advice on modelling mortality, as well as their recent PhD students, Terry Sithole and Marwa Khalaf-Allah, and their research assistant, Zoltan Butt, who have all worked on the subject of mortality trends and their impact on annuities and pensions. Steven Haberman would also like to thank Adrian Gallop from the Government Actuary’s Department for providing mortality data for England and Wales (by individual year of age and calendar year) that facilitated the modelling of trends by cohort. The financial support, provided through annual research grants, received from the Continuous Mortality Investigation Bureau of the UK Actuarial Profession is gratefully acknowledged. Annamaria Olivieri and Ermanno Pitacco would like to thank Enrico Biffis and Pietro Millossovich for stimulating exchanges and collaborations, Patrizia Marocco and Fulvio Tomè from Assicurazioni Generali for interesting discussions on various practical aspects of longevity, Marco Vesentini from Cattolica Assicurazioni, Verona, for providing useful material. The financial support from the Italian Ministero dell’Università e della Ricerca is gratefully acknowledged; thanks to the research project ‘Income protection against longevity and health risks: financial, actuarial and economic analysis of pension and health products. Market trends and perspectives’, coordinated by Ermanno Pitacco, various stimulating meetings have been held. Finally, special thanks go to all the participants of the Summer School of the Groupe Consultatif Actuariel Europeen on the topic ‘Modelling mortality dynamics for pensions and annuity business’ held twice in Italy (Trieste, 2005; Parma, 2006). Their feedback and comments have been very useful and such Continuing Professional Development initiatives offer to the lecturers involved exciting opportunities for the merging of theoretical approaches and practical issues, which we hope have been retained as a theme in this book.

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Contents

Preface

v

1

1

Life annuities 1.1 1.2

1.3

1.4

1.5

1.6

1.7

Introduction Annuities-certain versus life annuities 1.2.1 Withdrawing from a fund 1.2.2 Avoiding early fund exhaustion 1.2.3 Risks in annuities-certain and in life annuities Evaluating life annuities: deterministic approach 1.3.1 The life annuity as a financial transaction 1.3.2 Actuarial values 1.3.3 Technical bases Cross-subsidy in life annuities 1.4.1 Mutuality 1.4.2 Solidarity 1.4.3 ‘Tontine’ annuities Evaluating life annuities: stochastic approach 1.5.1 The random present value of a life annuity 1.5.2 Focussing on portfolio results 1.5.3 A first insight into risk and solvency 1.5.4 Allowing for uncertainty in mortality assumptions Types of life annuities 1.6.1 Immediate annuities versus deferred annuities 1.6.2 The accumulation period 1.6.3 The decumulation period 1.6.4 The payment profile 1.6.5 About annuity rates 1.6.6 Variable annuities and GMxB features References and suggestions for further reading

1 2 2 5 6 8 8 9 12 14 14 16 18 20 20 21 24 27 31 31 33 36 38 40 41 43

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2

The basic mortality model

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2.1 2.2

45 46 46 47 48 49 51 51 53 55 57 57 58 59 60 62 63 64 66 67 67 67 68 69 73 74 74 75 77 77 78

Introduction Life tables 2.2.1 Cohort tables and period tables 2.2.2 ‘Population’ tables versus ‘market’ tables 2.2.3 The life table as a probabilistic model 2.2.4 Select mortality 2.3 Moving to an age-continuous context 2.3.1 The survival function 2.3.2 Other related functions 2.3.3 The force of mortality 2.3.4 The central death rate 2.3.5 Assumptions for non-integer ages 2.4 Summarizing the lifetime probability distribution 2.4.1 The life expectancy 2.4.2 Other markers 2.4.3 Markers under a dynamic perspective 2.5 Mortality laws 2.5.1 Laws for the force of mortality 2.5.2 Laws for the annual probability of death 2.5.3 Mortality by causes 2.6 Non-parametric graduation 2.6.1 Some preliminary ideas 2.6.2 The Whittaker–Henderson model 2.6.3 Splines 2.7 Some transforms of the survival function 2.8 Mortality at very old ages 2.8.1 Some preliminary ideas 2.8.2 Models for mortality at highest ages 2.9 Heterogeneity in mortality models 2.9.1 Observable heterogeneity factors 2.9.2 Models for differential mortality 2.9.3 Unobservable heterogeneity factors. The frailty 2.9.4 Frailty models 2.9.5 Combining mortality laws with frailty models 2.10 References and suggestions for further reading

80 83 85 87

Contents

3

Mortality trends during the 20th century 3.1 3.2

3.3

3.4

3.5 3.6 4

Introduction Data sources 3.2.1 Statistics Belgium 3.2.2 Federal Planning Bureau 3.2.3 Human mortality database 3.2.4 Banking, Finance, and Insurance Commission Mortality trends in the general population 3.3.1 Age-period life tables 3.3.2 Exposure-to-risk 3.3.3 Death rates 3.3.4 Mortality surfaces 3.3.5 Closure of life tables 3.3.6 Rectangularization and expansion 3.3.7 Life expectancies 3.3.8 Variability 3.3.9 Heterogeneity Life insurance market 3.4.1 Observed death rates 3.4.2 Smoothed death rates 3.4.3 Life expectancies 3.4.4 Relational models 3.4.5 Age shifts Mortality trends throughout EU Conclusions

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Forecasting mortality: an introduction

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4.1 4.2

137 139

4.3

Introduction A dynamic approach to mortality modelling 4.2.1 Representing mortality dynamics: single-figures versus age-specific functions 4.2.2 A discrete, age-specific setting Projection by extrapolation of annual probabilities of death 4.3.1 Some preliminary ideas 4.3.2 Reduction factors

139 140 141 141 144

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4.4

4.5

4.6

4.7

4.8

4.9

4.3.3 The exponential formula 4.3.4 An alternative approach to the exponential extrapolation 4.3.5 Generalizing the exponential formula 4.3.6 Implementing the exponential formula 4.3.7 A general exponential formula 4.3.8 Some exponential formulae used in actuarial practice 4.3.9 Other projection formulae Using a projected table 4.4.1 The cohort tables in a projected table 4.4.2 From a double-entry to a single-entry projected table 4.4.3 Age shifting Projecting mortality in a parametric context 4.5.1 Mortality laws and projections 4.5.2 Expressing mortality trends via Weibull’s parameters 4.5.3 Some remarks 4.5.4 Mortality graduation over age and time Other approaches to mortality projections 4.6.1 Interpolation versus extrapolation: the limit table 4.6.2 Model tables 4.6.3 Projecting transforms of life table functions The Lee–Carter method: an introduction 4.7.1 Some preliminary ideas 4.7.2 The LC model 4.7.3 From LC to the Poisson log-bilinear model 4.7.4 The LC method and model tables Further issues 4.8.1 Cohort approach versus period approach. APC models 4.8.2 Projections and scenarios. Mortality by causes References and suggestions for further reading 4.9.1 Landmarks in mortality projections 4.9.2 Further references

145 146 147 148 149 149 151 152 152 153 155 156 156 160 162 163 165 165 166 167 169 169 171 172 173 173 173 175 175 175 178

Contents

5

Forecasting mortality: applications and examples of age-period models 5.1 5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

Introduction Lee–Carter mortality projection model 5.2.1 Specification 5.2.2 Calibration 5.2.3 Application to Belgian mortality statistics Cairns–Blake–Dowd mortality projection model 5.3.1 Specification 5.3.2 Calibration 5.3.3 Application to Belgian mortality statistics Smoothing 5.4.1 Motivation 5.4.2 P-splines approach 5.4.3 Smoothing in the Lee–Carter model 5.4.4 Application to Belgian mortality statistics Selection of an optimal calibration period 5.5.1 Motivation 5.5.2 Selection procedure 5.5.3 Application to Belgian mortality statistics Analysis of residuals 5.6.1 Deviance and Pearson residuals 5.6.2 Application to Belgian mortality statistics Mortality projection 5.7.1 Time series modelling for the time indices 5.7.2 Modelling of the Lee-Carter time index 5.7.3 Modelling the Cairns-Blake-Dowd time indices Prediction intervals 5.8.1 Why bootstrapping? 5.8.2 Bootstrap percentiles confidence intervals 5.8.3 Application to Belgian mortality statistics Forecasting life expectancies 5.9.1 Official projections performed by the Belgian Federal Planning Bureau (FPB) 5.9.2 Andreev–Vaupel projections 5.9.3 Application to Belgian mortality statistics

181 181 186 186 188 200 203 203 206 207 209 209 210 212 213 214 214 216 217 218 218 220 221 221 223 228 229 229 230 232 234 235 235 237

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5.9.4 Longevity fan charts 5.9.5 Back testing 6

Forecasting mortality: applications and examples of age-period-cohort models 6.1 6.2

6.3 6.4 6.5 7

Introduction LC age–period-cohort mortality projection model 6.2.1 Model structure 6.2.2 Error structure and model fitting 6.2.3 Mortality rate projections 6.2.4 Discussion Application to United Kingdom mortality data Cairns-Blake-Dowd mortality projection model: allowing for cohort effects P-splines model: allowing for cohort effects

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243 243 246 246 248 253 253 254 263 265

The longevity risk: actuarial perspectives

267

7.1 7.2

267 268 268

7.3

7.4

7.5

Introduction The longevity risk 7.2.1 Mortality risks 7.2.2 Representing longevity risk: stochastic modelling issues 7.2.3 Representing longevity risk: some examples 7.2.4 Measuring longevity risk in a static framework Managing the longevity risk 7.3.1 A risk management perspective 7.3.2 Natural hedging 7.3.3 Solvency issues 7.3.4 Reinsurance arrangements Alternative risk transfers 7.4.1 Life insurance securitization 7.4.2 Mortality-linked securities 7.4.3 Hedging life annuity liabilities through longevity bonds Life annuities and longevity risk 7.5.1 The location of mortality risks in traditional life annuity products 7.5.2 GAO and GAR 7.5.3 Adding flexibility to GAR products

270 273 276 293 293 299 303 318 330 330 332 337 343 343 346 347

Contents

7.6 7.7

7.8

Allowing for longevity risk in pricing Financing post-retirement income 7.7.1 Comparing life annuity prices 7.7.2 Life annuities versus income drawdown 7.7.3 The ‘mortality drag’ 7.7.4 Flexibility in financing post-retirement income References and suggestions for further reading

350 354 354 356 359 363 369

References

373

Index

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1 1.1

Life annuities

Introduction

Great attention is currently devoted to the management of life annuity portfolios, both from a theoretical and a practical point of view, because of the growing importance of annuity benefits paid by private pension schemes. In particular, the progressive shift from defined benefit to defined contribution pension plans has increased the interest in life annuities, which are the principal delivery mechanism of defined contribution pension plans. Among the risks which affect life insurance and life annuity portfolios, longevity risk deserves a deep and detailed investigation and requires the adoption of proper management solutions. Longevity risk, which arises from the random future trend in mortality at adult and old ages, is a rather novel risk. Careful investigations are required to represent and measure it, and to assess the relevant impact on the financial results of life annuity portfolios and pension plans. This book provides a comprehensive and detailed description of methods for projecting mortality, and an extensive introduction to some important issues concerning the longevity risk in the area of life annuities and pension benefits. Conversely, the present chapter mainly has an introductory role, aiming at presenting the basic structure of life annuity products. Moving from the simple model of the annuity-certain, typical features of life annuity products are presented (Section 1.2). From an actuarial point of view, the presentation progressively shifts from very traditional deterministic models (Section 1.3) to more modern stochastic models (Section 1.5). An appropriate stochastic approach allows us to capture the riskiness inherent in a life annuity portfolio, and in particular the risks arising from random mortality. Cross-subsidy mechanisms which work (or may work) in life annuity portfolios and pension plans are described in Section 1.4.

2

1 : Life annuities

The presentation of the actuarial structure of life annuities focusses on a very simple annuity model, namely the immediate life annuity. So, problems arising in the so-called accumulation phase (as well as problems regarding the annuitization of the accumulated amount) are initially disregarded. Conversely, in Section 1.6 a comprehensive description of a number of life annuity models is provided. In this section, actuarial aspects are just mentioned, in favour of more practical issues aiming in particular at paving the way for a following formal presentation. A list of references and some suggestions for further readings conclude the chapter (Section 1.7).

1.2 1.2.1

Annuities-certain versus life annuities Withdrawing from a fund

Assume that the amount S is available at a given time, say at retirement, and is used to build up a fund. Denote the retirement time with t = 0, and assume that the year is the time unit. In order to get her/his post-retirement income, the retiree withdraws from the fund at time t the amount bt (t = 1, 2, . . . ). Suppose that the fund is managed by a financial institution which guarantees a constant annual rate of interest i. Denote with Ft the fund at time t, immediately after the payment of the annual amount bt . Clearly: Ft = Ft−1 (1 + i) − bt

for t = 1, 2, . . .

(1.1)

with F0 = S. Thus, the annual variation in the fund is given by Ft − Ft−1 = Ft−1 i − bt

for t = 1, 2, . . .

(1.2)

Figure 1.1 illustrates the causes explaining the behaviour of the fund throughout time, formally expressed by equation (1.2). The behaviour of the fund throughout time obviously depends on the sequence of withdrawals b1 , b2 , . . .. In particular, if for all t the annual withdrawal is equal to the annual interest credited by the fund manager, that is, bt = Ft−1 i

(1.3)

then, from (1.1) we immediately find Ft = S

(1.4)

1.2

Annuities-certain versus life annuities

– Annual payment

Fund

Ft–1 Ft – Ft–1 Ft

+ Interest

t–1

t Time

Figure 1.1.

Annual variation in the fund providing an annuity-certain.

for all t, whence a constant withdrawal b = Si

(1.5)

follows. Conversely, if we assume a constant withdrawal b, b > Si

(1.6)

(as probably will be needed to obtain a reasonable post-retirement income) the drawdown process will exhaust, sooner or later, the fund. Indeed, from equation (1.2) we have F0 > F1 > · · · > Ft > · · ·

(1.7)

and we can find a time m such that Fm ≥ 0

and Fm+1 < 0

(1.8)

Clearly, the exhaustion time m depends on the annual amount b (and the interest rate i as well), as it can be easily understood from equation (1.2). The sequence of m constant annual withdrawals b (with m defined by conditions (1.8), and possibly completed by the exhausting withdrawal at time m + 1) constitutes an annuity-certain. Example 1.1 Assume S = 1000. Figure 1.2 illustrates the behaviour of the fund when i = 0.03 and for different annual amounts b. Conversely, Fig. 1.3 shows the behaviour of the fund for various interest rates i, assuming b = 100.

3

1 : Life annuities

2,000

b = 50 b = 75 b = 100 b = 125

1,000 0 Ft

0

5

10

15

20

25

30

35

–1,000

–2,000

–3,000

–4,000 Figure 1.2.

t

The fund providing an annuity-certain (i = 0.03).

1,200

i = 0.02 i = 0.03 i = 0.04 i = 0.05

1,000 800 600 400 Ft

4

200 0 0

2

4

6

8

10

12

14

16

–200 –400 –600 Figure 1.3.

t

The fund providing an annuity-certain (b = 100).

It is interesting to compare the exhaustion time m with the remaining lifetime of the retiree. Assume that her/his age at retirement is x, for example x = 65. Of course the lifetime is a random variable. Denote with Tx the remaining random life for a person age x. Let ω denote the maximum attainable age (or limiting age), say ω = 110. Hence, Tx can take all values between 0 and ω − x. If Tx < m then the amount Fm is available as a bequest. Conversely, if Tx > m there are ω − x − m years with no possibility of withdrawal (and hence no income).

1.2


In practice, the annual amount b (for a given interest rate i) could be chosen by comparing the related exhaustion time m with some quantity which summarizes the remaining lifetime. For example, a synthetic value is provided by the expected remaining lifetime E[Tx ]; another possibility is given by the remaining lifetime with the maximum probability, that is, the mode of the remaining lifetime, Mod[Tx ]. Note that, to find E[Tx ] or Mod[Tx ], assumptions about the probability distribution of the lifetime Tx are needed (see Section 1.3.2). For example, the value b may be chosen, such that m ≈ Mod[Tx ]

(1.9)

Thus, with a high probability the exhaustion time will coincide with the residual lifetime. Notwithstanding, events like Tx > m, or Tx < m, may occur and hence the retiree bears the risk originating from the randomness of her/his lifetime. Conversely, the choice m=ω−x

(1.10)

obviously removes the risk of remaining alive with no withdrawal possibility, but this choice would result in a low annual amount b.

1.2.2

Avoiding early fund exhaustion

Risks related to random lifetimes can be transferred from the annuitants to the annuity provider thanks to a different contractual structure, that is, the life annuity. To provide a simple introduction to technical features of life annuities, we adopt now a (very) traditional model; in the following sections, more modern and general models will be described. Consider the following transaction: an individual age x pays to a life annuity provider (e.g. an insurer) an amount S to receive a (life) annuity consisting in a sequence of annual benefits b, paid at the end of every year while she/he is alive. Assume that the same type of annuity is purchased at time t = 0 by a given number, say lx , of individuals all age x. Let lx+t denote an estimate (at time 0) of the number of individuals (annuitants) alive at age x + t (t = 1, 2, . . . , ω − x), out of the initial ‘cohort’ of lx individuals. As ω denotes the (integer) maximum age, we have by definition lω > 0 and lω+1 = 0. The following (estimated) cash flows of the annuity provider are then defined: (a) an income lx S at time 0; (b) a sequence of outgoes lx+t b at time t, t = 1, 2, . . . , ω − x.

5

6

1 : Life annuities

Let Vt denote the fund pertaining to a generic annuitant at time t. The total fund of the annuity provider is given by lx+t Vt , and is defined for t = 1, 2, . . . , ω − x as follows: lx+t Vt = lx+t−1 Vt−1 (1 + i) − lx+t b

(1.11)

clearly with lx V0 = lx S From (1.11), we find the following recursion describing the evolution of the individual fund: Vt =

lx+t−1 Vt−1 (1 + i) − b lx+t

(1.12)

with V0 = S. Recursion (1.12) can also be written as follows: Vt = Vt−1 (1 + i) +

lx+t−1 − lx+t Vt−1 (1 + i) − b lx+t

(1.13)

Thus, the annual variation in the fund is given by Vt − Vt−1 = Vt−1 i +

lx+t−1 − lx+t Vt−1 (1 + i) − b lx+t

(1.14)

It is worth noting from (1.14) that the annual decrement of the individual fund can be split into three contributions (see Figure 1.4): (a) a positive contribution provided by the interest Vt−1 i; (b) a positive contribution provided by the share of the funds released because of the death of lx+t−1 − lx+t annuitants in the t-th year, the share being credited to the lx+t annuitants alive at time t; (c) a negative contribution given by the benefit b. Contribution (b), which does not appear in the model describing the annuity-certain (see Figure 1.1), is maintained thanks to a cross-subsidy among annuitants, that is, the so-called mutuality effect. For more details, see Section 1.4.1. In the case of life annuities, the individual fund Vt (as defined by recursion (1.12)) is called the reserve.

1.2.3

Risks in annuities-certain and in life annuities

First, let us focus on the simple model of annuity-certain we have dealt with in Section 1.2.1, and consider the perspectives of the retiree and the financial institution providing the annuity.

1.2


– Annual payment Reserve

Vt–1 Vt – Vt–1

+ Interest

Vt + Mutuality

t

t–1 Time Figure 1.4.

Annual variation in the (individual) fund of a life annuity.

The provider of an annuity-certain does not bear any risk inherent in the random lifetime of the retiree, as, whatever this lifetime may be, the annuity will be paid up to the exhaustion of the fund. Conversely, the annuity provider takes financial risks which can be singled out looking at the two causes of change in the fund level (see Fig. 1.1). Risks are as follows: – market risk, more precisely interest rate risk, as we have assumed that i is the guaranteed interest rate which must be credited to the fund whatever the return from the investment of the fund itself may be; – liquidity risk, as the annual payment obviously requires cash availability. Conversely, the retiree does not take any financial risk thanks to the guaranteed interest rate, whereas she/he bears the risk related to her/his random lifetime, as seen above. Now, let us move to the life annuity. According to the structure of this product (at least as defined in Section 1.2.2), the annuitant does not bear any risk. Actually, the annuity is paid throughout the whole lifetime and the amount of the annual payment is guaranteed. Conversely, the annuity provider first bears the market risk and the liquidity risk as in the annuity-certain model. Further, if the actual lifetimes of annuitants lead to numbers of survivors greater than the estimated ones, the cross-subsidy mechanism (see Section 1.2.2 and Fig. 1.4) cannot finance the payments to the annuitants still alive. In other words, contribution (b), which is required to maintain the individual fund Vt , should be

7

8

1 : Life annuities

partially funded, in this case, by the annuity provider. Conversely, numbers of survivors less than the estimated ones lead to a provider’s profit. Hence, the annuity provider takes risks related to the mortality of the annuitants.

1.3 1.3.1

Evaluating life annuities: deterministic approach The life annuity as a financial transaction

Purchasing a life annuity constitutes a financial transaction, whose cash flows are – a price, or premium, paid by the annuitant to the annuity provider; – a sequence of amounts, namely the annuity, paid by the annuity provider to the annuitant while he/she is alive; the payment frequency may be monthly, quarterly, semi-annual, or annual. In what follows, we only refer to annual payments, hence disregarding annuities payable more frequently than once a year (which require special treatment; see the references cited in Section 1.7). Further, we will assume (if not otherwise specified) that payments are made at the end of each year (annuity in arrears). In the life annuity structure presented in Section 1.2.2, the amount S represents the premium paid against the annuity with b as the annual payment. Clearly, the life annuity structure we have described requires a single premium at time 0, as the annuity is an immediate one. Conversely, for other annuity models different premium arrangements are feasible, as we will see in Section 1.6. The relation between S and b is implicitly defined by recursion (1.11) (or (1.12)). Solving with respect to S (or b), when b (or S) has been assigned, leads to an explicit relation between the two amounts. In particular, S is the expected present value of the life annuity, as we will see in Section 1.3.2. Indeed, a reasonable starting point (but not necessarily the only one) for determining the single premium is given by the calculation of the expected present value of the life annuity. In particular, when the so-called equivalence principle is adopted, the single premium is set equal to the expected present value. Other premium calculation principles will be dealt with in Section 7.6.

1.3

1.3.2

Evaluating life annuities: deterministic approach

Actuarial values

For a given i and a given sequence lx , lx+1 , . . . , lω , from recursion (1.11), with lx V0 = lx S, we find lx S =

ω−x

b lx+t (1 + i)−t

(1.15)

t=1

and, referring to a single annuitant, S=

ω−x

b

t=1

lx+t (1 + i)−t lx

(1.16)

In formula (1.16), S turns out to be the present value of the sequence of amounts b ‘weighted’ with the ratios lx+t /lx . The numbers of survivors lx+t (and the interest rate i, as well) are assumed deterministic. Hence the model relying on these assumptions, and leading in particular to expression (1.16), is a deterministic one. Some comments can help in understanding the features of the deterministic model. First, a point in favour of the model is that, in spite of its deterministic nature, the risk borne by the life annuity provider, and arising from random lifetimes clearly emerges, although it is not explicitly accounted for (see Section 1.2.3). Second, equation (1.16) can be rewritten in ‘probabilistic’ terms, since lx+t /lx can be interpreted as the estimate of the probability of an individual age x being alive at age x+t. Denoting with t px this probability, it is formally defined as follows: t px

= P[Tx > t]

(1.17)

and we have S=b

ω−x

t px (1 + i)

−t

(1.18)

t=1

An alternative expression is provided by the following formula: S=b

ω−x h=1

where

ah h px qx+h

(1.19)

9

10

1 : Life annuities

– the symbol ah , defined as follows: ah =

1 − (1 + i)−h i

(1.20)

denotes the present value of a temporary annuity-certain consisting of h unitary annual payments in arrears; – the symbol qx+h denotes the probability of an individual age x + h dying within one year, formally qx+h = P[Tx+h < 1]

(1.21)

we note that, assuming ω as the maximum age, qω = 1; – hence, h px qx+h is the probability of an individual currently age x dying between ages x + h and x + h + 1; in symbols h px qx+h

= P[h ≤ Tx < h + 1]

(1.22)

Note that h px

= (1 − qx )(1 − qx+1 ) . . . (1 − qx+h−1 )

(1.23)

The equivalence of (1.18) and (1.19) can be proved using the following relation: t px

=1−

t−1

h px qx+h

(1.24)

h=0

where the sum expresses the probability of dying before age x + t. Clearly, the right-hand side of expression (1.19) represents the expected present value, or actuarial value, of the life annuity, thus: S = b E[aKx ]

(1.25)

where Kx denotes the curtate random remaining lifetime of an individual age x, namely the integer part of Tx . The quantities h px qx+h , h = 0, 1, . . . , ω−x, constitute the probability distribution of the discrete random variable Kx . With the symbol commonly used to denote the actuarial value of the life annuity, we have: S = b ax

(1.26)

where, according to (1.18), ax =

ω−x t=1

t px (1 + i)

−t

(1.27)

1.3


Finally, the quantity Vt can be interpreted as the mathematical reserve of the life annuity, whose evolution throughout time is described by recursion (1.12), namely, in probabilistic terms: Vt =

1 1 px+t−1

Vt−1 (1 + i) − b

(1.28)

It should be noted that recursion (1.28) expresses the reserve Vt as the result of the decumulation process, driven by financial items (the interest rate i and the payment b) and a demographic item (the probability 1 px+t−1 ). Under this perspective, the reserve Vt can be interpreted as assets pertaining to the generic annuitant. Conversely, the annuitant has the right to receive the annual amount b while she/he is alive. This obligation of the life annuity provider, viz. a liability, can be expressed as the expected present value at time t (and hence referred to the annuitant assumed to be alive at time t) of future annual payments: b ax+t = b

ω−x−t

h px+t

(1 + i)−h

(1.29)

h=1

It is easy to prove, replacing Vt and Vt−1 in equation (1.28) with b ax+t and b ax+t−1 respectively, that equation (1.28) itself is satisfied. Thus, Vt = b ax+t

(1.30)

whence the amount Vt can be interpreted as the amount of assets exactly meeting the provider’s liability. Note that the reserve Vt exhausts at the maximum age ω only. Example 1.2 In Fig. 1.5 the mathematical reserve Vt is plotted against time t. We have assumed S = 1000, i = 0.03, x = 65. The estimated numbers of survivors can be drawn from various data set. For example, assume that the probabilities qx+h , h = 0, 1, . . . , ω − x, where x is a given initial age of interest, have been assigned. From the qx+h ’s, the estimated number of survivors can be derived via the following recursion: lx+h+1 = lx+h (1 − qx+h )

(1.31)

starting from a (notional) initial value lx . For example, assume for qx+h the following expression:  x+h   GH if x + h < 110 qx+h = 1 + G H x+h (1.32)  1 if x + h = 110

11

1 : Life annuities

1,200

1,000

800

Vt

12

600

400

200

0 65

75

85

95

105

115

x+t Figure 1.5.

Mathematical reserve of a life annuity.

with the parameters G = 0.000002, H = 1.13451. From the data assumed, we obtain a65 = 14.173 and hence b = 70.559. Remark The first line on the right-hand side of (1.32) approximately expresses the mortality at older ages according to the first and second Heligman–Pollard laws, as we will see in Section 2.5.2.

1.3.3

Technical bases

The relation between S (the single premium) and b (the annual benefit) relies on the equivalence principle, as S is the expected present value of the sequence of annual amounts b. The adoption of this principle complies with common (but not necessarily sound) actuarial practice. Actually, when the equivalence principle is used for pricing insurance products and life annuities in particular, a safe-side technical basis (or prudential basis, or first-order basis) is chosen, namely an interest rate i lower than the estimated investment yield, and a set of probabilities expressing a mortality level lower than that expected in the life annuity portfolio. The estimated investment yield and the mortality actually expected constitute the scenario technical basis (or realistic basis, or second-order basis). For simplicity, assume a constant estimated investment yield i∗ ; denote with q∗x+h , h = 0, 1, . . . , ω − x the realistic probabilities of death. The survival probabilities, t p∗x , can be calculated from the q∗x+h as stated by relation

1.3


(1.23). The resulting actuarial value of the life annuity, a∗x , is clearly given (see (1.27)) by a∗x =

ω−x

∗ ∗ −t t px (1 + i )

(1.33)

t=1

The difference ax − a∗x can be interpreted as the expected present value (at time t = 0) of the profit generated by the life annuity contract. Note that, if i∗ > i, the yield from investment contributes to the profit. Usually profit participation mechanisms assign a (large) part of the investment profit to policyholders, and so the expected profit ax − a∗x should be taken as gross of the profit participation. Example 1.3 For example, assume i = 0.03 and the qx+h adopted in Example 1.1 as the items of the safe-side technical basis (i.e. the pricing basis); conversely, for the scenario basis assume i∗ = 0.05 as the estimated investment yield, and the mortality level described by probabilities q∗x+h given by the expression (1.32) implemented with the parameters G∗ = 0.0000023, H ∗ = 1.134. With these assumptions, we have that i∗ > i and q∗x+h > qx+h . We find that a∗65 = 11.442, and hence the expected present value of the profit produced by a life annuity with a unitary annual payment, that is, with b = 1, is a65 − a∗65 = 2.731. An appropriate choice of the first-order basis, for a given scenario basis, also provides the insurer with a safety loading in order to face an adverse mortality experience (and/or an adverse yield from investments). In other words, while the spread between the technical bases produces a (positive) profit if the insurer experiences a mortality and an investment yield as described by the scenario basis, the spread itself, increasing the single premium for a given annual payment or conversely reducing the annual payment for a given premium, can avoid losses when an adverse experience occurs. As regards the choice of the age-patterns of mortality, to adopt as the first-order and the second-order technical basis respectively, it should be kept in mind that life annuities may involve very long time intervals, say 25–30 years or even more. Indeed, survival probabilities (i.e. probabilities ∗ t px and t px ) should express reasonable mortality assumptions referring to the future lifetime of an individual who is currently at age x. Age-patterns of mortality are commonly available as the result of statistical observations, and usually express the mortality at various ages as it emerges at the time of the observation itself. As mortality is affected by evident trends (see Chapter 3), observed mortality (even when resulting from

13

14

1 : Life annuities

recent investigations) cannot be directly used to express long term future mortality, as required when dealing with life annuities. Thus, projection models (see Chapter 4) are needed to forecast future mortality.

1.4

Cross-subsidy in life annuities

Although insurance transactions can be analysed at an individual level (e.g. in terms of the equivalence principle), in practice these transactions usually involve a group of insureds transferring the same type of risk to an insurer. This is also the case for life annuity products, and actually these products have been introduced in Section 1.2.2 referring to a cohort of annuitants. Thanks to the existence of an insured population, money transfers inside the population itself (i.e. among the policyholders) are possible, causing a cross-subsidy among the insureds (or annuitants). The term cross-subsidy broadly refers to some arrangement adopted for sharing among a given population the cost of a set of benefits. However, various types of cross subsidy can be recognized. While mutuality underpins the management of any insurance portfolio (see Section 1.2.2, as regards life annuities), other types of cross subsidy are not necessarily involved, for example, solidarity. Further, special cross-subsidy structures may occur with particular policies; this is the case of tontine schemes in the context of life annuities. In the following parts of this section, we deal with cross-subsidy mechanisms (mutuality, solidarity, and tontines), focussing on life annuity portfolios.

1.4.1

Mutuality

The mutuality principle underpins the insurance process (whether or not it is run by a ‘mutual’ insurance company or by a proprietary insurance company, which is owned by shareholders), and arises from the pooling of a number of risks. Moreover, the mutuality effect also works in ‘mutual associations’ of individuals exposed to the same type of risk, even without resorting (at least in principle) to an insurance company. The mutuality effect leads to money transfers from insureds (or annuitants) who, in terms of actuarial value, paid premiums greater than the benefits received to insureds in the opposite situation. For example, in a non-life portfolio the insureds without claims transfer money to the insureds with claims.

1.4


Referring to a life annuity portfolio, it is interesting to focus on the annual equilibrium between assets available and liabilities. This equilibrium relies on an asset transfer among annuitants, namely, from annuitants dying in the year to annuitants alive at the end of the year. This clearly appears from recursion (1.11), where the accumulated fund pertaining to lx+t−1 annuitants alive at time t −1, whose amount is lx+t−1 Vt−1 (1 +i), is used to finance benefits to lx+t annuitants (out of the lx+t−1 ) alive at time t, namely, the payment of the amount lx+t b and the maintenance of the fund lx+t Vt for future payments. So, resources needed at time t are made available (also) thanks to this cross subsidy, namely the mutuality effect. Let us now look at the technical equilibrium under an individual perspective. Recursion (1.13) can be rewritten, in more compact terms, as Vt = Vt−1 (1 + i) (1 + θx+t ) − b

(1.34)

where θx+t =

lx+t−1 − lx+t lx+t

(1.35)

In terms of survival probabilities, as it emerges from (1.28), we have θx+t = (1/1 px+t−1 ) − 1. Looking at recursion (1.34), θx+t can be interpreted as an ‘extra-yield’ which is required to maintain the decumulation process of the individual reserve Vt , and hence can be interpreted as a measure of the mutuality effect. The extra yield θx+t is also called the mortality drag, or interest from mutuality. As already seen in Section 1.2.2, θx+t determines the share of the funds released because of the death of lx+t−1 − lx+t annuitants in the t-th year, and credited to the lx+t annuitants alive at time t. Remark The (annual) extra-yield provided by the mutuality effect is clearly a function of the current age x + t (see (1.35)). Referring to a given age interval (x, x + m), the sequence θx , θx+1 , . . . , θx+m can be summarized in an index, depending on x, m, and the interest rate i, called the implied longevity yield (ILY).1 This index plays an important role in the analysis of annuitization alternatives, as we will see in Section 7.7. Example 1.4 In Fig. 1.6 the quantity θx+t is plotted for x = 65 and t = 0, 1, . . .. The underlying technical basis is the first-order basis, with i = 0.03 and the qx+t defined in Example 1.2. It is interesting to note that, 1 The expression ‘Implied Longevity Yield’ and its acronym ‘ILY’ are registered trademarks and property of CANNEX Financial Exchanges.

15

1 : Life annuities

2 1.8 1.6 1.4 1.2 u

16

1 0.8 0.6 0.4 0.2 0 65

75

85

95

105

Age Figure 1.6.

A measure of the mutuality effect.

when moderately old ages are involved (say, in the interval 65–75), the values of θ are rather small. In such a range of ages, they could be ‘replaced’ with a higher yield from investments (provided that riskier investments can be accepted), and so, in that age interval, a withdrawal process could be preferred to a life annuity. Conversely, as the age increases, θ reaches very high values, which obviously cannot be replaced by investment yields. So, when old and very old ages are concerned, the life annuity is the only technical tool which guarantees a lifelong constant income. As regards theoretical results showing that the annuitization constitutes the optimal choice, see Section 1.7.

1.4.2

Solidarity

Assume that a population consisting of (potential or actual) insureds is split into risk classes. Each risk class groups individuals with the same probability of claim (or death, or survival, etc.). Risk classes could be directly referred to for pricing purposes, namely charging with a specific premium rate the individuals belonging to a given risk class. Conversely, two or more risk classes can be grouped leading to a rating class, which would be aimed at charging all individuals belonging to a given rating class with the same premium rate. The premium rate attributed to a rating class should be an appropriate weighted average of the premiums competing to the risk classes grouped into the rating class

1.4


itself. The weighting should reflect the expected numbers of (future) insureds belonging to the various risk classes. Assume that, as far as pricing is concerned, the population is split into rating classes rather than into risk classes. The rationale of this grouping may be, for example, a simplification in the tariff structure. When two or more risk classes are aggregated into one rating class, some insureds pay a premium higher than their ‘true’ premium, that is, the premium resulting from the risk classification, while other insureds pay a premium lower than their ‘true’ premium. Thus, the equilibrium inside a rating class relies on a money transfer among individuals belonging to different risk classes. This transfer is usually called solidarity (among the insureds). Clearly, such a premium system may cause adverse selection, as individuals forced to provide solidarity to other individuals can reject the policy, moving to other insurance solutions (or, more generally, risk management actions). The severity of this self-selection phenomenon depends on how people perceive the solidarity mechanism, as well as on the premium systems adopted by competitors in the insurance market. In any event, self-selection can jeopardize the technical equilibrium inside the portfolio, which depends on actual versus expected numbers of insureds belonging to the various risk classes grouped into a rating class. So, in practice, solidarity mechanisms can work provided that they are compulsory (e.g. imposed by insurance regulation) or they constitute a common market practice. As regards life annuities, risk classes are usually based on age and gender. In particular, it is well known that females experience a mortality lower than males and a higher expected lifetime. So, if for some reason the same premium rates (only depending on age) are applied to all annuitants, a solidarity effect arises, implying a money transfer from males to females. The solidarity effect is stronger when the number of rating classes is smaller, compared with the number of risk classes. In the private insurance field, an extreme case is achieved when one rating class only relates to a large number of underlying risk classes. Outside of the private insurance area, the solidarity principle is commonly applied in social security. In this field, the extreme case arises when the whole national population contribute to fund the benefits, even if only a part of the population itself is eligible to receive benefits; so, the burden of insurance is shared among the community. Finally, it is interesting to stress the implications of this argument. Mutuality affects the benefit (or claim) payment phase, so that ‘direction’ and

17

18

1 : Life annuities

‘measure’ of the mutuality effect in a portfolio are only known ex-post. Conversely, solidarity affects the premium income phase, and hence its direction and measure are known ex-ante.

1.4.3

‘Tontine’ annuities

Assume that each one of lx individuals, all aged x at time t = 0, pays at that time the amount S to a financial institution. Against the amount S = lx S, the financial institution will pay at the end of each year, that is, at times t = 1, 2, . . ., the (total) constant amount B , while at least one of the individuals of the group is alive. Each year the amount B is divided equally among the survivors. Hence, each individual (out of the initial lx ) alive at time t receives a benefit bt which depends on the actual number of survivors at that time. Denoting, as usual, with lx+t the estimated number of survivors, an estimate of bt is given by B /lx+t . Clearly,

B lx+1

≤

B lx+2

≤ ... ≤

B lx+t

≤ ···

(1.36)

The mechanism of dividing B among the survivors is called a tontine scheme, whereas the sequence (1.36) is called a tontine annuity. The relation between S (the initial income) and B (the annual payment) can be stated (at least in theory) on the basis of the equivalence principle. To this purpose, first note that the duration, K, of the annuity paid by the financial institution is random, being defined as follows: K = max{Kx(1) , Kx(2) , . . . , Kx(lx ) }

(1.37)

(j)

where Kx denotes the random curtate residual lifetime of the j-th individual. Hence, the equivalence principle requires

S = B E[aK ]

(1.38)

The calculation of E[aK ] is extremely difficult. In practice, a reasonable approximation could be provided by aω−x . While in general aω−x > E[aK ], the larger is lx the better is this approximation, as there is a higher probability that some individual reaches, or at least approaches, the maximum age ω. Example 1.5 The tontine annuity derives its name from Lorenzo Tonti (a Neapolitan banker living most of his life in Paris) who, around 1650, proposed a plan for raising monies to Cardinal Mazzarino, the Chief Minister

1.4


of France at the time of King Louis XIV. In this plan, a fund was raised by subscriptions. Let S denote the amount collected by the State. Then, the State had to pay each year the interest on S , at a given annual interest rate i. The constant annual payment S i was to be divided equally among the surviving members of the group and would terminate with the death of the last survivor. Thus, according to our notation, the duration of the annuity is K (see definition (1.37)), and we have B = S i. Note that 1 B =i= S a∞

(1.39)

where a∞ = 1/i is the present value of a perpetuity (given the discount rate i). As

S S S < < a∞ aω−x E[aK ]

(1.40)

(assuming that the same discount rate is used for all the present values), we find that original Tonti’s scheme did not fulfill the equivalence principle, whilst it is favourable to the issuer (i.e., to the State). Turning back to the general tontine scheme, two points should be stressed. (a) The tontine scheme clearly implies a cross subsidy among the annuitants, and in particular a mutuality effect arises as each dying annuitant releases a share of the amount B , which is divided among the surviving annuitants. (b) A basic difference between tontine annuities and ordinary life annuities should be recognized. In an ordinary life annuity, the annual (individual) benefit b is stated and guaranteed, in the sense that the life annuity provider has to pay the amount b to the annuitant for her/his whole residual lifetime, whatever the mortality experienced in the portfolio (or pension plan) may be. Conversely, in a tontine scheme the sequence of amounts b1 , b2 , . . . paid to each annuitant depends on the actual size of the surviving tontine group. Note that, when managing an ordinary life annuity portfolio the annuity provider takes the risk of a poor mortality experience in the portfolio (see Section 1.2.3), whereas in a tontine scheme the only cause of risk is the lifetime of the last survivor. Further, it should be noted that, for a given technical basis and a given amount S, the annual benefit b is likely to be much higher than the initial payments

19

20

1 : Life annuities

in a tontine scheme. Actually (using the approximation aω−x ), from

B=

S aω−x

=

lx S aω−x

we obtain, for small values of t (such that bt =

lx lx+t

(1.41)
t} x

(1.49)

where IE is the indicator function of event E. Note that the expected path E[Zt ], t = 1, 2, . . . , ω−x, can be immediately derived as

E[Zt ] = E[Zt−1 ] (1 + i∗ ) − E[Lx+t ] b

(1.50)

the expected numbers E[Lx+t ] being given by (1.47). Example 1.8 Figures 1.9(a) and (b) illustrate 10 paths of Zt , for t = 0, 1, . . . , 5 and t = 15, . . . , 20, respectively. The data already described have been assumed as the input of the simulation procedure, in particular i∗ = 0.05 and the q∗x+h ’s used in Example 1.3. Figures 1.10(a) and (b) illustrate the (simulated) statistical distribution of Z5 and Z20 respectively, based on a sample of 1000 simulated paths. Further interesting aspects may emerge from comparing the behaviour of the fund Zt with the (random) portfolio reserve, whose amount is Lx+t Vt , with Vt given by (1.30) (traditionally implemented with the first-order basis). As the assets actually available are given by Zt , the (random) quantity Mt = Zt − Lx+t Vt

(1.51)

represents the assets in excess of the level required (according to the firstorder basis) to meet expected future obligations. Example 1.9 Figures 1.11(a) and (b) represent the (simulated) statistical distribution of M5 and M20 respectively, based on the simulated sample previously adopted. The erratic behaviour in these figures (as well as in

23

1 : Life annuities (a) 100,000

(b) 45,000 40,000

95,000

35,000 90,000 Zt

Zt

30,000 25,000

85,000

20,000 80,000

15,000

75,000

10,000 0

1

2

3

4

5

15

t

Figure 1.9.

16

17

18

19

20

t

Some paths of Zt .

(a) 0.8

(b) 0.7

0.7

0.6

0.6

0.5

0.5

Frequency

0.4 0.3

0.4 0.3 0.2

0.2

0.1

0

0

10 0 ,0 0 20 0 ,0 0 30 0 ,0 0 40 0 ,0 0 50 0 ,0 0 60 0 ,0 0 70 0 ,0 0 80 0 ,0 0 90 0 ,0 00

0.1

10 0 ,0 0 20 0 ,0 0 30 0 ,0 0 40 0 ,0 00 50 ,0 0 60 0 ,0 0 70 0 ,0 0 80 0 ,0 0 90 0 ,0 00

Frequency

24

Z5

Figure 1.10.

Z20

Statistical distributions of Z5 and Z20 .

Figures 1.11(a), 1.11(b), 1.12(a), 1.12(b), and 1.14) is clearly due to the simulation procedure; smoother results can be obtained by increasing the number of simulations.

1.5.3

A first insight into risk and solvency

From the exercise developed in Examples 1.7–1.9, an important feature of stochastic models clearly emerges. Allowing for randomness provides us with a tool for assessing the ‘risk’ inherent in a life annuity portfolio or a pension plan. As we can see in Figure 1.9(a) and (b), random fluctuations affect the portfolio behaviour, and these are caused (in this example) by the randomness in the number of survivors throughout time. The risk we are

1.5

Evaluating life annuities: stochastic approach

0.18

0.16

0.16

0.14

0.14 Frequency

(b) 0.2

0.18

Frequency

(a) 0.2

0.12 0.1 0.08

0.06

0.04

0.04

0.02

0.02

–5

0, 0 –4 00 0, 0 –3 00 0, 00 –2 0 0, 00 –1 0 0, 00 0 10 0 ,0 0 20 0 ,0 0 30 0 ,0 0 40 0 ,0 00

0 10 0 ,0 0 20 0 ,0 0 30 0 ,0 0 40 0 ,0 00

0, 0 –4 00 0, 0 –3 00 0, 0 –2 00 0, 0 –1 00 0, 00 0

–5

0.1 0.08

0.06

0

M5 Figure 1.11.

0.12

M20

Statistical distributions of M5 and M20 .

now focussing on is usually named the risk of mortality random fluctuation, or the process risk due to mortality (see also Section 7.2). Figures 1.10(a) and (b) suggest measures which can be used for assessing the riskiness of a life annuity portfolio in terms of the ‘dispersion’ of the fund Zt . Analogous considerations emerge from Figure 1.11(a) and (b) in relation to the quantity Mt . For example, the variance or the standard deviation, estimated from the statistical distributions, can be used as (traditional) risk measures. The possibility of quantifying portfolio riskiness suggests ‘operational’ applications of our stochastic model, provided that it is properly generalized. For example, let us focus on the quantity Mt . From Figure 1.11(a) and (b) it emerges that, with a positive probability, M5 and M20 take negative values. Of course, the event Mt < 0 indicates an insolvency situation. So, probabilities of events like Mt < 0 for some t, at least within a stated time horizon, should be kept reasonably small. In particular, an initial allocation of (shareholders’) capital, leading to Z0 > S, clearly lowers the probability of being insolvent. Example 1.10 Allocating the amount M0 = 3000, so that Z0 = 100 S + 3000, leads to the distributions of M5 and M20 depicted in Figure 1.12(a) and (b), from which a smaller probability of insolvency clearly emerges. Of course, causes of risk other than mortality could be introduced into our model, and typically the investment risk, in particular arising from random fluctuations (i.e. ‘volatility’) in the investment yield. To this purpose, the sequence of annual investment yields must be simulated, on the basis of

25

1 : Life annuities

(b)

0.2

0.2 0.18

0.16

0.16

0.14

0.14 Frequency

0.18

0.12 0.1 0.08

0.12 0.1 0.08 0.04

0.02

0.02

0

0 –5

M20

M5 Figure 1.12.

10 0 ,0 0 20 0 ,0 0 30 0 ,0 0 40 0 ,0 00

0.04

0, 0 –4 00 0, 0 –3 00 0, 0 –2 00 0, 00 –1 0 0, 00 0

0.06

10 0 ,0 0 20 0 ,0 0 30 0 ,0 0 40 0 ,0 00

0.06

0, 0 –4 00 0, 00 –3 0 0, –2 000 0, 00 –1 0 0, 00 0

Frequency

(a)

–5

26

Statistical distribution of M5 and M20 .

an appropriate model for stochastic interest rates, and used in place of the estimated yield i∗ . We do not deal with these problems, which are beyond the scope of the present chapter. ()

Let us now refer to the random present value at time t = 0, Y0 , of future benefits in a portfolio consisting of one generation of life annuities. We have () Y0

=b

ω−x

Lx+t (1 + i)−t

(1.52)

t=1 ()

If we calculate the expected value of Y0 have

E[Y0() ]

=b

ω−x

E[Lx+t ] (1 + i)

−t

= b lx

t=1

using the first-order basis, we

ω−x

t px (1 + i)

−t

= lx V0 (1.53)

t=1

Formula (1.53) provides the (traditional) portfolio reserve, given by ()

V0

()

= E[Y0 ] = lx V0 ()

Obvious generalizations lead to Yt

()

and E[Yt

(1.54) ], for t ≥ 0.

However, in a stochastic context, the portfolio reserve can be defined in different ways, in particular in order to allow for the riskiness inherent in the life annuity portfolio. For example, the reserve can be defined as the

1.5


1−α

0 Figure 1.13.

E[Y0(Π)] ()

Probability distribution of Y0

yα

; α-percentile. ()

α-percentile of the probability distribution of Y0 (;α)

V0

(see Fig. 1.13):

= yα

(1.55)

with yα such that

P[Y0() > yα ] = 1 − α

(1.56)

Example 1.11 Using the data of the previous examples, from the simulated () distribution of Y0 (see Fig. 1.14) we find the results shown in Table 1.1. () () () Note that, conversely, we have P[Y0 > V0 ] = 0.209 (where V0 = E[Y0() ] = 100000). ()

It is worth noting that the calculation of the portfolio reserve V0 (and, () in general Vt ) according to (1.54) represents the traditional approach that is adopted in actuarial practice. In this context, the presence of risks is taken into account simply via the first-order basis adopted in implementing formula (1.54). Conversely, the reserving approach based on the probability () () distribution of Y0 (and Yt in general) and leading to the portfolio reserve (;α) (;α) V0 (Vt ) allows for risks via the choice of an appropriate percentile of the distribution itself.

1.5.4

Allowing for uncertainty in mortality assumptions

As already mentioned in Section 1.3.3, experience suggests that we should adopt projected mortality tables (or laws) for the actuarial appraisal of

27

1 : Life annuities

0.16 0.14 0.12

Probability

28

0.1 0.08 0.06 0.04 0.02 0 83,500

Figure 1.14.

88,500

93,500 Y0 ()

Statistical distribution of Y0

98,500

103,500

.

Table 1.1. Percentiles of the probability distribu() tion of Y0

α 0.75 0.90 0.95 0.99

yα 92067.033 101553.815 102608.253 104480.738

life annuities (and other living benefits), that is, use mortality assumptions which include a forecast of future mortality trends. Notwithstanding, whatever hypothesis is assumed, the future trend in mortality is random, and hence an uncertainty risk arises, namely a risk due to uncertainty in the representation of the future mortality scenario. Example 1.12 Assume the first-order basis already used in the previous examples. To describe the (future) mortality scenario, use the model (1.32) with the following alternative parameters: (1) G(1) = 0.0000025; H (1) = 1.13500 (2) G(2) = 0.0000023 (= G∗ ); H (2) = 1.13400 (=H ∗ ) (3) G(3) = 0.0000019; H (3) = 1.13300

1.5


We assume that scenario (2) (which coincides with the scenario adopted as the second-order basis in previous examples) represents the best estimate mortality hypothesis. Scenario (1) involves a higher mortality level and hence can be considered ‘optimistic’ from the point of view of the annuity provider. Conversely, scenario (3) expresses a lower mortality level and thus constitutes a ‘pessimistic’ mortality forecast. We obtain: (1)

a65 = 11.046,

(2)

a65 = 11.442,

(with obvious meaning of the notation).

(3)

a65 = 12.102

The coexistence of more than one mortality scenario (namely, three in Example 1.12) depicts a new modelling framework. When no uncertainty in future mortality trend is allowed for, and hence just one age-pattern of mortality is assumed (e.g. in terms of probability of dying), a deterministic actuarial value of the life annuity follows. Conversely, if we recognize uncertainty in the future pattern of mortality, randomness in actuarial values follows. Figure 1.15 illustrates three different approaches to uncertainty in mortality assumptions. The first approach (A) simply disregards uncertainty, so that the related result is a deterministic actuarial value of the life annuity. In the second case (B), a finite set of scenarios is used to express uncertainty, from which a finite set of actuarial values follows; clearly, this approach has been adopted in Example 1.12. Note that, according to this approach, each actuarial value should be regarded as an expected value conditional on a given scenario. Finally, the third approach (C) allows for uncertainty via a continuous set of scenarios and a consequent interval for the (conditional) actuarial value of the life annuity; this approach can be implemented, for example, assuming a given interval as the set of possible values for a parameter of the mortality law. Clearly, the uncertainty risk coexists with the risk of mortality random fluctuations. As regards the present value of a life annuity, random fluctuations lead to the probability distribution depicted for example, in Fig. 1.14 (see Example 1.11). When allowing also for uncertainty in future mortality, a set of probability distributions should be addressed. Thus, referring to approach B (see Fig. 1.15), a finite set of conditional distributions is involved, each one relating to an alternative mortality scenario (see Fig. 1.16). A comprehensive description of riskiness inherent in a life annuity product (still excluding financial risks arising from investment performance)

29

1 : Life annuities

(A)

Deterministic scenario Probability of dearth

Deterministic actuarial value of the life annuity

x (B)

Age

Value Uncertainty in the actuarial value of the life annuity

Uncertainty in scenario

(discrete setting)

(discrete setting)

Probability of dearth

(1) (2) (3)

x (C)

Age

Value

Uncertainty in scenario (continuous setting)

Uncertainty in the actuarial value of the life annuity (continuous setting)

Probability of dearth

30

x Age Figure 1.15.

Value

Mortality scenarios and actuarial values.

requires a further step. By assigning an appropriate probability description of the scenario space, we can move from conditional probability distributions to an unconditional distribution, which ‘summarizes’ both components of risk, namely the uncertainty risk and the risk of random fluctuations. This topic will be focussed in Chapter 7 while dealing with the assessment of longevity risk.

1.6

Types of life annuities

(1) (2) (3)

Present value of the life annuity Figure 1.16.

1.6

Conditional probability distributions of the random present value of the life annuity.


In the previous sections we have dealt with an immediate life annuity in arrears, that is a life annuity whose first payment is due one period (one year, according to our assumptions) from the date of purchase, while the last payment is made at the end of the period preceding the death of the annuitant. Although this structure is rather common, a number of other types of life annuities are sold on insurance markets, and paid by pension plans as well. So, the purpose of this section is to describe a range of annuity types, looking at features of both the accumulation period and the decumulation period (also called the liquidation period, or payout period); see Fig. 1.17.

1.6.1

Immediate annuities versus deferred annuities

Let us continue to focus on an immediate life annuity, and denote with b the annual benefit and S the net single premium (i.e., disregarding expense loadings). It is natural to look at the amount S as the result of an accumulation process carried out during (a part of) the working life of the annuitant. Let us now denote with x the age at the beginning of the accumulation process, that is, at time 0. The accumulation process stops at time n, so that x + n is the age at the beginning of the decumulation phase. The relation between S and b is given, according to the equivalence principle, by S = b ax+n (see (1.26) and the previous equations).

(1.57)

31

1 : Life annuities

Accumulation

Decumulation

S Fund/reserve

32

Annuity benefits/ withdrawals

Premiums/ savings

n–1 n n+1 x+n

0 1 2 x

Time and age Figure 1.17.

Accumulation and decumulation phases.

As regards the accumulation process, this can be carried out via various tools, for example insurance policies providing a survival benefit at maturity (time n). Some policy arrangements tools will be described in Section 1.6.2. Conversely, it is possible to look jointly at the accumulation and the decumulation phase, even in actuarial terms. Consider a deferred life annuity of one monetary unit per annum, with a deferred period of n years. Assume now that each annual payment is due at the beginning of the year (annuity in advance). The actuarial value at time 0, n| a¨ x , is given by ¨x = n| a

ω−x

(1 + i)−h h px

(1.58)

h=n

In this context, it is natural that the accumulation period coincides with the deferred period. In particular, the deferred annuity can be financed via a sequence of n annual level premiums P, paid at times 0, 1, . . . , n − 1. The annual premium for a deferred life annuity of b per annum, according to the equivalence principle, is then given by P=b

¨x n| a a¨ x:n

(1.59)

1.6


where a¨ x:n =

n−1

(1 + i)−h h px

(1.60)

h=0

Two important aspects of the actuarial structure of deferred life annuities financed by annual level premiums, as is apparent from equations (1.58) and (1.60), should be stressed: (a) Formulae (1.58) and (1.60) rely on the assumption that the technical basis is chosen at time 0, when the insured is aged x. If for example x = 40, this means that the technical rate of interest will be guaranteed throughout a period of, maybe, fifty years or even more. Further, the life table adopted should keep its validity throughout the same period. (b) In the case that the policyholder dies before time n, no benefit is due. This is, of course, a straight consequence of the policy structure, according to which the only benefit is the deferred life annuity. Feature (b) is likely to have a negative impact on the appeal of the annuity product. However, the problem can be easily removed by adding to the policy a rider benefit such as the return of premiums in case of death during the deferred period, or including some death benefit with term n. The problems arising from aspect (a) are much more complex, and require a re-thinking of the structure and design of the life annuity product. As a first step, we provide an analysis of the main features of life annuity products, addressing separately the accumulation period and the decumulation period.

1.6.2

The accumulation period

The deferred life annuity, as described above, can be interpreted as a pure endowment at age x with maturity at age x + n, ‘followed’ (in the case of survival at age x + n) by an immediate life annuity, with benefits due at the beginning of each year. In formal terms, from (1.58) we obtain ¨ x = (1 + i)−n n px n| a

ω−x−n

(1 + i)−h h px+n = n Ex a¨ x+n

(1.61)

h=0

(1 + i)−n

where n Ex = n px denotes the actuarial value of a pure endowment with a unitary amount insured. Clearly, relation (1.61) relies on the assumption that the same technical basis is adopted for both the period of accumulation and decumulation. As

33

1 : Life annuities

already noted, this implies a huge risk for the life annuity provider. So, an important idea is to address separately the two periods, possibly delaying the choice of the technical basis to be adopted for the life annuity. As regards the accumulation period, the pure endowment can be replaced by a purely financial accumulation, via an appropriate savings instrument. Then, the loss in terms of the mutuality effect is very limited when (part of) the working period is concerned. Hence, a very modest extra-yield can replace the mortality drag. Example 1.13 In Fig. 1.18, the function θ (see Section 1.4.1) is plotted against the age in the range 40–64. Note that θ is consistent with formula (1.34), with given values for the mathematical reserve, however with b = 0. The underlying technical basis is the first-order basis adopted in Example 1.4. It is interesting to compare the graph in Fig. 1.18 (noting the scale on the vertical axis) with the behaviour of the function θ throughout the decumulation period, illustrated in Fig. 1.6. Of course, various insurance products including a benefit in case of life at time n can replace the pure endowment throughout the accumulation period. Examples are given by the traditional endowment assurance policy, by various types of unit-linked endowments, and so on. In many cases, some minimum guarantee is provided: for example, the technical rate of interest in traditional insurance products like pure endowments and endowment

0.007 0.006 0.005 0.004 u

34

0.003 0.002 0.001 0 40

45

50

55 Age

Figure 1.18.

Mutuality effect during the accumulation period.

60

65

1.6


assurances, a minimum death benefit and/or a minimum maturity benefit in unit-linked products. Whatever the insurance product may be, the benefit at maturity can be used to purchase an immediate life annuity. However, the ‘quality’ of the insurance product used for the accumulation can be improved, from the perspective of the policyholder, including in the product itself an option to annuitize. This option is the possibility of converting the lump sum at maturity into an immediate life annuity, without the need to cash in the sum and pay the expense charges related to the underwriting of the life annuity. Clearly, when an option to annuitize is included in the policy, the insurer first takes the adverse selection risk, as the policyholders who choose the conversion into a life annuity will presumably be in good health, with a life expectancy higher than the average. However a further risk may arise, due to the uncertainty in the future mortality trend, that is, the longevity risk. If the annuitization rate, that is the quantity 1/äx+n , which is applied to convert the sum available at maturity into an immediate life annuity is stated (and hence guaranteed) just at maturity, the time interval throughout which the insurer bears the longevity risk clearly coincides with the time interval during which the life annuity is paid. However, more ‘value’ can be added to the annuity product if the annuitization rate is guaranteed during the accumulation period, the limiting case being represented by the annuitization rate guaranteed at time 0, that is, at policy issue. The opposite limit is clearly given by stating the guaranteed rate at time n, that is, at maturity. The so-called guaranteed annuity option (GAO) is a policy condition which provides the policyholder with the right to receive at retirement either a lump sum (the maturity benefit) or a life annuity, whose annual amount is calculated at a guaranteed rate. The annuity option will be exercised by the policyholder if the current annuity rate (i.e. the annuity rate applied by insurers at time n for pricing immediate life annuities) will be worse than the guaranteed one. As regards the accumulation period, the severity of the longevity risk borne by the life annuity provider can be reduced (with respect to the severity involved in a GAO with a guaranteed rate that is stated at the policy issue date) if the annuity purchase is arranged according to a singlerecurrent premium scheme. In this case, with the premium paid at time h (h = 0, 1, . . . , n − 1) a deferred life annuity of annual amount bh , with deferred period n − h, is purchased. In actuarial terms: Ph = bh n−h| a¨ [h] x+h

(1.62)

35

36

1 : Life annuities

Note that the actuarial value n−h| a¨ [h] is calculated according to the x+h technical basis adopted at time h. Hence, the annuity total benefit b, given by b=

n−1

bh

(1.63)

h=0

is ultimately determined and guaranteed at time n − 1 only. According to this step-by-step procedure, the technical basis, used in (1.62) to determine the amount bh purchased with the premium Ph , can change every year, so reflecting possible adjustments in the mortality forecast.

1.6.3

The decumulation period

Let us denote with n the starting point of the decumulation period, and with x + n the annuitant’s age. Let S be the amount, available at time n, to finance the life annuity. In the case of a deferred life annuity, S is given by the mathematical reserve at time n of the annuity itself. The relation between S and the annual payment b depends on the policy conditions which define the (random) number of payments, and hence the duration of the decumulation period. Let us denote with K the number of payments. Focussing on a life annuity in arrears only, the following cases are of practical interest: (a) If the number of annual payments is stated in advance, say K = m, we have an annuity-certain, that is, a simple withdrawal process. Then, the annual benefit b is defined by the following relation: S = b am

(1.64)

(b) In the case of a whole life annuity, the annual payments cease with the death of the annuitant. Thus, K = Kx+n (where Kx+n denotes the curtate remaining lifetime), and S = b ax+n

(1.65)

(c) The m−year temporary life annuity pays the annual benefit while the annuitant survives during the first m years. Then K = min{m, Kx+n }, and S = b ax:m =

m

(1 + i)−h h px+n

(1.66)

h=1

(d) If the annuitant dies soon after time n, neither the annuitant nor the annuitant’s estate receive much benefit from the purchase of the life

1.6


annuity. In order to mitigate (at least partially) this risk, it is possible to buy a life annuity with a guarantee period (5 or 10 years, say), in which case the benefit is paid for the guarantee period regardless of whether the annuitant is alive or not. Hence, for a guarantee period of r years we have K = max{r, Kx+n }, and S = b ar + b r| ax+n

(1.67)

We have so far assumed that the annuity payment depends on the lifetime of one individual only, namely the annuitant. However, it is possible to define annuity models involving two (or more) lives. Some examples (referring to two lives) follow: (e) Consider an annuity payable as long as at least one of two individuals (the annuitants) survives, namely a last-survivor annuity. Let now denote by y and z respectively the ages of the two lives at the annuity (1) (2) commencement, and with Ky , Kz their curtate remaining lifetimes. (1) (2) Thus, K = max{Ky , Kz }. The actuarial value of this annuity is usually denoted by ay,z , and can be expressed as (2)

ay,z = a(1) y + az − ay,z

(1.68)

where the suffices (1), (2) denote the life tables (e.g. referring to males and females respectively) used for the two lives, whereas ay,z denotes the actuarial value of an annuity of 1 per annum, payable while both individuals are alive (namely a joint-life annuity). Hence, (2)

S = b ay,z = b (a(1) y + az − ay,z )

(1.69)

Note that, if we accept the hypothesis of independence between the two random lifetimes, we have ay,z =

+∞

(2)

(1 + i)−h h p(1) y h pz

(1.70)

h=1

In equation (1.69) it has been assumed that the annuity continues with the same annual amount until the death of the last survivor. A modified form provides that the amount, initially set to b, will be reduced following the first death: to b if the individual (2) dies first, and to b if the individual (1) dies first. Thus (2)

S = b a(1) y + b az + (b − b − b ) ay,z

(1.71)

with b < b, b < b. Conversely, in many pension plans the last-survivor annuity commonly provides that the annual payment is reduced only if

37

38

1 : Life annuities

the retiree, say life (1), dies first. Formally, b = b (instead of b < b) in equation (1.71). (f) A reversionary annuity (on two individuals) is payable while a given individual, say individual (2), is alive, but only after the death of the other individual. In this case, the number of payments is K = (1) (2) max{0, Kz − Ky }, and the first payment (if any) is made at time (1) Ky + 1. Such an annuity can be used, for example, as a death benefit in pension plans, to be paid to a surviving spouse or dependant.

1.6.4

The payment profile

Level annuities (sometimes called standard annuities) provide an income which is constant in nominal terms. Thus, the payment profile is flat. A number of models of ‘varying’ annuities have been derived, mainly with the purpose of protecting the annuitant against the loss of purchasing power because of inflation. First, we focus on escalating annuities. (a) In the fixed-rate escalating annuity (or constant-growth annuity) the annual benefit increases at a fixed annual rate, α, so that the sequence of payments is b1 ,

b2 = b1 (1 + α),

b3 = b1 (1 + α)2 ,

...

Usually, the premium is calculated accounting for the annual increase in the benefit. Thus, for a given amount S (the single premium of the immediate life annuity), the starting benefit b1 is lower than the benefit the annuitant would get from a level annuity. Various types of index-linked escalating annuities are sold in annuity and pension markets. Two examples follow: (b) Inflation-linked annuities provide annual benefits varying in line with some index, for example a retail-price index (like the RPI in the UK), usually with a stated upper limit. An annuity provider should invest the premiums in inflation-linked assets so that these back the annuities where the payments are linked to a price index. (c) Equity-indexed annuities earn annual interest that is linked to a stock or other equity index (e.g., the Standard & Poor’s 500). Usually, the annuity promises a minimum interest rate. Moving to investment-linked annuities, we focus on the following models:

1.6


(d) In a with-profit annuity (typically in the UK market), the single premium is invested in an insurer’s with-profit fund. Annual benefits depend on an assumed annual bonus rate (e.g. 5%), and on the sequence of actual declared bonus rates, which in turn depend on the performance of the fund. In each year, the annual rate of increase in the annuity depends on the spread between the actual declared bonus and the assumed bonus. Clearly, the higher is the assumed bonus rate, the lower is the rate of increase in the annuity. The benefit decreases when the actual declared bonus rate is lower than the assumed bonus rate. Although the annual benefit can fluctuate, with-profit annuities usually provide a guaranteed minimum benefit. (e) Various profit participation mechanisms (other than the bonus mechanism described above in respect of with-profit annuities) are adopted, for example, in many European continental countries. A share (e.g. 80%) of the difference between the yield from the investments backing the mathematical reserves and the technical rate of interest (i.e. the minimum guaranteed interest, say 2% or 3%) is credited to the reserves. This leads to increasing benefits, thanks to the extrayield. (f) The single premium of a unit-linked life annuity is invested into unitlinked funds. Generally, the annuitant can choose the type of fund, for example medium risk managed funds, or conversely higher risk funds. Each year, a fixed number of units are sold to provide the benefit payment. Hence, the benefit is linked directly to the value of the underlying fund, and then it fluctuates in line with unit prices. Some unit-linked annuities, however, work in a similar way to with-profit annuities. An annual growth rate (e.g. 6%) is assumed. If the fund value grows at the assumed rate, the benefit stays the same. If the fund value growth is higher than assumed, the benefit increases, whilst if lower the benefit falls. Some unit-linked funds guarantee a minimum performance in line with a given index. We conclude this section addressing some policy conditions which provide a ‘final’ payment, namely some benefit after the death of the annuitant. The complete life annuity (or apportionable annuity) is a life annuity payable in arrears which provides a pro-rata adjustment on the death of the annuitant, consisting in a final payment proportional to the time elapsed since the last payment date. Clearly, this feature is more important if the annuity is paid annually, and less important in the case of, say, monthly payments.

39

40

1 : Life annuities

Capital protection represents an interesting feature of some annuity policies, usually called value-protected annuities. Consider, for example, a single-premium, level annuity. In the case of early death of the annuitant, a value-protected annuity will pay to the annuitant’s estate the difference (if positive) between the single premium and the cumulated benefits paid to the annuitant. Usually, capital protection expires at some given age (75, say), after which nothing is paid even though the difference above mentioned is positive. The capital protection benefit can be provided in two ways: – in a cash-refund annuity, the balance is paid as a lump sum; – in an instalment-refund annuity the balance is paid in a sequence of instalments. Adding capital protection clearly reduces the annuity benefit (for a given single premium). Remark Note that capital protection constitutes a death benefit, which is decreasing as the age at death increases and hence the number of annual benefits paid to the annuitant increases. For this reason, capital protection can help in building-up a (partial) ‘natural hedging’ of mortality-longevity risks inside the annuity product. See Section 7.3.2.

1.6.5

About annuity rates

The price of life annuities depend on several ‘risk factors’. In particular, (a) (b) (c) (d)

age at time of annuity purchase; gender; voluntary annuities versus pension annuities; information available to the insurer about the annuitant’s expected lifetime are important factors.

The importance of factor (a) is self-evident. Risk factor (b) is usually taken into account, because of the difference between the age-pattern of mortality in males and females. However, in uni-sex annuities the same annuity rate (for a given age at entry) is adopted for males and females. These annuities involve a solidarity effect (see Section 1.4.2) in the sense that men cross-subsidize women. The term voluntary annuities (see point (c)) usually denotes annuities bought as a consequence of individual choice, that is exercised on a voluntary basis. Conversely, the term pension annuities refers to benefits paid

1.6


to people as a direct consequence of their membership of an occupational pension plan, or to annuities bought because a compulsory purchase mechanism works. Voluntary annuities are usually purchased by people with a high life expectancy, whereas individuals who know that they have a low expected lifetime are unlikely to purchase an annuity. The consequence is that actual voluntary annuitants have a mortality pattern different from the population as a whole. This fact is known as adverse selection (from the point of view of the life insurer). In terms of annuity rates, adverse selection leads to higher premiums for voluntary annuities, compared with pension annuities. As regards point (d), insurers offer lower prices, that is, sell special-rate annuities, to people with an expected lifetime lower than the average one (or, equivalently, a higher annual benefit for a given single premium). In particular, – impaired-life annuities can be sold to people having health problems certified by a doctor (e.g. diabetes, chronic asthma, high blood pressure, cancer, etc.); – enhanced annuities can be purchased by people who self-certify the presence of some cause of a higher mortality level, like being overweight, or being a regular smoker. Remark Enhanced annuities should not be confused with enhanced pensions, which provide an uplift of the annual benefit if the annuitant enters a senescent disability state (namely, in the case of a ‘Long-Term Care’ claim).

1.6.6

Variable annuities and GMxB features

In the previous sections, various ‘guarantees’ have been addressed; for example: minimum guarantees like the guaranteed interest rate in the accumulation period (Section 1.6.2), a guaranteed minimum annual benefit in with-profit annuities, a minimum interest rate in equity-indexed annuities, a minimum performance in unit-linked annuities, a minimum total payout via capital protection mechanisms (Section 1.6.4). Packaging a range of guarantees is a feature of variable annuities. These products are unit-linked investment policies, providing deferred annuity benefits. The annuity can be structured as a level annuity or a unit-linked annuity (see Section 1.6.4). The guarantees, commonly referred to as GMxBs (namely, Guaranteed Minimum Benefits of type ‘x’), include minimum benefits both in case of

41

42

1 : Life annuities

death and in case of life. The GMxBs are usually defined in terms of the amount resulting from the accumulation process (the account value) at some point of time, compared with a given benchmark (which may be expressed in terms of the interest rate, a fixed benefit amount, etc.). One or more than one GMxB can be included in the policy as a rider to the basic variable annuity product. A brief description of some GMxBs follow: (a) GMDB = Guaranteed Minimum Death Benefit. The GMDB guarantees a minimum lump sum benefit payable upon the annuitant’s death. The GMDB can be defined in several ways; for example: – return of premiums consists in the payment of the greater of the amount of premiums paid and the account value; – highest anniversary value pays the greater of the highest account value at past anniversaries and the current account value (hence, according to a ratchet mechanism); – roll-up consists in the payment of the higher of an amount equal to the premiums paid accumulated at a given interest rate (say, 5%) and the account value. The GMDB typically expires either at the end of the accumulation period, or when a given time (say, 10 years) has elapsed since the commencement of the decumulation period. (b) GMAB = Guaranteed Minimum Accumulation Benefit. The GMAB can be exercised at pre-fixed dates (during the accumulation period); the policyholder receives, as the surrender value, a lump sum equal to the higher of the guaranteed amount and the account value. For example the guaranteed amount can be determined as the premiums paid accumulated at a given interest rate (say, 5%) according to a rollup rule, and can be paid for example at the 10-th anniversary (measured from the beginning of the accumulation period). (c) GMIB = Guaranteed Minimum Income Benefit. The term ‘income’ refers to (annual) amounts payable to the annuitant. The policyholder receives the higher of the guaranteed amount and the account value, payable as an annuity whose annual benefit is determined according to a given interest rate and life table. The guaranteed amount is typically calculated according to a roll-up accumulation or an annual ratchet. Hence, the GMIB guarantees a minimum annual income upon annuitization. (d) GMWB = Guaranteed Minimum Withdrawal Benefit. The policyholder receives the greater of return of premiums and the account value, payable as a sequence of periodic withdrawals throughout time. For

1.7

References and suggestions for further reading

example, the GMWB might guarantee that the policyholder will receive for 20 years an annual amount equal to 5% of the premiums paid. Some policies do not allow the policyholder to withdraw money after the commencement of the annuity payments. GMAB, GMIB, and GMWB are commonly referred to as GLB, namely Guaranteed Living Benefits. All GMxBs have option-like characteristics. However, the possible utilization of the GMDB follows the age-pattern of mortality, and hence can be assessed using a life table (together with assumptions about the performance of the financial market). Conversely, the utilization of a GLB depends on the policyholder’s behaviour, and hence the assessment of its impact is much more difficult.

1.7


In this section, we only quote textbooks and papers dealing with general aspects of life annuity products. Studies particularly devoted to longevity risk in life annuity portfolios and pension plans will be quoted in the relevant sections of the following chapters. Basic actuarial aspects of life annuities (namely expected present values, premium calculation, mathematical reserves) are dealt with in almost all of the main textbooks of actuarial mathematics and life insurance techniques. The reader can refer, for example, to Bowers et al. (1997), Gerber (1995), Gupta and Varga (2002), Rotar (2007). As regards the notation, the use of symbols like aKx (see (1.43)) can be traced back to de Finetti. Actually, de Finetti (1950, 1957) focussed on the random present value of insured benefits. For example, in the agecontinuous context, – the random present value of the whole life assurance (with a unitary sum assured) is (1 + i)−Tx , and then, according to usual actuarial notation, the expected present value is ¯ x = E[(1 + i)−Tx ] A – the random present value of the standard endowment is (1 + i)− min{Tx ,n} , and hence ¯ x,n = E[(1 + i)− min{Tx ,n} ] A

43

44

1 : Life annuities

As regards the stochastic approach to actuarial values, see also the seminal contribution by Sverdrup (1952). Mortality risks in life annuities are analysed by McCrory (1986). The objectives and main design features of life annuity products are extensively dealt with by Black and Skipper (2000). We have mainly referred to this textbook in Section 1.6. Various papers and reports have been recently devoted to innovation in life annuity products, especially addressing the impact of longevity risk. See, for example Cardinale et al. (2002), Department for Work and Pensions (2002), Retirement Choice Working Party (2001), Richard and Jones (2004), Wadsworth et al. (2001), Swiss Re (2007), Blake and Hudson (2000). Variable annuities are addressed in particular by Sun (2006) and O’Malley (2007). The book by Milevsky (2006) constitutes an updated reference in the context of life annuities and post-retirement choices. Great effort has been devoted to the analysis of life annuities from an economic perspective, in particular in the framework of wealth management and human life cycle modelling. We only cite the seminal contribution by Yaari (1965), whereas for other bibliographic suggestions the reader can refer to Milevsky (2006). The extra yield defined in Section 1.4.1 is the key element behind the seminal result of Yaari (1965). He shows that a risk averse, life cycle consumer facing an uncertain time of death would, under certain assumptions (e.g. the absence of bequest, and the absence of other sources of randomness), find it optimal to invest 100% of his/her wealth in an annuity (priced on an actuarially fair basis). An extensive discussion on the concepts of mutuality and solidarity (however with some terms used with a meaning different from that adopted in the present chapter) is provided by Wilkie (1997). Finally, some references concerning the history of life annuities and the related actuarial modelling follow. For the early history of life annuities the reader can refer to Kopf (1926). The paper by Hald (1987) is more oriented to actuarial aspects, and constitutes an interesting introduction to the early history of life insurance mathematics. Haberman (1996) provides extensive information about the history of actuarial science up to 1919, while in Haberman and Sibbett (1995) the reader can find the reproduction of a number of milestone papers in actuarial science. The papers by Pitacco (2004a) and Pitacco (2004c) mainly deal with the evolution of mortality modelling, ranging from Halley’s contributions to the awareness of longevity risk.

2

2.1

The basic mortality model

Introduction

Some elements of the basic mortality model underlying life insurance, life annuities and pensions have been already introduced in Chapter 1, while presenting the structure of life annuities; see in particular Sections 1.2 and 1.3. In Chapter 2, we consider the mortality model in more depth. We adopt a more structured presentation of the fundamental ideas, which means that some repetition of elements from Chapter 1 is unavoidable. However, new concepts are also introduced. In particular, an agecontinuous framework is defined in Section 2.3, in order to provide some tools needed when dealing with mortality projection models. Indices summarizing the probability distribution of the lifetime are described in Section 2.4, whereas parametric models (i.e. mortality ‘laws’) are presented in Section 2.5. Basic ideas concerning non-parametric graduation are introduced in Section 2.6. Transforms of the survival function are briefly addressed in Section 2.7. Less traditional topics, yet of great importance in the context of life annuities and mortality forecasts, are dealt with in Sections 2.8 and 2.9, respectively: mortality at very old ages (i.e. the problem of ‘closing’ the life table), and the concept of ‘frailty’ as a tool to represent heterogeneity in populations, due to unobservable risk factors. A list of references and suggestions for further readings (Section 2.10) conclude the chapter. As regards references to actuarial and statistical literature, in order to improve readability we have avoided the use of citations throughout the text of the first sections of this chapter, namely the sections devoted to traditional issues. Conversely, important contributions to more recent issues are cited within the text of Sections 2.8 and 2.9.

46

2 : The basic mortality model

2.2 2.2.1

Life tables Cohort tables and period tables

The life table is a (finite) decreasing sequence l0 , l1 , . . . , lω . The generic item lx refers to the integer age x and represents the estimated number of people alive at that age in a properly defined population (from an initial group of l0 individuals aged 0). The exact meaning of the lx ’s will be explained after discussing two approaches to the calculation of these numbers. First, assume that the sequence l0 , l1 , . . . , lω is provided by statistical evidence, that is by a longitudinal observation of the actual numbers of individuals alive at age 1, 2, . . . , ω, out of a given initial cohort consisting of l0 newborns. The (integer) age ω is the limiting age (say, ω = 115), that is, the age such that lω > 0 and lω+1 = 0. The sequence l0 , l1 , . . . , lω is called a cohort life table. Clearly, the construction of a cohort table takes ω + 1 years. Assume, conversely, that the statistical evidence consists of the frequency of death at the various ages, observed throughout a given period, for example one year. Assume that the frequency of death at age x (possibly after a graduation with respect to x) is an estimate of the probability qx . Then, for x = 0, 1, . . . , ω − 1, define lx+1 = lx (1 − qx )

(2.1)

with l0 (the radix) assigned (e.g. l0 = 100,000). Hence, lx is the expected number of survivors out of a notional cohort (also called a synthetic cohort) initially consisting of l0 individuals. The sequence l0 , l1 , . . . , lω , defined by recursion (2.1), is called a period life table, as it is derived from period observations. Remark Period observations are also called cross-sectional observations, as they analyse an existing population (in terms of the frequency of death) ‘across’ the various ages (or age groups). An important hypothesis underlying recursion (2.1) should be stressed. As the qx ’s are assumed to be estimated from mortality experience in a given period (say, one year), the calculation of the lx ’s relies on the assumption that the mortality pattern does not change in the future. As is well known, statistical evidence show that human mortality, in many countries, has declined over the 20th century, and in particular over its last decades (see Chapter 3). So, the hypothesis of ‘static’ mortality cannot be

2.2

Life tables

assumed in principle, at least when long periods of time are referred to. Hence, in life insurance applications, the use of period life tables should be restricted to products involving short or medium durations (5 to 10 years, say), like term assurances and endowment assurances, whilst it should be avoided when dealing with life annuities and pension plans. Conversely, these products require life tables which allow for the anticipated future mortality trend, namely projected tables constructed on the basis of the experienced mortality trend. For any given sequence l0 , l1 , . . . , lω it is usual to define dx = lx − lx+1 ;

x = 0, 1, . . . , ω

(2.2)

thus, dx is the expected number of individuals dying between exact age x and x + 1, out of the initial l0 individuals. Clearly, ω

dx = l0

(2.3)

x=0

2.2.2

‘Population’ tables versus ‘market’ tables

Mortality data, and hence life tables, can originate from observations concerning a whole national population, a specific part of a population (e.g. retired workers, disabled people, etc.), an insurer’s portfolio, and so on. Life tables constructed on the basis of observations involving a whole national population (usually split into females and males) are commonly referred to as population tables. Market tables are constructed using mortality data arising from a collection of insurance portfolios and/or pension plans. Usually, distinct tables are constructed for assurances (i.e. insurance products with a positive sum at risk, for example term and endowment assurances), annuities purchased on an individual basis, pensions (i.e. annuities paid to the members of a pension plan). The rationale for distinct market tables lies in the fact that mortality levels may significantly differ as we move from one type of insurance product to another. The case of different types of life annuities has been discussed in Section 1.6.5. Market tables provide experience-based data for premium and reserve calculations and for the assessment of expected profits. Population tables can provide a starting point when market tables are not available. Moreover, population tables usually reveal mortality levels higher than those expressed by market tables and hence are likely to constitute a prudential

47

48


(or ‘conservative’, or ‘safe-side’) assessment of mortality in assurance portfolios. Thus, population tables can be used when pricing assurances in order to include a profit margin (or an implicit safety loading) into the premiums. Indeed, in the early history of life insurance, population life tables were used in the calculation of premiums – and this prudential assessment of mortality led to many insurance companies making unanticipated profits.

2.2.3

The life table as a probabilistic model

We consider a person aged x, and denote by Tx the random variable representing his/her remaining lifetime. In actuarial calculations, probabilities like P[Tx > h] and P[h < Tx ≤ h + k] are usually involved. When a life table is available, these probabilities can be immediately derived from the life table itself, provided that the ages and durations are integers. In life insurance mathematics, a specific notation is commonly used for the probabilities of survival and death. The notation for the survival probability is as follows: (2.4) h px = P[Tx > h] where h is an integer. In particular 1 px can be simply denoted with px ; clearly 0 px = 1. The notation for the probability of death is as follows: h|k qx

= P[h < Tx ≤ h + k]

(2.5)

If h = 0 the notation k qx is used, and in particular, when h = 0 and k = 1, the symbol qx is commonly adopted. Trivially, 0 qx = 0. Note that, in all symbols, the right-hand side subscript denotes the age being considered. Conversely, the left-hand side subscript denotes some duration, whose meaning depends on the specific probability addressed. Starting from recursion (2.1), which defines the life table, and using well known theorems of probability theory, we can calculate probabilities of survival and death. Obviously, for the probability qx (called the annual probability of death) we have lx+1 dx qx = 1 − = (2.6) lx lx and hence, for the probability px (called the annual survival probability), px = 1 − qx

(2.7)

2.2

Life tables

Remark Sometimes the one-year probabilities qx and px are called ‘mortality rate’ and ‘survival rate’ respectively. We do not use these expressions to denote probability of death and survival, as the term ‘rate’ should be referred to a counter expressing the number of events per unit of time. In general, for the survival probability we have h px

= px px+1 . . . px+h−1 =

lx+h lx

(2.8)

while for the probabilities of dying we have k qx

and h|k qx

lx+k lx

(2.9)

lx+h − lx+h+k lx

(2.10)

= 1 − k px = 1 −

= h px k qx+h =

Note that the sequence 0|1 qx , 1|1 qx , . . . , ω|1 qx

(2.11)

constitutes the probability distribution of the random variable Kx , usually called the curtate remaining lifetime and defined as the integer part of Tx ; thus, the possible outcomes of Kx are 0, 1, . . . , ω − x. Further useful relations are as follows: h|k qx

= h+k qx − h qx

(2.12)

h|k qx

= h px − h+k px

(2.13)

When qx can be expressed as qx = φx /(1 + φx ), the function φx represents the so-called mortality odds, namely φx =

qx px

(2.14)

From 0 < qx < 1 (for x < ω), it follows that φx > 0. Thus, focussing on the odds, rather than the annual probabilities of dying, can make easier the choice of a mathematical formula fitting the age-pattern of mortality (see Section 2.5), as the only constraint is the positivity of the odds.

2.2.4

Select mortality

Consider, for example, a group of insureds, all age 45, deriving from a population whose mortality can be described by a given life table. Is q45 (drawn

49

50


from the assumed life table) a reasonable assessment of the probability of dying for each insured in the group? In order to answer this question, the following points should be addressed: (a) When starting a life insurance policy with an insurance company, an individual may be subject to medical screening and, possibly, to a medical examination. An individual, who passes such tests and who is not charged any extra premium, is often called a ‘standard risk’. (b) It has been observed that the mortality experienced by policyholders recently accepted (as standard risks) is lower than the mortality experienced by policyholders (of the same age) with a longer duration since policy issue. So, the answer to the above question is negative if the insureds have entered insurance in different years: it is reasonable to expect that an individual, who has just bought insurance, will be of better health than an individual who bought insurance several years ago. Hence, the attained age (45, in the example) should be split as follows: attained age = age at entry + time since policy issue The following notation is usually adopted to denote the annual probabilities of death for an insured age 45: q[45] , q[44]+1 , . . . , q[40]+5 , . . . where the number in square brackets denotes the age at policy issue, whereas the second number denotes the time since policy issue. In general, q[x]+u denotes the probability of an individual currently aged x + u, who bought insurance at age x, dying within one year. According to point (b), it is usual to assume: q[45] < q[44]+1 < · · · < q[40]+5 < · · · However, experience shows that it is reasonable to assume that the selection effect vanishes after some years, say r years after policy issue. So, in general terms, we can assume: q[x] < q[x−1]+1 < · · · < q[x−r]+r = q[x−r−1]+r+1 = · · · = qx

(2.15)

where qx denotes the probability of an individual currently age x, who bought insurance more than r years ago, dying within one year. The period r is called the select period.

2.3

Moving to an age-continuous context

Referring now to a person who bought insurance at age x, and assuming a select period of r = 3 years, the following probabilities should be used: q[x] , q[x]+1 , q[x]+2 , qx+3 , qx+4 , . . .

(2.16)

We denote by xmin and xmax the minimum and respectively the maximum age at entry. The set of sequences (2.16), for x = xmin , xmin +1 , . . . , xmax , is called a select table. In particular, the table used after the select period is called an ultimate life table. Conversely, life tables in which mortality depends on attained age only (as is the case for the life tables described in Section 2.2.1) are called aggregate tables. Select mortality also concerns life annuities. The person purchasing a life annuity is likely to be in a state of good health, and hence it is reasonable to assume that her/his probabilities of death, for a certain period after policy issue, are lower than the probabilities of other individuals with the same age. In this case, a self-selection effect works. Remark The selection effect, due to medical ascertainment (in the case of insurances with death benefit) or self-selection (in the case of life annuities), operates during the first years after policy issue, and the related age-pattern of mortality is often called issue-select. Another type of selection is allowed for, when some contingency can adversely affect the individual mortality. For example, in actuarial calculations regarding insurance benefits in the case of disability, the mortality of disabled policyholders is usually considered to be dependent on the time elapsed since the time of disablement inception (as well as on the attained age). In this case, the mortality is called inception-select.

2.3 2.3.1

Moving to an age-continuous context The survival function

Suppose that we have to evaluate the survival and death probabilities (like (2.8), (2.9) and (2.10)) when ages and times are real numbers. Tools other than the life table (as described in Section 2.2) are then needed. Assume that the function S(t), called the survival function and defined for t ≥ 0 as follows: S(t) = P[T0 > t]

(2.17)

51

52


has been assigned. Clearly, T0 denotes the random lifetime for a newborn. In the age-continuous framework, it is usual to assume that the possible outcomes of Tx lie in (0, +∞); nonetheless, we can assume that the probability measure outside the interval (0, ω) is zero, where ω is the limiting age. Consider the probability (2.4); we have

P[Tx > h] = P[T0 > x + h | T0 > x] = we then find h px

=

P[T0 > x + h] P[T0 > x]

S(x + h) S(x)

(2.18)

(2.19)

For probability (2.5), via the same reasoning, we obtain h|k qx

=

S(x + h) − S(x + h + k) S(x)

and, in particular k qx

=

S(x) − S(x + k) S(x)

(2.20)

(2.21)

Turning back to the life table, we note that, since lx is the expected number of people alive at age x out of a cohort initially consisting of l0 individuals,we have: lx = l0 P[T0 > x] (2.22) and, in terms of the survival function, lx = l0 S(x)

(2.23)

(provided that all individuals in the cohort have the same age-pattern of mortality, described by S(x)). Thus, the lx ’s are proportional to the values which the survival function takes on integer ages x, and so the life table can be interpreted as a tabulation of the survival function. Remark If a mathematical formula has been chosen to express the function S(t), ‘exact’ survival and death probabilities can be calculated, with ages and times given by real numbers. Conversely, when the survival function is tabulated at integer ages only, for example, derived from the life table setting S(x) = lx /l0 (see (2.23)), approximate methods are needed to calculate survival and death probabilities at fractional ages. Some of these methods are described in Section 2.3.5. Figure 2.1(a) illustrates the typical behaviour of the survival function S(x). This behaviour reflects results of statistical observations on mortality, as we will see in Chapter 3.

2.3

(a)


(b) 1 S(x)

S(x)

1

0

0 Age x

Figure 2.1.

Age x

Survival functions.

Figure 2.1(b) focusses on the dynamic aspects of mortality. In particular, two aspects (which emerge from mortality observations throughout time) can be singled out: – the survival curve moves (in a north easterly direction over time) towards a rectangular shape, and hence the term rectangularization is used to describe this feature; – the point of maximum downwards slope of the survival curve progressively moves towards the very old ages; this feature is called the expansion of the survival function. These aspects will be considered in more detail in Chapter 7, when dealing with longevity risk.

2.3.2

Other related functions

Other functions can be involved in age-continuous actuarial calculations. The most important is the force of mortality (or mortality intensity), dealt with in Section 2.3.3. In the present section we introduce the probability density function (pdf) and the distribution function of the random variable Tx , x ≥ 0. First, we focus on the random lifetime T0 . Let f0 (t) and F0 (t) denote, respectively, the pdf and the distribution function of T0 . In particular, F0 (t) expresses, by definition, the probability of a newborn dying within t years. Hence, F0 (t) = P[T0 < t] (2.24) or, according to the actuarial notation, F0 (t) = t q0

(2.25)

53


Of course, we have F0 (t) = 1 − S(t)

(2.26)

The following relation holds between the pdf f0 (t) and the distribution function F0 (t): t f0 (u) du (2.27) F0 (t) = 0

Usually it is assumed that, for t > 0, the pdf f0 (t) is a continuous function. Then, we have d d f0 (t) = F0 (t) = − S(t) (2.28) dt dt The pdf f0 (t) is frequently called the curve of deaths. Figure 2.2(a) illustrates the typical behaviour of the pdf f0 (t). Equation (2.28) justifies the relation between the curve of deaths and the survival curve (see Fig. 2.1(a)). In particular, we note that the point of maximum downward slope in the survival curve corresponds to the modal point (at adult-old ages) in the curve of deaths. Moving to the remaining lifetime at age x, Tx (x > 0), the following relations link the distribution function and the pdf of Tx with the analogous functions relating to T0 : Fx (t) = P[Tx < t] = d fx (t) = Fx (t) = dt

d dt

F0 (x + t) − F0 (x) P[x < T0 ≤ x + t] = P[T0 > x] S(x) F0 (x + t) S(x)

=

f0 (x + t) S(x)

(2.29) (2.30)

From functions Fx (t) and fx (t) (and in particular, via (2.29) and (2.30), from F0 (t) and f0 (t)), all of the probabilities involved in actuarial

(a)

(b)

mx

f0(x)

54

0

0 Age x Figure 2.2.

Probability density function and force of mortality.

Age x

2.3


calculations can be derived. For example: +∞ +∞ 1 p = 1 − F (t) = f (u) du = f0 (x + u) du t x x x S(x) t t

2.3.3

(2.31)

The force of mortality

We refer to an individual age x, and consider the probability of dying before age x + t (with x and t real numbers), namely t qx . The force of mortality (or mortality intensity) is defined as follows:

P[Tx ≤ t] t qx = lim t0 t0 t t

µx = lim

(2.32)

and hence it represents the instantaneous rate of mortality at a given age x. In reliability theory, this concept is usually referred to as the failure rate or the hazard function. From

P[Tx ≤ t] = Fx (t) = we obtain

F0 (x + t) − F0 (x) S(x)

f0 (x) F0 (x + t) − F0 (x) = t0 t S(x) S(x)

µx = lim or µx =

d − dx S(x)

S(x)

=−

d ln S(x) dx

(2.33)

(2.34)

(2.35)

Hence, once the survival function S(x) has been assigned, the force of mortality can be derived. Thus, the force of mortality does not add any information concerning the age-pattern of mortality, provided that this has been described in terms of S(x) (or f0 (x), or F0 (x)). Conversely, the role of the force of mortality is to provide a tool for a fundamental statement of assumptions about the behaviour of individual mortality as a function of the attained age. The Gompertz model for the force of mortality (see Section 2.5.1) provides an excellent example. Note that, as µx = f0 (x)/S(x) (see (2.34)), the relation between the graph of µx and the graph of f0 (x) (see Fig. 2.2) can be explained in terms of the behaviour of S(x). When S(x) is close to 1, the two graphs are quite similar, whereas as S(x) strongly decreases, µx definitely increases. From (2.35), with the obvious boundary condition S(0) = 1, we obtain:

x µu du (2.36) S(x) = exp − 0

55

56


As clearly appears from (2.36), the survival function S(x) can be obtained once the force of mortality has been chosen. Clearly, the possibility of finding a ‘closed’ form for S(x) strictly depends on the structure of µx . Relations between the force of mortality and the basic mortality functions relating to an individual age x can be easily found. For example, from (2.34) and (2.30), we obtain µx+t =

fx (t) f0 (x + t) = S(x + t) 1 − Fx (t)

(2.37)

and hence fx (t) = t px µx+t

(2.38)

Finally, the cumulative standard force of mortality (or cumulative hazard function) is defined as follows: x H(x) = µu du (2.39) 0

Remark A link between the quantities used in an age-discrete context (like lx , dx , etc.) and the quantities used in age-continuous circumstances (like S(x), f0 (x), etc.) may be of interest, especially when comparing and interpreting graphical representations of data provided by statistical experiences. The analogy between lx and S(x) immediately emerges from (2.23). As regards dx (see equation (2.2)), the analogy with the pdf f0 (x) follows from the fact that the former is minus the first-order difference of the function lx , while the latter is minus the derivative of the survival function S(x). Finally, an interesting link can be found between the probabilities and the pdf fx (t). The quantities h|1 qx

= h px qx+h ;

h|1 qx

h = 0, 1, . . . , ω

constitute the probability distribution of the curtate lifetime Kx (see (2.10) and (2.11)). Conversely, in age-continuous circumstances, the pdf of the probability distribution of Tx is given by fx (t) = t px µx+t ; t ≥ 0 (see (2.38)). The analogy between the right-hand sides of the two expressions is evident. Note, however, that fx (t) (as well as µx+t ) does not represent a probability, the probability of a person age x dying between age x + t and x + t + dt being given by fx (t) dt.

2.3

2.3.4


The central death rate

The behaviour of the force of mortality over the interval (x, x + 1) can be summarized by the central death rate at age x, which is usually denoted by mx . The definition is as follows: 1 mx =

0

S(x + u) µx+u du S(x) − S(x + 1) = 1 1 0 S(x + u) du 0 S(x + u) du

(2.40)

We note that mx is defined as the (age-continuous) weighted arithmetic mean of the force of mortality over (x, x + 1), the weighting function being the probability of being alive at age x + u, 0 < u ≤ 1, expressed in terms of the survival function S(x + u). 1 The integral 0 S(x+u) du can be approximated using the trapezoidal rule (and an approximation has to be used when only a life table is available). Then, we obtain an approximation to the central death rate: m ˜x=

S(x) − S(x + 1) (S(x) + S(x + 1))/2

(2.41)

Note that m ˜ x can also be expressed in terms of the annual probability of survival or the annual probability of death. Indeed, from (2.41) we immediately obtain: 1 − px 2 qx m ˜x =2 = (2.42) 1 + px 2 − qx

2.3.5

Assumptions for non-integer ages

Assume that a life table (as described in Section 2.2) is available. How to obtain the survival function for all real ages x, and probabilities of death and survival for all real ages x and durations t? In what follows, we describe three approximate methods widely used in actuarial practice: (a) Uniform distribution of deaths. Relation (2.23) suggests a practicable approach. First, set S(x) = lx /l0 for all integer x using the available life table. Then, for x = 0, 1, . . . , ω − 1 and 0 < t < 1, define S(x + t) = (1 − t) S(x) + t S(x + 1)

(2.43)

and assume S(x) = 0 for x > ω, and so the survival function is a piecewise linear function. It easy to prove that, from (2.43) we obtain in particular t qx = t qx , that is, a uniform distribution of deaths between

57

58


exact ages x and x + 1, whence the name of this approximation. It is also easy to prove that, from (2.43) and (2.35), µx+t =

qx 1 − t qx

(2.44)

so that µx+t is an increasing function of t in the interval 0 < t < 1. (b) Constant force of mortality. Let us assume, for 0 < t ≤ 1 µx+t = µ(x)

(2.45)

where µ(x) denotes a value estimated from mortality observations. It follows, in particular, t px = e−tµ(x) . This assumption, consisting in a piece-wise constant force of mortality, is frequently adopted in actuarial calculations. We note that, from (2.40), mx = µ(x)

(2.46)

(c) The Balducci assumption. Let us define, for 0 < t ≤ 1 t qx

=

t qx 1 − (1 − t) qx

(2.47)

The Balducci assumption has an important role in traditional actuarial techniques for constructing life tables from mortality observations. However, it is possible to prove that, from (2.47) and (2.35), µx+t =

qx 1 − (1 − t) qx

(2.48)

so that µx+t is a decreasing function of t in the interval 0 < t < 1: for most ages, this would be an undesirable consequence of the Balducci assumption.

2.4

Summarizing the lifetime probability distribution

Age-specific functions are usually needed in actuarial calculations. For example, in the age-discrete context functions like lx , qx , etc. are commonly used, whereas, for age-continuous calculations, the survival function S(x) or the force of mortality µx are the usual starting points. Nevertheless, the role of single-figure indices (or markers), summarizing the lifetime probability distribution, should not be underestimated. In particular, important features of past mortality trends can be singled out by focussing on the behaviour of some indices over time, as we will see in Chapter 3.

2.4

2.4.1


The life expectancy

In the age-continuous context, the life expectancy (or expected lifetime) for a newborn, denoted with e¯ 0 , is defined as follows: ∞ e¯ 0 = E[T0 ] = t f0 (t) dt (2.49) 0

integrating by parts, we also find in terms of the survival function: ∞ S(t) dt (2.50) e¯ 0 = 0

The definition can be extended to all (real) ages x. So, the expected remaining lifetime at age x is given by ∞ t fx (t) dt (2.51) e¯ x = E[Tx ] = 0

and also, integrating by parts, by e¯ x =

1 S(x)

∞ 0

S(x + t) dt

(2.52)

Note that, for an individual age x, the random age at death can be expressed as x + Tx , and then the expected age at death is given by x + E[Tx ] = x + e¯ x

(2.53)

For all x, x > 0, the following inequality holds: x + e¯ x ≥ e¯ 0

(2.54)

The expected lifetime is often used to compare mortality in various populations. In this regard, the following aspects should be stressed. The definition of e¯ x is based on the probability distribution of the lifetime conditional on being alive at age x. Thus, when x = 0 the probability distribution involved has the pdf f0 (t) (see (2.49)), and hence mortality at all ages contributes to the value of e¯ 0 , in particular, for example, the infant mortality. Conversely, if x > 0 the conditional pdf fx (t) is involved, and so the age-pattern of mortality beyond age x only determines the value of e¯ x . The expected value of the curtate lifetime Kx is called the curtate expectation of life at age x. It is usually denoted by ex , and is defined as follows: ω−x k k|1 qx (2.55) ex = E[Kx ] = k=0

59

60


From (2.55), the following simpler expression can be derived: ex =

ω−x

k px

(2.56)

k=1

Another interesting quantity is the so-called complete expectation of life at age x, defined as follows:

◦ ex = E Kx + 12 = ex + 12 (2.57) This quantity can be taken as an approximation to e¯ x , and is useful ◦ when only a life table is available. Thus, it is possible to prove that ex is an approximation to e¯ x by applying the trapezoidal rule to the integral in (2.52). Remark Age-specific functions (namely, functions of age x), like lx , qx , ex , etc. in the age-discrete context, and S(x), f0 (x), µx , e¯ x , etc. in the agecontinuous context, are frequently named biometric functions (or life table functions, even in the age-continuous context). It should be noted that, once one of certain of these functions has been assigned, the other functions (in the same context) can be derived. For example, in age-discrete calculations from the lx values we can derive the functions qx , ex , etc.; in the age-continuous framework, from the force of mortality µx the survival function can be calculated and then all of the probabilities of interest.

2.4.2

Other markers

As it is well known in probability theory, the expected value provides a location measure of a probability distribution, and this is also the case for the random lifetime T0 (or Tx in general). Other location measures can be used to summarize the probability distribution of the random lifetime. In particular: – the modal value (at adult ages) of the curve of death, Mod[T0 ], also called the Lexis point; – the median value of the probability distribution of T0 , Med[T0 ], or median age at death. A number of variability measures can be used to summarize the dispersion of the probability distribution of the lifetime. As we will see in Chapter 3, in a dynamic context interesting information about the rectangularization

2.4


process can be obtained from these characteristics. Some examples follow: – A traditional variability measure is provided by the variance of the random lifetime, Var[T0 ], or its standard deviation, σ0 = Var[T0 ] (2.58) – The coefficient of variation, defined as √ Var[T0 ] σ0 = CV[T0 ] = E[T0 ] e¯ 0 provides a relative measure of variability. – The entropy H[T0 ] is defined as follows: ∞ S(x) ln S(x) dx H[T0 ] = − 0 ∞ 0 S(x) dx

(2.59)

(2.60)

thus, the entropy is minus the mean value of ln S(x), weighted by S(x); it is possible to prove that, as deaths become more concentrated, the value of H declines and, in particular, H = 0 if the survival function has a perfectly rectangular shape. – As deaths become more concentrated in an increasingly narrow interval, the slope of the survival curve becomes steeper. A simple variability measure is thus the maximum downward slope of the graph of S(x) in the adult and old age range. Thus, a lower variability implies a steeper slope. Formally, the slope at the point of fastest decline is d max − S(x) = max{S(x) µx } = max{f0 (x)} (2.61) x x x dx Note that the point of fastest decline is Mod[T0 ], that is, the Lexis point. Further characteristics of the random lifetime follow: – the probability of a new born dying before a given age x1 , x1 q0

= 1 − S(x1 )

(2.62)

which, for x1 small (say 1, or 5), provides a measure of infant mortality; – the percentiles of the probability distribution of T0 ; in particular, the 10-th percentile, usually called endurance, is defined as the age ξ such that S(ξ) = 0.90

(2.63)

– the interquartile range is defined as follows: IQR[T0 ] = x − x

(2.64)

61

62


where x and x are respectively the first quartile (the 25-th percentile) and the third quartile (the 75-th percentile) of the probability distribution of T0 , namely the ages such that S(x ) = 0.75 and S(x ) = 0.25; note that the IQR decreases as the lifetime distribution becomes less dispersed. While most markers are referred to the probability distribution of T0 , it is also interesting to single out some characteristics referred to individuals alive at a chosen age x, that is, concerning the distribution of Tx , say with x = 65 (of obvious interest when analysing the age-pattern of mortality of annuitants and pensioners). An example is provided by the expected remaining lifetime at age x, e¯ x (or the expected age at death, x + e¯ x ) (see (2.52), (2.53)). Other examples are given by √ – the variance Var[Tx ], the standard deviation σx = Var[Tx ], and the coefficient of variation CV[Tx ]; – the interquartile range IQR[Tx ]. For example, the analysis of the values of IQR[T65 ] related to various subsequent mortality observations allows us to check whether the rectangularization phenomenon occurs even when only old ages are addressed. Figure 2.3 illustrate some markers of practical interest.

2.4.3

Markers under a dynamic perspective

Information provided by markers calculated on the basis of a period observation must be carefully interpreted, in particular keeping in mind mortality trends. Consider, in particular, the complete expectation of life at age x (see (2.57)), namely ω−x 1 ◦ ex = (2.65) k px + 2 k=1

Probabilities k px are derived from the qx ’s according to (2.7) and (2.8), and, in turn, the qx ’s are determined as the result of a (recent) period mortality ◦ observation. The quantity ex is usually called the (complete) period life expectancy. The life expectancy drawn from a period life table can be taken as a reasonable estimate of the remaining lifetime for an individual currently age x only if we accept the hypothesis that, from now on, the age-pattern of mortality will remain unchanged. See also the comments in Section 2.2.1 regarding the construction of the life table in terms of lx .

2.5

Mortality laws

f0(x) max{f0(x)} x

IQR[T0] x1q0

Age x x1

Endurance x'

x''

_

e0

Lexis _ e65+65

Figure 2.3.

Some markers.

When the hypothesis of unchanging future mortality is rejected, the cal◦ culation of period quantities like ex (as well as other markers) and the corresponding ‘cohort’ quantities requires the use of appropriate mortality forecasts, and hence of projected life tables. This aspect will be dealt with in Section 4.4.1.

2.5

Mortality laws

Since the earliest attempt to describe in analytical terms a mortality schedule (due to A. De Moivre and dating back to 1725), great effort has been devoted by demographers and actuaries to the construction of analytical formulae (or laws) that fit the age-pattern of mortality. When a mortality law is used to fit observed data, the age-pattern of mortality is summarized by a small number of parameters (two to ten, say, in the mortality laws commonly used in actuarial and demographical models). This exercise has the advantage of reducing the dimensionality of the problem – thus, we could replace the 120, say, items of a life table by a small number of parameters without sacrificing much information.

63

64


It is beyond the scope of this book to present an extensive list of mortality laws. Conversely we only focus on some important laws, which are interesting because of their possible use in a dynamic context, that is, to summarize observed mortality trends and to project the age-pattern of mortality in future years.

2.5.1

Laws for the force of mortality

A number of mortality laws refer to the force of mortality, µx (although some of them have been originally proposed in different terms, for example, in terms of the life table lx ). The Gompertz law, proposed in 1825, is as follows: µx = B c x

(2.66)

Sometimes the following equivalent notation is used: µx = α eβ x

(2.67)

It is interesting to look at the hypothesis underlying the Gompertz law. Assume that, moving from age x to age x+ x, the increment of the mortality intensity is proportional to its initial value, µx , and to the length of the interval, x; thus µx = β µx x (2.68) This assumption leads to the differential equation dµx = β µx , dx

β>0

(2.69)

and finally to (2.67), with α > 0. The Gompertz law is used to represent the age progression of mortality at the old ages, that is, the senescent mortality. The (first) Makeham law, proposed in 1867, is a generalization of the Gompertz law, namely (2.70) µx = A + B cx where the term A > 0 (independent of age) represents non-senescent mortality, for example, because of accidents. An interpretation in more general terms can be found in Section 2.5.3. The following equivalent notation is also used: µx = γ + α eβ x (2.71) The second Makeham law, proposed in 1890, is as follows: µx = A + H x + B c x and hence constitutes a further generalization of the Gompertz law.

(2.72)

2.5

Mortality laws

The Thiele law, proposed in 1871, can represent the age-pattern of mortality over the whole life span: 2

µx = A e−Bx + C e−D(x−E) + F Gx

(2.73)

The first term decreases as the age increases and represents the infant mortality. The second term, which has a ‘Gaussian’ shape, represents the mortality hump (mainly due to accidents) at young-adult ages. Finally, the third term (of Gompertz type) represents the senescent mortality. In 1932 Perks proposed two mortality laws. The first Perks law is as follows: α eβx + γ (2.74) µx = δ eβx + 1 Conversely, the second Perks law has the following more general structure: µx =

α eβx + γ δ eβx + e−βx + 1

(2.75)

As we will see in Section 2.8, Perks’ laws have an important role in representing the mortality pattern at very old ages (say, beyond 80); moreover, the first Perks law can be reinterpreted in the context of the ‘frailty’ models (see Section 2.9.5). The Weibull law, proposed in 1951 in the context of reliability theory, is given by µx = A xB (2.76) or, in equivalent terms: α µx = β

α−1 x β

(2.77)

The GM class of models (namely, the Gompertz-Makeham class of models), proposed by Forfar et al. (1988), has the following structure:   r−1 s−1 µx = αi xi + exp βj xj  (2.78) i=1

j=0

with the proviso that when r = 0 the polynomial term is absent, and when s = 0 the exponential term is absent. The general model in the class (2.78) is usually labelled as GM(r, s). Note that, in particular, GM(0, 2) denotes the Gompertz law, GM(1, 2) the first Makeham law and GM(2, 2) the second Makeham law. Models used by the Continuous Mortality Investigation Bureau in the UK to graduate the force of mortality µx are of the GM(r, s) type. In particular, models GM(0, 2), GM(2, 2), and GM(1, 3) have been widely used.

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66


2.5.2

Laws for the annual probability of death

Various mortality laws have been proposed in terms of the annual probability of death, qx , and in terms of the odds φx (see (2.14)). For example, Beard proposed in 1971 the following law: qx = A +

B cx E c−2x + 1 + D cx

(2.79)

Barnett proposed, in 1974, the following law for the odds: φx = A − H x + B cx

(2.80)

The odds can also be graduated using the following formula: φx = ePx

(2.81)

where Px is a polynomial in x. For example, with a first degree polynomial, we have φx = ea+b x (2.82) Heligman and Pollard (1980) proposed a class of formulae which aim to represent the age-pattern of mortality over the whole span of life (as made by Thiele, see (2.73)). The first Heligman–Pollard law, expressed in terms of the odds, is C

2

φx = A(x+B) + D e−E(ln x−ln F) + G H x

(2.83)

while the second Heligman–Pollard law, in terms of qx , is given by C

2

qx = A(x+B) + D e−E(ln x−ln F) +

G Hx 1 + G Hx

(2.84)

Note that, in both cases, at higher ages we have qx ≈

G Hx 1 + G Hx

(2.85)

The third Heligman–Pollard law, which generalizes the second one, is as follows: G Hx C 2 qx = A(x+B) + D e−E(ln x−ln F) + (2.86) 1 + K G Hx Another generalization of the second law is provided by the fourth Heligman–Pollard law, which is given by k

C

2

qx = A(x+B) + D e−E(ln x−ln F) +

G Hx

1 + G Hx

k

(2.87)

2.6

2.5.3

Non-parametric graduation

Mortality by causes

When various (say, r) causes or death are singled out, the force of mortality µx can be expressed in terms of ‘partial’ forces of mortality, each force pertaining to a specific cause: µx =

r

µ(k) x

(2.88)

k=1 (k)

where µx refers to the k-th cause of death. Makeham proposed a reinterpretation of his first law (see (2.70)) in terms of partial forces of mortality. Let A=

m

Ak

(2.89)

k=1

and

m+n

B=

Bk

(2.90)

k=m+1

whence µx =

m

Ak + c x

k=1

2.6 2.6.1

m+n k=m+1

Bk =

m+n

µ(k) x

(2.91)

k=1

Non-parametric graduation Some preliminary ideas

The term ‘graduation’ denotes an adjustment procedure applied to a set of estimated quantities, in order to obtain adjusted quantities which are close to a reasonable pattern and, in particular, do not exhibit an erratic behaviour. We note that previous experience and intuition suggest a smooth progression. In actuarial science, graduation procedures are typically applied to raw mortality rates which result from statistical observation. Graduated series of period mortality rates should exhibit a progressive change over a series of ages, without sudden and/or huge jumps, which cannot be explained by intuition or supported by past experience. A detailed analysis of the various aspects of graduation is beyond the scope of this book. So, we only focus on some topics which constitute starting points for projection models presented in Chapters 5 and 6.

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68


Various approaches to graduation can be adopted. In particular, two broad categories can be recognized: – parametric approaches, involving the use of mortality laws; – non-parametric approaches. According to a parametric approach, a functional form is chosen (e.g. Makeham’s law, Heligman–Pollard’s law, and so on; see Section 2.5), and the relevant parameters are estimated in order to find the parameter values which provide the best fit to the observed data, for example, to mortality rates. Various fitting criteria can be adopted for parameter estimation, for example maximum likelihood, based on a Generalized Linear Models formulation. The choice of a particular functional form is avoided when a nonparametric graduation method is adopted. Important methods in this category are: weighted moving average methods, kernel methods, the Whittaker–Henderson model, methods based on spline functions. In what follows, we restrict our attention to the latter two methods only.

2.6.2

The Whittaker–Henderson model

The Whittaker–Henderson approach to graduation is based on the minimization of an objective function. We denote by z1 , z2 , . . . , zn the observed values of a given quantity, and by y1 , y2 , . . . , yn the corresponding graduated values. For example, referring to the graduation of mortality rates, zh could represent the raw mortality rate at age xh , namely m ˆ xh , and yh the corresponding graduated value, mxh . The objective function (to be minimized with respect to y1 , y2 , . . . , yn ) is defined as follows: F(y1 , y2 , . . . , yn ) =

n

wh (yh − zh )2 + λ

h=1

n−k

( k yh )2

(2.92)

h=1

where – w1 , w2 , . . . , wn are weights attributed to the squared deviations; – k yh is the k-th forward difference of yh , defined as follows: k i k y (−1) yh = i h+k−i k

i=0

– λ is a (constant) parameter.

(2.93)

2.6


The first term on the right-hand side of formula (2.92) provides a measure of the discrepancy between observed and graduated values. The choice of each weight wh allows us to attribute more or less importance to the squared deviation related to the h-th observation. In particular, referring to the graduation of mortality rates, an appropriate choice of the weights should reflect a low importance attributed to the raw mortality rates concerning very old ages at which few individuals are alive, and hence the observed values could be affected by erratic behaviour. To this purpose, the weights can be chosen to be inversely proportional to the estimated variance of the observed mortality rates. The second term on the right-hand side of (2.92) quantifies the degree of roughness in the set of graduated values. Usually, the value of k is set equal to 2, 3, or 4. Finally, the parameter λ allows us to express our ‘preference’ regarding features of the graduation results: higher values of λ denote a stronger preference for a smooth behaviour of the graduated values, whereas lower values express more interest in the fidelity of the graduated values to the observed ones. The objective function can be generalized and modified. For example, it has been proposed to replace, in the first term of the right-hand side of (2.92), the squared deviations with other powers. As regards the second term, a mixture of differences of various orders can be used instead of the k-th differences only.

2.6.3

Splines

A spline is a function defined piecewise by polynomials. We denote by [a, b] an interval of real numbers, and by ξ0 , ξ1 , . . . , ξm , ξm+1 real numbers such that (2.94) a = ξ0 < ξ1 < · · · < ξm < ξm+1 = b Let s denote the spline function, and p0 , p1 , . . . , pm the polynomials. Thus, the spline function is defined as follows:   p0 (x); ξ0 ≤ x < ξ1    p (x); ξ ≤ x < ξ 1 1 2 s(x) = (2.95)  . . .    p (x); ξ ≤ x ≤ ξ m m m+1 The m + 2 numbers ξ0 , ξ1 , . . . , ξm+1 are called the knots. In particular, ξ1 , . . . , ξm are the internal knots. If the knots are equidistantly distributed in [a, b], the spline is called a uniform spline (a non-uniform spline otherwise).

69

70


As regards the behaviour of s(x) in a neighbourhood of the generic knot ξh , a measure of smoothness is provided by the maximum order of the derivative of the polynomials such that the polynomials ph−1 and ph have common derivative values; if the maximum order is k, the spline is said to have smoothness (or continuity) of class C k at ξh . When all polynomials have degree at most r, the spline is said to be of degree r. A spline of degree 0 is a step function. A spline of degree 1 is also called a linear spline. An example of a linear spline is provided by the piecewise linear survival function, constructed by assuming as knots all of the integer ages and adopting the hypothesis of uniform distribution of deaths over each year of age (see point (a) in Section 2.3.5). A spline of degree 3 is a cubic spline. In particular, a natural cubic spline has continuity C 2 at all of the knots, and the second derivatives of the polynomials equal to 0 in a and b; thus, the spline is linear outside the interval [a, b]. It can be proved that, for a given interval [a, b] and a given set of m internal knots, the set of splines of degree r constitutes a (real) vector space of dimension d = m + r + 1. A basis for this space is provided by the following d functions: 1, x, . . . , xr , [(x − ξ1 )+ ]r , . . . , [(x − ξm )+ ]r where

(x − ξh )+ =

0; x − ξh ;

x < ξh x ≥ ξh

(2.96)

(2.97)

for h = 1, . . . , m. The corresponding representation of the spline function is given by: r m αj xj + βh [(x − ξh )+ ]r (2.98) s(x) = j=0

h=1

where the αj ’s and the βh ’s are the coefficients of the linear combination. If d is the dimension of the space, then any basis consists of d elements. We denote by b1 , b2 , . . . , bd a basis. Hence, any spline s in the space can be represented as a linear combination of these functions, namely s(x) = γ1 b1 (x) + γ2 b2 (x) + · · · + γd bd (x)

(2.99)

where the coefficients γ1 , γ2 , . . . , γd are uniquely determined by the function s. The choice of a basis constitutes a crucial step in the graduation process through splines. The starting point of this process is the choice of the result

2.6


we want to achieve by using a spline function, and the related objective function to optimize. We assume that our target is a ‘best fit’ graduation, namely we require that the spline function is as close as possible (according to a stated criterion) to our data set, consisting of n points, (x1 , z1 ), (x2 , z2 ), . . . , (xn , zn )

(2.100)

with a ≤ xh ≤ b for h = 1, 2, . . . , n. For example, referring to actuarial applications, the xh ’s may represent ages, whereas the zh are the corresponding observed mortality rates (namely the m ˆ xh ’s referred to in Section 2.6.2). As regards the best-fit criterion, we focus on the weighted mean square error, expressed by the quantity n

wh [s(xh ) − zh ]2

(2.101)

h=1

where the wh ’s are positive weights. Using (2.99) to express the spline function, our best-fit problem can be stated as follows: find the coefficients γ1 , γ2 , . . . , γd which minimize the function  2  n d G(γ1 , γ2 , . . . , γd ) = wh  γj bj (x) − zh  (2.102) h=1

j=1

Although minimizing the function G is, in principle, a simple exercise which consists in solving a set of simultaneous equations, in practice computational difficulties may arise. However, the complexity of the minimization problem can be reduced if a particular basis is chosen in order to express the spline function s, namely the one consisting of the so-called B-splines. A formal definition of the B-splines and a detailed discussion of their use as a basis in graduation problems through splines is beyond the scope of this Section. The interested reader can refer, for example, to McCutcheon (1981). We just mention that the idea underlying the B-splines is to choose a basis such that each spline in the basis is zero outside a short interval. Typically, the basis consists of cubic polynomial pieces, smoothly joined together. In particular, when the spline function is uniform (i.e. the knots are equidistantly distributed), the B-splines are (for a given degree) just shifted copies of each other. The advantage provided by B-splines in the minimization problem (2.102) derives from the fact that, as each B-spline

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72


is zero outside a given short interval, the matrix involved by solving the related set of simultaneous equations has many entries equal to zero, and this improves the tractability of the best-fit problem. Spline functions can be introduced by adopting a different approach, namely the ‘variational approach’. Following Champion et al. (2004), we start by defining an interpolation problem. Assume that we need to find a function f interpolating the n data points (x1 , z1 ), (x2 , z2 ), . . . , (xn , zn ), that is, such that f (xh ) = zh ;

h = 1, 2, . . . , n

(2.103)

Among all functions f fulfilling condition (2.103), we are interested in those which have a continuous second derivative and a ‘limited’ oscillation (i.e. a smooth behaviour) in the interval [x1 , xn ]. We introduce the functional [f ] =

xn

x1

[f (x)]2 dx

(2.104)

(where f (x) denotes the second derivative of f ) as a measure of oscillation. Then, it is possible to prove that a cubic spline is the only function which minimizes the functional (2.104). We now shift from the interpolation problem to a graduation problem. To this purpose, we use the following functional in order to express our objective: xn n 2 [zh − f (xh )] + λ [f (x)]2 dx (2.105) [f ] = h=1

x1

Clearly, [f ] generalizes the functional (2.104). The first term on the righthand side of (2.105) provides a measure of the discrepancy between the data zh ’s and the graduated values f (xh )’s, whereas the second term can be interpreted as a measure of smoothness. The parameter λ allows us to express our preference in the trade-off between closeness to data and smoothness. The analogy with the structure of formula (2.92) is self-evident. It can be proved that, among all functions f with continuous second derivatives, there is a unique function which minimizes the functional (2.105). Finally, it is worth noting that the spline functions so far dealt with are ‘univariate’ splines, as their domains consist of intervals of real numbers. Extension to a bivariate context is possible; an example will be presented in Section 5.4, together with the more general concept of P-splines (namely, ‘Penalized’ splines).

2.7

2.7

Some transforms of the survival function

Some transforms of the survival function

Some transforms of life table functions may help us in reaching a better understanding of some aspects of the age-pattern of mortality (and of mortality trends as well). Two examples will be provided: the logit transform of the survival function S(x), and the so-called resistance function. Some aspects of their use in mortality projections will be addressed in Section 4.6.3. The logit transform of the survival function is defined as follows: 1 1 − S(x) (x) = ln (2.106) 2 S(x) Features of this transform have been analysed by Brass (see e.g. Brass (1974)). In particular, Brass noted empirically that (x) can be expressed in terms of the logit of the survival function describing the agepattern of mortality in a ‘standard’ population, ∗ (x), via a linear relation, that is, (x) = α + β ∗ (x) (2.107) whose parameters are (almost) independent of age. Figures 2.4–2.6 show the effect of various choices for the parameters α and β. A different transform of the survival function S(x) has been proposed by Petrioli and Berti (1979). The proposed transform is the so-called resistance function, defined as follows: ρ(x) =

(2.108)

(b)

(a) 7 6 a = 0; b = 1 5 a = –0.2; b = 1 4 a = 0.2; b = 1 3 2 1 0 –1 0 10 20 30 40 50 60 70 80 90 100 –2 –3 Figure 2.4.

S(x)/(ω − x) (1 − S(x))/x

1.25 1 0.75 a = 0; b = 1 a = –0.2; b = 1 a = 0.2; b = 1

0.5 0.25 0 30

Logit transforms and survival functions.

40

50

60

70

80

90

100

73

74


(b)

(a)

1.25

8 a = 0; b = 1 a = 0; b = 1.25 a = 0; b = 0.75

6 4

1 0.75

2

a = 0; b = 1 a = 0; b = 1.25 a = 0; b = 0.75

0.5

0 0 10 20 30 40 50 60 70 80 90 100

0.25

–2 0

–4

30

Figure 2.5.

40

50

60

70

80

90

100


(b)

(a) 8

1.25 a = 0; b = 1 a = –0.2; b = 1.25

6

1

4

0.75

2

0.5

0 0 10 20 30 40 50 60 70 80 90 100 –2

0.25

a = 0; b = 1 a = –0.2; b = 1.25

0 0 10 20 30 40 50 60 70 80 90 100

–4 Figure 2.6.


where ω denotes, as usual, the limiting age. Thus, the transform is the ratio of the average annual probability of death beyond age x to the average annual probability of death prior to age x (both probabilities being referred to a newborn).

2.8 2.8.1

Mortality at very old ages Some preliminary ideas

Several problems arise when analysing the mortality experience of very old population segments. A first problem obviously concerns the observed old-age mortality rates, which are heavily affected by random fluctuations because of their scarcity. In the past, mortality at very old ages was largely hypothetical and assumptions were normally made as the result of extrapolations from younger ages, based on models such as the Gompertz or the Makeham law. In recent times, mortality statistics have been improved

2.8

Force of mortality mx

Gompertz–Makeham–Thiele

Mortality at very old ages

Lindbergson

e.g. Logistic

Age x Figure 2.7.

Mortality at highest ages.

in many countries, and provide stronger evidence about the shape of the mortality curve at old and very old ages. In particular, it has been observed that the force of mortality is slowly increasing at very old ages, approaching a rather flat shape. In other words, the exponential rate of mortality increase at very old ages is not constant, as for example in Gompertz’s law (see (2.66)), but declines (see Fig. 2.7). However, a basic problem arises when discussing the appropriateness of mortality laws in representing the pattern of mortality at old ages: ‘what’ force of mortality are we dealing with? We will return on this important issue in Section 2.9.3. As classical mortality laws may fail in representing the very old-age mortality, shifting from the exponential assumption may be necessary in order to fit the relevant pattern of mortality.

2.8.2

Models for mortality at highest ages

Several alternative models have been proposed. In Section 2.5.2 we have addressed the Heligman–Pollard family of laws, which aim to represent the age-pattern of mortality over the whole span of life. As regards old ages, according to the first and the second Heligman–Pollard law, qx can G Hx be approximated by 1+G H x (see (2.85)). Conversely, the third Heligman– Pollard law when applied to old ages reduces to qx =

G Hx 1 + K G Hx

(2.109)

In Perks’ laws (see (2.74) and (2.75)), the denominators have the effect of reducing the mortality especially at old and very old ages. In particular the graph of the first law is a logistic curve.

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76


The logistic model for the force of mortality proposed by Thatcher (1999) assumes that δ α eβx +γ (2.110) µx = 1 + α eβx Its simplified version, used in particular for studying long-term trends and forecasting mortality at very old ages, has δ = 1 and hence has only three parameters, namely α, β, and γ: µx =

α eβx +γ 1 + α eβx

(2.111)

A modified version of the Makeham law has been proposed by Lindbergson (2001), replacing the exponential growth with a straight line at very old ages: a + b cx if x ≤ w (2.112) µx = w a + b c + d (x − w) if x > w The model proposed by Coale and Kisker (see Coale and Kisker (1990)) relies on the so-called exponential age-specific rate of change of central death rates, defined as follows: kx = ln

mx mx−1

(2.113)

The model assumes that kx is linear over the age of 85: kx = k85 − (x − 85) s

(2.114)

as documented by statistical evidence. The parameter s is determined assuming that k85 is calculated from empirical data, whereas a predetermined value is given to the mortality rate m110 . For given values of kx , x = 85, 86, . . . , 110, we find from (2.113)   x kh  (2.115) mx = m85 exp h=86

From (2.115) it follows that the Coale–Kisker model implies an exponentialquadratic function for central death rates at the relevant ages, that is, mx = exp(a x2 + b x + c) which is clearly in contrast with the Gompertz assumption.

(2.116)

2.9

2.9

Heterogeneity in mortality models


It is well known that any given population is affected by some degree of heterogeneity, as far as individual mortality is concerned. Heterogeneity in populations should be approached addressing two main issues: (i) detecting and modelling observable heterogeneity factors (e.g. age, gender, occupation, etc.); (ii) allowing for unobservable heterogeneity factors.

2.9.1

Observable heterogeneity factors

As regards observable factors, mortality depends on: (1) biological and physiological factors, such as age, gender, genotype; (2) features of the living environment; in particular: climate and pollution, nutritional standards (mainly with reference to excesses and deficiencies in diet), population density, hygienic and sanitary conditions; (3) occupation, in particular in relation to professional disabilities or exposure to injury, and educational attainment; (4) individual lifestyle, in particular with regard to nutrition, alcohol and drug consumption, smoking, physical activities and pastimes; (5) current health conditions, personal and/or family medical history, civil status, and so on. Item 2 affects the overall mortality of a population. That is why mortality tables are typically considered specifically for a given geographic area. The remaining items concern the individual and, when dealing with life insurance, they can be observed at policy issue. Their assessment is performed through appropriate questions in the application form and, as to health conditions, possibly through a medical examination. The specific items considered for insurance rating depend on the types of benefits provided by the insurance contract (see also Section 2.2.2). The aim of the insurer is to group people in classes within which insured lives bear the same expected mortality profile. Age is always considered, due to the apparent variability of mortality in this regard. Gender is usually accounted for, especially when living benefit are involved, given that females on average live longer than males. As far as genetic aspects are concerned, the evolving knowledge in this area has raised a lively debate (which is still running) on whether it is legitimate for insurance companies to resort to genetic tests for underwriting purposes. Applicants for living benefits are

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78


usually in good health condition, so a medical examination is not necessary; on the contrary, a proper investigation is needed for those who buy death benefits, given that people in poorer health conditions may be more interested in them and hence more likely to buy such benefits. When death benefits are dealt with, health conditions, occupation and smoking status lead to a classification into standard and substandard risks; for the latter (also referred to as impaired lives), a higher premium level is adopted, given that they bear a higher probability to become eligible for the benefit. In some markets, standard risks are further split into regular and preferred risks, the latter having a better profile than the former (e.g. because they never smoked); as such, they are allowed to pay a reduced premium rate. Mortality for people in poorer or better conditions than the average is usually expressed in relation to average (or standard) mortality. This allows us to deal only with one life table (or one mortality law), properly adjusted when substandard or preferred risks are dealt with. For the case of life annuities, usually specific tables are constructed for each subpopulation.

2.9.2

Models for differential mortality

Let us index with (S) standard mortality and with (D) a different (higher or lower) mortality. Below, some examples of differential mortality models follow.

qx(D) = aqx(S) + b

(2.117)

µx(D) = aµx(S) + b

(2.118)

( S)

qx(D) = qx+z ( S)

(2.119)

µx(D) = µx+z

(2.120)

qx(D) = qx(S) ϕ(x)

(2.121)

( D)

q[x−t]+t = qx(S) ρ(x − t, t) ( D)

( S)

( D)

( S)

q[x−t]+t = q[x−t]+t ν(t) q[x−t]+t = q[x−t]+t η(x, t)

(2.122) (2.123) (2.124)

In any case, x is the current age and t the time elapsed since policy issue (t ≥ 0), whence x − t is the age at policy issue.

2.9


Models (2.117) and (2.118) are usually adopted for substandard risks. ( S) ( S) Letting a = 1 and b = δqx−t , δ > 0, in (2.117) (b = δµx−t in (2.118)) the socalled additive model is obtained, where the increase in mortality depends on initial age. An alternative model is obtained choosing b = θ, θ > 0, that is, a mortality increase which is constant and independent of the initial age; such a model is consistent with extra-mortality due to accidents (related either to occupation or to extreme sports). Letting a = 1 + γ, γ > 0, and b = 0 the so-called multiplicative model is derived, where the mortality increase depends on current age. When risk factors are only temporarily effective (e.g. some diseases which either lead to an early death or have a short recovery time), parameters a, b may be positive up to some proper time τ; for t > τ, standard mortality is assumed, so that a = b = 0. Models (2.119) and (2.120) are very common in actuarial practice, both for substandard and preferred risks, due to their simplicity; they are called age rating or age shifting models. Model (2.120), in particular, can be formally justified, assuming the Gompertz law for the standard force of mortality and the multiplicative model for differential mortality. Actually, ( S) if µx = α eβx (see (2.67)), we have from (2.118), with a = 1 + γ and b = 0, ( S)

µx(D) = (1 + γ) α eβx = α eβ(x+z) = µx+z

(2.125)

where eβz = 1 + γ. In insurance practice, the age-shifting is often applied directly to premium rates. In (2.121), mortality is adjusted in relation to age. Such a choice is common when annuities are dealt with. For example, ϕ(x) may be a step-wise linear function. The other models listed above concern the effect on mortality of the time elapsed since policy issue, t (see Section 2.2.4). Model (2.122) expresses issue-select mortality in terms of aggregate mortality (so that, differential mortality simply means, in this case, select mortality). Conversely, models (2.123) and (2.124) express issue-select differential mortality through a transform of the issue-select standard probabilities of death; in particular, ν(t) and η(x, t) may be chosen to be linear. A particular implementation of model (2.117) (with b = 0) is given by the so-called numerical rating system, introduced in 1919 by New York Life Insurance and still adopted by many insurers. A set of m risk factors is referred to. The annual probability of death specific for a given individual is  1 + q(spec) = q(S) x x

m h=1

 γh 

(2.126)

79

80


where the parameters γh lead to a higher or lower death probability for the individual in relation to the values chosen risk factors massumed by the (S) (clearly, with the constraint −1 < h=1 γh < (1/qx ) − 1). Note that an additive effect of each of the risk factors is assumed.

2.9.3

Unobservable heterogeneity factors. The frailty

Heterogeneity of a population in respect of mortality can be explained by differences among the individuals; some of these are observable, as discussed in the previous section, whilst others (e.g. the individual’s attitude towards health, some congenital personal characteristics) are unobservable. When allowing for unobservable heterogeneity factors, two approaches can be adopted: – A discrete approach, according to which heterogeneity is expressed through a (finite) mixture of appropriate functions. – A continuous approach, based on a non-negative real valued variable, called the frailty, whose role is to include all unobservable factors influencing the individual mortality. The second approach is most interesting. We will deal with this approach only. In the following discussion, the term heterogeneity refers to unobservable risk factors only; in respect of the observable risk factors, the population is instead assumed to be homogeneous. In order to develop a continuous model for heterogeneity, a proper characterization of the unobservable risk factors must be introduced. In their seminal paper, Vaupel et al. (1979) extend the earlier work of Beard (1959, 1971) and define the frailty as a non-negative quantity whose level expresses the unobservable risk factors affecting individual mortality. The underlying idea is that those people with a higher frailty die on average earlier than others. Several models can be developed, which are susceptible to interesting actuarial applications. With reference to a population (defined at age 0, and as such closed to new entrants), we consider people current age x. They represent a heterogeneous group, because of the unobservable factors. Let us assume that, for any individual, such factors are summarized by a non-negative variable, viz the frailty. The specific value of the frailty of the individual does not change over time, but remains unknown. On the contrary, because of deaths, the distribution of people in respect of frailty does change with age, given that people with low frailty are expected to live longer; we denote by Zx the random frailty at age x, for which a continuous probability distribution

2.9


with pdf gx (z) is assumed. It must be mentioned that the hypothesis of unvarying individual frailty, which is reasonable when thinking of genetic aspects, seems weak when referring to environmental factors, which may change over time affecting the risk of death; however, there is empirical evidence which validates quite satisfactorily this assumption. For a person current age x with frailty level z, the (conditional) force of mortality (see (2.32)) is defined as

P[Tx ≤ t|Zx = z] t0 t

µx (z) = lim

(2.127)

Now the task is to look at possible relations between µx (z) and a standard force of mortality, given that mortality analysis requires the joint distribution of (Tx , Zx ). For brevity, conditioning on Zx = z will be denoted simply with z. In Vaupel et al. (1979) a multiplicative model for the force of mortality has been proposed: µx (z) = z µx (2.128) where µx represents the force of mortality for an individual with z = 1; µx is considered as the standard force of mortality. If z < 1, then µx (z) < µx , which suggests that the person is in good conditions; vice versa if z > 1. Note that (2.128) may be adopted also when the standard frailty level is other than 1. Let a, a = 1, be the standard frailty level and µx the standard force of mortality; according to the multiplicative model, µx (a) = µx , whence (replacing in (2.128)) µx = 1a µx . So, following (2.128), the force of mortality for a person age x and frailty level z may be written as z µ = z µx a x

µx (z) =

(2.129)

which coincides with (2.128) using an appropriate definition of the standard force of mortality and a scaling of the frailty level. A simple generalization may further be adopted to represent a mortality component independent of age and frailty (e.g. accident mortality). The model b µx (z)

= b + zµx

(2.130)

may be considered for this purpose. For brevity, in the following we refer just to (2.128). We denote with H(x) the cumulative standard force of mortality in (0, x) (see (2.39)). Let us refer to age 0. The survival function for a person with frailty z is S(x|z) = e−

x 0

µt (z)dt

= e−zH(x)

(2.131)

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82


The pdf of T0 conditional on a frailty level z, given by f0 (x|z) = S(x|z) µx (z), can be expressed as f0 (x|z) = e−zH(x) zµx = −

d S(x|z) dx

(2.132)

The joint pdf of (T0 , Z0 ), denoted by h0 (x, z), can then be easily obtained. We have h0 (x, z) = f0 (x|z) g0 (z) = S(x|z) µx (z) g0 (z) (2.133) Referring to the whole population, we can define the average survival function as ∞ ¯ S(x|z) g0 (z) dz (2.134) S(x) = 0

¯ Note that S(x) represents the share of people alive at age x out of the initial newborns. We now refer to a given age x, x ≥ 0. The pdf of Zx may be derived from the distribution of (T0 , Z0 ) considering that, as was mentioned earlier, the distribution of Zx changes because of a varying composition of the population due to deaths. We can then relate Zx to Z0 as follows: Zx = Z0 |T0 > x

(2.135)

For the pdf of Zx we obtain

P[z < Zx ≤ z + z] = z0 z P[T0 > x|z < Z0 ≤ z + z] P[z < Z0 ≤ z + z] = lim z0 z P[T0 > x]

gx (z) = lim

(2.136)

from which, under usual conditions, we obtain gx (z) =

S(x|z) g0 (z) S(x|z) g0 (z) = ∞ ¯S(x) S(x|z) g0 (z) dz

(2.137)

0

Note that the pdf of Zx is given by the pdf of Z0 , adjusted by the ratio ¯ S(x|z)/S(x) which updates at age x the proportion of people with frailty z. It is also interesting to stress that the assessment of gx (z) is based on an update of g0 (z) with regard to the number of survivors with frailty z compared to what would be expected over the whole population. We define the average force of mortality in the population as ∞ ∞ h0 (x, z) dz 0 µx (z) S(x|z) g0 (z) dz ∞ µ ¯x = = 0 ¯ S(x|z) g0 (z) dz S(x) 0

(2.138)

2.9


Thanks to (2.128) and (2.137) we obtain ∞ µ ¯ x = µx z gx (z) dz

(2.139)

0

that is, to µ ¯ x = µx z¯ x

∞

(2.140)

where z¯ x = 0 z gx (z) dz = E[Zx ] represents the expected frailty at age x. Note that the average force of mortality coincides with the standard one only if z¯ x = 1. A similar relation holds for model (2.130): we easily find µ ¯ x = b + µx z¯ x . It is easy to show that d z¯ x = −µx Var[Zx ] < 0 dx

(2.141)

Then, according to (2.140), µ ¯ x varies less rapidly than µx . This is due to the fact that those with a high frailty die earlier, therefore leading to a reduction of z¯ x with age. If one disregards the presence of heterogeneity, on average an underestimation of the force of mortality follows when one cohort only is addressed.

2.9.4

Frailty models

In order to get to numerical valuations (and further analytical results, as well), the distribution of Z0 must be chosen. In Vaupel et al. (1979), a Gamma distribution has been suggested, due to its nice features. Let then Z0 ∼ Gamma(δ, θ). The pdf g0 (z) is therefore g0 (z) =

θ δ zδ−1 −θz e (δ)

(2.142)

We have in particular

E[Z0 ] = z¯ 0 = Var[Z0 ] =

δ θ

δ θ2

(2.143) (2.144)

The coefficient of variation of Z0

√ 1 Var[Z0 ] =√ CV[Z0 ] = E[Z0 ] δ

(2.145)

shows that δ plays the role of measuring, in relative terms, the level of heterogeneity in population. If δ ∞, then CV[Z0 ] 0, that is, the

83

84


population can be considered homogeneous; for small values of δ, on the contrary, the value of CV[Z0 ] is high, representing a wide dispersion, that is, heterogeneity, in the population. It can be shown that also Zx , x > 0, has a Gamma distribution, with one of the two parameters updated to the current age. In order to check this, we need the expression of the average survival function at age x. Substituting (2.142) into (2.134), and using (2.131), we have ¯ S(x) =

θ θ + H(x)

δ

∞

0

(θ + H(x))δ zδ−1 −(θ+H(x))z e dz (δ)

(2.146)

δ δ−1

z e−(θ+H(x))z is the pdf of a random variable GammaNote that (θ+H(x)) (δ) distributed with parameters (δ, θ + H(x)); hence, the integral in (2.146) reduces to 1. Therefore,

¯ S(x) =

θ θ + H(x)

δ (2.147)

Replacing in (2.137) and rearranging we have gx (z) =

(θ + H(x))δ zδ−1 −(θ+H(x))z e (δ)

(2.148)

which is the pdf of a random variable Gamma(δ, θ+H(x)) Thus, the Gamma distribution has a self-replicating property, and the relevant parameters need to be chosen with reference to the distribution at age 0. So it follows that δ θ + H(x) δ Var[Zx ] = (θ + H(x))2 √ 1 Var[Zx ] =√ CV[Zx ] = E[Zx ] δ

E[Zx ] = z¯ x =

(2.149) (2.150) (2.151)

Note that whilst the expected value of the frailty reduces with age, its relative variability keeps constant. We can give an interesting interpretation for the average survival function. Rearranging (2.146) we find ¯ S(x) =

δ θ δ θ + H(x)

δ

=

z¯ x z¯ 0

δ (2.152)

2.9


and then we argue that the average survival function at age x, that is, the average probability of newborns attaining age x, depends on the comparison between the expected frailty level at age x and age 0; this result is independent of the particular mortality law that we adopt for the standard force of mortality, which actually has not yet been introduced, and is simply due to the properties of the Gamma distribution. The population force of mortality is µ ¯ x = µx

δ θ + H(x)

(2.153)

Usually, the initial values of the parameters of the Gamma distribution are chosen so that z¯ 0 = 1, that is, θ = δ. So we have µ ¯ x = µx

δ δ + H(x)

(2.154)

Only the parameter δ has to be assigned, in a manner which is consistent with the level of heterogeneity in the population. Finally, the unconditional pdf of Tx may be easily obtained from previous results. An alternative choice for the distribution of Z0 is the Gaussian-Inverse distribution. Like the Gamma, this distribution is self-replicating, so that Zx is Gaussian-Inverse for any age x; and hence, the relevant parameters need to be chosen only with reference to the distribution at age 0. When a Gaussian-Inverse distribution is used, in relative terms the variability of Zx decreases with age, which can be justified by the fact that as time passes those with a low (and similar) frailty keep on living, hence reducing the heterogeneity of the population. In this regard, the Gassian-Inverse hypothesis is more interesting than the Gamma. However, some authors (e.g. see Butt and Haberman (2004) and Manton and Stallard (1984)) note that individual frailty is unlikely to remain unchanged over the lifetime, but should increase with age. So the assumption that, within the population, the relative variability keeps constant can be accepted. In the following, we will mainly deal with the Gamma case.

2.9.5

Combining mortality laws with frailty models

Referring to adult ages, we can assume the Gompertz law (see (2.67)) for describing the standard force of mortality. So the cumulative standard force of mortality is x α H(x) = α eβt dt = (eβx − 1) (2.155) β 0

85

86


If we accept the Gamma assumption for Z0 , then the population force of mortality is αδeβx µ ¯x = (2.156) θ − βα + βα eβx Rearrange as 1 θ−

µ ¯x = Let

αδ θ−(α/β)

= α ,

α βθ−α

α β

αδeβx α 1 + βθ−α eβx

(2.157)

= δ ; so µ ¯x =

α eβx 1 + δ eβx

(2.158)

which is the first Perks law (see (2.74)), with γ = 0. Hence, (2.156) has a logistic shape; see Fig. 2.8. The logistic model for describing mortality within a heterogeneous population may be built also adopting a different approach (see Cummins et al. (1983); Beard (1971)). With reference to a heterogeneous population, assume that the individual force of mortality is Gompertz, with unknown ‘base’ mortality; hence µx = A eβx

(2.159)

where A (the parameter for base mortality) is a random quantity, specific to the individual, whilst β (the parameter for senescent mortality) is common to all individuals and known. Let ϕ(a) denote the pdf of A; the population force of mortality is then ∞ µ ¯x = a eβx ϕ(a) da = eβx E[A] (2.160) 0

2.5 x=0 x = 85

2 1.5 1 0.5 0 0.4

Figure 2.8.

0.6

0.8

1

1.2

1.4

1.6

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

Gomperiz Perks

65

Gamma distributions and forces of mortality.

75

85

95

105

115

2.10


If A ∼ Gamma(ρ, ν), then µ ¯ x = eβx Letting ρ =

α δβ ,

ν=

1 βx δ +e

β

ρ ν

(2.161)

, we find µ ¯x =

αeβx 1 + δeβx

(2.162)

which is still a particular case of (2.74), with γ = 0. Note, however, that this choice implies that the probability distribution of A depends on age. What we have just described can be easily classified under the multiplicative frailty model. Actually, if A in (2.159) is replaced with αz (with α certain and z random), one finds (2.128). The Perks model then follows by choosing a Gamma distribution for Z0 , with appropriate parameters. However, this approach is less elegant than that proposed by Vaupel et al. (1979), given that in (2.159) the distribution of A is not forced to depend on age. Actually, the multiplicative model allows for extensions and generalizations; further, it does not require a Gompertz force of mortality.

2.10


As regards the ‘traditional’ mortality model, that is, the model disregarding specific issues as mortality at very old ages and frailty, we restrict ourselves to general references. In some of these, the reader can find references to the original papers and reports, for example, by Gompertz, Makemam, Thiele, Perks, and so on. A number of textbooks of actuarial mathematics deal with life tables and mortality models, in both an age-discrete and an age-continuous context. The reader can refer for example to Bowers et al. (1997), Gerber (1995), Gupta and Varga (2002), Rotar (2007). The textbook by Benjamin and Pollard (1993) is particularly devoted to mortality analysis and mortality laws. The articles by Forfar (2004a) and Forfar (2004b) provide a compact and effective presentation of life tables and mortality laws respectively. Graduation methods are dealt with by many actuarial and statistical textbooks. Besides the textbook by Benjamin and Pollard (1993) already cited, the reader should consult, for example, London (1985), and the article by Miller (2004) which also provides an extensive list of references. As regards spline functions and their use to graduate mortality rates, the reader can

87

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refer to McCutcheon (1981), and Champion et al. (2004) and references therein. Historical aspects are dealt with by Haberman (1996), Haberman and Sibbett (1995) and Smith and Keyfitz (1977). In particular, in Haberman and Sibbett (1995) the reader can find the reproduction of milestone papers in mortality analysis up to 1919. In relation to mortality at old and very old ages, the deceleration in the rate of mortality increase is analysed in detail in the demographic literature. In particular, the reader can refer to Horiuchi and Wilmoth (1998), where the problem is attacked in the context of the frailty models. A discussion about non-Gompertzian mortality at very old ages is provided by Olshansky and Carnes (1997). Allowing for heterogeneity in population mortality (and, in particular, for non-observable heterogeneity) constitutes, together with mortality dynamics modelling, one of the most important issues in the evolution of survival models (see e.g. Pitacco (2004a)). Modelling frailty can suggest new ways to forecast mortality. Although the earliest contribution to this topic probably came from the actuarial field (Beard (1959) proposed the idea of individual frailty for capturing heterogeneity due to unobservable risk factors), the topic itself was ignored by actuaries up to some time ago. Conversely, seminal contributions have come from demography and biostatistics, also concerning the dynamics of mortality and longevity limits (see Vaupel et al. (1979), Hougaard (1984), and Yashin and Iachine (1997)). However, very recent contributions show interest for this topic within the actuarial community; see Butt and Haberman (2002), Butt and Haberman (2004), Olivieri (2006). Conversely, the interest of actuaries in observable factors (like gender, health condition, etc.) can be traced back to the first scientific models for life insurance. For example, see Cummins et al. (1983) as regards risk classification in life insurance and the numerical rating system in particular, that was pioneered by the New York Life Insurance company.

3 3.1

Mortality trends during the 20th century

Introduction

Life expectancy at birth among early humans was likely to be between 20 and 30 years as testified by evidence that has been glaned from tombstones inscriptions, genealogical records, and skeletal remains. Around 1750, the first national population data began being collected in the Nordic countries. At that time, life expectancy at birth was around 35–40 years in the more developed countries. It then rose to about 40–45 by the mid-1800s. Rapid improvements began at the end of the 19th century, so that, by the middle of the 20th century it was approximately 60–65 years. By the beginning of the 21st century, life expectancy at birth has reached about 70 years. The average life span has thus, roughly tripled over the course of human history. Much of this increase has happened in the past 150 years: the 20th century has been characterized by a huge increase in average longevity compared to all of the previous centuries. Broadly speaking, the average life span increased by 25 years in the 10,000 years before 1850. Another 25-year increase took place between 1850 and 2000. And there is no evidence that improvements in longevity are tending to slow down. The first half of the 20th century saw significant improvement in the mortality of infants and children (and their mothers) resulting from improvements to public health and nutrition that helped to withstand infectious diseases. Since the middle of the 20th century, gains in life expectancy have been due more to medical factors that have reduced mortality among older persons. Reductions in deaths due to the ‘big three’ killers (cardiovascular disease, cancer, and strokes) have gradually taken place, and life expectancy continues to improve. The population of the industrialized world underwent a major mortality transition over the course of the 20th century. In recent decades, the populations of developed countries have grown considerably older, because of two factors – increasing survival to older ages as well as the smaller numbers

90

3 : Mortality trends during the 20th century

of births (the so-called ‘baby bust’ which started in the 1970s). In this new demographic context, questions about the future of human longevity have acquired a special significance for public policy and fiscal planning. In particular, social security systems, which in many industrialized countries are organized according to the pay-as-you-go method, are threatened by the ageing of the population due to the baby bust combined with the increase in life expectancy. As a consequence, many nations are discussing adjustments or deeper reforms to address this problem. Thus, mortality is a dynamic process and actuaries need appropriate tools to forecast future longevity. We believe that any sound procedure for projecting mortality must begin with a careful analysis of past trends. This chapter purposes to illustrate the observed decline in mortality, on the basis of Belgian mortality statistics. The mortality experience during the 20th century is carefully studied by means of several demographic indicators which have been introduced in Chapter 2. Specifically, after having presented the different sources of mortality statistics, we compute age-specific death rates, life expectancies, median lifetimes and interquartile ranges, inter alia, as well as survival curves. We also compare statistics gathered by the insurance regulatory authorities with general population figures in order to measure adverse selection. A comparison between the mortality experience of some EU member countries is performed in Section 3.5. Before proceeding, let us say a few words about the notation used in this chapter. Here, we analyse mortality in an age-period framework. This means that we use two dimensions: age and calendar time. Both age and calendar time can be either discrete or continuous variables. In discrete terms, a person aged x, x = 0, 1, 2, . . ., has an exact age comprised between x and x+1. This concept is also known as ‘age last birthday’ (i.e., the age of an individual as a whole number of years, by rounding down to the age at the most recent birthday). Similarly, an event that occurs in calendar year t occurs during the time interval [t, t + 1]. This two-dimension setting is formally defined in Section 4.2.1; see Table 4.1. Otherwise, we follow the notation introduced in the previous chapters.

3.2

Data sources

In this chapter, we use three different sources of mortality data. Official data coming from a National Institute of Statistics or another governmental agency, data available from a scientific demographic database allowing for international comparisons, and market data provided by national regulatory authorities.

3.2

3.2.1

Data sources

Statistics Belgium

Statistics Belgium is the official statistical agency for Belgium. Formerly known as NIS-INS, Directoriate General Statistics Belgium is part of the Federal Public Service Economy. It is based in Brussels. Its mission is to deliver timely, reliable and relevant figures to the Belgian government, international authorities (like the EU), academics, and the public. For more information, we refer the reader to the official website at http://www.statbel.fgov.be. A national population register serves as the centralizing database in Belgium and provides official population figures. Statistics on births and deaths are available from this register by basic demographic characteristics (e.g. age, gender, marital status). Statistics Belgium constructs period life tables, separately for men and women. These life tables are available for the periods 1880–1890, 1928–1932, 1946–1949, 1959–1963, 1968–1972, 1979–1982, 1988– 1990, 1991–1993 and 1994–1996. After 1996, period life tables have been provided each year based on a moving triennium, starting from the 1997– 1999 life table, and continuing with the 1998–2000 life table, 1999–2001 life table, etc. The last available life table relates to the period 2002–2004. In each case, the mortality experienced by the Belgian population is represented as a set of one-year death probabilities qx (see Section 2.2.3 for a formal definition). Here, we use the life tables of the periods 1880– 1890, 1928–1932, 1968–1972, and 2000–2002 to investigate the long-term evolution of the mortality in Belgium. Even if the figures are computed from Belgian mortality experience, the analysis conducted in this chapter applies to any industrialized country and the findings would be very similar.

3.2.2

Federal Planning Bureau

The Federal Planning Bureau (FPB) is a public utility institution based in Brussels. The FPB makes studies and projections on socio-economic and environmental policy issues for the Belgian government. The population plays an important role in numerous themes examined by the FPB. This is why the FPB produces regularly updated projected life tables for Belgium. The official mortality statistics for Belgium come from FPB together with Statistics Belgium. Specifically, from 1948 to 1993, annual death probabilities were computed by FPB. From 1994, annual death probabilities are computed by Statistics Belgium and published on a yearly basis. The annual death probabilities are now available for calendar years

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t = 1948, 1949, . . . , 2004 and ages   0, 1, . . . , 100, x = 0, 1, . . . , 101,  0, 1, . . . , 105,

3.2.3

for t = 1948, 1949, . . . , 1993 for t = 1994, 1995, . . . , 1998 for t = 1999, 2000, . . .

Human mortality database

The Human mortality database (HMD) was launched in May 2002 to provide detailed mortality and population data to those interested in the history of human longevity. It has been put together by the Department of Demography at the University of California, Berkeley, USA, and the Max Planck Institute for Demographic Research in Rostock, Germany. It is freely available at http://www.mortality.org and provides a highly valuable source of mortality statistics. HMD contains original calculations of death rates and life tables for national populations, as well as the raw data used in constructing those tables. The HMD includes life tables provided by single years of age up to 109, with an open age interval for 110+. These period life tables represent the mortality conditions at a specific moment in time. We refer readers to the methods protocol available from the HMD website for a detailed exposition of the data processing and table construction. For Belgium, date were compiled by Dana Glei, Isabelle Devos and Michel Poulain. They cover the period starting in 1841 and ending in 2005. However, data are missing during World War I. This is why we have decided to restrict the study conducted in this chapter to the period 1920–2005.

3.2.4

Banking, Finance, and Insurance Commission

In addition to general population data, we also analyse mortality statistics from the Belgian insurance market. Any difference between the general population and the insured population is due to adverse selection, as explained in Section 1.6.5. Market data are provided by the Banking, Finance and Insurance Commission (BFIC) based in Brussels. BFIC has been created as a result of the integration of the Insurance Supervisory Authority (ISA) into the Banking and Finance Commission (BFC). Since the 1st of January 2004, it is the single supervisory authority for the Belgian financial sector. For more information, we refer readers to the official website http://www.cbfa.be.

3.3

Mortality trends in the general population

Annual tabulations of the number of deaths by age, by gender, and by policy type are made by the BFIC based on information supplied by insurance companies. Together with the number of deaths, the corresponding (central) risk exposure is also available in each case. These data allow us to calculate age-gender-type-of-product specific (central) death rates. We do not question the quality of the data provided by BFIC.

3.3 3.3.1

Mortality trends in the general population Age-period life tables

As explained in Section 2.2, life table analyses are based upon an analytical framework in which death is viewed as an event whose occurrence is probabilistic in nature. Life tables create a hypothetical cohort (or group) of, say, 100,000 persons at age 0 (usually of males and females separately) and subject it to age-gender-specific annual death probabilities (the number of deaths per 1,000 or 10,000 or 100,000 persons of a given age and gender) observed in a given population. In doing this, researchers can trace how the 100,000 hypothetical persons (called a synthetic cohort) would shrink in numbers due to deaths as the group ages. As stressed in Section 2.2.1, there are two basic types of life tables: period life tables and cohort life tables. A period life table represents the mortality experience of a population during a relatively short period of time, usually between one and three years. Life tables based on population data are generally constructed as period life tables because death and population data are most readily available on a time period basis. Such tables are useful in analysing changes in the mortality experienced by a population through time. These are the tables used in the present chapter. We analyse the changes in mortality as a function of both age x and calendar time t. This is the so-called age-period approach. In this chapter, we assume that the age-specific forces of mortality are constant within bands of age and time, but allowed to vary from one band to the next. This extends to a dynamic setting the constant force of mortality assumption (b) in Section 2.3.5. Specifically, let us denote as Tx (t) the remaining lifetime of an individual aged x at time t. Compared to Section 2.2.3, we supplement the notation Tx for the remaining lifetime of an x-aged individual with an extra index t representing calendar time. This individual will die at age x + Tx (t) in year t + Tx (t). Then, qx (t) is the probability that an x-aged individual in

93

94


calendar year t dies before reaching age x + 1, that is, qx (t) = P[Tx (t) ≤ 1]. Similarly, px (t) = 1 − qx (t) is the probability that an x-aged individual in calendar year t reaches age x + 1, that is, px (t) = P[Tx (t) > 1]. The force of mortality µx (t) at age x and time t is formally defined as µx (t) = lim

0

P[x < T0 (t − x) ≤ x + |T0 (t − x) > x]

(3.1)

Compare (3.1) to (2.32)–(2.34). Now, given any integer age x and calendar year t, we assume that µx+ξ1 (t + ξ2 ) = µx (t)

for 0 ≤ ξ1 , ξ2 < 1

(3.2)

This is best illustrated with the aid of a coordinate system that has calendar time as abscissa and age as coordinate as in Fig. 3.1. Such a representation is called a Lexis diagram after the German demographer who introduced it. Both time scales are divided into yearly bands, which partition the Lexis plane into square segments. Formula (3.2) assumes that the mortality rate is constant within each square, but allows it to vary between squares; see Fig. 3.1 for a graphical interpretation. Since life tables do not include mortality measures at non-integral ages or for non-integral durations, (3.2) can also be seen as a convenient interpolation method to expand a life table for estimating such values. Under (3.2), we have for integer age x and calendar year t that 1 px (t) = exp − µx+ξ (t + ξ) dξ = exp(−µx (t)) 0

Age

x+1

x

t–x–1 Figure 3.1.

t–x

t

t+1

Illustration of the basic assumption (3.2) with a Lexis diagram.

Time

(3.3)

3.3


which extends (2.36). For durations s less than 1 year, we have under assumption (3.2) that s µx+ξ (t + ξ) dξ s px (t) = exp − 0

= exp (−sµx (t)) = px (t)

s

(3.4)

Moreover, the forces of mortality and the central death rates (see Section 2.3.4 for formal definitions) coincide under (3.2), that is, µx (t) = mx (t). This makes statistical inference much easier since rates are estimated by dividing the number of occurrences of a selected demographic event in a (sub-) population by the corresponding number of person-years at risk (see next section).

3.3.2

Exposure-to-risk

When working with death rates, the appropriate notion of risk exposure is the person-years of exposure, called the (central) exposure-to-risk in the actuarial literature. The exposure-to-risk refers to the total number of ‘person-years’ in a population over a calendar year. It is similar to the average number of individuals in the population over a calendar year adjusted for the length of time they are in the population. Let us denote as ETRxt the exposure-to-risk at age x last birthday during year t, that is, the total time lived by people aged x last birthday in calendar year t. There is an easy expression for the average exposure-to-risk that is valid under (3.2). As in (1.45), let Lxt be the number of individuals aged x last birthday on January 1 of year t. Then, 1 ξ E[ETRxt |Lxt = l] = l px (t) dξ ξ=0

=− =

l 1 − px (t) µx (t)

−lqx (t) ln(1 − qx (t))

(3.5)

Hence, provided the population size is large enough, we get the approximation ETRxt ≈

−Lxt qx (t) ln(1 − qx (t))

(3.6)

that can be used to reconstitute the ETRxt ’s from the Lxt ’s and the qx (t)’s in the case where the ETRxt ’s are not readily available. This formula appears

95

96


to be useful since, in the majority of the applications to general population data, the exposure-to-risk is not provided. When the actuary works with market data, or with statistics gathered from a given insurance portfolio, the exposures-to-risk are easily calculated so that there is no need for the approximation formula (3.6).

3.3.3

Death rates

We consider the estimation of µx (t) under assumption (3.2). We will see that the maximum likelihood estimator of µx (t) is obtained by dividing the number of deaths recorded at age x in year t by the corresponding exposure-to-risk ETRxt . This is an expected result since µx (t) and mx (t) coincide under (3.2). To get this result in a formal way, let us associate to each of the Lxt individuals alive at the beginning of the period an indicator variable δi defined as 1, if individual i dies at age x δi = (3.7) 0, otherwise i = 1, 2, . . . , Lxt . Furthermore, let τi be the fraction of the year lived by individual i, and let Dxt be the number of deaths recorded at age x last birthday during calendar year t, from an exposure-to-risk ETRxt . We obviously have that Lxt Lxt δi = Dxt and τi = ETRxt (3.8) i=1

i=1

Note that the method of recording the calendar year of death and the age last birthday at death means that the death counts Dxt cover individuals born on January 1 in calendar year t−x−1 through December 31 in calendar year t − x (i.e., two successive calendar years) with a peak representation around January 1 in calendar year t − x. Under the assumption (3.2) and using (3.3), the contribution of individual i to the likelihood may be written as px (t) = exp(−µx (t))

(3.9)

if he survives, and τi px (t)µx+τi (t

+ τi ) = exp(−τi µx (t))µx (t)

(3.10)

if he dies at time τi during year t. Combining expressions (3.9)–(3.10), the contribution of individual i to the likelihood can be transformed into exp(−τi µx (t)) (µx (t))δi

(3.11)

3.3


If the individual lifetimes are mutually independent, the likelihood for the Lxt individuals aged x is then equal to Lxt ! L µx (t) = exp(−τi µx (t)) (µx (t))δi i=1

= exp(−µx (t)ETRxt ) (µx (t))Dxt

(3.12)

Note that this likelihood is proportional to the one based on the Poisson distributional assumption for Dxt . Setting the derivative of ln L µx (t) equal to 0, we find the maximum likelihood estimate " µx (t) of the force of mortality µx (t) that is given by " µx (t) =

Dxt " x (t) =m ETRxt

(3.13)

" x (t)’s are referred to as crude (i.e. unsmoothed) death rates for age The m x in calendar year t. The death rate is, thus, the proportion of people of a given age expected to die within the year, expressed in terms of the expected number of life-years rather than in terms of the number of individuals initially present in the group. Often, ETRxt is approximated by an estimate of the population aged x last birthday in the middle of the calendar year. This quantity is estimated by a national institute of statistics taking account of recorded births and deaths and net immigration. Formula (3.6) can also be used to reconstitute the exposure-to-risk under assumption (3.2). " x (t) for males and Figure 3.2 displays the logarithm of the death rates m females for four selected periods. They come from the official life tables constructed by Statistics Belgium, and cover the last 120 years. For each period, death rates are relatively high in the first year after birth, decline rapidly to a low point around age 10, and thereafter rise, in a roughly exponential fashion, before decelerating (or slowing their rate of increase) at the end of the life span. This is the typical shape of a set of death rates. From Fig. 3.2, it is obvious that dramatic changes in mortality have occurred over the 20th century. The striking features of the evolution of mortality are the downard trends and the substantial variations in shape. We see that the greatest relative improvement in mortality during the 20th century occurred at the young ages, which has resulted largely from the control of infectious diseases. The decrease over time at ages 20–30 for females reflects the rapid decline in childbearing mortality. The hump in mortality around ages 18–25 has become increasingly important, especially for young males. Accidents, injuries, and suicides account for the majority of the excess mortality of males over females at ages under 45 (this is why this hump is often referred to as the accident hump).

97


2000–2002 1968–1972 1928–1932 1880–1890

–2

ln mx

–4

–6

–8

0

20

40

60

80

100

60

80

100

x

2000–2002 1968–1972 1928–1932 1880–1890

–2

–4 ln mx

98

–6

–8

0

20

40 x

Figure 3.2. Death rates (on the log scale) for Belgian males (top panel) and Belgian females (bottom panel) from period life tables 1880–1890, 1928–1932, 1968–1972, and 2000–2002. Source: Statistics Belgium.

" x (t)’s for some selected ages is depicted The trend in the logarithm of the m in Figs 3.3 and 3.4. An examination of Fig. 3.3 reveals distinct behaviours for age-specific death rates affecting Belgian males. At age 20, a rapid reduction took place after a peak which occurred in the early 1940s due to World War II. A structural break seems to have occurred, with a relatively high level of mortality before World War II, and a much lower level after 1950. Since the mid-1950s, only modest improvements have occurred for the " 20 (t)’s. This is typical for ages around the accident hump, where male m mortality has not really decreased since the 1970s. At age 40, the same decrease after World War II is apparent, followed by a much slower reduction after 1960. The decrease after 1970 is nevertheless more marked than

–4.5 –5.0

ln m40(t)

ln m20(t)

–5.0 –5.5 –6.0

–5.5

–6.0

–6.5 –7.0 1920

1940

1960

1980

2000

1920

1940

1960 t

1980

2000

1920

1940

1960 t

1980

2000

t

–3.6

–1.8

–3.8 ln m80(t)

ln m60(t)

–2.0 –4.0

–2.2

–4.2 –2.4 –4.4 1920

Figure 3.3.

1940

1960 t

1980

2000

Trend in observed death rates (on the log scale) for Belgian males at ages 20, 40, 60, and 80, period 1920–2005. Source: HMD.

–5.0

–5.5 –6.0

–5.5 ln m40(t)

ln m20(t)

–6.5 –7.0

–6.0

–7.5 –6.5

–8.0 –8.5

–7.0 1940

1960 t

1980

2000

–3.8

–1.8

–4.0

–2.0

–4.2

–2.2 ln m80(t)

ln m60(t)

1920

–4.4 –4.6

1920

1940

1960 t

1980

2000

1920

1940

1960 t

1980

2000

–2.4 –2.6

–4.8

–2.8

–5.0

–3.0

–5.2 1920

1940

1960 t

Figure 3.4.

1980

2000

Trend in observed death rates (on the log scale) for Belgian females at ages 20, 40, 60, and 80, period 1920–2005. Source: HMD.

3.3


at age 20. At ages 60 and 80, mortality rates have declined rapidly after 1970, whereas the decrease during 1920–1970 was rather moderate. We note that the effect of World War II is much more important at younger ages than at older ages. This clearly shows that gains in longevity have been concentrated on younger ages during the first half of the 20th century, and have then moved to older ages after 1950. The analysis for Belgian females illustrated in Fig. 3.4 parallels that for males for ages 20 and 40, but with several differences. At age 20, modest improvements are visible after the mid-1950s. At age 40, more pronounced reductions occurred after 1960. At older ages, the rate of decrease is more regular, and has tended to accelerate after 1980. This acceleration is a feature seen in a number of Western European countries. Kannisto et al. (1994) report an acceleration in the late 1970s in rate of decrease of mortality rates at ages over 80 in an analysis of mortality rates for 9 European countries with reliable mortality data at these ages over an extended period.

3.3.4

Mortality surfaces

The dynamic analysis of mortality is often based on the modelling of the mortality surfaces that are depicted in Fig. 3.5. Such a surface consists of a " x (t)’s viewed as a function three-dimensional plot of the logarithm of the m of both age x and time t. Fixing the value of t, we recognize the classical shape of a mortality curve visible in Fig. 3.2. Specifically, along cross sections when t is fixed (or along diagonals when cohorts are followed), one observes relatively high mortality rates around birth, the well-known presence of a trough at about age 10, a ridge in the early 20s (which is less pronounced for females), and an increase at middle and older ages. Mortality does not vary uniformly over the age-year plane and the advantage of plots as in Fig. 3.5 is that they facilitate an examination of the way that mortality changes with year and cohort as well as with age. In addition to random deviation from the underlying smooth mortality surface, the surface is subject to period shocks corresponding to wars, epidemics, harvests, summer heat waves, etc. Roughness of the surface indicates volatility and ridges along cross sections at given years mark brief episodes of excess mortality. For instance, higher mortality rates are clearly visible for the years around World War II.

3.3.5

Closure of life tables

At higher ages (above 80), death rates displayed in Fig. 3.5 appear rather smooth. This is a consequence of the smoothing procedure implemented

101

102


–2 –4 –6 –8 1920 100

1940 80

1960

60

t 40

1980

x

20 2000

0

–2 –4 –6 –8 1920 1940

100 80

1960

t

60 40

1980

x

20 2000 0 Figure 3.5. Observed death rates (on the log scale) for Belgian males (top panel) and Belgian females (bottom panel), ages 0 to 109, period 1920–2005. Source: HMD.

3.3


in HMD. Death rates for ages 80 and above were estimated according to the logistic formula and were then combined with death rates from younger ages in order to reconstitute life tables. To have an idea of the behaviour of mortality rates at the higher ages, we have plotted in Fig. 3.6 the rough death rates observed for the Belgian population. As discussed in Section 2.8, we clearly see from Fig. 3.6 that data at old ages produce suspect results (because of small risk exposures): the pattern at old and very old ages is heavily affected by random fluctuations because of the scarcity of data. Sometimes, data above some high age are not available at all. Recently, some in-depth demographic studies have provided a more sound knowledge about the slope of the mortality curve at very old ages. It has been documented that the force of mortality is slowly increasing at very old ages, approaching a rather flat shape. The deceleration in the rate of increase in mortality rates can be explained by the selective survival of healthier individuals at older ages (see, e.g. Horiuchi and Wilmoth, 1998) for more details, as well as the discussion about frailty in Section 2.9.3). Demographers and actuaries have suggested various techniques for estimating the force of mortality at old ages and for completing the life table. See Section 2.8.2 for examples and references. Here, we apply a simple and powerful method proposed by Denuit and Goderniaux (2005). The starting point is standard: there is ample empirical evidence that the one-year death probabilities behave like the exponential of a quadratic polynomial at older ages, that is, qx (t) = exp(at + bt x + ct x2 ). Hence, a log-quadratic regression model of the form ln " qx (t) = at + bt x + ct x2 + xt

(3.14)

for the observed one-year death probabilities, with xt independent and Normally distributed with mean 0 and variance σ 2 , is fitted separately to each calendar year t (t = t1 , t2 , . . . , tm ) and to ages xt and over. Then, constraints are imposed to mimic the observed behaviour of mortality at old ages. First, a closure constraint q130 (t) = 1

for all t

(3.15)

which retains as working assumption that the limit age 130 will not be exceeded. Secondly, an inflection constraint # # ∂ qx (t)## =0 ∂x x=130

for all t

(3.16)

103

104


0 –2 –4 –6 –8 1950 1960

100 80

1970

t

60

1980 40

1990

x

20 2000 0

–2 –4 –6 –8 –10 –12 1950 1960 1970

t

100 80 60

1980

40 x

1990 20 2000 0

Figure 3.6. Observed death rates (on the log scale) for Belgian males (top panel) and Belgian females (bottom panel), period 1950–2004. Source: Statistics Belgium.

which is used to ensure that the behaviour of the ln qx (t)’s will be ultimately concave. This is in line with empirical studies that provide evidence of a decrease in the rate of mortality increase at old ages. One explanation proposed for this deceleration is the selective survival of healthier individuals to older ages, as noted above.

3.3


Note that both constraints are imposed here at age 130. In general, the closing age could also be treated as a parameter and selected from the data (together with the starting age xt , thereby determining the optimal fitting age range). These two constraints yield the following relation between the at ’s, bt ’s, and ct ’s for each calendar time t: at + bt x + ct x2 = ct (130 − x)2

(3.17)

for x = xt , xt + 1, . . . and t = t1 , t2 , . . . , tm . The ct ’s are then estimated on the basis of the series {" qx (t), x = xt , xt + 1, . . .} relating to year t from equation (3.14), noting the constraints imposed by (3.17). It is worth mentioning that the two constraints underlying the modelling of the qx (t) for high x are in line with empirical demographic evidence. Let us now apply this method to the data displayed in Fig. 3.6. The optimal starting age is selected from the age range 75–89. It turns out to be around 75 for all of the calendar years. Therefore, we fix it to be 75 for both genders and for all calendar years. The R2 corresponding to the fitted regression models (3.14), as well as the estimated regression parameters ct are displayed in Fig. 3.7. We keep the original " qx (t) for ages below 85 and we replace the death probabilities for ages over 85 with the fitted values coming from the constrained quadratic regression (3.14). The results for calendar years 1950, 1960, 1970, 1980, 1999, and 2000 can be seen in Fig. 3.8 for males and in Fig. 3.9 for females. The completed mortality surfaces are displayed in Fig. 3.10.

3.3.6

Rectangularization and expansion

Figure 3.11 shows the rectangularization phenomenon. It presents the population survival functions based on period life tables for, from left to right, 1880–1890, 1928–1932, 1968–1972, and 2000–2002. Survival functions have been formally introduced in Section 2.3.1. Broadly speaking, they give the proportion of individuals reaching the age displayed along the x-axis, where this proportion is computed on the basis of the set of age-specific mortality rates corresponding to the different period life tables. As we have noted in the introduction, considerable progress has been made in the 20th century towards eliminating the hazards to survival which existed at the young ages in the early 1900s. This is clearly visible from Fig. 3.11 where the proportion of the population still alive at some given age increases as we move forward in calendar time. As a consequence, the slope of the survival function has become more rectangular (less diagonal) through time. This is the so-called ‘curve squaring’ concept,

105


1.00

R2

0.98 0.96 0.94

Males Females

0.92 1950

1960

1970

1980

1990

2000

1980

1990

2000

t –0.0008

–0.0009

ct

106

–0.0010

–0.0011

Males Females

–0.0012 1950

1960

1970 t

Figure 3.7. Adjustment (3.14)–(3.17).

coefficients

and

estimated

regression

parameters

for

model

which has been the subject of passionate debate among demographers in recent years. Let us now consider the age corresponding to a value of 0.5 for the survival curve. This age is called median age at birth and is one of the standard demographic markers; see Section 2.4.2. Broadly speaking, the median is the age reached by half of a hypothetical population with mortality experience reflected by that particular period life table. Figure 3.12(top panel) shows the increasing trend in the median life at birth: median lifetimes are depicted by gender and calendar year, based on period life tables. Figure 3.12(bottom panel) is the analogue for the median remaining lifetime at age 65.

0

–2

–2

–2

–4

ln qx

0

ln qx

ln qx

0

–4

–6

–6

–6

–8

–8 0

20

40

60

80

100

0

120

20

40

60

80

100

0

120

20

40

0

–2

–2

–2

–4

–4

–4

ln qx

0

ln qx

0

–6

–6

80

100

120

80

100

120

–6 –8

–8

–8

60 x

x

x

ln qx

–4

–10 0

20

40

60

80 x

100

120

0

20

40

60

80 x

100

120

0

20

40

60 x

Figure 3.8. Completed life tables for Belgian males, years 1950, 1960, 1970, 1980, 1990, and 2000, together with empirical death probabilities (broken line), on the log-scale.

0

–2

–2

–2

–4 –6

ln qx

0

ln qx

ln qx

0

–4

–6

–6

–8

–8 0

20

40

60

80

100

120

–8 0

20

40

60

x

80

100

120

0

0

–2

–2

–2

–4

ln qx

0

–4 –6

–6

40

60

80 x

60

100

120

80

100

120

80

100

120

–4 –6

–8

–8 20

40

x

0

0

20

x

ln qx

ln qx

–4

–8 0

20

40

60

80 x

100

120

0

20

40

60 x

Figure 3.9. Completed FPB life tables for Belgian females, years 1950, 1960, 1970, 1980, 1990, and 2000, together with empirical death probabilities (broken line), on the log-scale.

3.3


0 −2 −4 −6 −8 1950 1960 100

1970 t

1980 50 x

1990 2000 0

0 −2 −4 −6 −8 1950 1960 100

1970 t

1980 50

1990

x

2000 0 Figure 3.10. Completed death rates (on the log scale) for Belgian males (top panel) and Belgian females (bottom panel), period 1920–2005.

109


1.0 0.8

S(x)

0.6 0.4 2000–2002 1968–1972 1928–1932 1880–1890

0.2 0.0 0

20

40

60

80

100

60

80

100

x 1.0 0.8 0.6 S(x)

110

0.4 2000–2002 1968–1972 1928–1932 1880–1890

0.2 0.0 0

20

40 x

Figure 3.11. Survival curves for Belgian males (top panel) and Belgian females (bottom panel) corresponding to the 1880–1890, 1928–1932, 1968–1972, and 2000–2002 period life tables. Source: Statistics Belgium.

Rectangularization of survival curves is associated with a reduction in the variability of age at death. As deaths become concentrated in an increasingly narrow age range, the slope of the survival curve in that range becomes steeper, and the curve itself begins to appear more rectangular. A simple measure of rectangularity is thus the maximum downward slope of the survival curve S in the adult age range that has been formally defined in (2.61). Increasing rectangularity according to this measure implies a survival curve which becomes increasingly vertical at older ages. Figure 3.13 displays the distribution of ages at death (empirical version of the theoretical probability density function f defined in (2.28)). It can be seen that the distribution of ages at death has shifted to the right and has

3.3


85 Males Females

Med[T65(t)]

80 75 70 65 60 1920

1940

1960 t

1980

2000

1940

1960 t

1980

2000

20 Males Females

Med[T65(t)]

18 16 14 12 10 1920

Figure 3.12. Observed median lifetimes at birth (top panel) and at age 65 (bottom panel), period 1920–2005. Source: HMD.

become less variable and less obviously bimodal. We clearly observe that the point of fastest decline increases with time, which empirically supports the expansion phenomenon.

3.3.7

Life expectancies

The index, life expectancy, has been formally defined in Section 2.4.1. Life expectancy statistics are very useful as summary measures of mortality, and they have an intuitive appeal. However, it is important to interpret data on life expectancy correctly when their computation is based on period life tables. Period life expectancies are calculated using a set of age-specific mortality rates for a given period (either a single year, or a run of years), with

111


2000–2002 1968–1972 1928–1932 1880–1890

0.03

f(x)

0.02

0.01

0.00 0

20

40

60

80

100

60

80

100

x

0.04

2000–2002 1968–1972 1928–1932 1880–1890

0.03 f(x)

112

0.02

0.01

0.00 0

20

40 x

Figure 3.13. Observed proportion of ages at death for Belgian males (top panel) and Belgian females (bottom panel) corresponding to 1880–1890, 1928–1932, 1968–1972, and 2000–2002 period life tables. Source: Statistics Belgium.

no allowance for any future changes in mortality. Cohort life expectancies are calculated using a cohort life table, that is, using a set of age-specific mortality rates which allow for known or projected changes in mortality at later ages (in later years). Period life expectancies are a useful measure of the mortality rates that have been actually experienced over a given period and, for past years, provide an objective means of comparison of the trends in mortality over time, between areas of a country and with other countries. Official life tables which relate to past years are generally period life tables for these reasons. Cohort life expectancies, even for past years, may require projected mortality rates for their calculation. As such, they are less objective because they are subject to substantial model risk and forecasting error.

3.3


In this chapter, we only compute period life expectancies. Cohort life expectancies will be derived in Chapter 5 using appropriate mortality pro↑ jection methods. Let ex (t) be the period life expectancy at age x in calendar year t. Here, we have used a superscript ‘↑’ to recall that we work along a vertical band in the Lexis diagram, considering death rates associated with ↑ a given period of time. Specifically, ex (t) is computed from the period life table for year t, given by the set µx+k (t), k = 0, 1, . . . . The formula giving ↑ ex (t), under assumption (3.2), is ξ e↑x (t) = exp − µx+η (t) dη dξ ξ≥0

0

1 − exp − µx (t) = µx (t)   k−1 ! 1 − exp −µx+k (t)   + exp −µx+j (t) µx+k (t) k≥1

(3.18)

j=0

In this formula, the ratio (1 − exp(−µx+k (t))/µx+k (t)) is the average fraction of the year lived by an individual alive at age x + k, and the product $k−1 ↑ j=0 exp(−µx+j (t)) is the probability k px (t) of reaching age x+k computed from the period life table. ↑

Figure 3.14 shows the trend in the period life expectancies at birth e0 (t) ↑ and at retirement age e65 (t) by gender. The period life expectancy at a particular age is based on the death rates for that and all higher ages that were experienced in that specific year. For life expectancies at birth, we observe a regular increase after 1950, with an effect due to World War II which is visible before that time (especially at the beginning and at the end ↑ of the conflict for e0 (t), and during the years preceding the conflict as well as ↑ during the war itself for e65 (t)). Little increase was experienced from 1930 to 1945. It is interesting to note that period life expectancies are affected by sudden and temporary events, such as a war or an epidemic.

3.3.8

Variability

Wilmoth and Horiuchi (1999) have studied different measures of variability for the distribution of ages at death. These authors favour the interquartile range for both its ease of calculation and for its straightforward interpretation. The interquartile range measures the distance between the lower and the upper quartiles of the distribution of ages at death in a life table. This range is formally defined as the difference between the age corresponding

113

3 : Mortality trends during the 20th century 85 Males Females

80

e0(t)

75 70 65 60 55 50 1920

1940

1960

1980

2000

1980

2000

t 20 Males Females

18

e65(t)

114

16 14 12 10 1920

1940

1960 t

Figure 3.14. Observed period life expectancies at birth (top panel) and at age 65 (bottom panel) for Belgian males (continuous line) and Belgian females (dotted line), period 1920–2005. Source: HMD.

to the value 0.25 of the survival curve minus the age corresponding to the value 0.75 of this curve; see (2.64). The former age (called the third quartile) is attained by 25% of the population whereas 75% of the population reaches the latter age (called the first quartile). The interquartile range is thus the width of the age interval containing the 50% central deaths in the population. As age at death becomes less variable, we would expect that this measure would decrease. It is very simple to calculate because it equals the difference between the ages where the survival curve S crosses the probability levels 0.25 and 0.75. Being the length of the span of ages containing the middle 50% of deaths, it possesses a simple interpretation. Note that the rectangularization of survival curves is associated with decreasing interquartile range.

3.3


Males

45

Females 40

IQR

35 30 25 20 15 1920

1940

1960 t

1980

2000

11.5

IQR

11.0

10.5 Males Females

10.0

1920

1940

1960 t

1980

2000

Figure 3.15. Observed interquartile range at birth (top panel) and at age 65 (bottom panel) for Belgian males (continuous line) and Belgian females (dotted line), period 1920–2005. Source: HMD.

Figure 3.15 depicts the interquartile range at birth and at age 65. Whereas the interquartile range at birth clearly decreases over time, there is an upward trend at age 65. This suggests that even if variability is decreasing for the entire lifetime, this may not be the case for the remaining lifetime at age 65.

3.3.9

Heterogeneity

Within populations, differences in life expectancy exist with regard to gender. Females tend to outlive males in all populations, and have lower mortality rates at all ages, starting from infancy. This is clear from all of the figures examined so far in this chapter. Another difference in life expectancy

115

116


occurs because of social class, as assessed through occupation, income, or education. In recent decades, population data have shown widening mortality differentials by socio-economic class. The mortality of the better off classes has improved more rapidly. The major cause of death responsible for the widening differential is cardiovascular disease: persons of higher social classes have experienced much larger declines in death due to cardiovascular disease than persons of lower classes. Other possible explanations include cigarette smoking (which is known to vary significantly according to social class) as well as differences in diet, selection mechanims, poorer quality housing conditions and occupation. In general, individuals with higher socio-economic status live longer than those in lower socio-economic groups. This heterogeneity can be accounted for as discussed in Section 2.9. We will see below that the effect of social class is significant for insurance market mortality statistics. Indeed, the act of purchasing life insurance products often reveals that the individual belongs to upper socio-economic class, which in turn yields lower mortality (even in the case of death benefits).

3.4 3.4.1

Life insurance market Observed death rates

Figure 3.16 displays the period life tables for the Belgian individual life insurance market, group life insurance market, and the general population observed in the calendar years 1995, 2000, and 2005. The variability in the set of death rates is clearly much higher for the insurance market, as exposures-to-risk are considerably smaller. This is why smoothing the market experience to make the underlying trend more apparent is desirable. This is achieved as explained below. The standardized mortality ratio (SMR) is a useful index for comparing mortality experiences: actual deaths in a particular population are compared with those which would be expected if ‘standard’ age-specific rates applied. Precisely, the SMR is defined as " x (t) (x, t)∈D ETRxt m (x, t)∈D Dxt SMR = = " stand " stand (t) (t) x x (x, t)∈D ETRxt m (x, t)∈D ETRxt m where D is the set of ages and calendar years under interest. Here are the SMRs by calendar year for the life insurance market: computed over 1993–2005, the estimated SMR is equal to 0.5377419 for ages

0

–2

–2

–2

–4

–4

–4

–6

–6

–6

–8

–8

–8

40

50

60

70 x

80

90

40

100

50

60

70 x

80

90

100

0

0

–2

–2

–2

–4

ln mx

0

ln mx

ln mx

ln mx

0

ln mx

ln mx

0

–4 –6

–6

–8

–8

–8

50

60

70 x

80

90

100

40

50

60

70 x

80

90

100

50

60

70 x

80

90

100

40

50

60

70 x

80

90

100

–4

–6

40

40

Figure 3.16. General population (broken line) and individual (circle) and group (triangle) life insurance market death rates (on the log scale) observed in 1995, 2000, and 2005 for Belgian males (top panel) and females (bottom panel). Source: HMD for the general population and BFIC for insured lives.

118


45–64 and to 0.3842981 for ages 65 and over for individual policies, and to 0.495525 and to 0.8042604 for group policies. The same values computed over 2000–2005 are equal to 0.4796451, 0.3699633, 0.4963897, and 0.8692767, respectively. Note that the values for group contracts, ages 45– 64 have been computed by excluding calendar year 2001, which appeared to be atypical for group life contracts before retirement age. We see that SMR’s are around 50% for individual and group life insurance contracts before retirement age, and then decrease to reach 40% for individual policies and increase to 80% for group life policies.

3.4.2

Smoothed death rates

It is clear from Fig. 3.16 that death rates based on market data exhibit considerable variations. This is why some smoothing is desirable in order to obtain a better picture of the underlying mortality experienced by insured lives. Since possible changes in underwriting practices or tax reforms are likely to affect market death rates, we smooth the death rates across ages by calendar year, as in Hyndman and Ullah (2007). To this end, we use local regression techniques. Local regression is used to model a relation between a predictor variable (or variables) x and a response Y, which is related to the predictor variable. Typically, x represents age in the application that we have in mind in this chapter, while Y is some (suitably transformed) demographic indicator such as the logarithm of the death rate or the logit of the death probability. The logarithmic and logit transformations involved in these models ensure that the dependent variables can assume any possible real values. As pointed out by Loader (1999), smoothing methods and local regression originated in actuarial science in the late 19th and early 20th centuries, in the problem of graduation. See Section 2.6 for an introduction to these concepts. Having observed (x1 , Y1 ), (x2 , Y2 ), . . ., (xm , Ym ), we assume a model of the form Yi = f (xi ) + i , i = 1, 2, . . . , m, where f (·) is an unknown function of x, and i is an error term, assumed to be Normally distributed with mean 0 and variance σ 2 . This term represents the random departures from f (·) in the observations, or variability from sources not included in the xi ’s. No strong assumptions are made about f , except that it is a smooth function that can be locally well approximated by simple parametric functions. For instance, invoking Taylor’s theorem, any differentiable function can be approximated locally by a straight line, and a twice differentiable function can be approximated locally by a quadratic polynomial. In order to estimate f at some point x, the observations are weighted in such a way that the largest weights are assigned to observations close to

3.4

Life insurance market

x. In many cases, the weight wi (x) assigned to (xi , Yi ) to estimate f (x) is obtained from the formula

xi − x wi (x) = W h(x)

(3.19)

where W(·) is choosen to be continuous, symmetric, peaked at 0 and supported on [−1, 1]. A common choice is the tricube weight function defined as 3 for −1 < u < 1 1 − |u|3 (3.20) W(u) = 0 otherwise The bandwidth h(x) defines a smoothing window (x − h(x), x + h(x)), and only observations in that window are used to estimate f (x). Within the smoothing window, f is approximated by a polynomial. The coefficients of this polynomial are then estimated via weighted least-squares. The bandwidth h(x) has a critical effect on the local regression. If h(x) is too small, insufficient data fall within the smoothing window and a noisy fit results. On the other hand, if h(x) is too large, the local polynomial may not fit the data well within the smoothing window, and important features of the mean function may be distorted or even lost. The nearest neighbour bandwidth is often used. Specifically, h(x) is selected so that the smoothing window contains a specified number of points. A high polynomial degree can always provide a better approximation to f than a low polynomial degree. But high order polynomials have large numbers of coefficients to estimate, and the result is increased variability in the estimate. To some extent, the effects of the polynomial degree and bandwidth are confounded. It often suffices to chose a low degree polynomial and to concentrate on choosing the bandwidth in order to obtain a satisfactory fit. The most common choices are local linear and local quadratic. A local linear estimate usually produces better fits, especially at the boundaries. The weight function W(·) has much less effect on the bias-variance trade-off, and the tricube weight function (3.20) is routinely used. Let us approximate f by a linear function β0 (x) + β1 (x)x in the smoothing window (x − h(x), x + h(x)). This leads to local linear regression. The coefficients β0 (x) and β1 (x) are estimated by minimizing the local residual sum of squares

OW (x) =

m i=1

wi (x) Yi − β0 (x) − β1 (x)xi

2

(3.21)

119

120


Denoting as

m wi (x)xi xw = i=1 m i=1 wi (x)

(3.22)

the weighted average of the xi ’s in the smoothing window, the minimization of the objective function OW (x) gives " β1 (x)x f (x) = " β0 (x) + " m m i=1 wi (x) xi − xw Yi i=1 wi (x)Yi = m + x − xw 2 m wi (x) xi − xw i=1 wi (x)

(3.23)

i=1

Let us give an interpretation to this expression for " f (x). The first term in " f (x) is the well-known Nadaraya–Watson kernel estimate that is obtained by approximating f by a constant in the smoothing window (x−h(x), x+h(x)). The second term is a correction for local slope of the data and skewness of the xi ’s. A local linear estimate would exhibit bias if the mean function f had a high curvature. Let us now apply this methodology to the life insurance market data. For a fixed calendar year t, we use the model " x (t) = f (x) + xt , ln m

x = 40, 41, . . . , 98

(3.24)

" x (t) is the observed death rate in the insurance market. Hence, the where m smoothed death rates are given by exp(" f (x)), x = 40, 41, . . . , 98. The model is fitted separately to males and females, and to group and individual mortality experiences. The result is visible in Fig. 3.17 which is the analogue of Figure 3.16, leading to smoothed mortality curves for the insurance market. We see that the individual life experience is consistently better than the general population mortality. The experience for group life contracts is better than the general population mortality before retirement age but then deteriorates and becomes comparable to the general population mortality after retirement. Remark Alternatively, f can be estimated by minimizing the objective function (2.105), that is, m 2 2 yi − f (xi ) + λ Oλ (f ) = (3.25) f "(u) du i=1

u∈R

The first term ensures that f (·) will fit the data as well as possible. The second term penalizes roughness of f (·); it imposes some smoothness on the estimated f (·). The factor λ quantifies the amount of smoothness: if λ +∞ then f " = 0 and we get a linear fit; and if λ 0 then f perfectly interpolates the data points.

–2 –3 –4 –5 –6 –7

–2 –3 –4 –5 –6

50

60

70 Age

80

90

100

40

50

60

70 Age

80

90

–3 –4 –5 –6 –7

–2 –3 –4 –5 –6

50

60

70 Age

80

90

100

–4 –5 –6

40

50

60

70 Age

80

90

100

40

50

60

70 Age

80

90

100

–2 –3 –4 –5 –6 –7

–7 40

–3

–1 Smooth log. death rate, year an

–2

–2

100


–1

–1

–7

–7 40

Smooth log. death rate, year an




–1

40

50

60

70 Age

80

90

100

Figure 3.17. General population (broken line) death rates and individual (circle) and group (triangle) life insurance market smoothed death rates (on the log scale) observed in 1994 for Belgian males (top panel) and females (bottom panel).

122


If x1 < x2 < · · · < xm then the solution " fλ is a cubic spline with knots x1 , x2 , . . . , xm ; see Section 2.6.3. This means that " fλ coincides with a thirddegree polynomial on each interval (xi , xi+1 ) and possesses continuous first and second derivatives at each xi . Remark Instead of working in a Gaussian regression model, we could also move to the generalized linear modelling framework by implementing a local likelihood maximization principle. Consider for instance the Bernoulli model where P[Yi = 1] = 1 − P[Yi = 0] = p(xi ). The contribution of the ith observation to the log-likelihood is l(Yi , p(xi )) = Yi ln p(xi ) + (1 − Yi ) ln(1 − p(xi )) p(xi ) = Yi ln + ln(1 − p(xi )) 1 − p(xi )

(3.26)

A local polynomial approximation for p(xi ) is difficult since the inequalities 0 ≤ p(xi ) ≤ 1 must be fulfilled. Therefore, we prefer to work on the logit scale, defining the new parameter from the logit transformation θ(x) = ln

p(x) 1 − p(x)

(3.27)

Note that θ(x) can assume any real value as p(x) moves from 0 to 1. The local polynomial likelihood at x is then m

wi (x) Yi β0 (x) + β1 (x)xi − ln 1 + exp(β0 (x) + β1 (x)xi )

(3.28)

i=1

The estimation of p(x) is then obtained from β1 (x)x exp " β0 (x) + " " p(x) = 1 + exp " β0 (x) + " β1 (x)x

3.4.3

(3.29)

Life expectancies

Figure 3.18 gives the life expectancy at age 65 for the general population and for insured lives, computed on the basis of observed death rates. We see that the life expectancies for the group life insurance market are close to the general population ones. This is due to the moderate adverse selection present in the collective contracts, where the insurance coverage is made compulsory by the employment contract, noting that there is a selection effect through being employed (the so-called ‘healthy worker effect’). On the contrary, the effect of adverse selection seems to be much stronger for individual policies. This is due to the particular situation prevailing

3.4


26

Life exp. 65

24 22 20 18 16 1994

1996

1998

2000 Year

2002

2004

1994

1996

1998

2000 Year

2002

2004

28

Life exp. 65

26 24 22 20 18

Figure 3.18. Life expectancy at age 65 for males (top panel) and females (bottom panel): General population (diamond) and individual (circle) and group (triangle) life insurance market. Source: HMD for the general population and BFIC for insured lives.

in Belgium, where no tax incentives are offered for buying life annuities or other life insurance products after retirement. This explains why only people with improved health status consider insurance products as valuable assets. Note that this situation has recently changed in Belgium, where purchasing life annuities at retirement age is now encouraged by the government.

3.4.4

Relational models

Actuaries are aware that the nominee of a life annuity is, with a high probability, a healthy person with a particularly low mortality in the first years of life annuity payment and, generally, with an expected lifetime higher than

123

124


average. In order to account for this phenomenon, Delwarde et al. (2004) have suggested a method for adjusting a reference life table to the experience of a given portfolio, based on non-linear regression models using local likelihood for inference. " HMD Denoting as m (t) the population death rates contained in the HMD, x BFIC " x (t) their analogue for the life insurance market computed from and as m BFIC statistics, we consider models of the form HMD " BFIC " ln m (t) = f ln m (t) + xt (3.30) x x for ages x = 40, 41, . . . , 98 and calendar years 1994–2005. The similarity with (3.24) is clearly apparent. Now, population death rates are used as explanatory variables, instead of age x. Note that both variables could enter the model as covariates, but we need here to establish a link between population and insurance market mortality statistics that will be exploited in Chapter 5. Figure 3.19 describes the result of the procedure for males, whereas Fig. 3.20 is the analogue for females. Figures 3.19 and 3.20 suggest that a linear relationship exists between population and market death rates (at least for older ages). If we fit the regression model " BFIC " HMD ln m (t) = a + b ln m (t) + xt x x

(3.31)

" BFIC " HMD (t), ln m (t)) that are available for ages to the observed pairs (ln m x x 60–98, and calendar years 1994 to 2005, we obtain estimated values for b that are significantly less than 1 (for group and individual policies, males and females). Moreover, the estimations are very sensitive to the age and time ranges included in the analysis. Let us briefly explain why b < 1 seems inappropriate. Mortality reduction factors express the decrease in mortality at some future time t + k compared with the current mortality experience at time t. They are widely used to produce projected life tables and are formally introduced in Section 4.3.2. The link between the regression model (3.31) and the mortality reduction factors for the insurance market is as follows. It is easily seen that if the linear relationship given above indeed holds true then mBFIC mHMD (t + k) (t + k) x x ln = b ln (3.32) mBFIC (t) mHMD (t) x x mBFIC (t + k) ⇔ x BFIC = mx (t)

(t + k) mHMD x HMD mx (t)

b (3.33)

3

Ind. ins. market

–2 –3 –4 –5 –6

–1

2

–2 Ind. ins. market

AM fit, ind. ins. market

–1

1 0 –1

–3 –4 –5 –6

–2

–7

–7 –6

–5

–4 –3 Gen. population

–2

–1

–6

–5


–2

–1

–6

–5


–2

–1

–6

–5


–2

–1

3

–4 –6 –8 –6

–5


–2

–1

–2

2

Group ins. market

AM fit, group ins. market

Group ins. market

–2

1 0 –1 –2

–4 –6 –8

–3 –6

–5


–2

–1

" HMD " BFIC Figure 3.19. Relational models for males: observed pairs (ln m (t), ln m (t)) are displayed in the left panels, the estimated functions f in (3.30) are x x displayed in the middle panels, and the resulting fits are displayed in the right panels, individual policies in the top panels, group policies in the bottom panels.

–1

–3 –4 –5 –6

–2

2

Ind. ins. market

AM fit, ind. ins. market

Ind. ins. market

–1

3

–2

1 0 –1

–7

–6

–5


–2

–1

–4 –5 –6

–2

–7

–3

–7 –7

–6

–5


–2

–1

–7

–6

–5


–2

–1

–7

–6

–5


–2

–1

Group ins. market

–4 –6 –8 –7

–6

–5


–2

–1

–2

2

Group ins. market

AM fit, group ins. market

4 –2

0 –2 –4

–4 –6 –8

–7

–6

–5


–2

–1

" HMD " BFIC Figure 3.20. Relational models for females: observed pairs (ln m (t), ln m (t)) are displayed in the left panels, the estimated functions f in (3.30) are x x displayed in the middle panels, and the resulting fits are displayed in the right panels, individual policies in the top panels, group policies in the bottom panels.

3.4


so that the mortality reduction factor for the market is equal to the mortality reduction factor for the general population raised to the power b. The same reasoning obviously holds for the group life insurance market. We note that the mortality reduction factors are less than 1 in the presence of decreasing trends in mortality rates. As socio-economic class mortality differentials have widened over time, we expect mortality improvements for assured lives to have been greater than in the general population. This statement is based on the fact that the socio-economic class mix of this group is higher than the population average. Of course, there may be distortion factors, like changes in underwriting practices, or reforms in tax systems. Considering that the estimated values for parameters b are less than 1, the interpretation is that the speed of the future mortality improvements in the insured population is somewhat smaller than the corresponding speed for the general population. This is not desirable and only reflects the changes in the tax regimes in Belgium, lowering adverse selection. This is why we now consider the following model: " BFIC ln m (t) = f (x) + ln" mHMD (t) + xt x x

(3.34)

" BFIC " HMD (t), ln m (t)) over calenWe fit (3.34) to the observed pairs (ln m x x dar years 1994–2005 and ages 60–98. This produces estimated SMR’s of the form exp(" f (x)) that can be used to adapt mortality projections to the insurance market. Note that in (3.34), we force the speed of mortality improvements to be equal to the one for the general population. The quality of the fit of (3.34) is remarkable, as it can be seen from the high values of the R2 ’s: 99.8% for males, individual policies, 97.2% for males, group policies, 99.8% for females, individual policies, and 97.8% for females, group policies. The estimated SMR’s are displayed in Fig. 3.21.

3.4.5

Age shifts

Another approach to quantify adverse selection consists in determining age shifts or Rueff’s adjustments. More details can be found in Section 4.4.3. Here, we determine the age shift (t) to minimize the objective function

Ot ( ) =

80 BFIC HMD " ex− (t) ex (t) −"

2

(3.35)

x=65

We select the optimal value of (t) by a grid search over {−10, −9, . . . , 10}. Then, the overall age shift is determined by minimizing O( ) = 2005 t=1994 Ot ( ). This gives the values displayed in Table 3.1.

127

3 : Mortality trends during the 20th century 0.50 0.8

SMR

SMR

0.45 0.40

0.7 0.6

0.35

0.5 60

70

80 x

90

0.60

60

70

80 x

90

60

70

80 x

90

1.00 0.95

0.55

0.90 SMR

SMR

128

0.50

0.85 0.80

0.45

0.75 0.70

0.40 60

70

80 x

90

Figure 3.21. Estimated SMR’s from (3.34) for males (top panels) and females (bottom panels), individual (left) and group (right) life insurance market. Table 3.1. Optimal age shifts obtained from the objective functions Ot in (3.35), t = 1994, 1995, . . . , 2005 and O = 2005 t=1994 Ot .

Year t

Ind., males

Ind., females

Group, males

Group, females

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 1994–2005

−8 −7 −9 −8 −6 −9 −8 −9 −9 −9 −8 −6 −9

−6 −6 −8 −5 −4 −6 −5 −8 −5 −4 −4 −3 −5

−4 −1 −2 −2 −1 −1 −1 −2 0 −1 1 1 −1

−1 0 −1 −1 −1 −1 0 0 1 0 3 1 0

Considering the period 1994–2005, we see that the actuarial computations for males, individual policies, should be based on general population life tables with age decreased by 9 years. The corresponding shift for group life policies is reduced to −1 year. For females, the values are −5 years for individual policies with no adjustment for group life contracts. Let us now briefly explain another approach to get these age shifts. To this end, we assume that the observed number of deaths at age x in calendar year

3.5

Mortality trends throughout EU

–125000 –130000

–124000

–135000

L

L

–122000

–140000

–126000

–145000

–128000

–150000 –130000 0

5

10 a

15

20

0

–74000

5

10 a

15

20

–60000

–76000 L

L

–65000

–78000

–70000

–80000

–75000 0

5

10 a

15

20

0

5

10 a

15

20

Figure 3.22. Log-likelihood L in function of the age shift for males (top panels) and females (bottom panels), individual (left) and group (right) life insurance market.

t in the insurance market DBFIC is Poisson distributed, with a mean equal to xt the product of the exposure to risk ETRBFIC of the market multiplied by the xt " HMD population death rate m (t) at age x− . This distributional assumption x− is not restrictive as the likelihood (3.12) has been seen to be proportional to a Poisson one. The age shift is then determined by maximizing the likelihood obtained by considering the DBFIC xt ’s as mutually independent, that is, by maximizing the objective function

L( ) =

!

" HMD exp −ETRBFIC xt m x− (t)

DBFIC xt HMD (t) −ETRBFIC " m xt x−

x,t

DBFIC xt !

over by a grid search. The results are displayed in Fig. 3.22 when calendar years 2005 and over are considered. The log-likelihood L = ln L is given in function of the age shift . We clearly see that the log-likelihoods peak around the age shifts given in Table 3.1.

3.5

Mortality trends throughout EU

This section compares the Belgian mortality experience with several other EU members: Sweden, Italy, Spain, West-Germany, France, and England & Wales. Even if the trend is comparable in these countries, regional

129

130


differences in death rates produce gaps in life expectancies, especially at retirement age. All the data used to perform the multinational comparison come from HMD and all the analysis is performed on a period basis. Figure 3.23 shows the trend in the period life expectancy at birth for males in Sweden, Italy, Spain, West-Germany, France, and England & Wales, compared with Belgium. Figure 3.25 is the analogue for females. Figures 3.24 and 3.26 display the period life expectancy at age 65. Sweden has had a complete and continually updated register of its population for more than 2 centuries. In general, life expectancies at birth increased from 1750 to present. In 1771–1772, a harvest failure lead to famine and epidemics, resulting in a drastic reduction of life expectancy at birth. Another increase in mortality took place during the first decade of the 19th century because of the Finnish war of 1808–1809 and related epidemics. The effect of the 1918 Spanish influenza epidemic is also clearly visible. Because Sweden has remained neutral during both world wars, life expectancies were minimally affected relative to other European countries. Compared to Belgium, we see that the life expectancy at birth is higher in Sweden for both genders, but that the gap tends to narrow over time. Considering age 65, we notice an important difference for males, with a clear advantage for Sweden over Belgium. Considering the Italian experience, the effect of the Spanish influenza epidemics is clearly visible, as well as the impact of World War II. The speed of longevity improvement seems to be better in Italy: until the 1950s, life expectancy at birth was higher in Belgium and this changed in the 1960s. Italy has now a slightly higher life expectancy at birth. The advantage of Italy is even more apparent for life expectancies at age 65. In addition to the marked effect of the 1918 influenza epidemics, the Spanish civil war (1936–1939) and post-war period (1941–1942) caused an important decline of life expectancy at birth. As has been observed with Italy, the Belgian advantage over Spain disappeared in the 1960s. Bigger differences exist for life expectancies at age 65, in favour of Spain. Let us now consider Germany. Instead of considering the whole country, we restrict ourselves to the territory of the former Federal Republic of Germany (called West Germany), starting after the end of World War II. We see that the mortality trends in Belgium and in West Germany are similar: life expectancies at birth and at age 65 closely agree in these two areas. The trends in life expectancies at birth in France and Belgium are almost identical, despite the fact that the effect of World War II is more pronounced in France. Note also that the conjuntion of World War I and the Spanish flu epidemics have a very strong effect on life expectancies in the second half

80

60 50 40 30

Life exp. at birth

Life exp. at birth

Life exp. at birth

70

70

70

60 50 40 30

20 1750

1800

1850 1900 Calendar year

1950

50 40 30

1880

2000

60

1900

1920 1940 1960 Calendar year

1980

1920

2000


1980

2000

76 70

72 70 68

70 Life exp. at birth

Life exp. at birth

Life exp. at birth

74

60 50 40

60 50 40

30

66 1960

1970

1980 Calendar year

1990

2000

1900

1920


1980

2000

1850


2000

Figure 3.23. Life expectancy at birth in the EU for males for, from upper left to lower right, Sweden, Italy, Spain, West-Germany, France and England & Wales, compared to Belgium (broken line). Source: HMD.

16

16

12 10 8

Life exp. at 65

Life exp. at 65

Life exp. at 65

16 14

14

12

10 1750

1800

1850

1900

1950

2000

14 12 10

1880

1900

1920

1940

1960

1980

2000

1920

Calendar year

Calendar year


1980

2000

18 16 15

15 14 13 12

Life exp. at 65

16 Life exp. at 65

Life exp. at 65

16

14

12


1990

2000

13 12 11 10

10 1960

14

1900

1920


1980

2000

1850


2000

Figure 3.24. Life expectancy at age 65 in the EU for males for, from upper left to lower right, Sweden, Italy, Spain, West-Germany, France and England & Wales, compared to Belgium (broken line). Source: HMD.

Life exp. at birth

Life exp. at birth

70 60 50 40

80

80

70

70

60 50

30

40

20

30 1750

1800


1950

Life exp. at birth

80

50 40 30

1880

2000

60

1900


1980

2000

1920


1980

2000

80 Life exp. at birth

Life exp. at birth

78 76 74

Life exp. at birth

80 80

70 60

70 60 50

50 40

72 1900 1960

1970

1980 Calendar year

1990

2000

1920


1980

2000

1850


2000

Figure 3.25. Life expectancy at birth in the EU for females for, from upper left to lower right, Sweden, Italy, Spain, West-Germany, France and England & Wales, compared to Belgium (broken line). Source: HMD.

18

18

16 14 12 10

20 18 Life exp. at 65

20 Life exp. at 65

Life exp. at 65

20

16 14 12

10

8 1800


1950

1880

2000

14 12

10 1750

16

1900


1980

2000

1920


1980

2000

22 18

18 17 16

Life exp. at 65

20 Life exp. at 65

Life exp. at 65

19

18 16 14

16 14 12

15

12

14

10 1960

1970

1980 Calendar year

1990

2000

1900

1920


1980

2000

1850


2000

Figure 3.26. Life expectancy at age 65 in the EU for females for, from upper left to lower right, Sweden, Italy, Spain, West-Germany, France and England & Wales, compared to Belgium (broken line). Source: HMD.

3.6

Conclusions

of the 1910s. Quite surprisingly, significant differences appear between the French and Belgian life expectancies at age 65, with a clear advantage for France. Mortality in England and Wales has been significantly influenced by the two world wars as well as by the 1918 flu epidemics. We see that the trend in the life expectancies at birth and at retirement age are very similar in Belgium and in England and Wales.

3.6

Conclusions

As clearly demonstrated in this chapter, mortality at adult and old ages reveals decreasing annual death probabilities throughout the 20th century. There is an ongoing debate among demographers about whether human longevity will continue to improve in the future as it has done in the past. Demographers such as Tuljapurkar and Boe (2000) and Oeppen and Vaupel (2002) argue that there is no natural upper limit to the length of human life. The approach that these demographers use is based on an extrapolation of recent mortality trends. The complexity and historical stability of the changes in mortality suggest that the most reliable method of predicting the future is merely to extrapolate past trends. However, this approach has come in for criticisms because it ignores factors relating to life style and the environment that might influence future mortality trends. Olshansky et al. (2005) have suggested that the future life expectancy might level off or even decline. This debate clearly indicates that there is considerable uncertainty about future trends in longevity. Mortality improvements are viewed as a positive change for individuals and as a substantial social achievement. Nevertheless, they pose a challenge for the planning of public retirement systems as well as for the private life annuities business. Longevity risk is also a growing concern for companies faced with off-balance-sheet or on-balance-sheet pension liabilities. More generally, all the components of social security systems are affected by mortality trends and their impact on social welfare, health care and societal planning has become a more pressing issue. And the threat has now become a reality, as testified by the failure of Equitable Life, the world’s oldest life insurance company, in the UK in 2001. Equitable Life sold deferred life annuities with guaranteed mortality rates, but failed to predict the improvements in mortality between the date the life annuities were sold and the date they came into effect. Despite the fact that the study of mortality has been core to the actuarial profession from the beginning, booming stock markets and high interest

135

136


rates and inflation have largely hidden this source of risk. In the recent past, with the lowering of inflation, interest rates, and expected equity returns, mortality risks have no longer been obscured. Low nominal interest rates have made increasing longevity a much bigger issue for insurance companies. When living benefits are concerned, the calculation of expected present values (which are needed in pricing and reserving) requires an appropriate mortality projection in order to avoid underestimation of future costs. This is because mortality trends at adult/old ages reveal decreasing annual death probabilities. In order to protect the company from mortality improvements, actuaries have to resort to life tables including a forecast of the future trends of mortality (the so-called projected tables). The building of such life tables will be the topic of the next chapters.

4

4.1

Forecasting mortality: An introduction

Introduction

This chapter aims at describing various methods proposed by actuaries and demographers for projecting mortality. Many of these have been actually used in the actuarial context, in particular for pricing and reserving in relation to life annuity products and pensions, and in the demographic field, mainly for population projections. First, the idea of a ‘dynamic’ approach to mortality modelling is introduced. Then, projection methods are presented starting from extrapolation procedures which are still widely used in current actuarial practice. More complex methods follow, in particular methods based on mortality laws, on model tables, and on relations between life tables. The Lee–Carter method, recently proposed, and some relevant extensions are briefly introduced, whereas a more detailed discussion, together with some examples of implementation, is presented in Chapters 5 and 6. The presentation does not follow a chronological order. In order to obtain an insight into the historical evolution of mortality forecasts the reader should refer to Section 4.9.1, in which some landmarks in the history of dynamic mortality modelling are identified. Allowing for future mortality trends (and, possibly, for the relevant uncertainty of these trends) is required in a number of actuarial calculations and applications. In particular, actuarial calculations concerning pensions, life annuities, and other living benefits (provided, e.g. by long-term care covers and whole life sickness products) are based on survival probabilities which extend over a long time horizon. To avoid underestimation of the relevant liabilities, the insurance company (or the pension plan) must adopt an appropriate forecast of future mortality, which should account for the most important features of past mortality trends. Various aspects of mortality trends can be captured looking at the behaviour, through time, of functions representing the age-pattern of

138

4 : Forecasting mortality: An introduction

mortality. The examples discussed in Chapter 3 clearly witness this possibility. Particular emphasis has been placed by many researchers on the behaviour, for each integer age x, of the quantity qx (i.e. the probability of dying within one year), drawn from a sequence of life tables relating to the same kind of population (e.g. males living in a given country, annuitants of an insurance company, etc.). The graph constructed plotting qx , for any given age x, against time is usually called the mortality profile. Mortality profiles are often declining, in particular at adult and old ages. Further, mortality experience over the last decades shows some aspects affecting the shape of curves representing the mortality as a function of the attained age, such as the curve of deaths (i.e. the graph of the probability density function of the random lifetime, in the age-continuous setting) and the survival function. In particular (see also Section 2.3.1): (a) an increasing concentration of deaths around the mode (at old ages) of the curve of deaths is evident; so the graph of the survival function moves towards a rectangular shape, whence the term rectangularization to denote this aspect; see Fig. 3.11 for an actual illustration, and Fig. 4.1(a) for a schematic representation; (b) the mode of the curve of deaths (which, owing to the rectangularization, tends to coincide with the maximum age ω) moves towards very old ages; this aspect is usually called the expansion of the survival function; see Fig. 3.13 for an actual illustration, and Fig. 4.1(b) for a schematic representation; (c) higher levels and a larger dispersion of accidental deaths at young ages (the so-called young mortality hump) have been more recently observed; see Fig. 3.2 for an illustration. (a)

Rectangularization

(b)

1

0

Figure 4.1.

Expansion 1

Age

v

0

Mortality trends in terms of the survival function.

Age

v

v⬘

4.2

A dynamic approach to mortality modelling

From the above aspects, the need for a dynamic approach to mortality assessment clearly arises. Addressing the age-pattern of mortality as a dynamic entity underpins, from both a formal and a practical point of view, any mortality forecast and hence any projection method.

4.2 4.2.1

A dynamic approach to mortality modelling Representing mortality dynamics: single-figures versus age-specific functions

When working in a dynamic context (in particular when projecting mortality), the basic idea is to express mortality as a function of the (future) calendar year t. When a single-figure representation of mortality is concerned (see Sections 2.4.1 and 2.4.2), a dynamic model is a real-valued function (t). For example, the expected lifetime for a newborn, denoted by e¯ 0 in a non-dynamic context, is represented by e¯ 0 (t), a function of the calendar year t (namely the year of birth), when the mortality trend is allowed for. Similarly, the general probability of death in a given population can be represented by a function q(t), where t denotes the calendar year in which the population is considered. In actuarial calculations, however, age-specific measures of mortality are usually needed. Then in a dynamic context, mortality is assumed to be a function of both the age x and the calendar year t. In a rather general setting, a dynamic mortality model is a real-valued or a vector-valued function (x, t). In concrete terms, a real-valued function may represent one-year probabilities of death, mortality odds, the force of mortality, the survival function, some transform of the survival function, etc. This concept has been already introduced in Section 3.3. Further, a vector-valued function would be involved when causes of death are allowed for. The projected mortality model is given by the restriction (x, t)|t > t , where t denotes the current calendar year, or possibly the year for which the most recent (reliable) period life table is available. The calendar year t is usually called the base year. The projected mortality model (and, in particular, the underlying parameters) is constructed by applying appropriate statistical procedures to past mortality experience. Although age-specific functions are needed in actuarial calculations, the interest in single-figure indexes as functions of the calendar year should not be underestimated. In particular, important features of past mortality trends can be singled out by focussing on the behaviour of some indexes that

139

140


are intended to be markers of the probability distribution of the random lifetime at birth, T0 , or at some given age x, Tx (see Section 2.4). In a dynamic context, all such markers should be noted to be functions of the calendar year t, for example, e¯ 0 (t), σ0 (t), ξ(t), etc.

4.2.2

A discrete, age-specific setting

Turning back to age-specific functions, we assume now that both age and calendar year are integers. Hence, (x, t) can be represented by a matrix whose rows correspond to ages and columns to calendar years. In particular, let (x, t) = qx (t), where qx (t) denotes the probability of an individual aged x in the calendar year t dying within one year (namely, the one-year probability of death in a dynamic context). The elements of the matrix (see Table 4.1) can be read according to three arrangements: (a) a vertical arrangement (i.e. by columns), q0 (t), q1 (t), . . . , qx (t), . . .

(4.1)

corresponding to a sequence of period life tables, with each table referring to a given calendar year t; (b) a diagonal arrangement, q0 (t), q1 (t + 1), . . . , qx (t + x), . . .

(4.2)

corresponding to a sequence of cohort life tables, with each table referring to the cohort born in year t; (c) a horizontal arrangement (i.e. by rows), . . . , qx (t − 1), qx (t), qx (t + 1), . . .

(4.3)

yielding the mortality profiles, with each profile referring to a given age x. Table 4.1.

0 1 ... x x+1 ... ω−1

One-year probabilities of death in a dynamic context

...

t−1

t

t+1

... ... ... ... ... ... ...

q0 (t − 1) q1 (t − 1) ... qx (t − 1) qx+1 (t − 1) ... qω−1 (t − 1)

q0 (t) q1 (t) ... qx (t) qx+1 (t) ... qω−1 (t)

q0 (t + 1) q1 (t + 1) ... qx (t + 1) qx+1 (t + 1) ... qω−1 (t + 1)

... ... ... ... ... ... ... ...

4.3

4.3 4.3.1

Projection by extrapolation of annual probabilities of death

Projection by extrapolation of annual probabilities of death Some preliminary ideas

An extrapolation procedure for mortality simply aims at deriving future mortality patterns (e.g. future probabilities of death) from a database that expresses past mortality experience. The database typically consists in crosssectional observations and, possibly, (partial) cohort observations. This idea is sketched in Figs. 4.2 and 4.3. However, a number of points should be addressed. In particular, consider the following: 1. How are the items in the database interpreted? Are they correctly interpreted as observed outcomes of random variables (e.g. frequencies of death), or, conversely, are they simply taken as ‘numbers’? 2. The projected table, resulting from the extrapolation procedure, is a two-dimensional array of numbers, providing point estimates of future mortality. How do we get further information, namely, interval estimates? If the answer to question (1) is ‘data are simply numbers’, then the extrapolation procedure does not allow for any statistical feature of the information available, as, for example, the reliability of the data. Conversely,

Past

Future t⬘

0 1

x

v–1 Figure 4.2.

The projected table.

qx( t ⬘)

Projected table

141

142


Projection

0 1

Projected table

x

v–1

Database Figure 4.3.

From the data set to the projected table.

when the data are interpreted as the outcomes of random variables, the extrapolation procedure must rely on sound statistical assumptions and, as a consequence, future mortality can be represented in terms of both point and interval estimates (whilst only point estimates can be provided by extrapolation procedures only based on ‘numbers’). Various traditional projection methods consist in extrapolation procedures simply based on ‘numbers’. First, we will describe these methods which, in spite of several deficiencies, offer a simple and intuitive introduction to mortality forecasts. Let us assume that several period observations (or ‘cross-sectional’ observations) are available for a given population (e.g. males living in a country, pensioners who are members of a pension plan, etc.). Each observation consists of the age-pattern of mortality for a given set X of ages, say X = {xmin , xmin + 1, . . . , xmax }. The observation referred to calendar year t is expressed by {qx (t)}x∈X = {qxmin (t), qxmin +1 (t), . . . , qxmax (t)}

(4.4)

Let us focus on the set of observation years T = {t1 , t2 , . . . , tn }. Then, we assume that the matrix {qx (t)}x∈X ; t∈T = {qx (t1 ), qx (t2 ), . . . , qx (tn )}x∈X

(4.5)

4.3


Future (Extrapolation)

qx (t)

Past (Graduation)

t' Time t Figure 4.4.

Extrapolation of the mortality profile

constitutes the data base for mortality projections. Note that each sequence on the right-hand side of (4.5) represents the observed mortality profile at age x. We assume that the trend observed in past years (i.e. in the set of years T ) can be graduated, for example, via an exponential function. Further, we suppose that the observed trend will continue in future years. Then, future mortality can be estimated by extrapolating the trend itself (see Fig. 4.4). Remark The choice of the set T is a crucial step in building up a mortality projection procedure. Even if a long sequence of cross-sectional observations is available (throughout a time interval of, say, more than 50 years), a choice restricted to recent observations (over, say, 30–50 years) may be more reasonable than the whole set of data. Actually, a very long statistical sequence can exhibit a mortality trend in which recent causes of mortality improvement have a relatively small weight, whereas causes of mortality improvement whose effect should be considered extinguished are still included in the trend itself (see Fig. 4.5). For more information, see Section 5.5. Extrapolation of the qx ’s (namely of the mortality profiles) represents a particular case of the horizontal approach for mortality forecasts (see Fig. 4.6). The horizontal approach can be applied to quantities other than the annual probabilities of death, for example, the mortality odds φx , the central death rates mx , etc. Adopting the horizontal approach means that extrapolations are performed independently for each qx (or other age-specific quantity), so that

143

qx (t)


t' Time t Figure 4.5.

Extrapolation results depending on the graduation period.

xmin

Age

144

x

xmax

Figure 4.6.

Calendar year th

t1

t2

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

⫻

c x min(t)

c x(t)

c x max(t)

The horizontal approach.

the result is a function ψx (t) for each age x. This may lead to inconsistencies with regard to the projected age-pattern of mortality, as we will see in Section 4.5.3.

4.3.2

Reduction factors

As far as future mortality is concerned, let us express the relation between probability of death at age x, referred to a given year t (e.g. t = tn ) and a generic year t (t > t ) respectively, as follows: qx (t) = qx (t ) Rx (t − t )

(4.6)

4.3


The quantity Rx (t − t ) is called the variation factor (and usually reduction factor, as it is expected to be less than 1 because of the prevailing downward trends in probabilities of death) at age x for the interval (t , t). A simplification can be obtained assuming that the reduction factor does not depend on the age x, that is, assuming for all t and x Rx (t − t ) = R(t − t )

(4.7)

Mortality forecasts can then be obtained through an appropriate modelling procedure applied to the reduction factor. The structure as well as the parameters of Rx (t − t ) should be carefully chosen. Then, projected mortality will be obtained via (4.6) (provided that we assume that the observed trend, on which the reduction factors are based, will continue in the future). Remark The approach to projection by extrapolation which we are describing is based on a mathematical formula, namely, the formula for the reduction factor (examples are provided in Sections 4.3.3–4.3.8). Conversely, extrapolation may be based on a graphical method. The graphical approach to extrapolation consists in drawing, for each age x, a smooth curve representing the past trend in probabilities of death, assumed to continue after the calendar year t , and then reading the projected probabilities from the extrapolated part of the curve.

4.3.3

The exponential formula

Let us suppose that the observed mortality profiles are such that the behaviour over time of the logarithms of the qx ’s is, for each age x, approximately linear (see Fig. 4.7). Then, we can find a value δx such that, for h = 1, 2, . . . , n − 1, we have approximately: ln qx (th+1 ) − ln qx (th ) ≈ −δx (th+1 − th )

(4.8)

qx (th+1 ) ≈ e−δx (th+1 −th ) qx (th )

(4.9)

qx (th+1 ) t −t ≈ rxh+1 h qx (th )

(4.10)

Hence

or, defining rx = e−δx :

Assume that, for each age x, the parameter δx (or rx ) is estimated, for example via a least squares procedure. So, the graduated probabilities qˆ x (t) can be calculated. The constraint qˆ x (tn ) = qx (tn ) is usually applied in the estimation procedure.

145


0

Time t qx (t)

ln qx (t)

146

0 Time t Figure 4.7.

Behaviour of the qx ’s along time

Relation (4.10) suggests a natural extrapolation formula. Set t = tn , and assume for t > t : qx (t) = qx (t ) rt−t (4.11) x from which we can express the reduction factor as follows:

= e−δx (t−t ) Rx (t − t ) = rt−t x

(4.12)

The extrapolation formula (4.11) (as well as, for instance, formula (4.17) in Section 4.3.5) originates from the analysis of the mortality profiles, and hence constitutes an example of the horizontal approach.

4.3.4

An alternative approach to the exponential extrapolation

For the calculation of parameters rx ’s (or δx ’s), procedures other than least squares estimation can be used. An example follows. Suppose, as above, that n period tables are available. For each age x and (h) for h = 1, 2, . . . , n − 1, calculate the quantities rx ’s as follows: r(h) x

%

qx (th+1 ) = qx (th )

&

1 th+1 −th

(4.13)

Then, for each x, we calculate rx as the weighted geometric average of (h) the quantities rx : n−1 ! wh r(h) (4.14) rx = x h=1

The weights n−1must, of course, fulfill the conditions: wh ≥ 0, h = 1, 2, . . . , n − 1; h=1 wh = 1.

4.3


Each weight, wh , should be chosen in a way to reflect both the length of the time interval between observations and the statistical reliability attaching to the observations themselves. Trivially, if we set wh = (th+1 − th )/(tn − t1 ) for all h, only the lengths of the time intervals are accounted for, and so expression (4.14) reduces to %

qx (tn ) rx = qx (t1 )

&

1 tn −t1

(4.15)

so that rx is determined only by the first and last values of qx (t) in the past data.

4.3.5

Generalizing the exponential formula

Let us turn back to the exponential formula. From (4.11) it follows that, if rx < 1, then qx (∞) = 0 (4.16) where qx (∞) = limt +∞ qx (t). Although the validity of mortality forecasts should be restricted to a limited time interval, it may be more realistic to assign a positive limit to the mortality at any age x. To this purpose, the following formula with an assigned asymptotic mortality can be adopted:

(4.17) qx (t) = qx (t ) αx + (1 − αx ) rt−t x where αx ≥ 0 for all x see Fig. 4.8. The reduction factor is thus given by Rx (t − t ) = αx + (1 − αx ) rt−t x

(4.18)

Clearly, (4.17) is a generalization of (4.11). From (4.17) we have: qx (∞) = αx qx (t )

(4.19)

The exponential formula expressed by equation (4.17) can be simplified by assuming that rx = r for all x, from which we obtain:

(4.20) qx (t) = qx (t ) αx + (1 − αx ) rt−t Although the mortality decline is not necessarily uniform across a given (wide) age range, this assumption can be reasonable when a limited set of ages is involved in the mortality forecast. This would be the case for mortality projections concerning annuitants or pensioners. In any case, some flexibility is provided by the parameters αx .

147

148


qx (t')

qx (t' )

qx (t' ) αx t'

t' Time t

Figure 4.8.

4.3.6

Time t

Asymptotic mortality in exponential formulae.

Implementing the exponential formula

An alternative version of the exponential formula (4.17) can help in directly assigning estimates to the parameters rx . Without loss of generality, we address the simplified structure represented by equation (4.20), so that r is independent of the age x. The total (asymptotic) mortality decline, from time t on, is given by qx (t ) − qx (∞), whereas the decline in the first m years is given by qx (t ) − qx (t + m). Let us define the ratio fx (m) as follows: fx (m) =

qx (t ) − qx (t + m) qx (t ) − qx (∞)

(4.21)

then, fx (m) is the proportion of the total mortality decline assumed to occur by time m. Dividing both numerator and denominator by qx (t ), we obtain: fx (m) =

(1 − αx )(1 − rm ) 1 − Rx (m) = = 1 − rm 1 − Rx (∞) 1 − αx

(4.22)

Note that, since we have assumed rx = r for all x, we have fx (m) = f (m). Hence 1 r = (1 − f (m)) m (4.23) The choice of the couple (m, f (m)) unambiguously determines the parameter r. Finally, we have Rx (t − t ) = αx + (1 − αx ) (1 − f (m))

t−t m

(4.24)

For example, if we assume that 60% of the total mortality decline occurs 1 in the first 20 years, we set (m, f (m)) = (20, 0.60), and so r = 0.40 20 = 0.9552.

4.3

4.3.7


A general exponential formula

The exponential formulae discussed in Sections 4.3.3 and 4.3.5 can be placed in a more general context. We assume the following expression for the annual probability of death: qx (t) = ax + bx cxt

(4.25)

in which the parameters ax , bx , cx depend on age x and are independent of the calendar year t. Thus, qx (t) is an exponential function of t. Equation (4.25) then represents a general exponential formula for projections via extrapolation. The projection formulae which are currently used in actuarial practice constitute particular cases of formula (4.25). For instance, with ax = 0, bx = qx (t ) rtx , cx = rx , we obtain formula (4.11). With ax = αx qx (t ), bx = (1 − αx ) qx (t ) rtx , cx = rx , we find the more general formula (4.17). The projection formula t−t

qx (t) = qx (t ) a x+b

(4.26) t−t

(called the Sachs formula) where a and b are constants and a x+b represents the reduction factor, also constitutes a particular case of (4.25), as can be easily proved. Note that formulae (4.11) and (4.17) (and some related expressions) explicitly refer to the base year t (usually related to the most recent observation, that is, t = tn ). Conversely, formula (4.25) as well as other formulae presented in Section 4.3.9 do not explicitly address a fixed calendar year. Nonetheless, a link with a given calendar year can be introduced via parameters, as illustrated, for example, by formula (4.26).

4.3.8

Some exponential formulae used in actuarial practice

Exponential formulae have been widely used in actuarial practice. Implementations of these formulae can be found, for instance, in the USA, Great Britain, Germany and Austria. Some examples follow. Example 4.1 In the UK, formula (4.11) was used for forecasting the mortality of life office pensioners and annuitants; see CMIB (1978). In particular, a simplified version with the same reduction factor at all ages (see (4.7)) was implemented, that is, qx (t) = qx (t ) rt−t

(4.27)

149

150


The approximation was considered acceptable from age x = 60 upwards. Example 4.2 Formula (4.20) was also proposed in the UK; see CMIB (1990). The reduction factor Rx (t − t ), with t = 1980 as the base year, is given by Rx (t − t ) = αx + (1 − αx )(1 − f )

t−t 20

with f = f (20) = 0.60 [see formula (4.24)] and:   0.50 if x < 60   x − 10 αx = if 60 ≤ x ≤ 110    100 1 if x > 110

(4.28)

(4.29)

It is easy to see that, for any year t, the reduction factor increases (i.e. the mortality improvement reduces) linearly with increasing age, between t−t

0.50 + 0.50 (0.40) 20 at age 60 and below, to unity at age 110 and above. For any given age x, the rate of improvement decreases as t increases. Further, following the analysis in Section 4.3.6, it is easy to prove that expression (4.28) for the reduction factor, with f = 0.60, implies that 60% of the total (asymptotic) mortality improvement (at any age x) is assumed to occur in the first 20 years. Example 4.3 A recent implementation of formula (4.17) by the Continuous Mortality Investigation Bureau is as follows (see CMIB (1999)). In this case, the reduction factor is given by Rx (t − t ) = αx + (1 − αx )(1 − fx )

t−t 20

The functions αx , fx have been chosen as follows:   c if x < 60   x − 110 fx = 1 + (1 − c) if 60 ≤ x ≤ 110  50   1 if x > 110  h    (110 − x) h + (x − 60) k αx =  50   k

(4.30)

(4.31)

if x < 60 if 60 ≤ x ≤ 110

(4.32)

if x > 110

where c = 0.13, h = 0.55, and k = 0.29. Parameters have been adjusted so that t = 1992 is the base year.

4.3


Example 4.4 An exponential formula has also been used in the United States. The Society of Actuaries published the 1994 probabilities on death as the base table and the annual improvement factors 1 − rx ; see Group Annuity Valuation Table Task Force (1995). The projected probabilities of death are determined as follows: qx (t) = qx (1994) rt−1994 x

(4.33)

The parameter rx varies from 0.98 to 1, being equal to 1 for x > 100, for both males and females.

4.3.9

Other projection formulae

Mortality improvements resulting from observed data may suggest assumptions other than the exponential decline of annual probabilities of death. Thus, a formula different from the exponential one (see (4.25)) can be used to express the probabilities qx (t). Conversely, the exponential formula can be used to express other life table functions, or a transform of a life table function, such as the odds φx (t). Below we present some formulae which have been suggested or used in applications: bx qx (t) = ax + (4.34) t p qx (t) = ax,h t h (4.35) qx (t) =

h=0 eGx (t)

1 + eGx (t)

(4.36)

where Gx (t) is, for each age x, a polynomial in t, that is, Gx (t) =

p

cx,h t h

(4.37)

h=0

Some comments about these formulae follow. Formula (4.35) with p = 1 represents the linear extrapolation method: qx (t) = ax,0 + ax,1 t

(4.38)

with ax,1 < 0 to express mortality decline. This formula is not usually adopted because of its obvious disadvantage that for large t a negative probability is predicted. The polynomial extrapolation formula (4.35) with p = 3 is called the Esscher formula.

151

152


Referring to formula (4.36), note that it can be expressed as follows: ln

qx (t) = Gx (t) px (t)

(4.39)

If observed mortality improvements suggest a linear behaviour of the logarithms of the odds, and thus an exponential behaviour of the odds, then we can use formula (4.36) with Gx (t) = cx,0 + cx,1 t

(4.40)

and so we have the following expression: qx (t) =

4.4 4.4.1

ecx,0 +cx,1 t 1 + ecx,0 +cx,1 t

(4.41)

Using a projected table The cohort tables in a projected table

A projected mortality table is a rectangular matrix {qx (t)}x∈X ; t≥t , where t is the base year. The appropriate use of the projected table requires that, in each year t, probabilities concerning the lifetime of a person age x in that year are derived from the diagonal qx (t), qx+1 (t + 1), . . .

(4.42)

that is, from the relevant cohort table (see also Section 4.2.2). Then, the probability of a person age x in year t being alive at age x + k is given by:

k px (t) =

k−1 !

[1 − qx+j (t + j)]

(4.43)

j=0

where the superscript recalls that we are working along a diagonal band in the Lexis diagram (see Section 3.3, and Fig. 3.1 in particular), or, similarly, along a diagonal of the matrix in Table 4.1 with the proviso that the ordering of the lines is inverted. Note that explicit reference to the year of birth τ is omitted, as this is trivially given by τ = t − x. For example, to calculate, in the calendar year t, the expected remaining lifetime of an individual age x in that year, the following formula should be adopted, rather than formula (2.65) (which relies on the assumption of unchanging mortality after the period observation from which the life table

4.4

was drawn): ◦ ex (t)

=

ω−x k=1

k px (t) +

Using a projected table

1 2

(4.44)

◦

The quantity ex (t) is usually called the (complete) cohort life expectancy, for a person age x in year t. If a decline in future mortality is expected (and hence represented by the projected cohort table), the following inequality holds: ◦ ◦ ex (t) > ex (4.45) ◦

where ex denotes the period life expectancy (see Section 2.4.3). Note that, in a dynamic framework, the period life expectancy should be denoted as follows: ω−x 1 ◦↑ ↑ ex (t) = (4.46) k px (t) + 2 k=1

with ↑ k px (t) =

k−1 !

[1 − qx+j (t)]

(4.47)

j=0

where the superscript ↑ recalls that we are working along a vertical band in the Lexis diagram, or, similarly, along a column of the matrix in Table 4.1. The same cohort-based approach should be adopted to calculate actuarial values of life annuities, for both pricing and reserving. Hence, various cohort tables should be simultaneously used, according to the year of birth of the individuals addressed in the calculations.

4.4.2

From a double-entry to a single-entry projected table

From a strictly practical point of view, the simultaneous use of various cohort tables may have some disadvantages. Moreover, probabilities concerning people with the same age x at policy issue vary according to the issue year t. These disadvantages have often led to the adoption, in actuarial practice, of one single-entry table only, throughout a period of some (say 5, or 10) years. The single-entry table must be drawn, in some way, from the projected double-entry table. Single-entry tables can be derived, in particular, as follows (see also Fig. 4.9): (1) A birth year τ¯ is chosen and the cohort table pertaining to the generation born in year τ¯ is only addressed; so, the probabilities qxmin (¯τ + xmin ), qxmin +1 (¯τ + xmin + 1), . . . , qx (¯τ + x), . . .

(4.48)

153

154


Future

Past t⬘

t

t

0 1

Projected table

qx( t ⬘)

x

v–1

(2) Figure 4.9.

(1)

Two approaches to the choice of a single-entry projected table.

where xmin denotes the minimum age of interest, are used in actuarial calculations. Thus, just one diagonal of the matrix {qx (t)} is actually used. The choice of τ¯ should reflect the average year of birth of annuitants or pensioners to whom the table is referred. (2) A (future) calendar year t¯ is chosen and the projected period table referring to year t¯ is only addressed; and so the probabilities qxmin (t¯), qxmin +1 (t¯), . . . , qx (t¯), . . .

(4.49)

are adopted in actuarial calculations. Thus, just one column of the matrix is used. The choice of t¯ should be broadly appropriate to the mix of life annuity business in force over the medium-term future. Following approach (1), and using the superscript [¯τ ] to denote reference to the cohort table for the generation born in year τ¯ , the probability of being alive at age x + k is given (for any year of birth τ = t − x) by [¯τ ] = k px

k−1 ! j=0

[1 − qx+j (¯τ + x + j)]

(4.50)

4.4

Using a projected table

Adopting approach (2), and denoting by [t¯] ↑ the reference to the period table for year t¯, the probability of being alive at age x+k is conversely given (for any year of birth τ = t − x) by [t¯]↑ k px

=

k−1 !

[1 − qx+j (t¯)]

(4.51)

j=0

Of course, both approaches lead to biased evaluations. Notwithstanding this deficiency, approach (1) can be ‘adjusted’ reducing such a bias. A common adjustment is described in the following section.

4.4.3

Age shifting

For people born in year τ = t −x, the probabilities (4.43) (which are related to the year of birth τ) should be used, whereas approach (1) leads to the use of probabilities (4.50), which are independent of the actual year of birth. To reintroduce a dependence on τ, at least to some extent, we use the following probabilities: qxmin +h(τ) (¯τ + xmin + h(τ)), qxmin +1+h(τ) (¯τ + xmin + 1 + h(τ)), . . . , qx+h(τ) (¯τ + x + h(τ)), . . .

(4.52)

Note that all the probabilities involved belong to the same diagonal referred to within approach (1). This adjustment (often called Rueff’s adjustment) involves an age-shift of h(τ) years. Assuming a mortality decline, the function h(τ) must satisfy the following relations:   ≥ 0 for τ < τ¯ h(τ) = 0 for τ = τ¯ (4.53)   ≤ 0 for τ > τ¯ The survival probability is then calculated as follows (instead of using formula (4.50)): [¯τ ; h(τ)] k px

=

k−1 !

[1 − qx+h(τ)+j (¯τ + x + h(τ) + j)]

(4.54)

j=0

where the superscript also recalls the age-shift. Probabilities given by formula (4.54) can be adopted to approximate the cohort life expectancy (see Section 4.4.1) as well as actuarial values of life annuities.

155

156

4 : Forecasting mortality: An introduction Table 4.2. Age-shifting function (table TPRV; τ¯ = 1950; i = 0)

τ

h(τ)

1901–1910 1911–1920 1921–1929 1930–1937 1938–1946 1947–1953 1954–1960 1961–1967 1968–1975 1976–1984 ≥ 1985

5 4 3 2 1 0 −1 −2 −3 −4 −5

As regards the determination of the age-shift function h(τ), various criteria can be adopted. We just mention that most criteria are based on the analysis of the actuarial values of life annuities calculated using the appropriate probabilities, given by (4.43), and, respectively, using the probabilities (4.54), with the aim of minimizing the ‘distance’ (conveniently defined) between the sets of actuarial values. When a criterion of this type is adopted, the function h(τ) depends on the interest rate used in calculating the actuarial values. Example 4.5 Table 4.2 shows the age-shifting function used in connection with the French projected table TPRV. The interest rate assumed for the construction of the function is i = 0. Remark It is worth noting that adjustments via an age-shifting mechanism are rather common in life insurance actuarial technique. For example, an increment in the insured’s age is often used to account for the effects of impairments on the age-pattern of mortality; see Section 2.9.1 and, in particular, formulae (2.119) and (2.120). A further example of age-shifting in the context of mortality projections is presented in Section 4.5.1 (see Example 4.8).

4.5 4.5.1

Projecting mortality in a parametric context Mortality laws and projections

When a mortality law is used to fit observed data, the age-pattern of mortality is summarized by some parameters (see Section 2.5). Then, the projection

4.5

Projecting mortality in a parametric context

procedure can be applied to the set of parameters (instead of the set of agespecific probabilities), with a dramatic reduction in the dimension of the forecasting problem, namely in the number of the ‘degrees of freedom’. Consider a law, for example, describing the force of mortality: µx = ϕ(x; α, β, . . . )

(4.55)

In a dynamic context, the calendar year t enters the model via its parameters µx (t) = ϕ(x; α(t), β(t), . . . )

(4.56)

Let T = {t1 , t2 , . . . , tn } denote the set of observation years. Hence, for a given set X of ages, the data base is represented by the set of observed values {µx (t)}x∈X ; t∈T = {µx (t1 ), µx (t2 ), . . . , µx (tn )}x∈X (4.57) For each calendar year th , we estimate the parameters to fit the model µx (th ) = ϕ(x; αh , βh , . . . )

(4.58)

(e.g. via least squares, or minimum χ2 , or maximum likelihood) so that a set of n functions of age x is obtained {µx (t1 ), µx (t2 ), . . . , µx (tn )}

(4.59)

Trends in the parameters are then graduated via some mathematical formula, and hence a set of functions of time t is obtained: α1 , α2 , . . . , αn ⇒ α(t) β1 , β2 , . . . , βn ⇒ β(t) ... (see Fig. 4.10). It is worth noting that the above projection procedure follows a vertical approach to mortality forecast, as the parameters of the chosen law are estimated for each period table based on the experienced mortality (see Fig. 4.11). Conversely, a diagonal approach can be adopted, starting from parameter estimation via a cohort graduation (see Fig. 4.12). In this case, parameters depend on the year of birth τ: µx (τ) = ϕ(x; γ(τ), δ(τ), . . . )

(4.60)

157


Past (Graduation)

Future (Extrapolation)

a

a(t)

ah

th

Figure 4.10.

t' Time t

Projection in a parametric framework.

Calendar year

xmin

Age

158

x

xmax

t1

t2

th

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

w (x; ah, bh, ...) w (x; a(t), b(t), ...) w (x; a2, b2, ...) w (x; a1, b1, ...) Figure 4.11.

The vertical approach.

For each year of birth τh , h = 1, 2, . . . , m, we estimate the parameters to fit the model µx (τh ) = ϕ(x; γh , δh , . . . )

(4.61)

so that a set of m functions of age x is obtained {µx (τ1 ), µx (τ2 ), . . . , µx (τm )}

(4.62)

4.5


Calendar year t1

Age

xmin

x

xmax

t2

th

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

w (x; gh, dh, ...) w (x; g(t), d(t), ...) w (x; g2, d2, ...) w (x; g1, d1, ...) Figure 4.12.

The diagonal approach.

Trends in the parameters are then graduated via some mathematical formula, and hence a set of functions of time τ is obtained: γ1 , γ2 , . . . , γm ⇒ γ(τ) δ1 , δ2 , . . . , δm ⇒ δ(τ) ... Example 4.6 A Makeham’s law (see (2.70)), representing mortality dynamics according to the vertical approach, can be defined as follows: µx (t) = A(t) + B(t) c(t)x

(4.63)

where t represents the calendar year. When the diagonal approach is adopted, the dynamic Makeham law is defined as follows: ¯ ¯ µx (τ) = A(τ) + B(τ) c¯ (τ)x (4.64) where τ = t − x denotes the year of birth of the cohort.

Example 4.7 In some law-based projection models it has been assumed that the age-pattern of mortality is represented by one of the Heligman– Pollard laws (see (2.83) to (2.87)), and that various relevant parameters are functions of the calendar year. Thus, according to a vertical approach,

159

160


functions A(t), B(t), C(t), . . . are used to express the dependency of the age-pattern of mortality on the calendar year t. Example 4.8 We assume that, for each past calendar year t, the odds φx (t) = qx (t)/px (t) are graduated using (2.81). Then, we have φx (t) = ePx (t)

(4.65)

where Px (t) denotes, for each t, a polynomial in x. Further, we assume that the odds are extrapolated, for t > t , via an exponential formula, that is, φx (t) = φx (t ) rs

(4.66)

where s = t − t and r < 1. As far as the age-pattern of mortality in the base year t is concerned, we assume: Px (t ) = α + βx (4.67) Then, from (4.66) we have: ln φx (t) = α + βx + s ln r

(4.68)

Defining w=−

ln r β

(4.69)

we finally obtain: ln φx (t) = α + β(x − w s) = Px−ws (t )

(4.70)

By assumption r < 1, and, given the behaviour of probabilities qx (t ) and px (t ) as functions of the age x, it is sensible to suppose β > 0. Then we find w > 0. Hence, a constant reduction factor applied to the odds leads to an age reduction w for each of the s projection years. If this result is transferred from the odds φx (t) to the probabilities qx (t), we have approximately: qx (t) ≈ qx−ws (t )

(4.71)

Formulae (4.70) and (4.71) provide examples of approximate evaluation via age-shifting. See also the Remark in Section 4.4.3.

4.5.2

Expressing mortality trends via Weibull’s parameters

Assume that the probability distribution of the random lifetime at birth, T0 , is represented (for a given cohort of lives) by the Weibull law, hence with

4.5


force of mortality given by (2.77). The corresponding pdf is then α x α−1 −( βx )α e ; α, β > 0 f0 (x) = β β

(4.72)

whereas the survival function is given by −( βx )α

S(x) = e

(4.73)

It is well known that, whilst the Weibull law does not fit well the agepattern of mortality throughout the whole life span (especially because of the specific features of infant and young-adult mortality), it provides a reasonable representation of mortality at adult and old ages. Moreover, the choice of the Weibull law is supported by the possibility of easily expressing, in terms of its parameters, the mode (at adult ages) of the distribution of the random lifetime T0 , that is, the Lexis point, 1 α−1 α ; α>1 (4.74) Mod[T0 ] = β α as well as the expected value and the variance, 1 +1 E[T0 ] = β α ' 2 ( 1 2 2 +1 − +1 Var[T0 ] = β α α

(4.75) (4.76)

where denotes the complete gamma function (see, e.g. Kotz et al. (2000)). Moments for the remaining lifetime at age x > 0, Tx , can similarly be derived. The above possibility facilitates the choice of laws which reflect specific future trends of mortality. When a dynamic mortality model is concerned, the force of mortality must be addressed as a function of the (future) calendar year t (according to the vertical approach), or the year of birth τ (diagonal approach). Hence, referring for example to the diagonal approach, we generalize formula (2.77) as follows: x α(τ)−1 α(τ) µx (τ) = (4.77) β(τ) β(τ) Functions α(τ) and β(τ) should be chosen in order to reflect the assumed trends in the rectangularization and expansion processes. To this purpose, formulae (4.74) to (4.76) provide us with a tool for checking the validity of a choice of the above functions.

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qx(t)

x1 x2 t' Figure 4.13.

4.5.3

t*

Time t

A possible inconsistency in mortality profile extrapolation.

Some remarks

Comparing mortality profile extrapolations (i.e. the horizontal approach) with law-based projections (i.e. the vertical and the diagonal approaches), we note the following point. First, when the projection consists in a straight extrapolation of the mortality profiles, inconsistencies may emerge as a result of the extrapolation itself. For example we may find that a future calendar year t ∗ exists, such that for t > t ∗ and for some ages x1 , x2 , with x1 < x2 , we find qx1 (t) > qx2 (t) (see Fig. 4.13), even at old ages. Hence, appropriate adjustments may be required. Conversely, simple calculation procedures have an advantage when extrapolating mortality profiles. A further disadvantage of mortality profile extrapolations is due to the fact that they do not ensure the representation of future sensible mortality scenarios. On the contrary, such outcomes can be rather easily produced by controlling the behaviour of projected parameters in a law-based context (see, in particular, Section 4.5.2). As already noted in Section 4.5.1, law-based mortality projections lead to a dramatic reduction in the dimension of the forecasting problem, namely in the number of the degrees of freedom. However, the age-pattern of mortality can be summarized without resorting to mathematical laws (and hence avoiding the choice of an appropriate mortality law). In particular, some typical values, or markers (see Section 4.2.1), of the mortality pattern can be used to this purpose; this aspect is dealt with in Section 4.6.2. Finally, many Authors note that the parameters of most mortality laws are often strongly dependent, for example the B and c parameters in Makeham’s law (see (2.70)). Hence, univariate extrapolation (as in the vertical and the diagonal approaches) may be misleading. Conversely, a multivariate

4.5


approach may provide a better representation of mortality trends, although problems in computational tractability may arise.

4.5.4

Mortality graduation over age and time

As seen in the previous sections, the construction of projected quantities (e.g. the one-year probabilities of death, or the force of mortality) is usually worked out in two steps separately. First, mortality tables are built up for various past calendar years and possibly graduated, in particular using mathematical formulae, for example, in order to obtain the force of mortality for each calendar year (see Section 4.5.1). Second, when no mortality law is involved, mortality profiles are analysed in order to construct a formula for extrapolating probabilities of death. Conversely, when a law-based projection model is used, the behaviour of the parameters over time is analysed, in order to obtain formulae for parameter extrapolation. In conclusion, the construction of the projected mortality is performed with respect to age and calendar year separately. The above approach is computationally straightforward, in particular thanks to the possibility of using well known techniques while performing the first step. Despite this feature, recent research work has shown that models which incorporate (simultaneously) both the age variation in mortality and the time trends in mortality have considerable advantages in terms of goodness-of-fit and hence, presumably, in terms of forecast reliability. Mortality projections based on models incorporating age variation and time trends represent the surface approach to mortality forecasts (see Fig. 4.14). We focus on the so-called Gompertz–Makeham class of formulae, denoted by GM(r, s) and defined in Section 2.5.1 (see (2.78)). Formulae of the GM(r, s) type can be included in models allowing for mortality trends. In this section, as an illustration, we introduce the model proposed by Renshaw et al. (1996), implemented also by Sithole et al. (2000), and Renshaw and Haberman (2003b), albeit in a modified form. Consider the following model:       s r s αi + µx (t) = exp βj Lj (x) ¯  exp γij Lj (x) ¯  t¯i  j=0

i=1

j=1

(4.78)

163


Calendar year

xmin

Age

164

x

xmax

t1

t2

th

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

⫻

⫻

⫻

⌽(x,t) Figure 4.14.

The surface approach.

with the proviso that some of the γij may be preset to 0. Lj (x) ¯ are Legendre polynomials. The variables x¯ and t¯ are the transformed ages and transformed calendar years, respectively, such that both x¯ and t¯ are mapped onto [−1, +1]. Note that the first of the two multiplicative terms on the right hand side is a graduation model GM(0, s + 1), while the second one may be interpreted as an age-specific trend adjustment term (provided that at least one of the γij is not preset to zero). Formula (4.78) has been proposed by Renshaw et al. (1996) for modelling with respect to age and time, noting that, for forecasting purposes, low values of r should be preferred – that is, polynomials in t with a low degree. A further implementation of this model has been carried out by Sithole et al. (2000). Trend analysis of UK immediate annuitants’ and pensioners’ mortality experiences (provided by the CMIB) suggested the adoption of the following particular formula (within the class of models (4.78)):   3 µx (t) = expβ0 + βj Lj (x) ¯ + (α1 + γ11 L1 (x)) ¯ t¯ (4.79) j=1

where we note that r = 1. Moreover, the reduction factor Rx (t − t ) related to the force of mortality (rather than to the probabilities of death) has been addressed: µx (t) = µx (t ) Rx (t − t )

(4.80)

4.6

Other approaches to mortality projections

where, as usual, t is the base year for the mortality projection. From (4.5.30) we obtain: % & t − t Rx (t − t ) = exp (α1 + γ11 x) ¯ (4.81) w where w denotes half of the calendar year range for the investigation period. Hence: ) * Rx (t − t ) = exp (a + b x) (t − t ) (4.82) (with a < 0 and b > 0, which result from the fitting of the observed data). Renshaw and Haberman (2003b) consider a regression-based forecasting model of the following simple structure: ln mx (t) = ax + bx (t − t )

(4.83)

Then, introducing a reduction factor that is related to the central death rate and interpreting the term ax as representing the central death rate for the base year mx (t ), we have that

and

mx (t) = mx (t ) Rx (t − t )

(4.84)

Rx (t − t ) = exp[bx (t − t )]

(4.85)

Renshaw and Haberman (2003b) also experiment with a series of breakpoint predictors (equivalent to linear splines) in order to model changes of slope in the mortality trend that have been observed in the past data. With one such term, the reduction factor would be Rx (t − t ) = exp[bx (t − t ) + bx (t − t0 )+ ]

(4.86)

where (t − t0 )+ = t − t0 for t > t0 , and 0 otherwise.

4.6 4.6.1

Other approaches to mortality projections Interpolation versus extrapolation: the limit table

From Sections 4.3 and 4.5, it clearly emerges that a number of projection methods are based on the extrapolation of observed mortality trends, possibly via the parameters of some mortality law. Important examples are provided by formulae (4.11), (4.20), and (4.63). Athough it seems quite natural that mortality forecasts are based on past mortality observations, different approaches to the construction of projected tables can be adopted.

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We suppose that an ‘optimal’ limiting life table can be assumed. The relevant age-pattern of mortality is to be interpreted as the limit pattern to which mortality improvements can lead. Let q˜ x denote the limit probability of death at age x, whereas qx (t ) denotes the current mortality. Then, we assume that the projected mortality qx (t) is expressed as follows: qx (t) = I[q˜ x , qx (t )]

(4.87)

where the symbol I denotes some interpolation model. Example 4.9 Adopting an exponential interpolation formula, we have: qx (t) = q˜ x + [qx (t ) − q˜ x ] rt−t

(4.88)

with r < 1. Note that formula (4.20) can be easily linked to (4.88), choosing αx such that qx (t )αx = q˜ x . Determining a limit table requires a number of assumptions about the trend in various mortality causes, so that an analysis of mortality by causes of death should be carried out as a preliminary step (see Section 4.8.2).

4.6.2

Model tables

As noted in Section 4.5.1, when a mortality law is used to fit mortality experience, the age-pattern of mortality is summarized by some parameters. Then, the projection procedure can be applied to each parameter (instead of each mortality profile), with a dramatic reduction in the dimension of the forecasting problem. However, the age-pattern of mortality can be summarized without resorting to mathematical laws, and, in particular, some markers of the mortality pattern can be used to this purpose (see Section 4.2.1). The possibility of summarizing the age-pattern of mortality by using some markers underpins the use of model tables. The first set of model tables was constructed in 1955 by the United Nations. A number of mortality tables was chosen, with the aim of representing the age-pattern of mortality corresponding to various degrees of social and economic development, health status, etc. The set was indexed ◦ on the expectation of life at birth, e0 , so that each table was summarized by the relevant value of this marker. Procedures based on model tables can be envisaged also for mortality forecasts relating to a given population. With this objective in mind, we choose a set of tables, representing the mortality in the population at several epochs, and assumed to represent also future mortality for that population.

4.6

Observed trend in markers

Other approaches to mortality projections

Extrapolation

Markers

Set of model tables

Life tables

The past

Future

In a given population Figure 4.15.

Model tables for mortality forecasts.

Trends in some markers are analysed and then projected, possibly using some mathematical formula, in order to predict their future values. Projected age-specific probabilities of death are then obtained by entering the system of model tables for the various projected values of the markers. The procedure is sketched in Fig. 4.15.

4.6.3

Projecting transforms of life table functions

A number of methods for mortality forecasts require that the projection procedure starts from the analysis of trends in mortality, in terms of oneyear probabilities of death or other fundamental life table functions, such as the force of mortality (in an age-continuous context) or the survival function. An alternative approach is to use some transforms of life table functions which may help us reach a better understanding of some features of mortality trends. Two examples will be provided: the relational method and the projection of the resistance function. The relational method was proposed by Brass (1974), who focussed on the logit transform of the survival function; see Section 2.7. For the purpose of forecasting mortality, equation (2.107) can be used in a dynamic sense. In a dynamic context, the Brass logit transform is particularly interesting when applied to cohort data, as the logits for successive birth-year cohorts seem to be linearly related (see Pollard (1987)). Hence,

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denoting by (x, τ) the logit of the survival function, S(x, τ), for the cohort born in the calendar year τ, we have: 1 − S(x, τ) 1 (4.89) (x, τ) = ln 2 S(x, τ) Referring to a pair of birth years, τk and τk+1 , we assume (x, τk+1 ) = αk + βk (x, τk )

(4.90)

So, the problem of projecting mortality reduces to the problem of extrapolating the two series αk and βk . Projected values of various life table functions can be derived from the inverse logit transform: S(x, τ) =

1 1 + exp[2 (x, τ)]

(4.91)

Figures 2.4–2.6 show how rectangularization and expansion phenomena, in particular, can be represented by choices of the parameter α and β. Application of the Brass transform to cohort-based projections requires a long sequence of mortality observations, in order to build up cohort survival functions. Further, inconsistencies may appear, since the method does not ensure that, for any year of birth τ, S(x1 , τ) > S(x2 , τ) for all pairs (x1 , x2 ) with x1 < x2 . So, negative values for mortality rates qx (τ + x) may follow, and hence appropriate adjustments in the linear extrapolation procedure are required. A different transform of the survival function S(x) has been addressed by Petrioli and Berti (see Petrioli and Berti (1979); see also Keyfitz (1982)). The proposed transform is the resistance function, defined in Setion 2.7 (see (2.108)). The resistance function has been graduated with the formula: ρ(x) = xα (ω − x)β eAx

2 +Bx+C

(4.92)

and, in particular, with the three-parameter formula: ρ(x) = k xα (ω − x)β

(4.93)

Model tables have been constructed using combinations of the three parameters, by focussing on the values of some markers. In a dynamic context, the mortality trend is represented by assuming that (some of) the parameters of the resistance function depend on the calendar year t. Thus, referring to equation (4.93), we have: ρ(x, t) = k(t) xα(t) (ω − x)β(t)

(4.94)

4.7

The Lee–Carter method: an introduction

Note that, when a model for the resistance function (see (4.92) and (4.93)) is assumed, the resulting projection model can be classified as an analytical model, even though it does not directly address the survival function. The Petrioli–Berti model has been used to project the mortality of the Italian population, and then has been adopted by the Italian Association of Insurers in order to build up projected mortality tables for life annuity business.

4.7 4.7.1

The Lee–Carter method: an introduction Some preliminary ideas

In general, most of the projection formulae presented in the previous sections do not allow for the stochastic nature of mortality. Actually, a number of projection methods used in actuarial practice simply consist in graduation–extrapolation procedures (see e.g. (4.11), (4.17), (4.63)). A more rigorous approach to mortality forecasts should take into account the stochastic features of mortality. In particular, the following points should underpin a stochastic projection model: – observed mortality rates are outcomes of random variables representing past mortality; – forecasted mortality rates are estimates of random variables representing future mortality. Hence, stochastic assumptions about mortality are required, that is, probability distributions for the random numbers of deaths, and a statistical structure linking forecasts to observations must be specified (see Fig. 4.16). In a stochastic framework, the results of the projection procedures consist in • Point estimates • Interval estimates

of future mortality rates (see Fig. 4.17) and other life table functions. Clearly, traditional graduation–extrapolation procedures, which do not explicitly allow for randomness in mortality, produce just one numerical value for each future mortality rate (or some other age-specific quantity). Moreover, such values can be hardly interpreted as point estimates, because of the lack of an appropriate statistical structure and model.

169

170


qx(t)

Sample= observed outcomes ofthe random mortality frequency

. . . .

Path of a stochastic process= possible future outcomes ofthe random mortality frequency

.

A model linking the probabilistic structure of the stochastic process to the sample

Time t

t' Figure 4.16.

From past to future: a statistical approach.

qx(t)

Interval estimation Graduation Point estimation

Observations

t⬘ Figure 4.17.

Time t

Mortality forecasts: point estimation vs interval estimation.

An effective graphical representation of randomness in future mortality is given by the so-called fan charts; see Fig. 4.18, which refers to the projection of the expected lifetime. The fan chart depicts a ‘central projection’ together with some ‘prediction intervals’. The narrowest interval, namely the one with the darkest shading, corresponds to a low probability prediction, say 10%, and is surrounded by prediction intervals with higher probabilities, say 30%, 50%, etc. See also Section 5.9.4. The Lee–Carter (LC) method (see Lee and Carter (1992); Lee (2000)) represents a significant example of the stochastic approach to mortality forecasts and constitutes one of the most influential proposals in recent times. A number of generalizations and improvements have been proposed, which follow and build on the basic ideas of the LC methodology.

4.7

The Lee–Carter method: an introduction

e65(t)

Prediction intervals

Central projection (point estimate)

t'

Figure 4.18.

4.7.2

Time t

Forecasting expected lifetime: fanchart.

The LC model

In order to represent the age-specific mortality we address the central death rate. Let mx (t) denote the central death rate for age x at time t, and we assume the following log-bilinear form: ln mx (t) = αx + βx κt + x,t

(4.95)

where the αx ’s describe the age-pattern of mortality averaged over time, whereas the βx ’s describe the deviations from the averaged pattern when κt varies. The change in the level of mortality over time is described by the (univariate) mortality index κt . Finally, the quantity x,t denotes the error term, with mean 0 and variance σ 2 , reflecting particular age-specific historical influence that are not captured by the model. Expression (4.95) constitutes the starting point of the LC method. It is worth stressing that the LC model differs from ‘parametric models’ (namely, mortality laws, see Section 2.5), because in (4.95) the dependence on age is non-parametric and is represented by the sequences of αx ’s and βx ’s. The model expressed by (4.95) cannot be fitted by simple regression, since there is no observable variable on its right-hand side. A least squares solution can be found by using the first element of the singular value decomposition. The parameter estimation is based on a matrix of available death rates, and we note that the system implied by (4.95) is undetermined without additional constraints. Lee and Carter (1992) propose the normalization β = 1, x x t κt = 0, which in turn forces each αx to be an average of the log-central death rates over calendar years. Once the parameters αx , βx and κt are estimated, obtaining the estimates αˆ x , βˆ x , κˆ t , mortality forecast follows by modelling the values of κt as a time

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172


series, for example, as a random walk with drift. Starting from a given year t , forecasted mortality rates are then computed, for t > t , as follows:

mx (t) = exp(αˆ x + βˆ x κt ) = mx (t ) exp βˆ x (κt − κˆ t )

(4.96)

It is worth noting that mx (t) is modelled as a stochastic process, that is driven by the stochastic process κt , from which interval estimates can be computed for the projected values of mortality rates.

4.7.3

From LC to the Poisson log-bilinear model

The LC method implicitly assumes that the random errors are homoskedastic. This assumption, which follows from the ordinary least squares estimation method that is used as the main statistical tool, seems to be unrealistic, as the logarithm of the observed mortality rate is much more variable at older ages than at younger ages, because of the much smaller number of deaths observed at old and very old ages. In Brouhns et al. (2002b) and Brouhns et al. (2002a), possible improvements of the LC method are investigated, using a Poisson random variation for the number of deaths. This is instead of using the additive error term x,t in the expression for the logarithm of the central mortality rate (see (4.95)). In terms of the force of mortality µx (t), the Poisson assumption means that the random number of deaths at age x in calendar year t is given by Dx (t) ∼ Poisson ETRx (t) µx (t)

(4.97)

where ETRx (t) is the central number of exposed to risk. In order to define the Poisson parameter ETRx (t) µx (t), Brouhns et al. (2002a) and Brouhns et al. (2002b) assume a log-bilinear force of mortality, that is, ln µx (t) = αx + βx κt

(4.98)

hence with the structure expressed by (4.95), apart from the error term. The meaning of the parameters αx , βx , κt is essentially the same as for the corresponding parameters in the LC model. The parameters are then determined by maximizing the log-likelihood based on (4.97) and (4.98). Brouhns et al. (2002b) do not modify the time series part of the LC method. Hence, the estimates αˆ x and βˆ x are used with the forecasted κt in order to generate future mortality rates (as in (4.96)), as well as other age-specific quantities.

4.8

4.7.4

Further issues

The LC method and model tables

An interesting example of projecting mortality patterns using the LC method is provided by Buettner (2002). The LC method is used to project mortality patterns on the basis of model tables that are indexed on the expec◦ tation of life at birth e0 (see Section 4.6.2). Since model tables do not contain any explicit time reference, the LC model has been implemented replacing the time index κt with an index reflecting the level of life expectancy. Then, the model is ln mx (e) = αx + βx κe + x,e (4.99) where the parameter κe represents the trend in the level of life expectancy at birth.

4.8

Further issues

In this section we address some issues of mortality forecasts part of which are, at least to some extent, beyond the main scope of this book, whereas others will be developed in the following chapters.

4.8.1

Cohort approach versus period approach. APC models

First, consider the following projection model referred to the mortality odds φx (t) = qx (t)/px (t): φx (t) = φx (t ) rt−t (4.100) where the first term on the right-hand side does not depend on t, whereas the second term does not depend on x. Denoting the first term with A(x) and the second term with B(t), equation (4.100) can be rewritten as follows: φx (t) = A(x) B(t)

(4.101)

Then, we consider the so-called K-K-K hypothesis (formulated in 1934 by Kermack, McKendrick, and McKinlay), according to which the following factorization is assumed: µx (τ) = C(x) D(τ)

(4.102)

where τ denotes, as usual, the year of birth of the cohort. In projection model (4.101), the future mortality structure is split into: – a factor A(x), expressing the age effect; – a factor B(t), expressing the year of occurrence effect or period effect.

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174


Conversely, in model (4.102) it is assumed that the future mortality structure can be split into: – a factor C(x), expressing the age effect; – a factor D(τ), expressing the year of birth effect or cohort effect. Recently, models including both the period effect and the cohort effect (as well as the age effect) have been proposed. These models are commonly called APC (Age-Period-Cohort) models. An APC model, referring to the force of mortality, can be expressed as follows: µx (t) = Q(x) R(t) S(t − x)

(4.103)

(where t − x = τ) or, in logarithmic terms: ln µx (t) = ln Q(x) + ln R(t) + ln S(t − x)

(4.104)

A slightly modified version of (4.104), referring to central death rates (see Willets (2004)), is as follows: ln mx (t) = m + αx + βt + γt−x

(4.105)

with finite sets for the values of x and t. Constraints are usually as follows: αx = βt = γt−x = 0 (4.106) x

t

t−x

The model can be estimated using Poisson maximum likelihood, or weighted least squares methods. However, no unique set of parameters result in an optimal fit because of the trivial relation cohort + age = period Further weak points can be found in APC models like (4.102) and (4.103). In particular, these models assume an age-independent period effect, or an age-independent cohort effect, whereas the impact of mortality improvements over time (or between cohorts) may vary with age. As far as statistical evidence is concerned, both period and cohort effects seem to impact on mortality improvements. In particular, it is reasonable that period effects summarize contemporary factors, for example, the general health status of the population, availability of healthcare services, critical weather conditions, etc. Conversely, cohort effects quantify historical factors, for example, World War II, diet, smoking habits, etc. From a practical point of view, the main difficulty in implementing projection models allowing for cohort effects obviously lies in the fact that statistical data for a very long period are required, and such data are rarely available.

4.9


Conversely, from a general point of view, the role of period and cohort effects in quantifying factors that affect mortality improvements suggests that we consider future likely scenarios and, in particular, causes of death.

4.8.2

Projections and scenarios. Mortality by causes

When projecting mortality, the collateral information available to the forecaster can be allowed for. Information may concern, for example, trends in smoking habits, trends in prevalence of some illness, improvements in medical knowledge and surgery, etc. Thus, projections can be performed according to an assumed scenario. The introduction of relationships between causes (e.g. advances in medical science) and effects (mortality improvements) underpins mortality projections which are carried out according to assumed scenarios. Obviously, some degree of arbitrariness follows, affecting the results. The projection methods that we have described refer to mortality in aggregate. Nonetheless, many of them can be used to project mortality by different causes separately. Projections by cause of death offer a useful insight into the changing incidence of the various causes. Conversely, some important problems arise when this type of projection is adopted. In particular, it should be stressed that complex interrelationships exist among causes of death, whilst the classic assumption of independence is commonly accepted. For example, mortality from heart diseases and lung cancer are positively correlated, as both are linked to smoking habits. A further problem concerns the difficult identification of the cause of death for elderly people. A final issue concerns the phenomenon in long-term projections by cause of death whereby the future projected mortality rate is dominated by the mortality trend for the cause of death where mortality rates are reducing at the lowest speed. For these reasons, many forecasters prefer to carry out mortality projections only in aggregate terms.

4.9


4.9.1 4.9.1.1

Landmarks in mortality projections The antecedents

As noted by Cramér and Wold (1935), the earliest attempt to project mortality is probably due to the Swedish astronomer H. Gyldén. In a work

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176


presented to the Swedish Assurance Association in 1875, he fitted a straight line to the sequence of general death rates of the Swedish population concerning the years 1750–1870, and then extrapolated the behaviour of the general death rate. A similar graphical fitting was proposed in 1901 by T. Richardt for sequences of the life annuity values a60 and a65 , calculated according to various Norwegian life tables, and then projected via extrapolation for application to pension plan calculations. Note that, in both of the proposals of Gyldén and Richardt, the projection of a single-figure index was concerned. Mortality trends and the relevant effects on life assurance and pension annuities were clearly identified at the beginning of the 20th century, as witnessed by various initiatives in the actuarial field. In particular, it is worth noting that the subject ‘Mortality tables for annuitants’ was one of the topics discussed at the 5th International Congress of Actuaries, held in Berlin in 1906. Nordenmark (1906), for instance, pointed out that improvements in mortality must be carefully considered when pricing life annuities and, in particular, cohort mortality should be addressed to avoid underestimation of the related liabilities. The 7th International Congress of Actuaries, held in Amsterdam in 1912, included the subject ‘The course, since 1800, of the mortality of assured persons’. As Cramér and Wold (1935) notes, a life table for annuities was constructed in 1912 by A. Lindstedt, who used data from Swedish population experience and, for each age x, extrapolated the sequence of annual probabilities of death, namely the mortality profile qx (t), hence adopting a horizontal approach. Probably, this work constitutes the earliest projection of an age-specific function. 4.9.1.2

Early seminal contributions

Blaschke (1923) proposed a Makeham-based projected mortality model (see Section 4.5.1). In particular he adopted a vertical approach, consisting in the estimation of Makeham’s parameters for each period table based on the experienced mortality, and then in fitting the estimated values. Hence, projected values for the three parameters were obtained via extrapolation. In 1924, the Institute of Actuaries in London proposed a horizontal method for mortality projection (see Cramér and Wold (1935)), assuming that probabilities of death are exponential functions of the calendar year, from which comes the name ‘exponential model’ frequently used to denote this approach to mortality projections. Various extrapolation formulae used by UK actuaries in recent times for annuitants and pensioners tables are particular cases of the early exponential model (see Section 4.3.8).

4.9


We now turn to the diagonal approach. In 1927 A. R. Davidson and A. R. Reid proposed a Makeham-based projection model, in which Makeham’s law refers to cohort mortality experiences. The relevant parameters were estimated via a cohort graduation (see Reid and Davidson (1927)). The use of Makeham-based projections is thoroughly discussed by Cramér and Wold (1935), dealing with the graduation and extrapolation of Swedish mortality. In particular, the vertical (i.e. period-based, see (4.63)) and the diagonal (i.e. cohort-based, see (4.64)) approaches are compared. Let µx (z) = γ(z) + α(z) β(z)x denote the force of mortality in both the vertical (with z = t) and the diagonal (with z = t − x) approach. For the graduation of the parameters, Cramér and Wold (1935) assumed that, in both the vertical and the diagonal approach, α(z) is linear while ln β(z) and ln γ(z) are logistic. The assumption formulated in 1934 by Kermack, McKendrick, and McKinlay constitutes another example of the diagonal approach to mortality projections. As Pollard (1949) notes, these authors showed that, for some countries, it was reasonable to assume that the force of mortality depended on the attained age x and the year of birth τ = t − x, and they deduced that µx (t) = C(x) D(τ), where C(x) is a function of age only and D(τ) is a function of the year of birth only; see also Section 4.8.1. 4.9.1.3

Some modern contributions

Seminal contributions to mortality modelling and mortality projections in particular have been produced by demographers, throughout the latter half of the 20th century. The ‘optimal’ life table, model tables and relational methods probably constitute three of the most influential proposals in recent times, in the framework of mortality analysis. The idea of an ‘optimal’ table (see Section 4.6.1) was proposed by Bourgeois-Pichat (1952). The question was: ‘can mortality decline indefinitely or is there a limit, and if so, what is this limit?’ While a number of projection methods are based on the extrapolation of observed mortality trends, focussing on optimal tables provides an alternative approach to mortality forecasts, as an interpolation procedure between past data and the limit table is required. The possibility of summarizing the age-pattern of mortality by using some markers underpins the use of ‘model tables’ in mortality projections (see Section 4.6.2). Model tables were first constructed by the United Nations,

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in 1955. Each table is summarized by the relevant value of the expectation of life at birth. A new way to mortality forecasts was paved by the ‘relational method’ proposed by W. Brass (see e.g. Brass (1974)), who focussed on the logit transform of the survival function (see Section 2.7). A different transform of the survival function, namely the ‘resistance function’, has been addressed by Petrioli and Berti (1979); see also Keyfitz (1982). In a dynamic context, the mortality trend is represented assuming that (some of) the parameters of the resistance function depend on the calendar year t. 4.9.1.4

Recent contributions

In the last decades of the 1900s, various mortality law-based projection models have been proposed. In particular, Forfar and Smith (1988) have fitted the Heligman–Pollard curve to the graduated English life tables ELT1 to ELT13, for both males and females, and then have analysed the behaviour of the relevant parameters. Mortality projections have been performed assuming that various parameters of the Heligman–Pollard law are functions of the calendar year (see Benjamin and Soliman (1993) and Congdon (1993) for examples). In the 1990s, a new method for forecasting the age-pattern of mortality was proposed and then extended by L. Carter and R.D. Lee (see Lee and Carter (1992) and Lee (2000)). The LC method addresses the central death rate to represent the age-specific mortality (see Section 4.7.2). While traditional projections models provide the forecaster with point estimates of future mortality rates (or other age-specific quantities), the LC method explicitly allows for random fluctuations in future mortality, representing the related effect in terms of interval estimates. The LC methodology constitutes one of the most influential proposals in recent times, in the field of mortality projections. Indeed, much research work as well as many recent applications to actuarial problems are directly related to this methodology (for detailed references see Section 4.9.2). Finally, frailty models in the context of mortality forecast have been addressed by Butt and Haberman (2004) and Wang and Brown (1998).

4.9.2

Further references

There are a number of both theoretical and practical papers dealing with mortality forecasts, produced by actuaries as well as by demographers. The reader interested in various perspectives on forecasting mortality should refer to Tabeau et al. (2001), and Booth (2006), in which a number of

4.9


approaches to mortality projections are discussed and several applications are described. Interesting reviews on mortality forecast methods can be found also in Benjamin and Pollard (1993), Benjamin and Soliman (1993), National Statistics - Government Actuary’s Department (2001), Olshansky (1988), Pollard (1987), and Wong-Fupuy and Haberman (2004). Mortality projections via reduction factors represent a practical and widely adopted approach to mortality forecast. As regards formulae used by UK actuaries, the reader should refer to CMIB (1978, 1990, 1999). Recent contributions to the modelling of reduction factors have been given by Renshaw and Haberman (2000, 2003a), and Sithole et al. (2000). In the field of law-based mortality projections, Felipe et al. (2002) have used the Heligman–Pollard law 2 for fitting and projecting mortality trends in the Spanish population. Also more traditional mortality laws have been used for analysing mortality trends and producing mortality forecasts. For example, Barnett (1960) has analysed mortality trends through the parameters of a modified Thiele’s formula, whereas Buus (1960) has used the Makeham law, focussing on the interdependence between the parameters. Poulin (1980) has proposed a Makeham-based projection formula, whereas Wetterstrand (1981) has used Gompertz’s law. Functions other than the force of mortality can also be addressed. For example, Beard (1952) built up a projection model by fitting a Pearson Type III curve to the curve of deaths, and then taking some parameters (in particular the maximum age) as functions of the year of birth. The Weibull law has been used by Olivieri and Pitacco (2002a) and Olivieri (2005), in order to express, via the relevant parameters, various assumptions about the expansion and rectangularization of the survival function. The use of a law-based approach to mortality forecasting is rather controversial. For interesting discussions on this issue, the reader should consult Keyfitz (1982) and Pollard (1987). Brouhns et al. (2002b) stress that the estimated parameters are often strongly dependent. Hence, univariate extrapolation of the parameters may be misleading, whereas a multivariate time series for the parameters is theoretically possible but can lead to computational intractability. Of course, a distribution-free approach to mortality projections avoids these problems. Very important examples of the distribution-free approach are provided by the LC model and several models aiming to improve the LC methodology. The practical use of projected tables deserves special attention, especially when just one cohort table is actually adopted in pricing and reserving (see Section 4.4.2). In particular, the optimal choice of the age-shifting function

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(see Section 4.4.3) has been dealt with by Delwarde and Denuit (2006); see also Chapter 3. Considerable research work has been recently devoted to improve and generalize the LC methodology. In particular, the reader should refer to Carter (1996), Alho (2000), Renshaw and Haberman (2003a, b, c), Brouhns and Denuit (2002), Brouhns et al. (2002b). See also the list of references in Lee (2000). Among the extensions of the LC method, we note the following developments. Carter (1996) incorporates in the LC methodology uncertainty about the estimated trend of mortality kt , through a specific model for the trend itself. Renshaw and Haberman (2003c) have noted that the standard LC methodology fails to capture and then project recent upturn in crude mortality rates in the age range 20–39 years. So, an extension of the LC methodology is proposed, in order to incorporate in the LC model specific age differential effects. Booth et al. (2002) have developed systematic methods for choosing the most appropriate subset of the data to use for modelling – the graduation subset of Fig. 4.4. The importance of ensuring that the estimates αˆ x and βˆ x are smooth with respect to age so that irregularities are not magnified via extrapolations into the future has been discussed by Renshaw and Haberman (2003a), Renshaw and Haberman (2003c), De Jong and Tickle (2006), and Delwarde et al. (2007). A cause-of-death projection study was proposed by Pollard (1949), based on Australian population data. As regards scenario-based mortality forecasts, Gutterman and Vanderhoof (1998) stress that a projection methodology should allow for relationships between causes (e.g. advances in medical science) and effects (mortality improvements).

5

5.1

Forecasting mortality: applications and examples of age-period models

Introduction

As explained in Chapter 4, actuaries working in life insurance and pension have been using projected life tables for some decades. But the problem confronting actuaries is that people have been living much longer than they were expected to according to the life tables being used for actuarial computations. What was missing was an accurate estimation of the speed of the mortality improvement: thus, most of the mortality projections performed during the second half of the 20th century have underestimated the gains in longevity. The mortality improvements seen in practice have quite consistently exceeded the projected improvements. As a result, insurers have, from time to time, been forced to allocate more capital to support their inforce annuity business, with adverse effects on free reserves and profitability. From the point of view of the actuarial approach to risk management, the major problem is that mortality improvement is not a diversifiable risk. Traditional diversifiable mortality risk is the random variation around a fixed, known life table. Mortality improvement risk, though, affects the whole portfolio and can thus not be managed using the law of large numbers (see Chapter 7 for a detailed discussion of systematic and non-systematic risks). In this respect, longevity resembles investment risk, in that it is non-diversifiable: it cannot be controlled by the usual insurance mechanism of selling large numbers of policies, because they are not independent in respect of that source of uncertainty. However, longevity is different from investment risk in that there are currently no large traded markets in longevity risk so that it cannot easily be hedged. The reaction to this problem is twofold. First, actuaries are trying to produce better models for mortality improvement, paying more attention to the levels of uncertainty involved in the forecasts. The second part of the reaction is to look to the

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5 : Age-period projection models

capital markets to share the risk, through the emergence of mortality-linked derivatives or longevity bonds. This kind of securitization will be discussed in Chapter 7. As explained in the preceding chapter, there is a variety of statistical models used for mortality projection, ranging from the basic regression models, in which age and time are viewed as continuous covariates, to sophisticated robust non-parametric models. Mortality forecasting is a hazardous yet essential enterprise for life annuity providers. This chapter examines the problem in the favourable circumstances encountered in developed countries, where extensive historical data are often easily available. A statistical model (in the form of a regression or a time series) is used to describe historical data and extrapolate past trends to the future. In this chapter, we first consider the log-bilinear projection model pioneered by Lee and Carter (1992) that has been introduced in Section 4.7.2. The method describes the log of a time series of age-specific death rates as the sum of an age-specific component that is independent of time and another component that is the product of a time-varying parameter reflecting the general level of mortality, and an age-specific component that represents how rapidly or slowly mortality at each age varies when the general level of mortality changes. This model is fitted to historical data. The resulting estimate of the time-varying parameter is then modelled and projected as a stochastic time series using standard Box-Jenkins or ARIMA methods. From this forecast of the general level of mortality, the future death rates are derived using the estimated age effects. The key difference between the classical generalized linear regression model approach (see Section 4.5.4) and the method pioneered by Lee and Carter (1992) centers on the interpretation of time which in the logbilinear approach is modelled as a factor and under the generalized linear regression approach is modelled as a known covariate. The model proposed by Lee and Carter (1992) has now been widely adopted. However, it is of course not the only candidate for extrapolating mortality to the future. It should be stressed that some models are designed to project specific demographic indicators, and that the forecast horizon may depend on the type of model. In this respect, the model proposed by Lee and Carter (1992) is typically meant for long-term projections of aggregate mortality indicators like life expectancies. It is not intended to produce reliable forecasts of series of death rates for a particular age. This is why this model is so useful for actuaries, who are interested in life annuity premiums and reserves, which are weighted versions of life expectancies (the weights being the financial discount factors). Some extensions incorporating features specific to each cohort are proposed in the next chapter.

5.1

Introduction

In addition to the Lee–Carter model, we also consider a powerful alternative mortality forecasting method proposed by Cairns et al. (2006a). It includes two time factors (whereas only one time factor drives the future death rates in the Lee–Carter case) with a smoothing of age effects using a logit transformation of one-year death probabilities. Specifically, the logit of the one-year death probabilities is modelled as a linear function of age, with intercept and slope parameters following some stochastic process. Compared with the Lee–Carter approach, the Cairns–Blake–Dowd model includes two time factors. This allows the model to capture the imperfect correlation in mortality rates at different ages from one year to the next. This approach can also be seen as a compromise between the generalized regression approach and the Lee–Carter views of mortality modelling, in that age enters the Cairns–Blake–Dowd model as a continuous covariate whereas the effect of calendar time is captured by a couple of factors (time-varying intercept and slope parameters). The Cairns–Blake–Dowd model is fitted to historical data. The resulting estimates for the time-varying parameters are then projected using a bivariate time series model. From this forecast of the future intercept and the slope parameters, the future one-year death probabilities are computed in combination with the linear age effect. Mortality forecasts performed by demographers are traditionally based on the forecaster’s subjective judgements, in the light of historical data and expert opinions. This traditional method has been widely used for official mortality forecasts, and by international agencies. A range of uncertainty is indicated by high and low scenarios (surrounding the medium scenario which is meant to be the best estimate), which are also constructed through subjective judgements. In the hands of a skilled and knowledgeable forecaster, the traditional method has the advantage of drawing on the full range of relevant information for the medium forecast and the high–low range. However, it also has certain deficiencies. First, mortality projections in industrialized countries have been found to under-predict mortality declines and gains in life expectancy when compared to subsequent outcomes, as pointed out by Lee and Miller (2001). Thus, a systematic downward bias has been observed for this traditional approach during the 20th century. A second difficulty is that it is not clear how to interpret the high–low range of a variable unless a corresponding probability for the range is stated. We will come back to this issue in Section 5.8. Both the Lee–Carter and the Cairns–Blake–Dowd models greatly reduce the role of subjective judgement since standard diagnostic and statistical

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modelling procedures for time series analysis are followed. Nonetheless, decisions must be taken about a number of elements of these models–for example, how far back in history to begin, or exactly what time series model to use. It should be noted that the models investigated in this chapter do not attempt to incorporate assumptions about advances in medical science or specific environmental changes: no information other than previous history is taken into account. The (tacit) underlying assumption is that all of the information about the future is contained in the past observed values of the death rates. This means that this approach is unable to forecast sudden improvements in mortality due to the discovery of new medical treatments, revolutionary cures including antibiotics, or public health innovations. Similarly, future deteriorations caused by epidemics, the appearance of new diseases or the aggravation of pollution cannot enter the model. The actuary has to keep this in mind when he uses this model and makes decision on the basis of the outputs, for example, in the setting of a reinsurance programme. Some authors have severely criticized the purely extrapolative approach because it seems to ignore the underlying mechanisms of a social, economic or biological nature. As pointed out by Wilmoth (2000), such a critique is valid only insofar as such mechanisms are understood with sufficient precision to offer a legitimate alternative method of prediction. Since our understanding of the complex interactions of social and biological factors that determine mortality levels is still imprecise, we believe that the extrapolative approach to prediction is particularly compelling in the case of human mortality. The R software has been found convenient to perform the analysis described in this chapter (as well as those in Chapter 3). R is a free language and environment for statistical computing and graphics. R is a GNU project which is similar to the S language and environment which was developed at Bell Laboratories (formerly AT&T, now Lucent Technologies) by John Chambers and colleagues. For more details, we refer the interested reader to http://www.r-project.org/. In addition to our own R code, we have benefitted from the demography package for R created by Rob J. Hyndman, Heather Booth, Leonie Tickle, and John Maindonald. This package contains functions for various demographic analyses. It provides facilities for demographic statistics, modelling and forecasting. In particular, it implements the forecasting model proposed by Lee and Carter (1992) and several variations of it, as well as the forecasting model proposed by Hyndman and Ullah (2007).

5.1

Introduction

After the Crédit Suisse longevity index based on the expectation of life derived from US data, the more comprehensive JPMorgan LifeMetrics has innovated by producing publicly available indices on population longevity. LifeMetrics is a toolkit for measuring and managing longevity and mortality risk. LifeMetrics advisors include Watson Wyatt and the Pensions Institute at Cass Business School. LifeMetrics Index provides mortality rates and period life expectancy levels across various ages, by gender, for each national population covered. Currently the LifeMetrics Index publishes index values for the United States, England & Wales, and The Netherlands. All of the methodology, algorithms and calculations are fully disclosed and open. The LifeMetrics toolkit includes a set of computer based models that can be used in forecasting mortality and longevity. These models have been evaluated in the research paper ‘A quantitative comparison of eight stochastic mortality models using data from England & Wales and the United States’ by Cairns et al. (2007). The R source code required to run the forecast models is available for download along with a user guide. We also mention two other resources which are available from the web (but which were not used in the present book). Federico Girosi and Gary King offer the YourCast software that makes forecasts by running sets of linear regressions together in a variety of sophisticated ways. This open source software is freely available from http://gking.harvard.edu/yourcast/. It implements the methods introduced in Federico Girosi and Gary King’s manuscript on Demographic Forecasting, to be published by Princeton University Press. Further, we note the recent initiative of the British CMIB (Continuous Mortality Investigation Bureau), that is, the bureau affiliated to the UK actuarial profession, with the function of producing mortality tables for use by insurers and pension plans. The CMIB has made available software running on R with the aim of illustrating the P-Spline methodology for projecting mortality. CMIB software now allows the fitting of the Lee-Carter model as well, but with restricted ARIMA specifications. For more details, please consult http://www.actuaries.org.uk. Before embarking in the presentation of the Lee–Carter and the Cairns– Blake–Dowd approaches, let us say a few words about the material not included in the present chapter. First, we do not consider possible cohort effects, and limit our analysis to the age and period dimensions. For countries like Belgium, cohort effects are weak enough and can be neglected. However, for countries like the UK, cohort effects are significant and must be accounted for. Chapter 6 is devoted to the inclusion of cohort effects in the Lee–Carter and Cairns–Blake–Dowd models discussed here.

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We also do not consider continuous-time models for mortality, that are inherited from the interest rate management and credit risk literature. We refer the reader to the works by Biffis and Millossovich (2006a), Biffis and Millossovich (2006b), Biffis and Denuit (2006), and Biffis (2005) for more information and further references about this approach. See also Chapter 7 in this book.

5.2 5.2.1

Lee–Carter mortality projection model Specification

Lee and Carter (1992) proposed a simple model for describing the secular change in mortality as a function of a single time index. Throughout this chapter, we assume that assumption (3.2) is fulfilled, that is, that the age-specific mortality rates are constant within bands of age and time, but allowed to vary from one band to the next. Recall that under (3.2), the force of mortality µx (t) and the death rate mx (t) coincide. Lee and Carter (1992) specified a log-bilinear form for the force of mortality µx (t), that is, ln µx (t) = αx + βx κt

(5.1)

The specification (5.1) differs structurally from parametric models given that the dependence on age is non-parametric, and represented by the sequences of αx ’s and βx ’s. Interpretation of the parameters is quite simple: exp αx is the general shape of the mortality schedule and the actual forces of mortality change according to an overall mortality index κt modulated by an age response βx (the shape of the βx profile tells which rates decline rapidly and which slowly over time in response of change in κt ). The parameter βx represents the age-specific patterns of mortality change. It indicates the sensitivity of the logarithm of the force of mortality at age x to variations in the time index κt . In principle, βx could be negative at some ages x, indicating that mortality at those ages tends to rise when falling at other ages. In practice, this does not seem to happen over the long-run, except sometimes at the very oldest ages. There is also some evidence of negative βx estimates for males at young adult ages in certain industrialized countries. This has been attributed to an increase in mortality due to AIDS in the late 1980s and 1990s. In a typical population, age-specific death rates have a strong tendency to move up and down together over time. The specification (5.1) uses this tendency by modelling the changes over time in age-specific death rates as

5.2

Lee–Carter mortality projection model

driven by a scalar factor κt . This strategy implies that the modelled death rates are perfectly correlated across ages, which is the strength but also the weakness of the approach. As pointed by Lee (2000), the rates of decline in the ln µx (t)’s at different ages are given by βx (κt − κt−1 ) so that they always maintain the same ratio to one another over time. In practice, the relative speed of decline at different ages may vary. In such a case, the extended version of the Lee–Carter model introduced by Booth et al. (2002) – see equation (5.14) – or the Cairns–Blake–Dowd approach might be preferable. Remark Hyndman and Ullah (2007) extend the principal components approach by adopting a functional data paradigm combined with nonparametric smoothing (penalized regression splines) and robust statistics. Univariate time series are then fitted to each component coefficient (or level parameter). The Lee-Carter method then appears to be a particular case of this general approach. Remark Many models produce projected death rates that tend to 0. Hence, some constraint should be imposed on the long-term behaviour of the death rates. In that respect, limit life tables that have been discussed in Section 4.6.1 may be specified, or we can use a forecast that incorporates a theoretical maximum achievable life expectancy. This feature implies a slowdown in the rate of mortality decline as the theoretical maximum life expectancy is reached. If we denote as µ∞ x the limiting force of mortality, ∞ the model becomes ln µx (t) − µx = αx + βx κt . Remark Considering the global convergence in mortality levels, and the common trends evidenced in Section 3.5 of Chapter 3, it may seem appropriate to prepare mortality forecasts for individual national populations in tandem with one another. Li and Lee (2005) have modified the original projection model of Lee and Carter (1992) for producing mortality forecasts for a group of populations. To this end, the central tendencies for the group are first identified using a common factor approach, and national historical particularities are then taken into account. Note that the most direct application of this approach is to forecast in a population for the two sexes mortality. The same βx and κt can be used for both males and females, letting the αx ’s depend on gender, as in Li and Lee (2005). Alternatively, Carter and Lee (1992) used the same κt ’s for males and females but allowed the αx ’s and βx ’s to be gender-specific. Delwarde et al. (2006) have analysed the pattern of mortality decline in the G5 countries (France, Germany, Japan, UK, and USA). Each G5 country is viewed as the value of a covariate. This model allows us to analyse the

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level and age pattern of mortality by country, the general time pattern of mortality change, and the speed and age pattern of mortality change by country. As for the Lee–Carter model, the extrapolation of estimates " κt gives future mortality rates for given gender, age, time, and country. The main interest of this method lies in the estimation of a unique time series (or two if each gender is treated separately) which gives mortality rates for all countries and age-time categories. As expected, the analysis conducted by Delware et al. (2006) reveals that age is the most important factor determining mortality rate. The time effect is more relevant than the country effect if weights are taken into account, which is a sign of convergence. In other words, the time horizon is more important than the country, but since the country effect is not negligible, the differences between country-specific death rates increase with time. These results allow us to compare the mortality experience observed in the G5 countries through the same model and also to produce forecasts. An estimated average death rate and a common index of mortality decline can be obtained from the analysis, which is essential for economists. Most financial and insurance decisions are taken on the basis of a worldwide view, more than on a regional or particular location. From this analysis, one can obtain baseline mortality forecasts from the pooled G5 population, but at the same time, one can see the influence of each gender, age, time trend, and country on the mortality forecast. In this way, the observed past behaviour of the G5 is summarized in a single model and the identification and comparison of each country specific effect become much easier.

5.2.2 5.2.2.1

Calibration Identifiability constraints

Let us assume that we have observed data for a set of calendar years t = t1 , t2 , . . . , tn and for a set of ages x = x1 , x2 , . . . , xm . On the basis of these observations, we would like to estimate the corresponding αx ’s, βx ’s, and κt ’s. However, this is not possible unless we impose additional contraints. In (5.1), the αx parameters can only be identified up to an additive constant, the βx parameters can only be identified up to a multiplicative constant, and the κt parameters can only be identified up to a linear transformation. Precisely, if we replace βx with cβx and κt with κt /c for any c = 0 or if we replace αx with αx − cβx and κt with κt + c for any c, we obtain the same values for the death rates. This means that we cannot distinguish between the two parametrizations: different values of the parameters produce the same mx (t)’s. To see that two constraints are needed to ensure

5.2


identification, note that if (5.1) holds true, we also have ln µx (t) = + αx + + βx+ κt

(5.2)

with + αx = αx +c1 βx , + βx = βx /c2 and+ κt = c2 (κt −c1 ). Therefore, we need to impose two constraints on the parameters αx , βx , and κt in order to prevent the arbitrary selection of the parameters c1 and c2 . A pair of additional constraints are thus required on the parameters for estimation to circumvent this problem. To some extent, the choice of the constraints is a subjective one, although some choices are more natural than others. In the literature, the parameters in (5.1) are usually subject to the constraints xm tn κt = 0 and βx = 1 (5.3) t=t1

x=x1

ensuring model identification. Under this normalization, βx is the proportion of change in the overall log mortality attributable to age x. We also note that other constraints can be found in the literature, for instance, msets of 2 = 1. κtn = 0 or xx=x β 1 x Note that the lack of identifiability of the Lee-Carter model is not a real problem. It just means that the likelihood associated with the model has an infinite number of equivalent maxima, each of which would produce identical forecasts. Adopting the constraints (5.3) consists in picking one of these equivalent maxima. The important point is that the choice of constraints has no impact on the quality of the fit, or on forecasts of mortality. Some care is needed, however, in any bootstrap procedures used for simulation (see Section 5.8). 5.2.2.2

Least-squares estimation

Statistical model The model classically used to estimate the αx ’s, βx ’s, and κt ’s is " x (t) = αx + βx κt + x (t) ln m (5.4) " x (t) denotes the for x = x1 , x2 , . . . , xm and t = t1 , t2 , . . . , tn , where m observed force of mortality at age x during year t computed according to (3.13), and where the x (t)’s are homoskedastic centered error terms. The error term x (t), with mean 0 and variance σ 2 reflects any particular agespecific historical influences that are not captured in the model. Note that the errors have the same variance over age, which is sometimes a questionable assumption: the logarithm of the observed force of mortality is usually much more variable at the older ages than at the younger ages because of the much smaller absolute number of deaths at the older ages. However,

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if the mortality surface has been previously completed (i.e. extrapolated to the oldest ages using a parametric model), the homoskedasticity assumption is not a problem provided that the actuary restricts the age range for modelling to 50 and over, say, in order to avoid the instability around the accident hump. It is worth mentioning that model (5.4) is not a simple regression model, since there are no observed quantities on the right-hand side. Specifically, age x and calendar time t are treated as factors and the effect on mortality is quantified by the sequences αx1 , αx2 , . . . , αxm and βx1 , βx2 , . . . , βxm for age, and by the sequence κt1 , κt2 , . . . , κtn for calendar time. Note that the model (5.4) is particularly useful when the actuary has only a set of death " x (t) at his disposal. In the case where more detailed information rates m is available, the Poisson approach described in the next section makes an effective use of observations of death counts and exposure-to-risk. Objective function The model (5.4) is fitted to a matrix of age-specific observed forces of mortality using singular value decomposition. Specifically, the " αx ’s, " βx ’s, and " κt ’s are such that they minimize

OLS (α, β, κ) =

xm tn

" x (t) − αx − βx κt ln m

2

(5.5)

x=x1 t=t1

This is equivalent to maximum likelihood estimation provided that the x (t)’s obey the Normal distribution. Remark Wilmoth (1993) suggested a weighted least-squares procedure for estimating the (α, β, κ) parameters. Specifically, the objective function (5.5) is replaced with

OWLS (α, β, κ) =

xm tn

2 " x (t) − αx − βx κt wxt ln m

(5.6)

x=x1 t=t1

Empirical studies reveal that using the observed dxt ’s as weights (i.e. wxt = dxt ) has the effect of bringing the parameters estimated into close agreement with the Poisson-response-based estimates (discussed below). However, the choice of the death counts as weights is questionable, and the Poisson maximum likelihood approach described in the next section has better statistical properties, and should therefore be preferred for inference purposes. The reason is that a valid weighted least-squares approach must use exogeneous weights, but obviously the number of deaths is a random variable. As such, estimates resulting from the minimization of OWLS have no known statistical properties and can be strongly biased.

5.2


Effective computation: Singular value decomposition equal to 0 yields tn

Since

tn

t=t1

t=t1 κt

tn

" x (t) = (tn − t1 + 1)αx + βx ln m

Setting

κt

∂ ∂αx OLS

(5.7)

t=t1

= 0 by the constraint (5.3), we get tn 1 " x (t) " αx = ln m tn − t1 + 1 t=t

(5.8)

1

The minimization of (5.5) thus consists in taking for " αx the row average " x (t)’s. When the model (5.4) is fitted by ordinary least-squares, of the ln m " x (t) over time t so the fitted value of αx exactly equals the average of ln m that exp αx represents the general shape of the mortality schedule. We then obtain the " βx ’s and" κt ’s from the first term of a singular value decomposition " x (t) − " of the matrix ln m αx . Specifically, death rates can be combined to form a matrix   mx1 (t1 ) · · · mx1 (tn )   .. .. .. M=  . . . mxm (t1 )

(5.9)

· · · mxm (tn )

of dimension (xm − x1 + 1) × (tn − t1 + 1). Model (5.1) is then fitted so that it reproduces M as closely as possible. Now, let us create the matrix " −" Z = ln M α  " x1 (t1 ) − " αx1 ln m  . . = . " xm (t1 ) − " αxm ln m

 " x1 (tn ) − " · · · ln m αx1  .. ..  . . " xm (tn ) − " · · · ln m αxm

(5.10)

of dimension (xm −x1 +1)×(tn −t1 +1). Approximating the zxt ’s with their Lee–Carter expression βx κt indicates that the absence of age-time interactions is assumed, that is, the βx ’s are fixed over time and the κt ’s are fixed over ages. Most data sets do not comply with the time-invariance of the βx ’s, unless the optimal fitting period has been selected as explained below. κt ’s are such that they minimize Now, the " βx ’s and " +LS (β, κ) = O

xm tn

zxt − βx κt

2

(5.11)

x=x1 t=t1

The solution is given by the singular value decomposition of Z. More precisely, let us define the square matrices Z T Z of dimension (tn − t1 + 1) ×

191

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(tn − t1 + 1) and ZZ T of dimension (xm − x1 + 1) × (xm − x1 + 1). Let u1 be the eigenvector corresponding to the largest eigenvalue of Z T Z. Let v 1 be the corresponding eigenvector of ZZ T . The best approximation of Z in the least-squares sense is known to be (5.12) Z ≈ Z = λ1 v 1 u T 1 from which we deduce v1

" β = x −x +1 m 1 j=1

v1j

  xm −x 1 +1 and " κ = λ1  v1j  u1

(5.13)

j=1

xm −x1 +1 v1j = 0. The constraints (5.3) are then satisfied by provided that j=1 the " βx ’s and" κt ’s. Note that the second and higher terms of the singular value decomposition together comprise the residuals. Typically, for low mortality populations, the first order approximation (5.12) behind the Lee–Carter " x (t)’s. model accounts for about 95% of the variance of the ln m Remark As pointed out by Booth et al. (2002), the original approach by Lee and Carter (1992) makes use of only the first term of the singular value decomposition of the matrix of centered log death rates. In principle, the second-and higher-order terms could be incorporated in the model. The full expanded model is r [j] [j] " x (t) = αx + ln m β x κt (5.14) j=1 [j] [j]

where r is the rank of the ln mx (t)−αx matrix. In this case, βx κt is referred to as the jth order term of the approximation. Any systematic variation in the residuals from fitting only the first term would be captured by the second and higher terms. In their empirical illustration, Booth et al. (2002) find a diagonal pattern in the residuals that was interpreted as a cohortperiod effect. We will come back to the modelling of cohort effects in the next chapter. Brouhns et al. (2002b) have tested whether the inclusion of a second log-bilinear term significantly improves the quality of the fit, and this was not the case in their empirical illustrations. Renshaw and Haberman (2003a) report on the failure of the first-order Lee–Carter model to capture important aspects of the England and Wales mortality experience (despite explaining about 95% of the total variance) together with the presence of noteworthy residual patterns in the secondorder term. As a consequence, Renshaw and Haberman (2003b) have investigated the feasibility of constructing mortality forecasts on the basis of the first two sets of SVD vectors, rather than just on the first set of such

5.2


vectors, as in the Lee–Carter approach. Whereas Renshaw and Haberman (2003b) have applied separate univariate ARIMA processes to the first two period components, Renshaw and Haberman (2005) have used a bivariate time series. Effective computation: Newton–Raphson The estimations for the parameters αx , βx and κt can also be obtained recursively using a Newton– Raphson algorithm avoiding singular value decomposition. The system to solve in order to obtain the estimated values of the parameters αx , βx and κt is obtained by equating to 0 the partial derivative of OLS (α, β, κ) given in (5.5) with respect to αx , κt and βx , that is, 0=

tn

" x (t) − αx − βx κt , ln m

x = x1 , x2 , . . . , xm

t=t1

0=

xm

" x (t) − αx − βx κt , βx ln m

t = t1 , t2 , . . . , tn

(5.15)

x=x1

0=

tn

" x (t) − αx − βx κt , κt ln m

x = x1 , x2 , . . . , xm

t=t1

Each of these equations is of the form f (ξ) = 0, where ξ is one of the parameters αx , βx , and κt . The idea is to update each parameter in turn using a univariate NewtonRaphson recursive scheme. Starting from some initial value ξ (0) , the (k+1)th iteration gives ξ (k+1) from ξ (k) by ξ (k+1) = ξ (k) −

f (ξ (k) ) f (ξ (k) )

Each time one of the Lee–Carter parameters αx , βx and κt is updated, the already revised values of the other parameters are used in the iterative formulas. The recurrence relations are thus as follows: tn (k) (k) (k) " x (t) − " αx − " βx " κt t=t1 ln m (k+1) (k) " αx =" αx + tn − t 1 + 1 xm (k+1) (k) (k) " (k) ln m " x (t) − " αx −" βx " κt x=x1 βx (k+1) (k) " κt =" κt + (5.16) xm (k) 2 " β x x=x1 tn (k+1) (k+1) (k) (k+1) " x (t) − " κt αx −" βx " κt ln m t=t1 " (k+1) (k) " " = βx + βx tn (k+1) 2 κt t=t1 "

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This alternative to singular value decomposition does not require a rectangular array of data (it suffices to let the summation indices range over the available observations). Further estimation can proceed in the presence of empty cells, as these would receive a zero weight and are then simply excluded from the computations. Identifiability constraints The estimates for αx , βx , and κt produced by the methods described above (the singular value decomposition or the NewtonRaphson procedure (5.16)) do not satisfy the constraints (5.3). To fullfill the identifiability constraints, we replace " αx with " κt −κ)" β• , αx +" βx κ," κt with (" and " βx with " βx /" β• where " β• is the sum of the " βx ’s coming out of the singular value decomposition or the Newton–Raphson procedure (5.16), and κ is the average of the " κt ’s coming out of the singular value decomposition or the Newton–Raphson procedure (5.16). Adjustment of the κt ’s by refitting to the total observed deaths Instead of keeping the " κt ’s obtained from singular value decomposition or Newton– Raphson algorithm, Lee and Carter (1992) suggested that the " κt ’s (taking " the " αx ’s and βx ’s asgiven) be adjusted in order to reproduce the observed m number of deaths xx=x Dxt in year t. This avoids discrepancies arising 1 from modelling on the logarithmic scale. Since it is desirable that the differences between the actual and expected total deaths in each year are zero, as in the construction and graduation of period life tables, the adjusted " κt ’s solve the equation xm x=x1

Dxt =

xm

ETRxt exp(" αx + " βx ζ)

(5.17)

x=x1

in ζ. So, the κt ’s are reestimated in such a way that the resulting death rates (with the previously estimated " αx and " βx ), applied to the actual risk exposure, produce the total number of deaths actually observed in the data for the year t in question. There are several advantages to making this second stage estimate of the parameters κt . In particular, it avoids sizable discrepancies between predicted and actual deaths (which may occur because the model (5.4) is specified by means of logarithms of death rates). We note that no explicit solution is available for (5.17), which has thus to be solved numerically (using a Newton–Raphson procedure, for instance). It is worth mentioning that more than one solution for (5.17) may arise when all the " βx ’s do not have the same sign. A nonuniform sign for the " βx ’s implies that mortality is increasing at some ages and decreasing at others. This is not normally expected to happen, except sometimes at advanced ages

5.2


(but the phenomenon disappears when the actuary starts the modelling by closing the life tables). Therefore, solving (5.17) usually does not pose any problem. Adjustment of the κˆ t ’s by refitting to the observed period life expectancies Whereas Lee and Carter (1992) have suggested that the " κt be adjusted as in (5.17) by refitting to the total observed deaths, Lee and Miller (2001) have proposed an adjustment procedure in order to reproduce the period life expectancy at some selected age (instead of the total number of deaths recorded during the year). In practice, the actuary first selects an age x0 . In population studies, it is common to take x0 = 0 but in mortality projections for annuitants, taking x0 = 60 or 65 may be more meaningful. Considering (3.18), the estimated " κt is adjusted to match the observed life expectancy at age x0 in year t given the estimated αx ’s and βx ’s obtained from the singular value decomposition or from the Newton–Raphson algorithm. Thus, the adjusted " κt ’s solve the equation 1 − exp − exp " αx0 + " βx0 ζ ex↑0 (t) = exp " αx + " βx0 ζ   0 k−1 !  exp − exp " αx +j + " βx +j ζ  + 0

k≥1

0

j=0

βx0 +k ζ 1− exp − exp " αx0 +k + " × exp " αx0 +k + " βx0 +k ζ

(5.18)

in ζ. The advantage of this second adjustment procedure is that it does not require exposures-to-risk nor death counts and is thus generally applicable. Note that, as before, numerical problems may arise when the " βx ’s do not have the same sign, but we believe that this problem is unlikely to occur in practice. Adjustment of the κˆ t ’s by refitting to the observed age distribution of deaths Booth et al. (2002) have suggested another procedure for the adjusting m " κt ’s. Rather than fitting the yearly total number of deaths xx=x D as in xt 1 (5.17), this variant fits to the age distribution of deaths Dxt assuming the Poisson distribution for the age-specific death counts and using the deviance statistic to measure the goodness-of-fit. Specifically, for a fixed calendar year t, the Dxt ’s are considered as independent random variables obeying

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the Poisson distribution with respective mean ETRxt exp " αx + " βx κt , where the values of the " αx ’s and " βx ’s are those coming from either the singular value decomposition or the Newton–Raphson iterative method, and where κt has to be determined in order to make the observed Dxt ’s as likely as possible. This means that κt maximizes the Poisson log-likelihood xm

Dxt ln ETRxt exp " αx + " βx ζ

− ETRxt exp " αx + " βx ζ

(5.19)

x=x1

over ζ, or equivalently, minimizes the deviance

D=2

xm x=x1

Dxt " xt Dxt ln − Dxt − D " xt D

(5.20)

" xt = ETRxt exp " αx +" βx ζ is the expected number of deaths, keeping where D " the " αx ’s and βx ’s unchanged. Identifiability constraints The identifiability constraints (5.3) are no κt with " κt − κ longer satisfied by the adjusted " κt . Therefore, we replace " and " αx with " αx + " βx κ, where κ is the average of the adjusted " κt ’s. This simple method only works because we are dealing with an identification constraint (not a model restriction). Poisson maximum likelihood estimation Statistical model Let us now assume that the actuary has at his/her disposal observed death counts Dxt and corresponding exposures ETRxt . Then, the least-squares approach can be applied to the ratio of the death numbers to the expo" x (t) = Dxt /ETRxt ’s as explained above). The method sure (i.e. to the m presented in this section better exploits the available information, and does " x (t)’s is the same whatever the age not assume that the variability of the m x. Specifically, we assume that the number of deaths at age x in year t has a Poisson random variation. To justify this approach, we prove that assumption (3.2) is compatible with Poisson modelling for death counts. To this end, let us focus on a particular pair: age x – calendar year t. We observe Dxt deaths among Lxt individuals aged x on January 1 of year t. We assume that the remaining lifetimes of these individuals are independent and identically distributed. The likelihood function (3.12) is proportional to the Poisson likelihood, that is, the one obtained under the assumption that Dxt is Poisson distributed with mean ETRxt µx (t) = ETRxt exp(αx + βx κt )

5.2


where the parameters are still subjected to the constraints (5.3). Therefore, provided that we resort to the maximum likelihood estimation procedure, working on the basis of the ‘true’ likelihood (3.12) or working on the basis of the Poisson likelihood are equivalent, once the assumption (3.2) has been made. Objective function The parameters αx , βx , and κt are now estimated by maximizing the log-likelihood based on the Poisson distributional assumption. This is given by L(α, β, κ) =

xm tn

Dxt (αx + βx κt ) − ETRxt exp(αx + βx κt ) + constant.

x=x1 t=t1

(5.21) Equivalently, the parameters are estimated by minimizing the associated deviance defined as D = −2(L(α, β, κ) − Lf ) (5.22) where Lf is the log-likelihood of the full or saturated model (characterized by equating the fitted and actual numbers of deaths). Effective computation Because of the presence of the bilinear term βx κt , it is not possible to estimate the proposed model with commercial statistical packages that implement Poisson regression. We can nevertheless easily solve the likelihood equations with the help of a uni-dimensional or elementary Newton–Raphson method implemented in (5.16) in the least-squares case. (0) (0) The updating scheme is as follows: starting with " αx = 0, " βx = 1, and (0) (k) (k) " κt = 0 (random values can also be used), the sequences of " αx , " βx , and (k)

" κt

are obtained from the formulas tn (k) (k) (k) " D − ETR exp " α + β " κ xt xt x x t t=t 1 =" α(k) " α(k+1) x x − n (k) (k) (k) " − tt=t ETR exp " α + β " κ xt x x t 1 xm (k+1) "(k) (k) (k) " D βx −ETR exp " α + β " κ xt xt x x t x=x 1 (k+1) ( k) " κt =" κt − (5.23) xm (k+1) "(k) (k) (k) 2 " − x=x1 ETRxt exp " αx +βx " κt βx tn (k+1) (k+1) (k) (k+1) " − ETR exp " α + β " κ D " κt xt xt x x t t=t 1 " βx(k+1) = " βx(k) − n (k+1) 2 (k+1) (k) (k+1) " " κ − tt=t ETR exp " α + β " κ xt x x t t 1

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The criterion used to stop the procedure is a relative increase in the loglikelihood function that is smaller than a pre-selected sufficiently small fixed number. The maximum likelihood estimations of the parameters coming out of (5.23) have to be adapted in order m to fulfill the constraints xm (5.3): specifically, " " " " we replace " κt with (" β κt − κ) xx=x , β with β / αx with x x x=x1 βx , and " 1 x " " αx + βx κ. Remark As pointed out by Renshaw and Haberman (2006), the error structure can be imposed by specifying the second moment properties of the model, as in the framework of generalized linear modelling. This allows for a range of options for the choice of the error distribution, including Poisson, both with and without dispersion, as well as Gaussian, as used in the original approach by Lee and Carter (1992). Remark In contrast to the classical least-squares approach to estimating the parameters, the error applies directly on the number of deaths in the Poisson regression approach. There is, thus, no need for a second-stage estimation like (5.17) for the κt ’s. Note that differentiating the log-likelihood (5.21) with respect to αx gives the equation tn tn Dxt = ETRxt exp(" αx + " βx" κt ) (5.24) t=t1

t=t1

which is similar to (5.17) except that the sum is now over calendar time instead of age. So, the estimated κt ’s are such that the resulting death rates applied to the actual risk exposure produce the total number of deaths actually observed in the data for each age x. Sizable discrepancies between predicted and actual deaths are thus avoided. 5.2.2.3

Alternative estimation procedures for logbilinear models

Brillinger (1986) showed that under reasonable assumptions about the processes governing births and deaths, the Poisson distribution is a good candidate to model the numbers of deaths at different ages. This provides a sound justification for the Poisson model for estimating the (α, β, κ) parameters. There are nevertheless (at least) two alternatives for estimating the parameters. Binomial maximum likelihood estimation Cossette and Marceau (2007) have proposed a Binomial regression model for estimating the parameters in logbilinear mortality projection models. The annual number Dxt

5.2


of recorded deaths is then assumed to follow a Binomial distribution, with a death probability qx (t), which is expressed as a function of the force of mortality (5.1) via qx (t) = 1 − exp(−µx (t)). The number of deaths Dxt at age x during year t has a Binomial distribution with parameters Lxt and qx (t). The specification for µx (t) gives (5.25) qx (t) = 1 − exp − exp αx + βx κt To ensure identifiability, we adhere to the set of constraints (5.3). Assuming independence, the likelihood for the entire data is the corresponding product of binomial probability factors. The log-likelihood is then given by L(α, β, κ) =

xm tn

dxt ln 1 − " qx (t) + dxt ln " qx (t) + constant

(5.26)

t=t1 x=x1

As in the Poisson case, the presence of the bilinear term βx κt makes commercial statistical packages that implement linear Binomial regression useless. An iterative procedure has been proposed in Cossette et al. (2007) for estimating the parameters. A parallel analysis is provided by Haberman and Renshaw (2008) with an investigation of a number of alternative specifications to (5.25). Overdispersed Poisson and Negative Binomial maximum likelihood estimation Poisson modelling induces equidispersion. We know from Section 3.3.9 that populations are heterogeneous with respect to mortality. Heterogeneity tends to increase the variance compared to the mean (a phenomenon termed as overdispersion), which rules out the Poisson specification and favours a mixed Poisson model. Besides gender, age x and year t, there are many other exogeneous factors affecting mortality. It is, therefore, natural to extend the Lee–Carter model in order to take this feature into account. One approach advocated by Renshaw and Haberman (2003b), Renshaw and Haberman (2003c), Renshaw and Haberman (2006), is to postulate that the random number of deaths Dxt has an overdispersed Poisson distribution. Thus, it is suggested that

Var[Dxt ] = φE[Dxt ]

(5.27)

where φ is a parameter that measures the degree of overdispersion. Clearly, φ = 1 reduces to the standard Poisson case. An alternative approach is to take the exogeneous factors into account by adding a random effect xt super-imposed on the Lee–Carter predictor αx + βx κt , exactly as in (5.4). More precisely, Delwarde et al. (2007b)

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200


have suggested the replacement of the Poisson model with a Mixed Poisson one. Given xt , the number of deaths Dxt is assumed to be Poisson distributed with mean ETRxt exp(αx + βx κt + xt ). Unconditionally, Dxt obeys a mixture of Poisson distributions. The xt ’s are assumed to be independent and identically distributed. A prominent example consists in taking the Dxt ’s to be Negative Binomial distributed. See also Renshaw and Haberman (2008). Mortality data from the life insurance market often exhibit overdispersion because of the presence of duplicates. It is common for individuals to hold more than one life insurance or annuity policy and hence to appear more than once in the count of exposed to risk or deaths. In such a case, the portfolio is said to contain duplicates, that is, the portfolio contains several policies concerning the same lives. It is well known that the variance becomes inflated in the presence of duplicates. Consequently, even if the portfolio (or one of its risk class) is homogeneous, the presence of duplicates would increase the variance and cause overdispersion. The overdispersed Poisson and Negative Binomial models for estimating the parameters of logbilinear models for mortality projections are thus particularly promising for actuarial applications.

5.2.3

Application to Belgian mortality statistics

Before embarking on a mortality projection case study, we have to decide about the type of mortality statistics that will be used. In some countries (like in the UK), extensive data are available for policyholders, according to the type of contract. In such a case, we might wonder whether the forecast should be based on population or market data. Using market data allows us to take adverse selection into account. However, basing mortality projections on market data implicitly would mean that no structural breaks have occurred because of changes to the character of the market, or modifications in the tax system or in the level of adverse selection, for instance. Thus, this is not always the best strategy. Assume, for example, that the government starts offering incentives to individuals from the lower socio-economic classes to buy life annuities in order to supplement public pensions. Using market data would result in a worsening in mortality because of a modification in the profile of the insured lives (as lower socio-economic classes usually experience higher mortality rates). Hence, this will artificially modify the mortality trends for the market. It is, thus, impossible to separate long-term mortality trends from modifications in the structure of the insured population. If, however, we need to undertake forecasts based on market data, covariates are often helpful, like

5.2


the amount of the annuity (reflecting individuals’ socio-economic class), for instance. Actuaries sometimes weight their calculations by policy size to account for socio-economic differentials amongst policyholders. These ‘amountbased’ measures usually produce lower mortality rates than their ‘livesbased’ equivalents due to the tendency for wealthier policyholders to live longer. The pension size is thus used as a proxy of socio-economic group. However, this approach is somewhat ad hoc, and the amount of pension should better be included explicitly as a covariate in the regression models used for mortality projections. For the reason given above, we prefer to use general population data for mortality forecasting. Relational models introduced in Section 3.4.4 would allow us to take adverse selection into account, and to exploit the long-term changes in population mortality. Specifically, the overall mortality trend would be estimated from the general population, and a regression model then used to switch from the general population to the insurance market. Proceeding in this way would separate the long-term mortality trends from the particular features of the insured population. We begin by fitting the log-bilinear model to the HMD data set by the least-squares method. We only consider males; the analysis for females is similar. The calendar years 1920–2005 and ages 0–104 are included in the analysis. The reason for restricting the highest age to 104 is that the Belgian 2002–2004 population life table that will serve as the basis for the forecast (as explained in below) does not extend beyond this age. Note that the data at high ages have been processed in the HMD, so that the independence assumption is no more valid at these ages and the corresponding results have to be interpreted with care. Figure 5.1 (top panels) plots the estimated αx ’s, βx ’s, and κt ’s. The estimated αx ’s exhibit the typical shape of a set of log death rates with relatively high values around birth, a decrease at infant ages, the accident hump, and finally the increase at adult ages with an ultimately concave behaviour. The estimated βx ’s appear to decrease with age, suggesting that most of the mortality decreases are concentrated on the young ages. The estimated κt ’s are adjusted to reproduce the observed period life expectancies at birth. The estimated κt ’s are affected by World War II, with comparatively higher values in the early 1940s. We note that the model explains 92.09% of the total variance. We now restrict ourselves to ages above 60. Figure 5.1 (bottom panels) plots the estimated αx , βx , and κt . The model now explains 90.18% of the total variance. Note that compared to the case where all the ages 0–104 were included in the analysis, the adjusted " κt ’s are much more similar to the initial ones coming from singular value decomposition.

201

0.030

–4

0.020 Kappa

Beta

Alpha

50

0.025

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0.015

–50

0.010

–6

0

0.005 –100 0

20

40

60

80

100

0

20

Age

40

60

80

1920

100

–0.5

1960 Time

1980

2000

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1980

2000

10 0.030

–1.5

0.025

–2.0 –2.5

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–1.0

Beta

Alpha

1940

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0.020

–10

0.015

–3.0

–5

–15

–3.5

0.010 –20

–4.0 60

70

80 Age

90

100

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70

80 Age

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100

1920

Figure 5.1. Estimated αx , βx , and κt (from left to right), x = 0, 1, . . . , 104 (top panels) and x = 60, 61, . . . , 104 (bottom panels), t = 1920, 1921, . . . , 2005, obtained with HMD data by minimizing the sum of squares (5.5) with the estimated κt ’s adjusted by refitting to the period life expectancies at birth or at ages 60 (for the estimated κt ’s, the values before adjustment are displayed in broken line).

5.3

Cairns–Blake–Dowd mortality projection model

It is important to mention that the sole use of the proportion of the total temporal variance (as measured by the ratio of the first singular value to the sum of singular values) is not a satisfactory diagnostic indicator. An examination of the residuals is needed to check for model adequacy (see below). The fitted mortality surfaces are depicted in Fig. 5.2. These surfaces should be compared with Fig. 5.3. The mortality experience appears rather smooth, with some ridges around 1940–1945. We now fit the log-bilinear model to the HMD data set by the method of Poisson maximum likelihood. All of the ages 0–104 are included in the analysis. Figure 5.3 (top panels) plots the estimated αx , βx and κt . The estimated parameters are compared with those obtained by minimizing the sum of the squared residuals (5.5). We see that the least-squares and Poisson maximum likelihood procedures produce very similar sets of estimated parameters αx , βx , and κt . As above, we restrict ourselves to ages above 60. Figure 5.3 (bottom panels) plots the estimated αx , βx , and κt . The estimated parameters are compared with those obtained by minimizing least squares. We observe sizeable discrepancies between the " βx ’s produced by the least-squares and Poisson maximum likelihood procedures, whereas the " αx ’s and " κt ’s remain similar.

5.3 5.3.1

Cairns–Blake–Dowd mortality projection model Specification

Empirical analyses suggest that ln qx (t)/px (t) is reasonably linear in x for fixed t (sometimes with a small degree of curvature in the plot of x versus ln qx (t)/px (t)), except at younger ages. This is why Cairns et al. (2006a) assume that [1] [2] + κ x exp κ t t qx (t) ln = κt[1] + κt[2] x ⇔ qx (t) = (5.28) [1] px (t) 1 + exp κ + κ[2] x t

t

where κt[1] and κt[2] are themselves stochastic processes. This specification does not suffer from any identifiability problems so that no constraints need to be specified. We see that age is now treated as a continuous covariate and enters the model in a linear way on the logit scale. The intercept κt[1] and slope κt[2] parameters make up a bivariate time series the future path of which governs the projected life tables. The intercept period term κt[1] is generally declining

203

–2

–4 –6 –8

1920 100

1940 80 1960

60

t 40

1980

x

20 2000 0

–1

–2

–3 –4 1920 100

1940 90

1960

t

80 1980

x 70 2000 60

Figure 5.2. Fitted death rates (on the log scale) for Belgian males, ages 0–104 (top panel) and ages 60–104 (bottom panel), period 1920–2005.

0

0.035 50

0.030

–4

Kappa

0.025 Beta

Alpha

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0.00

–4 60

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Figure 5.3. Estimated αx , βx , and κt (from left to right), x = 0, 1, . . . , 104 (top panels) and x = 60, 61, . . . , 104 (bottom panels), t = 1920, 1921, . . . , 2005, obtained with HMD data by maximizing the Poisson log-likelihood (5.21) (the values obtained by least-squares are displayed in broken line).

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over time, which corresponds to the feature that mortality rates have been decreasing over time at all ages. Hence, the upward-sloping plot of the logit of death probabilities against age is shifting downwards over time. If during the fitting period, the mortality improvements have been greater at lower ages than at higher ages, the slope period term κt[2] would be increasing over time. In such a case, the plot of the logit of death probabilities against age would be becoming more steep as it shifts downwards over time. Sometimes, the logit of the death probabilities qx (t) plotted against age x exhibits a slight curvature after retirement age. This curvature can be modelled by including a quadratic term in age in the Cairns–Blake–Dowd model. However, the dynamics of the time factor associated with this quadratic effect often remains unclear and when combined with the quadratic age term, its contribution to mortality dynamics is highly complex. The Cairns–Blake–Dowd model has two time series κt[1] and κt[2] which affect different ages in different ways. This is a fundamental difference compared with the 1-factor Lee–Carter approach where a single time series induces perfect correlation in mortality rates at different ages from one year to the next. There is empirical evidence to suggest that changes in the death rates are imperfectly correlated, which supports the Cairns–Blake– Dowd model or the 2-factor Lee–Carter model represented by equation (5.14) with r = 2. Compared to the 1-factor Lee–Carter model, the Cairns– Blake–Dowd model thus allows changes in underlying mortality rates that are not perfectly correlated across ages. Also, the longer the run of data that the actuary uses, the better does the 2-factor model relative to its 1-factor counterpart. For example, if we consider the entire 20th century, mortality improvements concentrate on younger ages during the first half of the century and on higher ages during the second half. We need a 2-factor model to capture these two different dynamics. Note, however, that the restriction to the optimal fitting period in the Lee–Carter case favours recent past history so that the inclusion of a second factor may not be needed. Note that the switch from a unique time series to a pair of time-dynamic factors has far-reaching consequences when we discuss securitization, as the existence of an imperfect correlation structure implies, for example, that hedging longevity-linked liabilities would require more than one hedging instrument.

5.3.2

Calibration

We assume that we have observed data for a set of calendar years t = t1 , t2 , . . . , tn and for a set of ages x = x1 , x2 , . . . , xm . On the basis of

5.3

Cairns–Blake–Dowd mortality projection model

these observations, we would like to estimate the intercept κt[1] and slope κt[2] parameters. This can be done by least-squares. This means that the regression model " qx (t) ln = κt[1] + κt[2] x + x (t) (5.29) " px (t) is fitted to the observations of calendar year t, where the " qx (t)’s are the crude one-year death probabilities, and where the error terms x (t) are independent and Normally distributed, with mean 0 and constant variance σ 2 . The objective function

Ot (κ) =

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x=x1

" qx (t) − κt[1] − κt[2] x " px (t)

(5.30)

has to be minimized for each calendar year t, giving the estimations of the κt[1] and κt[2] parameters. Note that, in contrast to the Lee–Carter case, where the estimated time index κt depends on the observation period, the time indices κt[1] and κt[2] are estimated separately for each calendar year t in the Cairns–Blake–Dowd model. The Cairns–Blake–Dowd model can also be calibrated in a number of alternative ways, as was the case for the Lee–Carter model. For instance, a Poisson regression model can be specified by assuming that the observed death counts are independent and Poisson distributed, with a mean equal to the product of the exposure-to-risk times the population death rate of the form µx (t) = − ln(1 − qx (t)) = ln 1 + exp κt[1] + κt[2] x (5.31) Estimation based on a Binomial or Negative Binomial error structure can also be envisaged.

5.3.3


As for the implementation of the Lee–Carter approach, we fit the CairnsBlake-Dowd model by least-squares to the HMD data set, using Belgian males from the general population. The results of the fit are displayed in Fig. 5.4. The top panels of Fig. 5.4 display the results when all of the ages 0–104 are included in the analysis. Note that the Cairns–Blake–Dowd model was never designed to cover all ages, certainly not down to age 0. The linearity in x means that this model is not able to capture the levelling off around age 30 and the accident hump around age 20. From left

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Figure 5.4. Estimated κt and κt parameters together with the values of the adjustment coefficient by calendar year (from left to right), for ages x = 0, 1, . . . , 104 (top panels) and x = 60, 61, . . . , 104 (bottom panels), t = 1920, 1921, . . . , 2005, obtained with HMD data by least-squares.

5.4

Smoothing

to right, we see the estimated κt[1] ’s, the estimated κt[2] ’s, and the value of the adjustment coefficient R2 (t) for each calendar year t. The bottom panels give the corresponding results for the restricted age range 60, 61, . . . , 104. When all of the ages are considered, the estimated κt[1] ’s exhibit a downward trend, which expresses the improvement in mortality rates over time for all ages. A peak around 1940–1945 indicates a higher mortality experience during World War II. The estimated κt[2] ’s tend to increase over time, indicating that mortality improvements have been comparatively greater at younger ages over the period 1920–2005. We note that World War II also affected the estimated κt[2] ’s, with a decrease in the early 1940s. The values of the adjustment coefficient R2 (t) indicate that the Cairns-Blake-Dowd model explains from about 80% of the variance in 1920 to about 95% in the early 2000s. If we restrict the age range to 60, 61, . . . , 104, we see that the goodnessof-fit is greatly increased, with adjustment coefficients larger than 99%. The Cairns–Blake–Dowd model takes advantage of the approximate linearity in age (on the logit scale) at higher ages to provide a parsimonious representation of one-year death probabilities. The adjustment coefficients close to 1 demonstrate the ability of the Cairns–Blake–Dowd model to describe the mortality experienced in Belgium. The trend in the estimated intercept and slope parameters is less clear, unless we restrict our interest to the latter part of the 20th century, where the estimated κt[1] ’s and κt[2] ’s become markedly linear (with a decreasing trend for the former, and an increasing one for the latter).

5.4 5.4.1

Smoothing Motivation

Actuaries use projected life tables in order to compute life annuity prices, life insurance premiums as well as reserves that have to be held by insurance companies to enable them to be able to pay the future contractual benefits. Any irregularities in these life tables would then be passed on to the price list and to balance sheets, which is not desirable. Therefore, as long as these irregularities do not reveal particular features of the risk covered by the insurer, but are likely to be caused by sampling errors, actuaries prefer to resort to statistical techniques to produce life tables that exhibit a regular progression, in particular with respect to age.

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5.4.2

P-splines approach

Durban and Eilers (2004) have smoothed death rates with P-splines in the context of a Poisson model. The P-spline approach is an example of a regression model and is similar to the generalized linear modelling discussed in Section 4.5.4. But unlike generalized linear models, P-splines allow for more flexibility in modelling observed mortality. Regression models take a family of basis functions, and choose a combination of them that best fits the data according to some criterion. The P-spline approach uses a spline basis, with a penalty function that is introduced in order to avoid oversmoothing. P-splines are related to B-splines which have been discussed in Section 2.6.3. Recall that univariate, or unidimensional, B-splines are a set of basis functions each of which depends on the placement of a set of ‘knot’ points providing full coverage of the range of data. Defining B-splines in two dimensions is straightforward. We define knots in each dimension, and each set of knots gives rise to a univariate B-spline basis. The two-dimensional B-splines are then obtained by multiplying the respective elements of these two bases. Durban and Eilers (2004) have suggested a decomposition of µx (t) as follows: ln µx (t) = θij Bij (x, t) (5.32) i,j

for some prespecified two-dimensional B-splines Bij in age x and calendar time t, with regularly-spaced knots, and where the θij ’s are parameters to be estimated from historical data. If we use a large number of knots in the year and age dimensions, then we can obtain an extremely accurate fit. However, such a fit does not smooth the random variations present in the data and the resulting death rates become less reliable. Switching to P-splines helps to overcome this problem, because of the presence of the penalty function. The method of P-splines suggested by Eilers and Marx (1996) is now well-established as a method of smoothing in Generalized Linear Models. It consists of using B-splines as the basis for the regression and in modifying the log-likelihood by a difference penalty that relates to the regression coefficients. The inclusion of a penalty with appropriate weight means that the number of knots can be increased without radically altering the smoothness of the fit. Penalties can be calculated separately in each dimension, involving sums of (θij − 2θi−1, j + θi−2, j )2 in the age dimension, sums of (θij − 2θi, j−1 + θi, j−2 )2 in the calendar year dimension, and sums of

5.4

Smoothing

(θi+1, j−1 − 2θij + θi−1, j+1 )2 across cohorts. The CMI Bureau in the UK has suggested the use of age and cohort penalties (see also Chapter 6). Each of these penalties involves an unknown weight coefficient that has to be selected from the data. Note that there is a difference in the structural assumption behind the Pspline approach, compared with the Lee–Carter and Cairns–Blake–Dowd alternative approaches: the P-spline approach assumes that there is smoothness in the underlying mortality surface in the period effects as well as in the age and cohort effects. Some further extensions have recently been proposed to account for period shocks. The P-splines approach is a powerful smoothing procedure for the observed mortality surface. Using the penalty to project the θij ’s to the future, it is also possible to use this tool to forecast future mortality rates, by extrapolating the smooth mortality surface. However, as pointed out by Cairns et al. (2007), the P-spline approach to mortality forecasting is not transparent. Its output is a smooth surface fitted to historical data and then projected into the future. An important difference (compared with the Lee–Carter and Cairns–Blake–Dowd alternatives) is that forecasting with the P-splines approach is a direct consequence of the smoothing process. The choice of the penalty then corresponds to a view of the future pattern of mortality. In contrast, the two stages of fitting the data and extrapolating past trends are kept separate in the Lee–Carter annd Cairns–Blake–Dowd approaches. This is an advantage for actuarial applications, since it allows for more flexibility. Moreover, the form of the penalty is usually difficult to infer from the data, whereas it entirely drives the P-spline mortality forecast (a similar feature occurs in period-based mortality graduation using splines when mortality rates are extrapolated beyond the data to the oldest ages). The degree of smoothing in empirical applications depends on the variability of the observed death rates. The size of the population under study, as well as the range of ages considered, thus, both influence the smoothing coefficient and, possibly, the choice of the penalty. In the Lee–Carter and Cairns–Blake–Dowd approaches, these features of the data do not directly affect the projection of the time index. As the order of the penalty has no discernible effect on the smoothness of the observed data, it is hard to deduce it from the observed data. The choice of the penalty, in fact, corresponds to a view of the future pattern of mortality: future mortality continuing at a constant level, future mortality improving at a constant rate or future mortality improving at an accelerating (quadratic) rate.

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5.4.3

Smoothing in the Lee–Carter model

As can be seen from Fig. 5.1, the estimated βx ’s exhibit an irregular pattern. This is undesirable from an actuarial point of view, since the resulting projected life tables will also show some erratic variations across ages. Bayesian formulations assume some sort of smoothness of age and period effects in order to improve estimation and facilitate prediction. A Bayesian treatment of mortality projections has been proposed by Czado et al. (2005). Note that the estimated αx ’s are usually very smooth, since they represent an average effect of mortality at age x (however, Renshaw and Haberman (2003a) experiment with different choices for αx , representing different averaging periods and hence different levels of smoothing, as well as explicit graduation of the αx estimates). The estimated κt ’s are often rather irregular, but the projected κt ’s, obtained from some time series model (as explained below), will be smooth. Hence, we only need to smooth the βx ’s in order to get projected life tables with mortality varying smoothly across the ages. This can be achieved by penalized least-squares or maximum likelihood methods. The estimated Lee–Carter parameters are traditionally obtained by minimizing (5.5). This has produced estimated βx ’s and κt ’s with an irregular shape in the majority of empirical studies. In order to smooth the estimated βx ’s we can use the objective function

OPLS (α, β, κ) =

xm tn

ln " µx (t) − αx − βx κt

2

x=x1 t=t1

+πβ

xm

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2

(5.33)

x=x1

where πβ is the smoothing parameter. This is the penalized least-squares approach proposed in Delwarde et al. (2007a). The second term penalizes irregular βx ’s. The objective function can therefore be seen as a compromise between goodness-of-fit (first term) and smoothness of the βx ’s (second term). The penalty involves the sum of the squared second-order differences of the βx ’s, that is, the sum of the squares of βx+2 − 2βx+1 + βx . Secondorder differences penalize deviations from the linear trend. The trade off between fidelity to the data (governed by the sum of squared residuals) and smoothness (governed by the penalty term) is controlled by the smoothing parameters πβ . The larger the smoothing parameters the smoother the resulting fit. In the limit (πβ ∞) we obtain a linear fit. The choice of the smoothing parameters is crucial as we may obtain quite different

5.4

Smoothing

fits by varying the smoothing parameters πβ . The choice of the optimal πβ is based on the observed data, using cross-validation techniques. See Delwarde et al. (2007a) for more details. We note that equation (5.33) is similar to the objective function used in Whittaker–Henderson graduation discussed in Section 2.6.2, a non-parametric graduation method that has been commonly used in the United States. If the parameters are estimated using Poisson maximum likelihood, the penalized least-squares method becomes a penalized log-likelihood approach. Specifically, following Delwarde et al. (2007a) the log-likelihood (5.21) is replaced with L(α, β, κ) −

xm 1 πβ βx+2 − 2βx+1 + βx 2 x=x

2

(5.34)

1

As above, the selection of the optimal value for the roughness penalty coefficient πβ is based on cross validation. Here, we adopt a very simple strategy in our case study: instead of fitting the Lee–Carter model to the rough mortality surface, we first smooth it using the methods described in Section 3.4.2 and then we fit the model to the resulting surface. Remark An alternative aproach to smoothing the βx ’s has also been sugested. It is more ad hoc in nature than the above, in that it introduces an extra stage in the modelling process. Thus, Renshaw and Haberman (2003a,c) smooth the Lee–Carter βx estimates using linear regression as well as cubic B-splines and natural cubic splines and the methods of least-squares. Remark An advantage of the Cairns–Blake–Dowd model is that it automatically produces smooth projected life tables, because future death probabilities depend on age in a linear way, and on the projected time indices κt[1] and κt[2] .

5.4.4


We fit the Lee–Carter model to the set of smoothed HMD death rates by least-squares to ages 0–104 and the years 1920–2002. Figure 5.5 (top panels) plots the estimated αx , βx , and κt . The estimated κt ’s are then adjusted in order to reproduce the observed period life expectancies at birth. The values obtained without smoothing (i.e. those displayed in Fig. 5.1) are plotted using a broken line. We see that the prior smoothing of the death rates does not impact on the estimated αx ’s, except just before the accident hump, nor on the estimated κt ’s (mainly because of the adjustment procedure). Prior

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smoothing does, however, impact on the estimated βx ’s which now appear to behave very regularly with age. The model explains 93.70% of the total variance. We now restrict ourselves to ages above 60. Figure 5.5 (bottom panels) plots the estimated αx , βx , and κt . We see that prior smoothing has almost no impact on the estimated αx ’s nor on the estimated κt ’s, whereas the estimated βx ’s are smoothed in an appropriate way. The model now explains 91.37% of the total variance. The estimated αx ’s and κt ’s closely agree, while the estimated βx ’s are smoothed appropriately.

5.5 5.5.1

Selection of an optimal calibration period Motivation

Many actuarial studies have based the projections of mortality on the statistics relating to the years from 1950 to the present. The question then becomes why the post-1950 period better represents expectations for the future than does the post-1900 period, for example. There are several justifications for the use of the second half of the 20th century. First, the pace of mortality decline was more even across all ages over the 1950–2000 period than over the 1900–2000 period. Second, the quality of mortality data, particularly at the older ages, for the 1900–1950 period is questionable. Third, infectious diseases were an uncommon cause of death by 1950, while heart disease and cancer were the two most common causes, as they are today. This view seems to imply that the diseases affecting death rates from 1900 through 1950 are less applicable to expectations for the future than the dominant causes of death from 1950 through 2000. According to Lee and Carter (1992), the length of the mortality time series was not critical as long as it was more than about 10–20 years. However, Lee and Miller (2001) obtained better fits by restricting the start of the calibration period to 1950 in order to reduce structural shifts. Specifically, in their evaluation of the Lee–Carter method, Lee and Miller (2001) have noted that for US data the forecast was biased when using the fitting period 1900–1989 to forecast the period 1990–1997. The main source of error was the mismatch between fitted rates for the last year of the fitting period (1989 in their study) and actual rates in that year. This is why a bias correction is applied. It was also noted that the βx pattern did not remain stable over the whole 20th century. In order to obtain more stable βx ’s, Lee and Miller (2001) have adopted 1950 as the first year of the fitting period. Their conclusion is that restricting the

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fitting period to 1950 on avoids outlier data. Similarly, Lundström and Qvist (2004) have reduced the 1901–2001 period to the past 25 years with Swedish data. Baran and Pap (2007) have applied the Lee–Carter method to forecast mortality rates in Hungary for the period 2004–2040 on the basis of either mortality data between 1949 and 2003 or on a restricted data set corresponding to the period 1989–2003. The model fitted to the data of the period 1949–2003 forecasts increasing mortality rates for men between ages 45 and 55, indicating that the Lee–Carter method may not be applicable for countries where mortality rates exhibit trends as peculiar as in Hungary. However, models fitted to the data for the last 15 years both for men and women forecast decreasing trends, which are similar to the case of countries where the method has been successfully applied. This clearly shows that the selection of an optimal fitting period is of paramount importance.

5.5.2

Selection procedure

Booth et al. (2002) have designed procedures for the selection of an optimal calibration period which identifies the longest period for which the estimated mortality index parameter κt is linear. Specifically, these authors seek to maximize the fit of the overall model by restricting the fitting period in order to maximize the fit to the linearity assumption. The choice of the fitting period is based on the ratio of the mean deviances of the fit of the underlying Lee–Carter model to the overall linear fit. This ratio is computed by varying the starting year (but holding the jump-off year fixed) and the chosen fitting period is that for which the ratio is substantially smaller than for periods starting in previous years. More specifically, Booth et al. (2002) assume, a priori, that the trend in the adjusted " κt ’s is linear, based on the ‘universal pattern’ of mortality decline that has been identified by several researchers, including Lee and Carter (1992) and Tuljapurkar and Boe (2000). When the " κt ’s depart from linearity, this assumption may be better met by appropriately restricting the fitting period. As noted above, the ending year is kept equal to tn and the fitting period is then determined by the starting year (henceforth denoted as tstart ). Restricting the fitting period to the longest recent period (tstart , tn ) for which the adjusted " κt ’s do not deviate markedly from linearity has several advantages. Since systematic changes in the trend in " κt are avoided, the uncertainty in the forecast is reduced accordingly. Moreover, the βx ’s are likely to satisfy better the assumption of time invariance.

5.5

Selection of an optimal calibration period

Finally, the estimate of the drift parameter more clearly reflects the recent experience. An ad hoc procedure for selecting tstart has been suggested in Denuit and Goderniaux (2005). Precisely, the calendar year tstart ≥ t1 is selected in such a way that the series {" κt , t = tstart , tstart + 1, . . . , tn } is best approximated by a straight line. To this end, the adjustment coefficient R2 (which is the classical goodness-of-fit criterion in linear regression) is maximized (as a function of the number of observations included in the fit). Note that in Denuit and Goderniaux (2005), the κt ’s are replaced by a linear function of t and a parametric regression model (using a linear effect term for the continuous covariate calendar time with an interaction with the categorical variable age, together with a term for the categorical variable age) is then used. Even if this approach produces almost the same projections as the Lee–Carter method, it underestimates the uncertainty in mortality forecasts. The resulting confidence intervals are then artificially narrow because of the imposition of the linear trend in the κt ’s. The situation is slightly different in the Cairns–Blake–Dowd model. As the time-varying parameters are estimated separately for each calendar year, they remain unaffected if we modify the range of calendar years under interest. Considering Fig. 5.4, we clearly see that the slope and intercept parameters become linear only in the last part of the observation period (especially for ages 60 and over). Therefore, it is natural to extrapolate their future path on the basis of recent experience only. The approach suggested by Denuit and Goderniaux (2005) is easily extended to the Cairns-BlakeDowd setting, by selecting the starting year as the maximum of the starting years for each time factor. The deviance approach proposed by Booth et al. (2002) can also easily be adapted to the Cairns–Blake–Dowd model. Note, however, that the selection of the optimal fitting period is subject to criticisms, in the sense that it could lead to an underestimation of the uncertainty in forecasts, and artificially favours the Lee–Carter specification. The same comment applies in the Cairns–Blake–Dowd approach. We do not share this view, and we believe that the selection of the optimal fitting period is an essential part of the mortality forecast.

5.5.3


We first consider the Lee–Carter fit. Applying the method of Booth et al. (2002) gives tstart = 1978. The ad hoc method suggested in Denuit and Goderniaux (2005) roughly confirms this choice. Restricting the age range

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to 60 and over yields tstart = 1974. Again, the ad-hoc method agrees with this choice. Whereas the common practice would consist of taking all of the available data 1920–2005, we discard here observations for the years 1920–1977 when all of the ages are considered, and observations for the years 1920– 1973 when the analysis is restricted to ages 60 and over. Here, short-term trends are preferred even if long-term forecasts are needed for annuity pricing. The reason is that past long-term trends are not expected to be relevant to the long-term future. Note that the fact that the optimal fitting period is selected on the basis of goodness-of-fit criteria to the linear model results in relatively small deviations from this short-term linear trend, but the shorter fitting period results in a more rapid widening of the confidence intervals. The final estimates based on observations comprised in the optimal fitting period are displayed in Fig. 5.6 which plots the estimated αx , βx , and κt . We see that the estimated αx ’s and κt ’s obtained with and without prior smoothing closely agree whereas the estimated βx ’s are smoothed in an appropriate way. The model explains 67.70% of the total variance for males on the basis of unsmoothed data, 90.57% of the total variance for males on the basis of smoothed data for ages 0–104. The model explains 92.62% of the total variance for males on the basis of unsmoothed data, 95.74% of the total variance for males on the basis of smoothed data for ages 60 and over. For the Cairns–Blake–Dowd model, the optimal projection periods now become 1969–2005 when all of the ages are included in the analysis and 1979–2005 when ages are restricted to the range 60–104. Note that the estimated time indices are not influenced by the restriction of the time period, so that those displayed in Fig. 5.4 remain valid.

5.6 5.6.1

Analysis of residuals Deviance and Pearson residuals

Since we work in a regression framework, it is essential to inspect the residuals. Model performance is assessed in terms of the randomness of the residuals. A lack of randomness would indicate the presence of systematic variations, such as age–time interactions. We note that the adjustment of the " κt ’s in the Lee–Carter case may have introduced systematic changes to

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the residuals so that the examination of model performance is in fact based on the residuals computed with the adjusted " κt ’s. When the parameters are estimated by least-squares, Pearson residuals have to be inspected. In the Lee–Carter case, these residuals are given by rxt = .

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" x (t) xm

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+" κt[2] x).

If the residuals rxt exhibit some regular pattern, this means that the model is not able to describe all of the phenomena appropriately. In practice, looking at (x, t) → rxt , and discovering no structure in those graphs ensures that the time trends have been correctly captured by the model. With a Poisson, Binomial, or Negative Binomial random component, it is more appropriate to consider the deviance residuals in order to monitor the quality of the fit. These residuals are defined as the signed square root of the contribution of each observation to the deviance statistics. These residuals should also be displayed as a function of time at different ages, or as a function of both age and calendar year.

5.6.2


We find that the residuals computed from the model fitted to ages 0–104 reveal systematic patterns and comparatively large values at young ages. In the Lee–Carter case, the fit around the accident hump is very poor, with large negative residuals for ages below 20. The residuals are positive for all of the higher ages. The same phenomenon appears with the Cairns–Blake– Dowd fit, with huge positive residuals around age 0. Overall, we find that the inclusion of young ages significantly deteriorates the quality of the fit at the higher ages. The presence of a trend in the residuals violates the independence assumption and homoskedasticity does not hold as the graph presents clustering. The large residuals before the accident hump suggest that the Lee–Carter and Cairns–Blake–Dowd approaches are not able to account for the particular mortality dynamics at younger ages. Since older ages are the most relevant in pension and annuity applications, we restrict the analysis to ages 60 and over.

5.7

Mortality projection

Residuals can be displayed as a function of both age and calendar time, and inspected with the help of maps as displayed in Fig. 5.7 for the Lee– Carter fit (top panel) and for the Cairns–Blake–Dowd fit (bottom panel), for ages 60–104. The particular patterns at oldest ages come from the closing procedure applied to HMD mortality statistics, and do not invalidate the fit. The residuals are unstructured, except for a moderate cohort effect for generations reaching age 60 around 1980. Thus, apart from these cohorts born just after World War I and the 1918–1920 influenza epidemics, there is no significant diagonal pattern in the residuals. We find that the cohort effect revealed in the residuals is too weak to reject the age-period Lee–Carter model. In some countries, the cohort effects are stronger and need to be included in the mortality modelling. This is the case for instance in the United Kingdom, as will be seen in the next chapter where cohort effects will be included in the models discussed in the present chapter. We now turn to the residuals for the Cairns–Blake–Dowd model. The residuals are less dispersed than those for the Lee–Carter fit. The generations born around 1920 again emerge as a notable feature in the residuals plot. We now observe a clustering of negative residuals for the generations born after this particular one, whereas positive residuals are associated with the older generations. This suggests that the inclusion of a cohort effect could be envisaged in the Cairns–Blake–Dowd setting. We postpone the analysis of this kind of effect to the next chapter.

5.7 5.7.1

Mortality projection Time series modelling for the time indices

An important aspect of both the Lee–Carter and the Cairns–Blake–Dowd methodology is that the time factor (κt in the Lee–Carter case, and (κt[1] , κt[1] ) in the Cairns–Blake–Dowd case) is intrinsically viewed as a stochastic process. Box-Jenkins techniques are then used to estimate and forecast the time factor within an ARIMA time series model. These forecasts in turn yield projected age-specific mortality rates, life expectancies and single premiums for life annuities. In the Lee–Carter model, the estimated κt ’s are viewed as a realization of a time series that is modelled using the classical autoregressive integrated moving average (ARIMA) models. Such models explain the dynamics of a time series by its history and by contemporaneous and past schocks.

221

5 : Age-period projection models 4.6179 4.1200 3.5400 2.9600 2.3800 1.8000 1.2200 0.6400 0.0600 –0.5200 –1.1000 –1.6800 –2.2600 –2.8400 –3.4200 –4.0753

100

Age

90

80

70

60 1975

1980

1985

1990 Time

1995

2000

2005

3.1197 2.7199 2.3202 1.9205 1.5208 1.1211 0.7213 0.3216 –0.0780 –0.4777 –0.8774 –1.2771 –1.6769 –2.0766 –2.4763 –2.8760

100

90

Age

222

80

70

60 1980

1985

1990

1995

2000

2005

Time Figure 5.7. Residuals for Belgian males, Lee–Carter model, ages x = 60, 61, . . . , 104 (top panel), and Cairns–Blake–Dowd model, ages x = 60, 61, . . . , 104 (bottom panel).

The dynamics of the κt ’s is described by an ARIMA(p, d, q) process if it is stationary and ∇ d κt = φ1 ∇ d κt + · · · + φp ∇ d κt + ξt + ψ1 ξt−1 + · · · + ψq ξt−q

(5.37)

with φp = 0, ψq = 0, and where ξt is a Gaussian white noise process such that σξ2 > 0.

5.7


There are a few basic steps to fitting ARIMA models to time series data. The main point is to identify the values of the autoregressive order p, the order of differencing d, and the moving average order q. If the time index is not stationary, then a first difference (i.e. d = 1) can help to remove the time trend. If this proves unsuccessful then it is standard to take further differences (i.e. investigate d = 2 and so on). Preliminary values of p and q are chosen by inspecting the autocorrelation function and the partial autocorrelation function of the κt ’s. More details can be found in standard textbooks devoted to time series analysis. The appropriateness of the Lee–Carter approach has been questioned by several authors. The rigid structure imposed by the model necessitates the selection of an optimal fitting period (which is also conservative in the context of life annuities, that is, it tends to overstate the expected value of annuities). The Gaussian distributional assumption imposed on the κt ’s means that large jumps are unlikely to occur. This feature can be problematic for death benefits, where negative jumps correspond to events which threaten the financial strength of the insurance company. For instance, insurers currently are worrying about an avian influenza pandemic which could cause the death of many policyholders. On the basis of vital registration data gathered during the 1918–1920 influenza pandemic, extrapolations indicate that if the mortality were concentrated in a single year, it would increase global mortality by 114%. However, neglecting such jumps is conservative for life annuities. Positive jumps corresponding to sudden improvements in mortality thanks to the availability of new medical treatments are considered to be unlikely to occur, since it would take some time for the population to benefit from these innovative treatments. Hence, the assumptions behind the Lee–Carter model are compatible with mortality projections for life annuity business, and we do not need to acknowledge explicitly period shocks in the stochastic mortality model. We note also that the optimal fitting period, that has been widely used, has tended to start after the three pandemics of the 20th century (1918–1920, 1957–1958, and 1968–1970).

5.7.2 5.7.2.1

Modelling of the Lee–Carter time index Stationarity

Time series analysis procedures require that the variables being studied be stationary. We recall that a time series is (weakly) stationary if its mean and variance are constant over time, and the covariance for any two time periods (t and t + k, say) depends only on the length of the interval between the two time periods (here k), not on the starting time (here t).

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Nonstationary series may be the result of two different data-generating processes: 1. The non-stationarity can reflect the presence of a deterministic component. Such a trending series can be rendered stationary by simply setting up a regression on time and working on the resulting residuals. These series are said to be trend stationary. 2. The non-stationarity can result from a ‘non-discounted’ accumulation of stochastic shocks. In this case, stationarity may be achieved by differencing the series one or more times. These series are said to be difference stationary. A first check for stationarity consists of displaying the data in graphic form and in looking to see if the series has an upward or downward trend. We have observed a gradually decreasing underlying trend in the estimated κt ’s. The series of the estimated κt ’s is, thus, clearly not stationary: it tends to decrease over time on average. Figure 5.8 displays the estimated autocorrelation function of the estimated κt ’s (on the left panel). The classic signature for a nonstationary series is a set of very strong correlations that decay slowly as the lag length increases. Specifically, if the time series is stationary, then its autocorrelation function declines at a geometric rate. As a result, such processes have short-memory since observations far apart in time are essentially independent. Conversely, if the time series needs to be differenced once, then its autocorrelation function declines at a linear rate and observations far apart in time are not independent. The sample autocorrelation coefficients of the " κt ’s in Fig. 5.8 clearly exhibit a linear decay which supports nonstationarity. In addition to these graphical procedures, several formal tests for (non)stationarity have been developed. Stationarity tests are for the null hypothesis that a time series is trend stationary. Taking the null hypothesis as a stationary process and differencing as the alternative hypothesis is in accordance with a conservative testing strategy: if we reject the null hypothesis then we can be confident that the series indeed needs to be differenced (at least once). The Kwiatowski–Philips–Schmidt–Shin test with a linear deterministic trend has a test-statistic equal to 0.168 with 3 lags, and 0.1529 with 9 lags. This leads to rejecting trend stationarity for males (at 5%) and to the conclusion that the κt ’s need to be differenced. Since the estimated κt ’s are difference stationary, we compute the first differences of the estimated κt ’s for males and females. In order to check whether a second difference is needed, we test the resulting series for (non)stationarity using unit root tests. The Augmented Dickey–Fuller pvalue is less than 1%, so that we conclude that the first differences of the κt ’s are stationary and so do not need further differencing.

5.7


1.0 0.8 0.6 ACF

0.4 0.2 0.0 –0.2 –0.4 0

5

10

15

Lag

0.8

Partial ACF

0.6 0.4 0.2 0.0 –0.2 2

4

6

8 Lag

10

12

14

Figure 5.8. Autocorrelation function (on the left) and partial autocorrelation function (on the right) of the estimated κt ’s obtained with completed data for the ages 60 and over.

5.7.2.2

Random walk with drift model for the time index

As no autocorrelation coefficient nor partial autocorrelation coefficient of the differenced time index appears to be significantly different from 0, an ARIMA(0,1,0) process seems to be appropriate for the estimated κt ’s. The Ljung–Box–Pierce test supports this model. Running a Shapiro– Wilk test yields a p-value of 23.08%, which indicates that the residuals seem to be approximately Normal. The corresponding Jarque-Bera p-value equals 48.27%, which confirms that there is no significant departure from Normality. The previous analysis suggests that for Belgian mortality statistics, a random walk with drift model is suitable for modelling the estimated κt ’s (as is the case in many of the empirical studies in the literature). In this case,

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the dynamics of the estimated κt ’s are given by κt = κt−1 + d + ξt

(5.38)

where the ξt ’s are independent and Normally distributed with mean 0 and variance σ 2 , and where d is known as the drift parameter. In this case, κtn +k = κtn + kd +

k

ξtn +j

(5.39)

j=1

The point forecast of the time index is thus κ˙ tn +k = E[κtn +k |κt1 , κt2 , . . . , κtn ] = κtn + kd

(5.40)

which follows a straight line as a function of the forecast horizon k, with slope d. The conditional variance of the forecast is

Var[κtn +k |κt1 , κt2 , . . . , κtn ] = kσ 2

(5.41)

Therefore, the conditional standard errors for the forecast increase with the square root of the distance of the forecast horizon k. Using the random walk with drift model for forecasting κt is equivalent to forecasting each age-specific death rate to decline at its own rate. Indeed, it follows from (5.38) that the differences in expected log-mortality rates between times t + 1 and t is ln µx (t + 1) − ln µx (t) = βx E[κt+1 − κt ] = βx d

(5.42)

The ratio of death rates in two subsequent years of the forecast is equal to exp(βx d) and is thus invariant over time. The product βx d is therefore equal to the rate of mortality change over time at age x. In such a case, the parameter βx can be interpreted as a normalized schedule of age-specific rates of mortality change over time. It is important to notice that the future mortality age profile produced by the Lee–Carter model always becomes less smooth over time, as pointed out by Girosi and King (2007). This explains why this approach has been designed to forecast aggregate demographic indicators, such as life expectancies (or actuarial indicators like annuity values), and not future period or cohort life tables. This comes from the fact that the forecast of the log-death rates is linear over time from (5.42): as the βx ’s vary with age, the age profile of log-mortality will eventually become less smooth over time, since the distance between log-mortality rates in adjacent age groups can only increase. Each difference in βx is amplified as we forecast further

5.7


into the future. Sometimes, the forecast lines converge for a period, but after converging they cross and the age profile pattern becomes inverted. The dynamics (5.38) ensures that κt −κt−1 , t = t2 , t3 , . . . , tn , are independent and Normally distributed, with mean d and variance σ 2 . The maximum likelihood estimators of d and σ 2 are given by the sample mean and variance of the κt − κt−1 ’s, that is, the maximum likelihood estimators of the model parameters are " d=

tn 1 " κt − " κt1 (" κt − " κt−1 ) = n tn − t1 t=t tn − t1

(5.43)

2

and " σ2 =

tn 2 1 κt−1 − " d " κt − " tn − t1 t=t

(5.44)

2

This gives " d = −0.5867698 and " σ 2 = 0.3985848 for Belgian males for the optimal fitting period 1974–2005. This approach is known as the ruler method of forecast as it connects the first and last points of the available data with a ruler and then extends the resulting line further in order to produce a forecast. Considering the expression for " d, the actuary has to check the value of " κtn for reasonableness. For instance, if a summer heat wave occurs during calendar year tn , producing excess mortality at older ages, then" κtn might be implausibly high, resulting in a too small " d, and biasing downwards the future improvements in longevity (as noted by Lee (2000), Renshaw and Haberman (2003a), Renshaw and Haberman (2003c)). As note above, Lee and Carter (1992) did not prescribe the random walk with drift model for all situations. However, this model has been judged to be appropriate in very many cases. For instance, Tuljapurkar et al. (2000) find that the decline in the κt ’s is in accordance with the random walk with drift model for the G7 countries. Even when a different model is indicated, the more complex model is found to give results which are close to those obtained with the random walk with drift. Remark Building on the random walk with drift model for the κt ’s, Girosi " x (t)’s using and King (2007) propose that we should model directly the ln m a multivariate random walk with drift model. In this reformulation of the Lee–Carter model, the drift vector and the covariance matrix of the innovations are arbitrary. Remark Carter (1996) has developed a method in which the drift d in the random walk forecasting equation for κt is itself allowed to be a random

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variable. This is done using state-space methods for modelling time series. Nevertheless, it is noteworthy that the forecast and probability intervals remain virtually unchanged compared to the simple random walk with drift model. Remark Booth et al. (2006) compare the original Lee–Carter method with the different adjustments for the estimated κt ’s, as well as the extensions proposed by Hyndman and Ullah (2007) and De Jong and Tickle (2006). They find that, from the forecasting point of view, there are no significant differences between the five methods. See also Booth et al. (2005).

5.7.3

Modelling the Cairns-Blake-Dowd time indices

The analysis of each time index in isolation parallels the analysis performed for the Lee–Carter time index. These preliminary results have now to be supplemented with a bivariate analysis of the time series κ t = (κt[1] , κt[2] )T that goes beyond the scope of this book. When fitted to data, the changes over time in κ t have often been approximately linear, at least in the recent past. This suggests that the dynamics of the time factor κ t could be appropriately described by a bivariate random walk with drift of the form

[1] κt[1] = κt−1 + d1 + ξt[1]

(5.45)

[2] κt[2] = κt−1 + d2 + ξt[2]

where d1 and d2 are the drift parameters, and ξ t = (ξt[1] , ξt[2] )T are independent bivariate Normally distributed random pairs, with zero mean and variance-covariance matrix 2 σ1 σ12 = (5.46) σ12 σ22 The drift parameters are estimated as " κt[i] −" κt[i] n 1 " di = , tn − t 1

i = 1, 2

(5.47)

the marginal variances are estimated as " σi2

tn 2 1 [i] " κt[i] − " = κt−1 −" di , tn − t1 t=t 2

i = 1, 2

(5.48)

5.8


and the covariance is estimated as " σ12 =

t n −1 t n −1 1 [1] [2] " κs+1 κt+1 −" κs[1] − " d1 " −" κt[2] − " d2 tn − t1 s=t t=t 1

(5.49)

1

d2 = 0.0007619443, " σ12 = 0.01563272, This gives " d1 = −0.0757558, " " σ22 = 3.3048 × 10−6 , and " σ12 = −0.0002247978 for Belgian males for the period 1979-2005. While a bivariate random walk with drift model has been used in connection with the Cairns–Blake–Dowd approach to mortality forecasting, mean reverting alternatives might have a stronger biological justification. Andrew Cairns pointed out in a personal communication that negative autocorrela[2] tion coefficients between the " κt[2] −" κt−1 ’s indicate that at higher ages good years and bad years alternate. This can be explained as follows: if a flu epidemic kills a lot of the unhealthy older people, it leaves the healthy older and then the next year mortality is low.

5.8 5.8.1

Prediction intervals Why bootstrapping?

The projections made so far, while interesting, reveal nothing about the uncertainty attached to the future mortality. In forecasting, it is important to provide information on the error affecting the forecasted quantities. In the traditional demographic approach to mortality forecasting, a range of uncertainty is indicated by high and low scenarios, around a medium forecast that is intended to be a best estimate. However, it is not clear how to interpret this high-low range unless a corresponding probability distribution is specified. In this respect, prediction intervals are particularly useful. This section explains how to get such margins on demographic indicators in the Lee– Carter setting. The ideas are easily extended to the Cairns–Blake–Dowd setting. In the current application, it is impossible to derive the relevant prediction intervals analytically. The reason for this is that two very different sources of uncertainty have to be combined: sampling errors in the parameters αx , βx , and κt , and forecast errors in the projected κt ’s. An additional complication is that the measures of interest – life expectancies or life annuities premiums and reserves – are complicated non-linear functions of the parameters αx , βx , and κt and of the ARIMA parameters. The key idea behind the

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bootstrap is to resample from the original data (either directly or via a fitted model) in order to create replicate data sets, from which the variability of the quantities of interest can be assessed. Because this approach involves repeating the original data analysis procedure with many replicate sets of data, it is sometimes called a computer-intensive method. Bootstrap techniques are particularly useful when, as in our problem, theoretical calculation with the fitted model is too complex. If we ignore the other sources of errors, then the confidence bounds on future κt ’s can be used to calculate prediction intervals for demographic indicators. Even if for long-run forecasts (over 25 years), the error in forecasting the mortality index clearly dominates the errors in fitting the mortality matrix, prediction intervals based on κt alone seriously understate the errors in forecasting over shorter horizons. We know from Lee and Carter (1992), Appendix B, that prediction intervals based on κt alone are a reasonable approximation only for forecast horizons greater than 10–25 years. If there is a particular interest in forecasting over the shorter term, then we cannot make a precise analysis of the forecast errors. Because of the importance of appropriate measures of uncertainty in an actuarial context, the next sections aim to derive prediction intervals taking into account all of the sources of errors. To fix the ideas, we will con sider a cohort life expectancy ex (t) as defined in Section 4.4.1 or in (5.57) below, but the approach is easily adapted to other demographic or actuarial indicators.

5.8.2

Bootstrap percentiles confidence intervals

To avoid any (questionable) Normality assumption, we use the bootstrap percentile interval to construct confidence interval for the predicted life expectancy. The bootstrap procedure yields B samples of αx , βx , and κt parameters, denoted as αbx , βxb , and κtb , b = 1, 2, . . . , B. This procedure can be carried on in several ways: Monte Carlo simulation Brouhns et al. (2002b), Brouhns et al. (2002a) sample directly from the approximate multivariate Normal distribution of the Poisson maximum likelihood estimators (" α, " β," κ ), that is, those obtained by maximizing the log-likelihood (5.21). Invoking the large sample properties of the maximum likelihood estimators, we know that (" α, " β," κ ) is asymptotically multivariate Normally distributed, with mean (α, β, κ) and covariance matrix given by the inverse of the Fisher information matrix, whose elements equal minus the expected value of the

5.8


second derivatives of the log-likelihood with respect to the parameters of interest. We refer to Appendix B in Brouhns et al. (2002b) for the expression of the information matrix and how to sample values from the multivariate Normal distribution of interest. As reported by Renshaw and Haberman (2008), the result of this first approach heavily rely on the identifiability constraints. Given that the choice of constraints is not unique and that this choice materially affects the resulting simulations, this first approach should not be used for risk assessment purposes unless there are compelling reasons for selecting a particular set of identifiability constraints. Poisson bootstrap Starting from the observations (ETRxt , Dxt ), Brouhns et al. (2005b) create B bootstrap samples (ETRxt , Dbxt ), b = 1, 2, . . . , B, where the Dbxt ’s are realizations from the Poisson distribution with mean ETRxt " µx (t) = Dxt . The bootstrapped death counts Dbxt are thus obtained by applying a Poisson noise to the observed numbers of deaths. For each bootstrap sample, the αx ’s, βx ’s, and κt ’s are estimated. Residuals bootstrap Another possibility is to bootstrap from the residuals of the fitted model, as suggested by Koi et al. (2006). The residuals should be independent and identically distributed (provided that the model is well specified). From these, it is possible to reconstitute bootstrapped residuals, and then bootstrapped mortality data. A good fit resulting in a set of pattern-free random residuals for sampling repeatedly with replacement is a basic requirement for this approach. When this is not the case, distortions can occur in the simulated histogram of the quantity of interest. Specifically, we create the matrix R of residuals, with elements rxt as defined in Section 5.6. Then, we generate B replications Rb , b = 1, 2, . . . , B by sampling with replacement the elements of the matrix R. The inverse formula for the residuals is then used to obtain the corresponding matrix of death counts Dbxt ; we refer the reader to Koissi et al. (2006) for further explanation about the inverse of the residuals, as well as to Renshaw and Haberman (2008) for further comments. This leads to the computations of B sets of estimated parameters " αbx , " βxb , b and " κt . We then estimate the time series model using the κtb as data points. This yields a new set of estimated ARIMA parameters. We can then generate a projection κtb , t ≥ tn + 1 using these ARIMA parameters. The future errors ξtb are sampled from a univariate Normal distribution with a mean of 0 and a standard deviation of σεb . Note that the κt ’s are projected on the basis of the reestimated ARIMA model. Note that we do not select a new ARIMA model but keep the ARIMA model selected on the basis of the original

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data. Nevertheless, the parameters of these models are reestimated with the bootstrapped data. The first step is meant to take into account the uncertainty in the parameters αx ’s, βx ’s, and κt ’s. The second step deals with the fact that the uncertainty in the ARIMA parameters depends on the uncertainty in the αx ’s, βx ’s and κt ’s parameters. The third step ensures that the uncertainty of the forecasted κt ’s not only depends on the ARIMA standard error, but also on the uncertainty of the ARIMA parameters themselves. Finally, in the computation of the relevant measures in step four, all sources of uncertainty are taken into account. This yields B realizations αbx , βxb , κtb and projected κtb on the basis of which

we can compute the measure of interest ex (t). Assume that B bootstrap b estimates ex (t), b = 1, 2, . . . , B, have been computed. The (1−2α) percentile b(ζ)

b(α) b(1−α) interval for ex (t) is given by (ex (t), ex (t)), where ex (t) is the 100 ×

ζth empirical percentile of the bootstrapped values for ex (t), which is equal to the (B×ζ)th value in the ordered list of replications exb (t), b = 1, 2, . . . , B. For instance, in the case of B = 1, 000 bootstrap samples, the 0.95th and the 0.05th empirical percentiles are, respectively, the 950th and 50th numbers

in the increasing ordered list of 1,000 replications of ex (t). Note that these bootstrap procedures account for parameter uncertainty as well as Arrowian uncertainty (also known as risk, in which the set of future outcomes is known and probabilities can be assigned to each of the possible outcomes based on a known model with known parameters generating the distribution of future outcomes). Knightian uncertainty, by comparison, ackowledges the presence of both model uncertainty and parameter uncertainty. Allowing for model uncertainty would require the consideration of several mortality projection models and the assignment to these of probabilities in line with their relative likelihoods. Remark Empirical studies conducted in Renshaw and Haberman (2008) reveal varying magnitudes of the Monte Carlo based confidence and prediction intervals under different sets of identifiability constraints. Such diverse results are attributed by these authors to the over parametrization present in the model rather than to the non-linearity of the parametric structure.

5.8.3


In the approach proposed by Lee and Carter (1992), future age-specific death rates are obtained using extrapolated κt ’s and fixed αx ’s and βx ’s, that is, the pointwise projections κ˙ tn +s of the κtn +s ’s, s = 1, 2, . . ., are inserted

5.8


into the formulas giving the force of mortality and provide αx + " βx κ˙ tn +s ) µ ˙ x (tn + s) = exp("

(5.50)

In this case, the jump-off rates (i.e. the rates in the last year of the fitting period or jump-off year) are fitted rates. The basic Lee–Carter method has been criticized by Bell (1997) for the fact that a discontinuity is possible between the observed mortality rates and life expectancies for the jumpoff year and the forecast values for the first year of the forecast period. The bias arising from this discontinuity would then persist throughout the forecast. As suggested by Bell (1997), Lee and Miller (2001), Lee (2000), Renshaw and Haberman (2003a), Renshaw and Haberman (2003c), the forecast could be started with observed death rates rather than with fitted ones. This would help to eliminate a jump between the observed and forecasted death rates in the first year of the forecast as the model does not fit agespecific death rates exactly in the last year. If the fitting period is sufficiently long, then the difference between the observed and the fitted death rates can be appreciable. Specifically, the forecast mortality rates are aligned to the latest available empirical mortality rates as " x (tn ) exp " µ ˙ x (tn + s) = m βx κ˙ tn +s − " κtn " x (tn )RF(x, tn + s) =m

(5.51)

" x (tn ) denotes the death rate observed at age x in year tn Note that here, m and not the fitted one, and RF denotes the reduction factor as introduced in equation (4.6). If the latest empirical mortality rates were judged to be atypical in level or shape, an alternative would be to average across a few years at the end of the observation period, or to resort to a recent population life table, as advocated by Renshaw and Haberman (2003a,c). In the example, here, we use the Belgian 2002–2004 population life table released in 2006 by Statistics Belgium as the base for the mortality forecast. Here, we bootstrap the residuals displayed in Figure 5.7 (top panel). With 10,000 replications, we obtain the histograms displayed in Figure 5.9 for the

cohort life expectancies e65 (2006) for males. The point estimate is 18.17. The mean of the bootstrapped values is 18.05, with a standard deviation of 0.3802. The bootstrap percentiles confidence interval at level 90% is (17.41183,18.66094). We have also applied a Poisson bootstrap. The results are shown in Fig. 5.9, lower panel. The mean and the standard deviations are

233


1000

Frequency

800 600 400 200 0 16.5

17.0

17.5

18.0

18.5

19.0

19.5

1000 800 Frequency

234

600 400 200 0 16.5

17.0

17.5

18.0

18.5

19.0

19.5

Figure 5.9. Histograms for the 10,000 bootstrapped values of the cohort life expectancies at age 65 in year 2006 for the general population, males: residuals bootstrap in the top panel, Poisson bootstrap in the bottom panel.

almost equal to those of the residuals bootstrap (respectively, 18.03 and 0.3795). The bootstrap percentiles confidence interval at level 90% is (17.40993,18.65580). The histograms obtained with the Poisson bootstrap and with the residuals bootstrap have very similar shapes, and the confidence intervals closely agree.

5.9

Forecasting life expectancies

In this section, we consider the computation of projected life expectancies at retirement age 65, obtained from the Lee–Carter and Cairns–Blake–Dowd approaches, by replacing the death rates with their forecasted values. Moreover, the results are then compared with other projections performed for the Belgian population.

5.9

5.9.1


Official projections performed by the Belgian Federal Planning Bureau (FPB)

The FPB was asked in 2003 by the Pension Ministry to produce (in collaboration with Statistics Belgium) projected life tables to be used to convert pension benefits into life annuities in the second pillar. A working party was set up by the FPB with representatives from Statistics Belgium, BFIC, the Royal Society of Belgian Actuaries and UCL. The results are summarized in the Working Paper 20-04 available from http://www.plan.be. The FPB model specifies qx (t) = exp(αx + βx t) where αx = ln qx (0) and βx is the rate of decrease of qx (t) over time. Thus, each age-specific death probability is assumed to decline at its own exponential rate. The αx ’s and βx ’s are first estimated by the least-squares method, that is, minimizing the objective function xm tn

ln " qx (t) − αx − βx t

2

(5.52)

x=x1 t=t1

Then, the resulting " βx ’s are smoothed using geometric averaging. Finally, the " αx ’s are adjusted to represent the recent mortality experience. This methodology is similar to the generalized linear modelling regression-based approach proposed by Renshaw and Haberman (2003b). " x (t) and the death probabilities " The death rates m qx (t) are typically very close to one another in value. This is why we would expect that the FPB approach would lead to similar projections to the Lee–Carter method once the optimal fitting period has been selected. However, no such selection is performed in the FPB analysis, which may result in some differences in the forecasts.

5.9.2

Andreev–Vaupel projections

The method used by Andreev and Vaupel (2006) is based on Oeppen and Vaupel (2002). Plotting the highest period female life expectancy attained for each calendar year from 1840 to 2000, Oeppen and Vaupel (2002) have noticed that the points fall close to a straight line, starting at about 45 years in Sweden and ending at about 85 years in Japan. They find that record female life expectancy has increased linearly by 2.43 years per decade from 1840 to 2000 (with an adjustment coefficient R2 = 99.2%). The record male life expectancy has increased linearly from 1840 to 2000 at a rate of 2.22 years per decade (with R2 = 98%). Moreover, there is no indication of either an acceleration or deceleration in the rates of change. If the trend

235

236


continues, they predict that female record life expectancy will be 97.5 by mid-century and 109 years by 2100. Life expectancy can be forecast for a given country by considering the gap between national performance and the best-practice level. See also Lee (2003). Andreev and Vaupel (2006) combine the approach due to Oeppen and Vaupel (2002) with the Lee–Carter model to gain stability over the long run. More precisely, they assume that the linear trend in the best practice female life expectancy continues into the future and also that the difference between the life expectancy of a particular country and the general trend stays constant over time. Then, the life expectancy at birth can be forecast as ↑ ↑ e0 (t) = e0 (tn ) + s(t − tn ) (5.53) where s is the pace of increase in the best practice life expectancy over time ↑ that has been estimated by Oeppen and Vaupel (2002) and e0 (t) is the life expectancy at birth in the particular country. Andreev and Vaupel (2006) do not use separate values of s for males and females but the female value of 0.243 for both genders. Andreev and Vaupel (2006) consider ages 50 and over so that they need to ↑ ↑ deduce the value of e50 (t) from e0 (t). To do so, they start with a forecast of death rates by the linear decline model (according to which each age-specific death rate is forecasted to decline at its own independent rate) along the lines of µ ˙ x (tn + s) = " µx (tn ) exp(−gx s) (5.54) where gx is the annual rate of decline for the mortality rate at age x. Then, the forecasted death rates are multiplied by a constant factor so that ↑ the life expectancy at birth matches the e0 (t) values coming from (5.53). ↑ The value of e50 (t) is then obtained from these adjusted death rates. ↑

Given the estimated value of e50 (t), we need to calculate the set of mortality rates at ages over 50 that correspond to this value. Andreev and Vaupel (2006) use the Kannisto model µx (t) =

at exp(bt x) 1 + at exp(bt x)

(5.55)

which is fitted to data for ages 50 and over by the method of Poisson maximum likelihood. The at ’s are then projected into the future from the linear model ln at = β0 + β1 t (5.56) ↑

Then, for each t ≥ tn + 1, the parameter bt is determined to match e50 (t) given the value of at obtained from (5.56).

5.9


This method may produce a jump in death rates. To avoid this drawback, the death rates can be blended with the death rates produced by the Lee– Carter method over a short period of time. Specifically, the Lee–Carter model is fitted to data for ages 50 and over, and the estimated κt ’s are ↑ adjusted by refitting to the e50 (t)’s. The bias correction ensures that the forecasted death rates closely agree with the latest available death rates in the first years of the forecast. The weight assigned to the Lee–Carter death rates is 1 − k/n + 1 for years tn + k, k = 1, 2, . . . , n, where n is the length of the blending period. The value of n ranges from 10 for countries where the model (5.55) provides a good fit to 40 where this is not the case.

5.9.3


Life expectancies are often used by demographers to measure the evolution

of mortality. Specifically, ex (t) is the average number of years that an x-aged individual in year t will survive, allowing for the evolution of mortality rates

with time after t. We, thus, expect that this person will die in year t + ex (t)

at age x + ex (t). The formula giving ex (t) under (3.2) is e x (t)

=

ξ≥0

exp −

0

ξ

µx+η (t + η) dη dξ

1 − exp − µx (t) = µx (t)   k−1 ! 1 − exp −µx+k (t + k)   + exp −µx+j (t + j) . (5.57) µx+k (t + k) k≥1

j=0

It is interesting to compare (5.57) with the expression (3.18) previously obtained for the period life expectancy. The actual computation of the projected cohort life expectancies at (retirement) age 65 is made using formula (5.57) where the future death rates are replaced with their forecast values. First, the cohort life expectancies obtained in the Lee–Carter model are compared with the values coming from the Cairns–Blake–Dowd forecast. Then, the Lee–Carter projections are compared with two projections performed for the Belgian population, by the Federal Planning Bureau and by Andreev and Vaupel (2006). Figure 5.10 displays the values of the cohort life expectancies at age 65 obtained from the Lee–Carter and Cairns–Blake–Dowd mortality projections. For the Lee–Carter forecast, we present also a 90% prediction interval. We see that the values obtained from the two approaches are in

237


21

20 e65(t)

238

19

18

2010

2015 t

2020

2025

Figure 5.10. Forecast of cohort life expectancies at age 65 for the general population (circle) with 90% confidence intervals (gray-shaded area), together with values obtained from the Cairns– Blake–Dowd model (triangle).

close agreement, with slightly larger values coming from the Lee–Carter approach. Figure 5.11 displays the cohort life expectancies at age 65 resulting from the Lee–Carter forecast for the general population, together with the official FPB values and the corresponding values obtained by Andreev and Vaupel (2006). The small differences (of 0 . 0, ETRxt = 0

The updating of a typical parameter θ proceeds according to / ∂D ∂2 D updated(θ) = u(θ) = θ − ∂θ ∂θ 2

(6.16)

(6.17)

249

250

6 : Forecasting mortality

where D is the deviance of the current model. Table 6.1 provides fuller details. Effective starting values, conforming to the usual LC constraints (6.2) are κˆ t = 0, βˆ x = 1/k, coupled with the SVD estimate αˆ x =

tn 1 ∧ ln mx (t) tn − t1 + 1 t=t

(6.18)

1

so that αx is estimated by the logarithm of the geometric mean of the empirical mortality rates. The model has ν = (k − 1)(n − 2) degrees of freedom. Table 6.1.

Parameter updating relationships

Gaussian

Poisson

LC

u(αˆ x ) = αˆ x +

u(αˆ x ) = αˆ x +

wxt (yxt − yˆ xt )βˆ x wxt βˆ x2

u(ˆκt ) = κˆ t +

wxt (yxt − yˆ xt )ˆκt wxt κˆ t2

u(βˆ x ) = βˆ x +

t


wxt (yxt − yˆ xt ) t wxt

x


x

t

t

APC

x,t t−x=z

u(ˆιz ) = ˆιz +

2 wxt βˆ x(0)

(0)

(0)

u(βˆ x ) = βˆ x + u(ˆκt ) = κˆ t +

t

=

(1) βˆ x

− yˆ xt )βˆ x(1)

wxt (yxt x

+

t

wxt yˆ xt βˆ x2

wxt (yxt t

2 wxt yˆ xt βˆ x(0)

(0)

(0)


(1)2

wxt βˆ x

wxt (yxt − yˆ xt )ˆκt wxt κˆ t2

wxt (yxt t t

x

(1) u(βˆ x )

=

(1) βˆ x

− yˆ xt )ˆιt−x

wxt yˆ xt ˆι2t−x

wxt (yxt − yˆ xt )βˆ x(1)

x


+

2 wxt yˆ xt βˆ x(1)

wxt (yxt t

t

AC

− yˆ xt )ˆκt

x,t t−x=z

u(ˆιz ) = ˆιz +

t

(1) u(βˆ x )

− yˆ xt )βˆ x

x,t t−x=z

wxt (yxt − yˆ xt )ˆιt−x wxt ˆι2t−x

x

wxt yˆ xt

wxt (yxt x x

− yˆ xt )

wxt yˆ xt κˆ t2 t wxt (yxt − yˆ xt )βˆ x(0)

x,t t−x=z

t

wxt (yxt − yˆ xt )βˆ x(0)

wxt (yxt t

t

− yˆ xt )ˆκt

wxt yˆ xt κˆ t2

u(αˆ x ) computed as above wxt (yxt − yˆ xt )βˆ x

u(αˆ x ) computed as above wxt (yxt − yˆ xt )βˆ x

x,t t−x=z

x,t t−x=z

u(ˆιz ) = ˆιz +

u(ˆιz ) = ˆιz +

x,t t−x=z


wxt βˆ x2

t

wxt (yxt − yˆ xt )ˆιt−x wxt ˆι2t−x t

wxt yˆ xt βˆ x2


x,t t−x=z

wxt (yxt t t

− yˆ xt )ˆιt−x

wxt yˆ xt ˆι2t−x

6.2

LC age–period–cohort mortality projection model

This iterative fitting process generates maximum likelihood estimates under the Poisson error structure presented in (6.9) and (6.10) on setting yxt = dxt , yˆ xt = dˆ xt = ext exp(αˆ x + βˆ x κˆ t ) D(yxt , yˆ xt ) =

dev(x, t) =

x,t

=

yxt 2wxt

x,t

2wxt yxt ln

x,t

yxt yˆ xt

yˆ xt

(6.19)

yxt − u du V(u)

− yxt − yˆ xt

(6.20)

As noted by Renshaw and Haberman (2003a, 2003b, 2006), we can attribute the iterative method for estimating log-linear models with bilinear terms to Goodman (1979). Table 6.1 provides full details of the parameter updating relationships. 6.2.2.3

Fitting the LC model by Method B

Following James and Segal (1982), we use the iterative procedure: Set starting values βˆ x ↓ given βˆ x , update αˆ x , κˆ t given κˆ t , update αˆ x , βˆ x compute D(yxt , yˆ xt ) ↓ Repeat the updating cycle; stop when D(yxt , yˆ xt ) converges Given βˆ x or κˆ t , updating is by selecting the desired generalized linear model and fitting the predictor, which is linear in the respective remaining parameters. Thus, log-link Poisson responses yxt = dxt with offsets ln ETRxt are set in order to generate the same results as in the iterative fitting process of Section 5.2.2.3. The respective predictors are declared by accessing the model formulae (design matrices), a feature which is available in GLIM (Francis et al., 1993), for example, and other software packages. In specifying the model formulae, we impose the constraints κt1 = 0, βx = 1, (6.21) x

and then revert back to the standard LC constraints (6.2) once convergence is attained.

251

252


6.2.2.4

Fitting the APC LC model

It is well known that APC modelling is problematic, since the three factors are constrained by the relationship cohort = period − age To ensure a unique set of parameter estimates, we resort to a twostage fitting strategy in which αx is estimated first, typically as in (6.18) corresponding to the original LC SVD approach. Then, the remaining parameters, those of the reduction factor RF, may be estimated by suitably adapting Method B by declaring log-link Poisson responses yxt = dxt and the augmented offsets ln ext + αˆ x and adapting the design matrices, together with the constraints βx(0) = 1, βx(1) = 1 and either ιt1 −xk = 0 (or κt1 = 0). (6.22) x

x

Obvious simplifications to the design matrices are needed when fitting the (0) (1) associated sub-models with βx = 1 or βx = 1, while the iterative element (0) in the fitting procedure is redundant when fitting the model with βx = (1) βx = 1 for all x. We note that the APC model has ν = k(n − 3) − 2(n − 2) degrees of freedom (excluding any provision for the first-stage modelling (0) (1) of αx ). We find that effective starting values are βx = βx = 1/k. Fitting is also possible under Method A, once αx has been estimated, using the extended definitions of yˆ xt and adapting the core of the iterative cycle in accordance with the relevant updating relationships (Table 6.1). Effective (0) (1) starting values may be obtained by setting βx = βx = 1 and fitting this restricted version of the APC model to generate starting values for ιz and κt . 6.2.2.5

Fitting the AC LC model

Model identification is conveniently achieved by means of the parameter constraints βx(0) = 1 (6.23) ιt1 −xk = 0, x

(0)

Model fitting is then possible by reformulating Method A in terms of αx , βx and ιt−x . Thus, ιt−x instead of κt is updated in the core of the iterative cycle (subject to the adjustment ιt1 −xk = 0), using the replacement updating relationships of Table 6.1. Fitting is also possible using Method B by replacing κt with ιt−x and modifying the design matrices accordingly. A possible strategy (0) for generating starting values is to set βˆ x = 1 and additionally fit the main effects structure αx +ιt−x in accordance with the distributional assumptions under Method A. There are ν = (k − 1)(n − 3) degrees of freedom in this model.

6.2

6.2.3

LC age–period–cohort mortality projection model

Mortality rate projections

Projected mortality rates ·

∧

·

mx (tn + s) = mx (tn ) RF(x, tn + s),

s>0

(6.24) ∧

are computed by alignment with the latest available mortality rates mx (tn ). Here, · RF(x, tn + s) = exp βˆ x(0) ˜ιtn −x+s − ˆιtn −x + βˆ x(1) κ˙ tn +s − κˆ tn , s > 0 (6.25) for which ˙ tn + s) = 1 lim RF(x, s→0

(i) (i) is based on the parameter estimates βˆ x , ˆιz , κˆ t and the time series forecasts 1 0 1 0 ˆιz : z ∈ [t1 − xk , tn − x1 ] → ˙ιtn −x1 +s : s > 0 0 1 0 1 κˆ t : t ∈ [t1 , tn ] → κ˙ tn +s : s > 0 (6.26)

where ˜ιtn −x+s

, ˆι = tn −x+s ˙ιtn −x+s ,

0 < s ≤ x − x1 s > x − x1

As we have seen in Section 5.7, the time series forecasts are typically generated using univariate ARIMA processes. The random walk with drift (or ARIMA(0,1,0) process) features prominently in many of the published applications of the LC model. If no provision for alignment with the latest available mortality rates is made (as in equation (6.24)), the extrapolated mortality rates decompose multiplicatively as · m ˙ x (tn + s) = exp αˆ x + βˆ x(0) ˆιtn −x + βˆ x(1) κˆ tn RF(x, tn + s), s > 0 (6.27) which has the same functional form as (6.8), and can be directly compared with (6.24). This was the approach originally proposed in Lee and Carter (1992).

6.2.4

Discussion

By specifying the second moment distributional properties when defining the model error structure, the choice of distribution is not restricted to the Poisson and Gaussian distributions, and may indeed be expanded by selecting different variance functions (within the exponential family of distributions). Empirical evidence suggests that, for all practical purposes,

253

254


maximum likelihood estimates obtained for the LC model using the iterative fitting processes under the Gaussian error structure given by (6.12), are the same as those obtained under fitting by SVD. Unlike modelling by SVD, however, the choice of weights (6.16) means that estimation can proceed, in the presence of empty data cells, under the Gaussian, Poisson, and any of the other viable error settings. Wilmoth (1993) uses weights wxt = dxt in combination with the Gaussian error setting. Empirical studies reveal that this has the effect of bringing the parameter estimates into close agreement with the Poisson-response-based estimates. When comparing a range of results obtained under both modelling approaches (with identical model structures), we have found that the same number of iterations is required to induce convergence. However, convergence is slow when fitting the APC model. As discussed in Section 5.6, diagnostic checks on the fitted model are very important. For consistency with the model specification, we consider plots of the standardized deviance residuals . rxt = sign(yxt − yˆ xt ) dev(x, t)/φˆ (6.28) where

D(yxt , yˆ xt ) (6.29) ν The sole use of the proportion of the total temporal variance, as measured by the ratio of the first singular value to the sum of singular values under SVD, is not a satisfactory diagnostic indicator. However, this index is widely quoted in the demographic literature: see, for example, Tuljapurkar et al. (2000). φˆ =

The parameters αx are estimated simultaneously with the parameters of the reduction factor RF in both the LC and AC models. A two-stage estimation process is necessary, however, in which αx is estimated separately to condition on the estimation of RF, when fitting the APC model (and its substructures). This two-stage approach can also be applied when fitting the LC and AC models. In the case of the former, empirical studies show that this has little practical material effect, due to the robust nature of the αx estimate (6.18).

6.3

Application to United Kingdom mortality data

To explore the potential of the APC model, we present results for the United Kingdom 1961–2000 mortality experiences for each gender, with crossclassification by individual year of age from 0 to 99. This data set has been

6.3


provided by the Government Actuary’s Department – the availability of data cross-classified by single year of age and by individual calendar year facilitates the construction of cohort-based mortality measures. We make a direct comparison with the standard age–period LC and the AC models. In this application, all of the models are fitted under the Poisson error setting, represented by equations (6.9) and (6.10). The implications of the choice of model structure are immediately apparent from the respective residual plots, illustrated for the UK female experience (Fig. 6.1). Here the distinctive ripple effects in the year-of-birth residual plots under LC modelling (Fig. 6.1(a), RH frame), signify a failure of the model to capture cohort effects. This is then transferred to the calendar-year residual plots under AC modelling (Fig. 6.1(b), LH frame) signifying a reciprocal failure to capture period effects. However, these distinctive ripple effects are largely removed under APC modelling (Fig. 6.1(c)) – this feature indicates that the model captures in a relatively successful manner all of the three main effects and represents a significant improvement

(a)

3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 1965 1970 1975 1980 1985 1990 1995 2000 Calendar year

(b)

3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0

0 10 20 30 40 50 60 70 80 90 100 Age

3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0

0 10 20 30 40 50 60 70 80 90 100 Age

1880 1900 1920 1940 1960 1980 2000 Year of birth 3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0

3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 1965 1970 1975 1980 1985 1990 1995 2000 Calendar year

1880 1900 1920 1940 1960 1980 2000 Year of birth 3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0

3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 1965 1970 1975 1980 1985 1990 1995 2000 Calendar year

(c)

3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0

3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0

0 10 20 30 40 50 60 70 80 90 100 Age

1880 1900 1920 1940 1960 1980 2000 Year of birth

Figure 6.1. Female mortality experience: residual plots for (a) LC model; (b) AC model; and (c) APC model.

255


over the fitted LC model. Similar patterns are observed in the residual plots for the UK male experience (not reproduced here but the details are available from the authors). Turning first to the parameter estimates for the APC modelling approach (Fig. 6.2), we believe that it is helpful and informative to compare matching frames between the sexes. Thus, the main age–effect plots (αˆ x vs x) display the familiar characteristics, including the ‘accident’ humps, of static crosssectional life-tables (on the log scale), with a more pronounced accident hump and heavier mortality for males than for females. We recall that these effects are estimated separately, by averaging crude mortality rates over t for each x, to condition for both period and cohort effects. The main period effects plot (ˆκt vs t) is linear for females but exhibits mild curvature for males, which can be characterized as piece-wise linear with a knot or hinge positioned in the first-half of the 1970s. This effect is also present in the separate LC analysis of mortality data of the G7 countries

–1

0.035

–2

0.030

–3

0.025 b0

a

–4 –5 –6

b1

(a)

0.020 0.015

–7

0.010

–8

0.005 0

10 20 30 40 50 60 70 80 90 Age

0

0 –10 –20 –30 –40 –50 –60 –70 –80 –90 –100 1960 19651970 1975 1980 1985 1990 1995 2000 2005 20102015 2020 Calendar year

–2 –3 b0

a

–4 –5 –6 –7 –8 0

10 20 30 40 50 60 70 80 Age

40 i

20 –0 –20 –40 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 Year or birth

0.060 0.055 0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005

90

0.014 0.012 0.010 0.008 0.006 0.004 0.002 0

10 20 30 40 50 60 70 80 90 Age

0 –10 –20 –30

100

–40

40

0 10 20 30 40 50 60 70 80 90 Age

80

i

60

–50 –60 –70

20 –0

–80 1960 1965 1970 1975 1980 19851990 19952000 20052010 2015 2020 Calendar year

Figure 6.2.

0 10 20 30 40 50 60 70 80 90 Age

60

b1

–1

10 20 30 40 50 60 70 80 90 Age

0.022 0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002

80

k

(b)

k

256

–20 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 Year of birth

Parameter estimates and forecasts for the APC model: (a) females; (b) males.

6.3


(Tuljaparkar et al., 2000) and has been discussed further for the United Kingdom by Renshaw and Haberman (2003a). The forecasts for κt are based on the auto-regressive time series model yt = d + φ1 yt−1 + εt

where yt = κt − κt−1

(6.30)

which is the equivalent of ARIMA(1, 1, 0) modelling. There are noteworthy (1) differences in the βˆ x patterns, which control the rate of decline by period of the age specific rates of mortality in the projections. In particular, the (1) trough in the male βˆ x pattern in the 20–40 age range is consistent with similar findings from trends in the male England and Wales mortality rates (Renshaw and Haberman, 2003a). The plots of the main cohort effects (ˆιz vs z = t−x) are particularly revealing. Thus, noteworthy discontinuities occur corresponding to the ending of World Wars I and II. While it is possible to identify the first of these with the 1919 influenza epidemic, we are not aware of the likely cause of the second discontinuity. (The 1886–1887 discontinuity can be traced to a set of outliers, and is possibly due to mis-stated exposures for this particular cohort.) The pronounced decline in the ˆιz profile in the inter-war years is consistent with the reported rapid mortality improvements experienced by generations born between 1925 and 1945 (for both sexes) and discussed at the start of this chapter. The apparent stable linear trends in the ˆιz profiles, present since the late 1940s, form the basis of the depicted time series forecasts, generated using an ARIMA (0, 1, 0) process for females and an (0) ARIMA(1, 1, 0) process for males. The βˆ x patterns, which control the agespecific cohort contributions to the mortality projections, are similar, for both sexes, for ages up to 65. We illustrate the implications of these projections in Fig. 6.3 by plotting current log crude mortality rates (for the calendar year 2000) against age for each gender and comparing these with projections for the calendar year 2020 under three different models: the LC (or standard Lee-Carter) model, AC model, and APC model. In Fig. 6.3(a), we show the LC–AC comparison and in Fig. 6.3(b) the LC–APC comparison. We note the marked mortality reductions projected for 2020 under the AC and APC models at ages which correspond to the cohorts identified at the start of this chapter (based on descriptive analyses): those born between 1925 and 1945 and hence aged 75–95 in 2020. In order to illustrate the impact of such diverse projections under ageperiod (LC) and age-period-cohort (APC) modelling, we have calculated complete life expectancies e65 (t) at age 65 (Fig 6.4) and immediate life annuity values a65 (t) at age 65 assuming a 5% pa fixed interest rate (Fig. 6.5)

257

258


(a)

UK female population study

UK male population study –1

–1 –2

–2

Age-cohort (2020) Age-period (2020) Latest (2000)

–3 –4 –5

–4 –5

–6

–6

–7

–7

–8 –9 –10

–8 –9 –10 0

10

20

(b)

30

40

50 Age

Age-cohort (2020) Age-period (2020) Latest (2000)

–3

60

70

80

0

90

10

20

UK female population study

30

40

50 Age

60

70

80

90

80

90

UK male population study –1

–1 –2

–2

Age-period-cohort (2020) Age-period (2020) Latest (2000)

–3 –4 –5

–4 –5

–6

–6

–7

–7

–8 –9 –10

–8 –9 –10 0

10

20

30

40

50 Age

60

Age-period-cohort (2020) Age-period (2020) Latest (2000)

–3

70

80

90

0

10

20

30

40

50 Age

60

70

Figure 6.3. Current (2000) and projected (2020) ln µx (t) age profiles: (a) LC and AC models; (b) LC and APC models.

for a range of years t using both the cohort and period method of computing. (We note that the annuity values represent the expected present value of an income of one paid annually in arrears while the individual initially aged 65 remains alive.) For the cohort method of computing, we use the following formulae, which are analogous to (5.57): 1 h≥0 lx+h (t + h){1 − 2 qx+i (t + h)} ex (t) = , lx (t) h h≥1 lx+i (t + h)v (6.31) ax (t) = lx (t) where qx (t) ≈ 1 − exp(−µxt ),

lx+1 (t + 1) = {1 − qx (t)}lx (t)

(6.32)

with annual discount factor v, where v = 1/(1 + i) is calculated using a constant interest rate. Thus, in the cohort versions, we allow fully for the dynamic aspect of the mortality rates, with the summations proceeding (diagonally) along a cohort. We illustrate values up to the year 2005 calculated by the cohort method and this requires extrapolation up to the year

6.3


UK female population study 26 by period: LC by period: APC by cohort: LC by cohort: APC

24 22

e(65,t)

20 18 16 14 12 1960

1965

1970

1975

1980

1985

1990 1995 Period t

2000

2005

2010

2015

2020

2000

2005

2010

2015

2020

UK male population study 26 by period: LC by period: APC by cohort: LC by cohort: APC

24

e(65,t)

22 20 18 16 14 12 1960

1965

1970

1975

1980

1985

1990 1995 Period t

Figure 6.4. Projected life expectancies at age 65, computed by period and by cohort methods for age-period (LC) and age-period-cohort (APC) models.

2040. In contrast, under the period method of calculation, the mortality rates are treated as a sequence of (annual) static life tables, and computing proceeds by suppressing the variation in t in expressions (6.31) and (6.32), with (marginal) summation over age (≥x) for each fixed t, as for example in (3.18). We illustrate values up to the year 2020 using this method based on the empirical mortality rates µ ˆ x (t) = m ˆ x (t) in the period up to 2000 and requiring extrapolation for subsequent years up to 2020. The period method of computation fails to capture the full dynamic impact of the evolving mortality rates under the modelling assumptions and generates less uncertainty than the cohort method of calculation. The latter necessarily requires

259


UK female population study 14 13

a(65,t)

12

by period: LC by period: APC by cohort: LC by cohort: APC

11 10 9 8 7 1960 1965

1970 1975

1980

1985

1990 1995 Period t

2000

2005

2010

2015

2020

2005

2010

2015

2020

UK male population study 14 13 12 a(65,t)

260

by period: LC by period: APC by cohort: LC by cohort: APC

11 10 9 8 7 1960 1965

1970 1975

1980

1985

1990 1995 Period t

2000

Figure 6.5. Projected life annuity values at age 65 (calculated using a 5% per annum fixed interest rate), computed by period and by cohort under age-period (LC) and age-period-cohort (APC) models.

more lengthy extrapolations and this contributes a source of increasing uncertainty. One means of quantifying this uncertainty is through the adoption of boot-strapping simulation methods, as described in Section 5.8, in the context of LC modelling. This and other methods are currently under investigation for the case of the APC model. The reserves that insurance companies selling life annuities and pension funds would have to hold in order to meet their future contractual liabilities are directly related to terms like a65 (t); see Booth et al. (2005). The financial implications of the upward trends in cohort-based life annuity values (which are the most relevant for pricing and reserving calculations) in Fig. 6.5 are clear and significant and

6.3


e(65, t) computed by period 6 5

t = 2020

4

t = 2016

3

t = 2012

2

t = 2008

1

t = 2004

Age-period (LC) Age-period-cohort (APC) Age 65

0 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 Life expectancy e(x, 2000) computed by cohort 6 x = 85

5

x = 80

4

x = 75

3

x = 70

2 Age-period (LC) Age-period-cohort (APC) Period 2000

1

x = 65

0 4

5

6

7

8

9

10

11 12 13 Life expectancy

14

15

16

17

18

19

20

Figure 6.6. E +W male mortality: comparison of life expectancy predictions using (i) age-periodcohort and (ii) age-period Poisson structures. Predictions with intervals by bootstrapping the time series prediction error in the period (and cohort) components, and selecting the resulting 2.5, 50, 97.5 percentiles.

indicate the burden that increasing longevity may place on such financial institutions. As we have discussed in Section 5.8, it is important to be able to qualify any projections of key mortality indices with measures of the error or uncertainty present. Because of the complexities of the structure of the APC LC model, the indices of interest are non-linear functions of the parameters αx , βx , κt , it−x and hence analytical deviations of prediction intervals are not possible. It is therefore necessary to employ bootstrapping techniques. In Figs. 6.6 and 6.7, we use the LC and APC models fitted to England and Wales male mortality rates over the period 1961–2000 in order to compare estimates of life expectancy and 95% prediction intervals. Specifically, we show in Fig. 6.6(a) computations of the period life expectancy at age 65 for various future periods (equivalent to the median of the simulated distributions) and the corresponding 2.5 and 97.5 percentiles from the simulated

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a(65, t) computed by period 6 5 t = 2020 4 t = 2016 3 t = 2012 2 t = 2008 Age-period (LC) Age-period-cohort (M) Age 65

1 t = 2004

0 10.0 10.2 10.4 10.6 10.8 11.0 11.2 11.4 11.6 11.8 12.0 12.2 12.4 12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.0 4% Annuity a(x, 2000) computed by cohort 6 5 4

x = 85 x = 80 x = 75

3

x = 70

2 1

Age-period (LC) Age-period-cohort (M) Period 2000

x = 65

0 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 4% Annuity Figure 6.7. E + W male mortality: comparison of 4% fixed rate annuity predictions using (i) ageperiod-cohort and (ii) age-period Poisson structures. Predictions with intervals by bootstrapping the time series prediction error in the period (and cohort) components, and selecting the resulting 2.5, 50, 97.5 percentiles.

distributions. In this case, the simulated distributions have been calculated by bootstrapping only the time series prediction error as experiments reveal that this is the most important component of the uncertainty in the model (as originally suggested by Lee and Carter, 1992). In this way, we avoid using the detailed bootstrapping strategies discussed in Section 5.8 which can be rather slow to converge. The results in Fig. 6.6 (upper frame) show that the central estimates for life expectancy are higher when using the APC model as against using the LC model (as illustrated in Fig. 6.4) and that the prediction intervals are wider for the APC models for each year. As we move forward in time from 2004 to 2020, we note that both pairs of prediction intervals become wider, indicating a greater level of uncertainty present in the estimates for future years. Thus, the calculations for 2004 involve 4 years of forward projections whereas the calculations for 2020 involve 20 years of projections.

6.4

Cairns–Blake–Dowd mortality projection model: allowing for cohort effects

The results in Fig. 6.6 (lower frame) show the corresponding figures for cohort life expectancy for five cohorts of males aged 65, 70, 75, 80, and 85 in 2000. The younger cohorts have estimates of cohort life expectancy that are higher under the APC model than under the LC model (as in Fig. 6.4). The prediction intervals under the APC model are much wider for the younger cohorts. As we consider the older cohorts, we note that the central estimates and the prediction intervals become more similar under the two models indicating the particular incidence of the cohort effect which affects those aged 55–75 in 2000. Expectedly, under both models, the prediction intervals are wider for the cohorts aged 65 and 70 in 2000 than for the older cohorts, and the width decreases in stages as age in 2000 increases. This reflects the underlying level of projection involved in the calculations – if we regard age 110 as approximately the terminal age in the underlying survival model, then the cohort estimates at age 65 would involve 45 years of projected quantities while the cohort estimates at age 85 would involve only 25 years of projections. Figure 6.7 reproduces the calculations of Fig. 6.6 but for immediate life annuities calculated using a constant interest instant rate of 4% per annum. We can regard Fig. 6.7 as extending the results of Fig. 6.5 by including prediction intervals and a more detailed comparison. Figure 6.7 shows the same principal features as Fig. 6.6.

6.4

Cairns–Blake–Dowd mortality projection model: allowing for cohort effects

In Section 5.3, we introduced the Cairns–Blake–Dowd mortality projection model which is motivated by the empirical observation that logitqx (t) is a reasonably linear function of x for fixed t. The model introduced by Cairns et al. (2007) is the following: ln

qx (t) (1) (2) = κt + κ t x px (t)

(6.33)

which can be regarded as a specific example of a more general class of models qx (t) (1) (2) = βx(1) κt + βx(2) κt (6.34) ln px (t) Responding to the need to consider the cohort effect observed in the historic mortality data for a number of countries, Cairns et al. (2007) introduce an AC term into the predictor as follows, in an analogous manner to the

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Renshaw and Haberman (2006) enhancement of the original Lee-Carter model. Thus, Cairns et al. (2007) propose the following family of models: ln

qx (t) (1) (2) = βx(1) κt + βx(2) κt + βx(3) ιt−x px (t)

(6.35)

where the it−x term represents a cohort effect as in (6.6). Having considered goodness-of-fit of this family of models to historic data from England and Wales and the USA, Cairns et al. (2007) investigate two specific versions in some detail. The special cases are I. βx(1) = 1, βx(2) = x − x, II. βx(1) = 1, βx(2) = x − x,

βx(3) = 1

(6.36)

and βx(3) = xc − x

(6.37)

and

where x is the average age in the data set and xc is a constant parameter that needs to be estimated. As with the APC version of the Lee Carter model in Section 6.2, we need to introduce some identifiability constraints to ensure that the parameters can be uniquely estimated. Version II is motivated by the observation that in the applications of the APC model of Section 6.2, the coefficient of the cohort term it−x is often found to be a decreasing function (3) of age: (6.37) incorporates the simplest such specification of βx . Cairns et al. (2007) fit the models by the method of maximum likelihood, assuming that Dxt has a Poisson distribution as assumed in earlier Sections 5.2.2.3 and 6.2.2. For England and Wales data comprising the calendar years 1961–2004 inclusive and ages 60–89, they find that the best-fitting model is (6.37). For US data comprising the calendar years 1968–2003 inclusive and ages 60–89 (although only data for ages 85–89 are used for 1980–2003), they find that the best-fitting model is (6.6). When robustness to the choice of fitting period is considered, the best fits to the historic data from both countries are obtained for an augmented version of (6.35) viz. qx (t) (1) (2) (3) = βx(1) κt + βx(2) κt + βx(3) κt + βx(4) it−x (6.38) px (t) (1) (2) (3) ¯ 2 − σˆ x2 and with the specific choices βx = 1, βx = x − x, βx = (x − x) (4) βx = 1. Here σˆ x2 is the average value of (x − x) ¯ 2 . This development of the model is inspired by the observation that there is some curvature in the age-profile of log it qx (t) in the United States data. ln

As in Sections 5.3 and 6.2, we could use the Cairns–Blake–Dowd class of models for projection purposes. This would require models to be postulated and estimated for the dynamics of the period and cohort effects

6.5

P-splines model: allowing for cohort effects

terms in (6.35)–(6.38). An obvious approach would be to employ standard time series methods based on ARIMA models, as discussed earlier. A complication with the models represented by equations (6.35)–(6.38) is that they involve three or four stochastic (time series) terms. We could follow Section 6.2 and postulate that and (for different values of i) are independent. Still, the presence of two or three terms would mean that we would need to consider multivariate time-series modelling to estimate the underlying dependency structure. This would be likely to involve vector autoregressive models and co-integration techniques as discussed by Hamilton (1994).

6.5

P-splines model: allowing for cohort effects

As noted in Section 5.4.2, Currie et al. (2004) have introduced a twodimensional graduation methodology based on B-splines, which is fitted to observational data using a regression framework. The two-dimensional version of univariate B-splines is obtained by multiplying the respective elements of the univariate B-splines in the age- and time-dimensions. Thus, the model is ln µx (t) = θij Bij (x, t) (6.39) i, j

where Bij (x, t) = Bi (x) · Bj (t)and where Bij and the θij are parameters to be estimated from the data and Bi and Bj are the respective univariate splines. In reality, B-splines can provide a very good fit to the data if we employ a large number of knots in the year and age dimensions. But this excellent level of goodness of fit is achieved by sacrificing smoothness in the resulting fit. The method of P-splines (or penalized splines) has been suggested by Eilers and Marx (1996) to overcome this problem: in this case, the log-likelihood is adjusted by a penalty function, with appropriate weights. Schematically, the penalized by likelihood would have the following form for an LC model: PL(θ) = L(θ) − λx Px (θ) − λt Pt (θ)

(6.40)

where λx and λt are weighting parameters and Px (θ) is a penalty function in the age dimension and Pt (θ) is a penalty function in the calendar time dimension. An alternative formulation would involve an AC model: PL(θ) = L(θ) − λx Px (θ) − λz Pz (θ)

(6.41)

where, as in Section 6.2, we use z = t − x to index cohorts. The λ’s are estimated from the data. As noted in Section 5.4.2, typical choices for quadratic

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penalty functions would be Px (θ) = (θij − 2θi−1, j + θi−2, j )2

(6.42)

i, j

Pt (θ) =

(θij − 2θi, j−1 + θi, j−2 )2

(6.43)

(θi+1,j−1 − 2θij + θi−1, j+1 )2

(6.44)

i, j

Pz (θ) =

i, j

Thus, the B-splines are used as the basis for the underlying regression and the log likelihood is modified by penalty functions like the above which depend on the smoothness of the θij parameters. The idea of using P-spline regression not just for graduating mortality data but also for mortality projections was first suggested by CMIB (2005). In this application, that is, projecting mortality rates, the choice of the P(θ) function plays a critical role in extending the mortality surface beyond the range of the data so that projections are a direct consequence of the smoothing process. Thus, a quadratic penalty function effectively leads to linear extrapolation – in the age and time dimensions, for (6.40) combined with the choices (6.42) and (6.43); or in the age and year of birth dimensions for (6.41) combined with the choices (6.42) and (6.44). Different choices for P(θ) would be possible, and these may have little impact on the quality of fit to the historic data and hence would be difficult to infer from the data. However, the impact on the projected mortality surface is considerable. The choice of P(θ) corresponds to a decision on the projected trend. We have seen the implications of a quadratic penalty. Similarly, a linear penalty function would lead to constant log mortality rates being projected in the appropriate dimensions and a cubic penalty function would lead to quadratic log mortality rates being projected in the appropriate dimensions. Detailed applications of the P-spline methodology indicate that it is better suited to graduation and smoothing of historic observational data than to projection: see, for example, Cairns et al. (2007) and Richards et al. (2007). Further, we should note that P-spline models can be used to generate percentiles for the measurement of uncertainty but unlike, the LC and Cairns–Blake–Dowd families of models, P-spline models are not able to generate sample paths. In many asset–liability modelling applications in insurance and pensions, the production of sample paths is an important feature and could be useful elsewhere such as in the pricing of longevity-linked financial instruments – see Chapter 7.

7

7.1

The longevity risk: actuarial perspectives

Introduction

In this chapter we deal with the mortality risks borne by an annuity provider, and in particular with the longevity risk originating from the uncertain evolution of mortality at adult and old ages. The assessment of longevity risk requires a stochastic representation of mortality. Possible approaches are described in Section 7.2, which is also devoted to an analysis of the impact of longevity risk on the risk profile of the provider. In Section 7.3 and 7.4 we take a risk management perspective, and we investigate possible solutions for risk mitigation. In particular, risk transfers as well as capital requirements for the risk retained are discussed. Policy design and the pricing of life annuities allowing for longevity risk are dealt with in Section 7.5 and 7.6; such aspects, owing to commercial pressure and modelling difficulties, are rather controversial. We do not develop an in-depth analysis, but we instead remark on the main issues. To reach a proper arrangement of the policy conditions of a life annuity, the possible behaviour of the annuitant in respect of the planning of her/his retirement income has to be considered. In Section 7.7 we describe possible choices available to the annuitant in this respect. The topics dealt with in this chapter are rather new and not wellestablished either in practice or in the literature. So the chapter is based on recent research. To give a comprehensive view of the available literature, most contributions are cited in Section 7.8, which is devoted to comments on further readings; for some specific issues, however, references are also quoted in the previous sections. In this chapter, we refer usually to annuitants and insurers. Such terms are anyhow used in a generic sense. The discussion could also be referred to pensioners, with a proper adjustment of the parameters of the relevant

7 : The longevity risk: actuarial perspectives

mortality models, and to annuity providers other than an insurer. Just for brevity, only annuitants and insurers are mentioned.

7.2 7.2.1

The longevity risk Mortality risks

Mortality risk may emerge in different ways. Three cases can in particular be envisaged. (a) One individual may live longer or less than the average lifetime in the population to which she/he belongs. In terms of the frequency of deaths in the population, this may result in observed mortality rates higher than expected in some years, lower than expected in others, with no apparent trend in such deviations. (b) The average lifetime of a population may be different from what is expected. In terms of the frequency of deaths, it turns out that mortality rates observed in time in the population are systematically above or below those coming from the relevant mortality table. (c) Mortality rates in a population may experience sudden jumps, due to critical living conditions, such as influenza epidemics, severe climatic conditions (e.g. hot summers), natural disasters and so on. In all the three cases, deviations in mortality rates with respect to what is expected are experienced; an illustration is sketched in Fig. 7.1 where, with reference to a given cohort, in each panel dots represent mortality rates observed along time, whereas the solid line plots their forecasted level.

Figure 7.1.

Mortality rates

Mortality rates

Case (a) is the well-known situation of possible deviations around expected mortality rates; the mortality risk here comes out as a risk of

Mortality rates

268

Time

Time

Time

Case (a)

Case (b)

Case (c)

Experienced (dots) vs expected (solid line) mortality rates for a given cohort.

7.2

The longevity risk

random fluctuations, which is traditional in the insurance business, in both the life and the non-life area (actually, it is the basic grounds of the insurance business). It is often named process risk or also insurance risk. It concerns the individual position, and as such its severity reduces as the single position becomes negligible in respect of the overall portfolio. The process risk can be hedged through the realization of a proper pooling effect, since it reduces as soon as the portfolio is made of similar policies and its size is large enough, as well as through traditional risk transfer arrangements. Under case (b), deviations are from expected values, rather than around them, hence their systematic nature. This may be the result of either a misspecification of the relevant mortality model (e.g. because the time-pattern of actual mortality differs from that implied by the adopted mortality table) or a biased assessment of the relevant parameters (e.g. due to a lack of data). The former aspect is referred to as the model risk, the latter as the parameter risk. The term uncertainty risk is often used to refer to model and parameter risk jointly, meaning uncertainty in the representation of a phenomenon (e.g. future mortality). When adult-old ages are concerned, uncertainty risk may emerge in particular because of an unanticipated reduction in mortality rates (as is presented in the mid-panel of Fig. 7.1, where the mortality profile of the cohort is better captured by the dashed line rather than by the solid line). In this case, the term longevity risk is used instead of uncertainty risk. It must be stressed that longevity risk concerns aggregate mortality; so pooling arguments do not apply for its hedging. In case (c), a catastrophe risk emerges, namely the risk of a sudden and short-term rise in the frequency of deaths. Similar to case (b), aggregate mortality is concerned; however, when compared with longevity risk, the time-span involved by the emergence of the risk needs to be stressed: shortterm in case (c), long-term (possibly, permanent) in case (b). Clearly, a proper hedging of catastrophe risk is required when death benefits are dealt with (whilst when dealing with life annuities, profit arises because of the higher mortality experienced). The usual pooling arguments do not apply; however, diversification effects may be realized and risk transfers can be conceived as well. Some remarks in this regard are given in Sections 7.3.2 and 7.4.2. Apart from some short remarks on the management of process and catastrophe risk, in the following we focus on longevity risk. Before moving to the relevant discussion, it is necessary to make a comment on terminology. The vocabulary introduced above for mortality risks is commonly acknowledged in the literature. In some risk classification systems, however, the meaning of some terms may be different, and this may lead to possible misunderstandings. We mention in particular the evolving Solvency

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2 system, where (see CEIOPS (2007)) both the mortality and the longevity risks are meant to result from uncertainty risk. Mortality risk addresses possible situations of extra-mortality; concern here is for a business with death benefits. On the contrary, longevity risk addresses the possible realization of extra-survivorship; clearly, in this case concern is for a business with living benefits, life annuities in particular. In the following, we disregard this meaning; reference is therefore to what we have described under items (a)–(c) above and the relevant remarks.

7.2.2

Representing longevity risk: stochastic modelling issues

Whenever we aim at representing a risk, a stochastic valuation is required. In general terms, a stochastic mortality model should allow for the several types of possible deviations in the frequency of death in respect of the forecasted mortality rate, namely: (a) random fluctuations (to represent process risk); (b) deviations due to the shape of the time-pattern implied by the mortality model, in respect of both age and calendar year (model risk); (c) deviations due to the level of the parameters of the mortality model (parameter risk); (d) shocks due to period effects (catastrophe risk). As to the shape of the time-pattern of mortality rates in respect of calendar year, we recall that by longevity risk we mean the risk of an unanticipated decrease in mortality rates at adult ages (see Section 7.2.1); hence, some projection must be adopted. Except for the Lee–Carter model, projected mortality models do not allow explicitly for risk (see Chapter 4). So, given the purpose of this chapter we need to attack mortality modelling in a new perspective. Embedding four sources of randomness in the mortality model is a tricky business. So some modelling choices are required. In this section, we explore general aspects of stochastic modelling. Notation is stated in general terms. More specific examples are then presented in Section 7.2.3. Let Y be the random number of deaths in a given cohort at a given age. We assume that Y depends on two input variables, say X1 , X2 ; so Y = φ(X1 , X2 ). The quantity X1 could be meant to represent the probability of death or the force of mortality in the cohort in a given year in the absence of extreme situations. Possible shocks are then represented by X2 .

7.2

The longevity risk

Various approaches can be conceived for investigating Y. A graphical illustration is provided by Fig. 7.2. Approach 1 is purely deterministic. Assigning specific values x1 , x2 to the two input variables, the corresponding outcome y of the result variable is simply calculated as y = φ(x1 , x2 ). In our example, x1 is a normal (projected) probability of death or force of mortality, whilst x2 is a given shock (possibly set to zero). It is interesting to note that classical actuarial calculations follow this approach, replacing random variables with their expected or best-estimate value. In a more modern perspective, this approach is adopted for example when performing stress testing (assigning to some variables ‘extreme’ values), or scenario testing. Randomness in input variables is, to some extent, acknowledged when approach 2 is adopted. Reasonable ranges for the outcomes of the input variables are chosen (e.g. the interval of possible values for a shock in a given year), and consequently a range (ymin , ymax ) for Y is derived. As far as X1 is concerned, the range of possible values may represent randomness due to random fluctuations, as well as to the unknown trend of the cohort. Note, however, that the valuation is fundamentally deterministic; the main difference between approach 1 and approach 2 is the number of possible outcomes which is considered. Approach 3 provides a basic example of stochastic modelling, typically adopted for assessing the impact of process risk. The probabilistic structure assigned to X1 is meant here to represent the intrinsic stochastic nature of mortality, that is, random fluctuations. Assuming a continuous probability distribution function, the probability density function fX1 can be obtained, for example, first assigning the probability distribution function of the lifetime of each individual (based on some projected mortality model with given parameters), then aggregating the relevant results. Note that setting fixed parameters for the mortality model, a deterministic assumption for trend is understood. The probability distribution of Y (and X1 as well) can be found using only analytical tools just in very simple (or simplified) circumstances. Numerical methods or stochastic simulation procedures help in most cases. Approach 4 addresses, albeit in a naive manner, the risk of systematic deviations. The three probability distributions assigned to X1 are intended to be based on alternative models for the lifetime of each individual. In practical terms, the same mortality projection model may be assumed, but with alternative sets of values chosen for the relevant parameters to represent alternative mortality trends. Approach 4 then simply consists of iterating the procedure implied by approach 3, each iteration corresponding to a specific assumption about the probability distribution of an input variable.

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272


Input

Output

X1 1

X2 Y

X1 2

X2 Y

fX1 3 X2 fY A1 A2 A3 fX1|Ah 4 X2 fY|Ah

fX1|Ah

+

5 X2 fY

fX1|Ah 6

+

fX2

fY

Figure 7.2.

Modelling approaches to stochastic valuations.

7.2

The longevity risk

A set of conditional distributions of Y is determined. Note that in respect of systematic deviations a representation similar to approach 2 is gained; the difference concerns process risk, which is explicitly addressed under approach 4. Under approach 5 a probability distribution is assigned over the set of trend assumptions. Hence, the unconditional distribution of the output variable Y can be calculated. Note that, this way, both process and uncertainty risk are allowed for. In the graphical representation of Fig. 7.2 a discrete setting is considered in respect of uncertainty risk; more complex models may attack the problem within a continuous framework. Finally, under approach 6 a probabilistic structure is assigned to all of the input variables. In this case, either the joint distribution may be directly addressed or the marginal distributions of the input variables as well as the relevant correlation assumptions (as is depicted in Fig. 7.2). The problem can be handled just through stochastic simulation; difficulties arise with reference to the choice of the probability distribution of the uncertainty risk, of the catastrophe risk, as well as with regard to the dependencies among the various sources of randomness.

7.2.3

Representing longevity risk: some examples

We now specifically address longevity risk. Clearly, approach 5 (or 6) in Fig. 7.2 is required, but some insights into the problem may be gained also from approach 4. Let (x, t) denote a projected mortality quantity, where x is the age attained in calendar year t by the cohort born in year τ = t − x. The projected quantity may either be the probability of death, qx (t), the mortality odds, qx (t)/px (t), the force of mortality, µx (t), and so on. To develop approach 4, alternative hypotheses about future mortality evolution must be chosen. Such alternative assumptions may originate from different sets of the relevant parameters of the projection model; in this way, parameter risk is addressed. Otherwise, the alternative assumptions may be given by mortality projections obtained under different procedures; in this case, also model risk would be addressed. However, it is intrinsically difficult to perform an explanatory comparison of different models (e.g. it is not easy to state whether the different outcomes of two models are mainly due to the implied time-pattern or to the relevant parameters). For this reason, we focus in the following discussion on parameter risk. In any case, unless it is explicitly addressed (as it was depicted in Fig. 7.2), the risk of catastrophe mortality is not considered.

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Let A(τ) denote a given assumption about the mortality trend for people born in year τ, and A(τ) the set of such assumptions. The notation (x, τ + x|A(τ)) refers to the projected mortality quantity conditional on the specific assumption A(τ). The set of all mortality projections is denoted as the family {(x, τ + x|A(τ)); A(τ) ∈ A(τ)}. In principle, the set A(τ) can be either discrete or continuous. The former case is anyhow more practicable. Examples may be found in the projections developed by CMIB, addressing the cohort effect and assuming three hypotheses about the persistence in the future of such an effect; see CMI (2002) and CMI (2006). Let us then suppose that a discrete set has been designed for A(τ). A scenario testing, and possibly a stress testing, can be performed. In particular, the sensitivity of some quantities, such as reserves, profits, and so on, in respect of future mortality trends can be investigated. As mentioned in Section 7.2.2, process risk can be explicitly appraised through the probability distribution function of the lifetime of all the individuals in the cohort, conditional on a given trend assumption. However, the approach in respect of parameter risk is deterministic. Some examples are described in Section 7.2.4. A step forward consists of assigning a (non-negative and normalized) weighting structure to A(τ) (see approach 5 in Fig. 7.2). In this way, unconditional valuations can be performed, thus accounting explicitly for parameter risk. Let

A(τ) = {A1 (τ), A2 (τ), . . . , Am (τ)}

(7.1)

be the set of alternative mortality assumptions; then, let ρh be the weight attached to assumption Ah (τ), such that 0 ≤ ρh ≤ 1 for h = 1, 2, . . . , m and m ρ = 1. The set h h=1 {ρh }h = 1,2,...,m

(7.2)

can be intended as a probability distribution on A(τ). Unfortunately, experience providing data for estimating such weights is rarely available and so personal judgement is often required. See Section 7.2.4 for some examples. We now address possible ways of attacking the problem within a continuous setting. To define A(τ) as a continuous set, a continuous probability distribution must be assigned to the parameters of the mortality model. Difficulties, here, concern the appropriate correlation assumptions among the parameters and hence the complexity of the overall model is clearly greater than in the discrete case. Because there is likely a paucity of data allowing us to make a reliable estimate of the correlations, simplifying hypotheses

7.2

The longevity risk

would have to be accepted. Hence, the setting would not necessarily be more powerful than the discrete one. For this reason, we do not provide examples in respect of a continuous approach. Whatever setting is referred to, either discrete or continuous, the framework discussed above can, to some extent, be classified as a static one. Actually, the notation indicates that the set A(τ) is fixed. Uncertainty is expressed in terms of which of the assumptions A(τ) ∈ A(τ) is the better one for describing the aggregate mortality behaviour of the cohort, that is, the relevant prevailing trend. Irrespective of the setting, either discrete or continuous, no future shift from such a trend is allowed for in the probabilistic distribution (see also panels 5 and 6 in Fig. 7.2). A critical aspect can be found in the fact that assumptions about the temporal correlation of changes in the probabilities of death are implicitly involved; see, for example, Tuljapurkar and Boe (1998). Further, we note that mortality shocks are not embedded into the static representation, which is not a serious problem given that we are addressing the longevity risk. Finally, we mention that, while keeping the setting as a static one, possible updates to the weights (7.2) based on experience could be introduced; an example in this respect is described in Section 7.2.4. According to a dynamic probabilistic approach, either the probability of death or the force of mortality (or possibly some other quantity) is modelled as a path of a random process. In this context, the probabilistic model consists of assumptions concerning the random process and the relevant parameters. In the current literature, many authors have been focussing on this approach. Most investigations, which are, in particular, motivated by the problem of setting a pricing structure for longevity securities, move from assumed similarities between the force of mortality and interest rates or simply from the assumption that the market for longevity securities should behave like other aspects of the capital market. The application to mortality of some stochastic models developed originally for financial purposes is then tested. In particular, interest rate models and credit risk models have been considered. However, financial models are not necessarily suitable for describing mortality; actually, the force of mortality and interest rates do not necessarily behave in a similar manner. Therefore, the basic building blocks of the new theory still require careful discussion and investigation. Some examples are quoted in Section 7.6. It is important to note that the Lee–Carter model (see Chapters 5 and 6) is an early example of mortality modelled as a stochastic process. In its original version, deviations originating from sampling errors are in particular addressed, and hence process risk is considered. Indeed, when stochastic processes are adopted, certainly the intrinsic stochastic nature of mortality,

275

276


that is, random fluctuations, must be acknowledged. To represent also aggregate mortality risk, a second source of randomness must be introduced. So, in recent proposals, mortality is described as a doubly stochastic process. In particular when, moving from financial modelling, diffusion processes are considered for the force of mortality, unexpected movements in the mortality curve may be accounted for through stochastic jumps. See Section 7.8 for some references.

7.2.4

Measuring longevity risk in a static framework

In this section we highlight the impact of longevity risk. With reference to a portfolio comprising one cohort of annuitants, the distribution of both the present value of future payments and annual outflows is investigated. What follows can be referred also to a cohort of pensioners, with a proper adjustment of the parameters of the mortality model; in either case, a homogeneous group is considered. As mentioned in Section 7.1, for brevity explicit reference is to annuitants only. Similarly, the provider could be an insurer or a pension fund; however, we refer explicitly just to an insurer. A static representation is considered for evolving mortality and, in particular, parameter risk is addressed. To understand better the impact of longevity risk, a comparison is made with process risk. We assume

qx (t) = G(τ) (K(τ))x px (t)

(7.3)

where τ = t − x is the year of birth of the cohort. Hence, the third term of the first Heligman–Pollard law, that is, the one describing the old-age pattern of mortality, is adopted to express the time-pattern of mortality (see Section 2.5.2). Note, in particular, that the relevant parameters are cohort-specific. Whilst the age-pattern of mortality for cohort τ is accepted to be logistic, namely G(τ) (K(τ))x qx (t) = (7.4) 1 + G(τ) (K(τ))x see (2.85) (see also the second Heligman–Pollard law in Section 2.5.2), uncertainty concerns the level of parameters G(τ), K(τ). Actually, our investigation focusses on parameter risk and we note that such uncertainty may, in particular, arise from an underlying unknown cohort effect. We define five alternative sets of parameters, quoted in Table 7.1 which also shows the expected lifetime (E[T65 |Ah (τ)]) and the standard deviation ( Var[T65 |Ah (τ)]) of the lifetime at age 65 conditional on a given set of

7.2 Table 7.1.

The longevity risk

Parameters for the Heligman–Pollard law

A1 (τ) G(τ) 6.378E-07 K(τ) 1.14992 E[T65 |Ah (τ)] 20.170 Var[T65 |Ah (τ)] 7.796

A2 (τ)

A3 (τ)

A4 (τ)

A5 (τ)

3.803E-06 1.12347 20.743 8.780

2.005E-06 1.13025 21.849 8.707

1.060E-06 1.13705 22.887 8.602

3.149E-06 1.11962 24.187 9.910

parameters. It emerges that, in terms of the survival function itself, the alternative assumptions imply different levels of rectangularization (i.e. squaring of the survival function, as it is witnessed by Var[T65 |Ah (τ)]) and expansion (i.e. forward shift of the adult age at which most deaths occur, which is then reflected in the value of E[T65 |Ah (τ)]) (see Sections 3.3.6 and 4.1 for the meaning of rectangularization and expansion). The relevant survival functions and curves of deaths are plotted in Fig. 7.3. Assumption A3 (τ) will be referred to as the best-estimate description of the mortality trend for cohort τ; its parameters have been obtained by fitting (7.3) to the current market Italian projected table for immediate annuities (named IPS55). When comparing the values taken by E[T65 |Ah (τ)] and Var[T65 |Ah (τ)] (quoted in Table 7.1) under the various assumptions, it turns out that in respect of A3 (τ) at age 65: – assumption A1 (τ) implies a lighter expansion (i.e. lower expected lifetime) joint with a stronger rectangularization (i.e. lower standard deviation of the lifetime); – assumption A2 (τ) implies a lighter expansion and rectangularization as well; – assumption A4 (τ) implies a stronger expansion and rectangularization as well; – assumption A5 (τ) implies a stronger expansion joint with a lighter rectangularization. In each case, the maximum attainable age has been set equal to 117, according to the reference projected table. The portfolio we refer to consists of one cohort of immediate life annuities. We assume that all annuitants are aged x0 at the time t0 of issue. To shorten the notation, time t will be recorded as the time elapsed since the policy issue, that is, it is the policy duration; hence, at policy duration t the underlying calendar year is t0 + t. The lifetimes of annuitants are assumed, conditional on any given survival function, to be independent of each other and identically distributed. Since our objective is the

277


1 0.9

Number of survivors

0.8 0.7 0.6

A1 A2 A3 A4 A5

0.5 0.4 0.3 0.2 0.1 0

65

75

85

95

105

115

Age 0.06

0.05

Number of deaths

278

0.04 A1 A2 A3 A4 A5

0.03

0.02

0.01

0 65

75

85

95

105

115

Age Figure 7.3. Survival functions (top panel) and curves of deaths (bottom panel) under the Heligman–Pollard law.

measurement of longevity risk only, we disregard uncertainty in financial markets; hence, a given flat yield curve is considered. All of the annuitants are entitled to a fixed annual amount (participating mechanisms are not allowed for). Finally, we focus on net outflows; therefore, expenses and related expense loadings are not accounted for.

7.2

The longevity risk

Let Nt be the random number of annuitants at time t, t = 0, 1, . . . , with N0 a specified number (viz the initial size of the portfolio). Whenever the current size of the portfolio is an observed quantity, we will denote it as nt ; so N0 = n0 . Quantities relating to the generic annuitant are labelled with (j) on the top, j = 1, 2, . . . , n0 . The in-force portfolio at policy time t is defined as (j) t = {j|Tx0 > t} (7.5) Quantities relating to the portfolio are then labelled with () on the top. Annual outflows for the portfolio are defined, for t = 1, 2, . . . , as () b(j) (7.6) Bt = j:j∈t

where b(j) is the annual amount to annuitant j. The present value of future payments at time t, t = 0, 1, . . . , may at first be defined for one annuitant as (j)

Yt = b(j) aK(j)

(7.7)

x0

(see Section 1.5.1). By summing up in respect of in-force policies, we obtain the present value of future payments for the portfolio (j) () Yt (7.8) Yt = j:j∈t ()

()

We are interested in investigating some typical values of Bt and Yt , as well as the coefficient of variation and some percentiles. We will in particular consider the impact of longevity risk in relation to the size of the portfolio. So, unless otherwise stated, a homogeneous portfolio in respect of annual amounts is considered; that is, we set b(j) = b for all j. Note that in this case (7.6) may be rewritten as () Bt = b Nt (7.9) whilst the present value of future payments for the portfolio may also be expressed as () Yt

=

ω−x 0 h=t+1

() Bh (1 + i)−(h−t)

=

ω−x 0

b Nh (1 + i)−(h−t)

(7.10)

h=t+1

where i is the annual interest rate. For a homogeneous portfolio, in the (1) following Yt is used to denote the present value of future payments to a generic annuitant.

279

280


We first adopt approach 4 described in Section 7.2.2 (see also Fig. 7.2). All valuations are then conditional on a given mortality assumption. We have E[Yt() |Ah (τ), nt ] = nt E[Yt(1) |Ah (τ)] (7.11) Because we are assuming independence of the annuitants’ lifetimes, conditional on a given mortality trend, the following results hold:

Var[Yt() |Ah (τ), nt ] = nt Var[Yt(1) |Ah (τ)] . Var[Yt(1) |Ah (τ)] 1 () CV[Yt |Ah (τ), nt ] = √ nt E[Yt(1) |Ah (τ)]

(7.12) (7.13)

where nt is the size of the in-force portfolio, observed at the valua(1) (1) tion policy time t. (Expressions for E[Yt |Ah (τ)] and Var[Yt |Ah (τ)] are straightforward and therefore omitted.) The coefficient of variation, in particular, allows us to investigate the effect of the size of the portfolio on the overall riskiness. Expression (7.13) shows that, in relative terms, the riskiness of the portfolio decreases as nt increases. Thus, we have ()

lim CV[Yt

nt ∞

|Ah (τ), nt ] = 0

(7.14)

This represents the well-known result that the larger is the portfolio, the less risky it is, since with high probability the observed values will be close () to the expected ones. The quantity CV[Yt |Ah (τ), nt ] is sometimes called the risk index. Conditional on a given mortality assumption and because of the independence among the lifetimes of the annuitants and the assumption of () homogeneity of annual amounts, the percentiles of Yt could be assessed through a process of convolution. In practice, however, due the number () of random variables constituting Yt (i.e. due to the magnitude of nt ), analytical calculations are not practicable and so we must resort to stochas() tic simulation. The ε-percentile of the distribution of Yt conditional on assumption Ah (τ) and an observed size of the in-force portfolio nt at time t is defined as

# () yt,ε [Ah (τ), nt ] = inf u ≥ 0 # P Yt ≤ u|Ah (τ), nt > ε (7.15) ()

In particular, we are interested in investigating the right tail of Yt therefore, high values for ε should be considered. ()

;

As far as the distribution of annual outflows Bt is concerned, simi() lar remarks to those for Yt can be made. Thus, due to independence

7.2

The longevity risk

Table 7.2. Expected present value of future payments, conditional on a given scenario, per policy in-force at time ()

t:

E[Yt

|Ah (τ),nt ] (1) = E[Yt |Ah (τ)] nt

Assumption Time t 0 5 10 15 20 25 30 35

A1 (τ)

A2 (τ)

A3 (τ)

A4 (τ)

A5 (τ)

14.462 12.004 9.504 7.102 4.962 3.221 1.944 1.099

14.651 12.374 10.076 7.862 5.846 4.127 2.766 1.765

15.259 12.956 10.599 8.294 6.167 4.336 2.877 1.807

15.817 13.500 11.097 8.714 6.484 4.543 2.988 1.849

16.413 14.238 11.981 9.724 7.570 5.626 3.980 2.681

()

and homogeneity, the random variables Bt have (under the information available at time 0) a binomial distribution, with parameters n0 and the survival probability from issue time to policy duration t calculated under the given mortality assumption. For reasons of space, we omit the relevant expressions (which are straightforward). Example 7.1 In the following tables, we provide an example, in which the age at entry is x0 = 65, the interest rate is 3% p.a., the annual amount () of each annuity is b(j) = 1. It then follows that Bt = Nt . In Table 7.2, the expected present value of future payments is presented, per annuitant (having set, under each assumption Ah (τ), nt = E[Nt |Ah (τ)] for each valuation time t, t = 0, 5, . . . , 35). As was clear from the assumptions (see also Table 7.1), at the time of issue the five assumptions (ordered from 1 to 5) imply an increasing expected present value of future payments. The comparison may change in later years, due to the shape of the survival function for a given assumption (actually, some survival functions cross over each other; see Fig. 7.3, top panel). From these results, we get an idea about the possible range of variation of the current value of liabilities, due to uncertainty about the mortality trend. In Table 7.3, we present the variance of the present value of future payments, per annuitant. The illustrated variability is a consequence of the rectangularization level implied by the different assumptions. We recall that only process risk is accounted for in this assessment; so when addressing longevity risk such information is not of intrinsic interest, but is helpful for comparison with the impact of longevity risk. To compare longevity risk with process risk, we make some further calculations involving process risk only. Thus, Tables 7.4 and 7.5 show,

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282

7 : The longevity risk: actuarial perspectives Table 7.3. Variance of the present value of future payments, conditional on a given scenario, per policy in-force ()

at time t:

Var[Yt

|Ah (τ),nt ] (1) = Var[Yt |Ah (τ)] nt

Assumption Time t 0 5 10 15 20 25 30 35

A1 (τ)

A2 (τ)

A3 (τ)

A4 (τ)

A5 (τ)

20.838 20.858 18.963 15.314 10.777 6.550 3.479 1.677

25.301 24.858 22.607 18.780 14.092 9.497 5.771 3.217

23.804 23.994 22.375 19.008 14.505 9.855 5.969 3.277

22.250 22.985 21.970 19.095 14.838 10.181 6.159 3.337

25.315 26.102 25.229 22.581 18.493 13.726 9.198 5.594

Table 7.4. Coefficient of variation of the present value of future payments, () conditional on the best-estimate scenario: CV[Yt |A3 (τ), nt ]

Initial portfolio size Time t 0 5 10 15 20 25 30 35

n0 = 1

n0 = 100

n0 = 1 000

n0 = 10 000

…

n0 ∞

31.974% 38.514% 47.039% 58.973% 77.647% 111.894% 189.580% 424.200%

2.982% 3.618% 4.452% 5.626% 7.469% 10.853% 18.541% 41.832%

0.969% 1.156% 1.397% 1.734% 2.259% 3.218% 5.379% 11.815%

0.027% 0.030% 0.038% 0.056% 0.103% 0.250% 0.883% 5.202%

… … … … … … … …

0% 0% 0% 0% 0% 0% 0% 0%

respectively, the coefficient of variation for some initial sizes of the portfolio and some percentiles of the present value of future payments, per unit of expected value. Only the best-estimate assumption is considered. As far as the coefficient of variation is concerned, we note that at any given time it decreases rapidly as the size of the portfolio increases, as we commented on earlier. For a given initial portfolio size, the coefficient of variation increases in time; this is due to the decreasing residual size of the portfolio and to annuitants becoming older as well. A similar result is found when analysing the right tail of the distribution, as it emerges in Table 7.5. Tables 7.6 and 7.7 give a highlight on the distribution of annual outflows. In particular, Table 7.6 quotes the expected value of annual outflows under the different assumptions; we recall that, having set b(j) = 1, what is shown is the expected number of annuitants (not rounded, to avoid too many approximations). Remarks are similar to those discussed for the present value of future payments.

7.2

The longevity risk

Table 7.5. Some percentiles of the present value of future payments, conditional on the best-estimate scenario, per unit of expected value: yt,ε [A3 (τ),nt ] ()

E[Yt

|A3 (τ),nt ]

Probability Time t

ε = 0.75

ε = 0.90

ε = 0.95

ε = 0.99

4.983% 6.266% 7.904% 9.857% 12.898% 20.338% 33.982% 83.067%

6.739% 8.554% 11.289% 13.801% 17.866% 26.503% 47.540% 130.409%

1.631% 1.934% 2.423% 2.854% 3.781% 5.443% 9.765% 22.253%

2.286% 2.668% 3.386% 4.472% 6.223% 7.967% 14.334% 31.641%

0.523% 0.609% 0.786% 0.967% 1.239% 1.765% 3.090% 6.048%

0.733% 0.850% 1.051% 1.385% 1.761% 2.705% 4.309% 10.430%

Initial portfolio size: n0 = 100 0 5 10 15 20 25 30 35

2.159% 2.500% 3.074% 3.738% 5.418% 8.319% 13.658% 32.386%

3.995% 4.863% 6.110% 7.161% 10.393% 15.577% 26.115% 63.107%

Initial portfolio size: n0 = 1,000 0 5 10 15 20 25 30 35

0.635% 0.820% 0.898% 1.131% 1.354% 2.117% 3.638% 9.155%

1.286% 1.531% 1.923% 2.221% 2.692% 4.281% 7.355% 18.426%


0.200% 0.238% 0.332% 0.415% 0.518% 0.670% 1.165% 2.323%

0.407% 0.461% 0.622% 0.739% 0.968% 1.414% 2.317% 4.661%

We now assign the (naive) probability distribution (7.2) on the set A(τ). ˜ The unknown mortality trend, assumed to lie in A(τ), is denoted by A(τ). For the unconditional expected present value of future payments, the following relations hold (the suffix ρ denotes that the underlying probability distribution is given by (7.2)): (1) ˜ ˜ E[Yt() |nt ] = Eρ [E[Yt() |A(τ), nt ]] = nt Eρ [E[Yt |A(τ)]]

= nt

m h=1

E[Yt(1) |Ah (τ)] ρh = nt E[Yt(1) ]

(7.16)

283

284

7 : The longevity risk: actuarial perspectives Table 7.6. Expected value of annual outflows, conditional on a () given scenario: E[Bt |Ah (τ)]; initial portfolio size: n0 = 1,000

Assumption A1 (τ)

A2 (τ)

A3 (τ)

A4 (τ)

A5 (τ)

963.105 893.255 768.675 570.930 319.516 105.929 14.160

954.252 877.810 756.711 581.960 367.226 165.716 43.221

963.630 900.177 794.479 632.539 418.752 200.690 55.764

971.087 918.558 826.872 678.377 468.784 237.508 70.089

969.636 918.556 835.463 707.957 530.954 323.526 139.572

Time t 5 10 15 20 25 30 35

Table 7.7. Coefficient of variation of annual outflows, () conditional on the best-estimate scenario: CV[Bt |A3 (τ)]; initial portfolio size: n0 = 1,000

(1)

Time t

n0 = 100

n0 = 1,000

n0 = 10,000

5 10 15 20 25 30 35

1.943% 3.330% 5.086% 7.622% 11.782% 19.957% 41.150%

0.614% 1.053% 1.608% 2.410% 3.726% 6.311% 13.013%

0.194% 0.333% 0.509% 0.762% 1.178% 1.996% 4.115%

The

m

(1) h=1 E[Yt |Ah (τ)] ρh . () unconditional variance of Yt can

where E[Yt ] =

be calculated as

() ˜ ˜ Var[Yt() |nt ] = Eρ [Var[Yt() |A(τ), nt ]] + Varρ [E[Yt |A(τ), nt ]] (1)

(1)

˜ ˜ + nt2 Varρ [E[Yt |A(τ)]] = nt Eρ [Var[Yt |A(τ)]] = nt

m

Var[Yt(1) |Ah (τ)] ρh

h=1

+ nt2

m 2 E[Yt(1) |Ah (τ)] − E[Yt(1) ] ρh

(7.17)

h=1

The first term in the expression for the variance reflects deviations around the expected value; so it can be thought of as a measure of process risk. The second term, instead, reflects deviations from the expected value (i.e. systematic deviations) and so it may be thought of as a measure of longevity (namely parameter, in our example) risk. Under the unconditional

7.2

The longevity risk

valuation, the coefficient of variation now takes the following expression: . Var[Yt() ] () CV[Yt |nt ] = E[Yt() ] 2 3 3 1 Eρ [Var[Y (1) |A(τ)]] ˜ ˜ Varρ [E[Yt(1) |A(τ)]] t + (7.18) =4 nt E2 [Yt(1) ] E2 [Yt(1) ] The first term under the square root shows that random fluctuations represent a pooling risk, since (in relative terms) their effect is absorbed by the size of the portfolio. This result is similar to that obtained under the valuation conditional on a given mortality trend (see (7.13)). The second term, instead, shows that systematic deviations constitute a non-pooling risk, which is not affected by changes in the portfolio size. In particular, the asymptotic value of the risk index 2 3 3 Varρ [E[Y (1) |A(τ)]] ˜ () t lim CV[Yt |nt ] = 4 (7.19) (1) nt ∞ E2 [Yt ] can be thought of as a measure of that part of the mortality risk which is not affected by simply changing the size of the portfolio. ()

The ε-percentile of the unconditional probability distribution of Yt under an observed size of the in-force portfolio nt at time t is defined as

# () (7.20) yt,ε [nt ] = inf u ≥ 0 # P Yt ≤ u|nt > ε To assess this quantity, stochastic simulation is required, through which first the mortality trend is randomly picked up from A(τ), and then the lifetimes of annuitants are generated. In regard of annual amounts, similar valuations and comments can be performed. Example 7.2 We now describe a numerical example of the results presented above. We consider the same inputs of Example 7.1. We assign to A(τ) the weights quoted in Table 7.8. The best-estimate assumption (A3 (τ)) has been given the highest weight. The residual weight has been spread out uniformly on the remaining assumptions. Table 7.9 shows the unconditional expected value of future payments. Its magnitude is driven by the best-estimate assumption, as seen by comparison with the results in Table 7.2.

285

286

7 : The longevity risk: actuarial perspectives Table 7.8. Probability distribution on A(τ)

Assumption

Weight ρh

A1 (τ) A2 (τ) A3 (τ) A4 (τ) A5 (τ)

0.1 0.1 0.6 0.1 0.1

Table 7.9. (Unconditional) expected present value of future payments, per policy in-force at time t

Time t 0 5 10 15 20 25 30 35

()

E[Yt

nt

|nt ]

(1)

= E[Yt ]

15.290 12.985 10.625 8.317 6.187 4.353 2.894 1.824

()

In Table 7.10, the unconditional variance of Yt for some portfolio sizes is shown, split into the pooling and non-pooling components. For comparison with the conditional valuation, also the case n0 = 1 is quoted. We note the increase in the magnitude of the variance, due to the nonpooling part, as the portfolio size increases. Whenever the portfolio is large at policy issue, the non-pooling component remains important relative to the pooling component even at high policy durations. The behaviour of the coefficient of variation in respect of the portfolio size is illustrated in Table 7.11. When compared with the case allowing for process risk only (see Table 7.4), the risk index decreases more slowly as the portfolio size increases. We note, in particular, its positive limiting value, which is evidence of the magnitude of the systematic risk. In Table 7.12 the right tail of the distribution of the present value of future payments is investigated, for some portfolio sizes. We note that the tail is rather heavier than in the case allowing for process risk only (see Table 7.5). Finally, in Tables 7.13–7.15 the distribution of annual outflows is investigated. Similar remarks hold to those made above for the distribution of future payments.

7.2

The longevity risk

Table 7.10. (Unconditional) variance of the present value of future payments per policy in-force at time t, and components

Time t

Variance ()

Var[Yt

|nt ]

nt

Pooling part ()

Eρ [Var[Yt

˜ |A(τ),n t ]]

()

Var[Yt

|nt ]

Non-pooling part ()

Varρ [E[Yt

˜ |A(τ),n t ]]

()

Var[Yt

|nt ]


23.916 23.303 20.369 15.322 9.331 4.202 1.221 0.187

98.90% 98.73% 98.56% 98.43% 98.45% 98.75% 99.30% 99.79%

1.10% 1.27% 1.44% 1.57% 1.55% 1.25% 0.70% 0.21%


50.026 52.493 49.436 39.209 23.670 9.391 2.062 0.226

47.28% 43.83% 40.61% 38.46% 38.81% 44.18% 58.81% 82.60%

52.72% 56.17% 59.39% 61.54% 61.19% 55.82% 41.19% 17.40%


287.390 317.858 313.680 256.365 154.023 56.568 9.707 0.580

8.23% 7.24% 6.40% 5.88% 5.96% 7.33% 12.49% 32.20%

91.77% 92.76% 93.60% 94.12% 94.04% 92.67% 87.51% 67.80%


2 661.023 2 971.508 2 956.118 2 427.922 1 457.548 528.334 86.159 4.120

0.89% 0.77% 0.68% 0.62% 0.63% 0.79% 1.41% 4.53%

99.11% 99.23% 99.32% 99.38% 99.37% 99.21% 98.59% 95.47%

We finally address the problem of choosing the weights (7.2). As we have already mentioned, data to estimate such weights are available rarely. However, some numerical tests suggest that the weights do not deeply affect the results of the investigation, unless only process risk is allowed for. We show this effect in Example 7.3.

287

288

7 : The longevity risk: actuarial perspectives Table 7.11. (Unconditional) coefficient of variation of the present value of future () payments: CV[Yt |nt ]


n0 = 1

n0 = 100

n0 = 1,000

n0 = 10,000

…

n0 ∞

31.985% 38.579% 47.188% 59.241% 78.060% 112.448% 190.275% 425.348%

4.626% 5.790% 7.351% 9.477% 12.432% 16.811% 24.726% 46.751%

3.506% 4.506% 5.856% 7.663% 10.029% 13.048% 16.965% 23.680%

3.374% 4.356% 5.685% 7.457% 9.756% 12.610% 15.983% 19.957%

… … … … … … … …

3.359% 4.339% 5.665% 7.434% 9.725% 12.560% 15.870% 19.499%

Table 7.12.

Some (unconditional) percentiles of the present value of

future payments, per unit of expected value:

yt,ε [nt ] ()

E[Yt

|nt ]

Probability Time t

ε = 0.75

ε = 0.90

ε = 0.95

ε = 0.99

7.796% 10.041% 14.492% 18.412% 25.518% 30.693% 46.114% 92.929%

11.465% 14.405% 19.236% 25.884% 34.996% 52.379% 75.150% 138.271%

7.357% 9.654% 12.471% 16.712% 22.206% 29.772% 37.774% 48.299%

8.591% 11.276% 14.889% 19.332% 25.939% 34.880% 47.734% 72.891%

7.298% 9.695% 12.743% 16.754% 22.162% 28.886% 36.976% 47.056%

7.726% 10.062% 13.304% 17.552% 23.146% 30.326% 40.782% 52.805%


3.473% 3.453% 4.638% 5.293% 7.968% 14.135% 17.964% 33.817%

6.175% 8.010% 10.290% 13.350% 17.498% 23.243% 37.964% 70.810%


1.057% 1.378% 1.878% 2.108% 2.797% 3.417% 5.100% 14.933%

5.164% 6.300% 7.501% 9.426% 11.838% 14.771% 19.791% 27.796%


0.261% 0.189% 0.254% 0.316% 0.461% 0.799% 1.571% 2.902%

4.124% 4.800% 5.417% 6.102% 6.499% 6.417% 7.292% 13.794%

7.2

The longevity risk

Table 7.13. (Unconditional) expected value of annual out() flows: E[Bt ]; initial portfolio size: n0 = 1,000 ()

Time t

E[Bt

]

5 10 15 20 25 30 35

963.986 900.924 795.459 633.446 419.899 203.682 60.162

Table 7.14. Components of the (unconditional) variance of annual outflows

Pooling part Time t

()

Eρ [Var[Bt

˜ |A(τ)]]

()

Var[Bt

]

Non-pooling part ()

Varρ [E[Bt

˜ |A(τ)]]

()

Var[Bt

]

Initial portfolio size: n0 = 100 5 10 15 20 25 30 35

99.488% 98.652% 97.119% 94.229% 89.724% 85.729% 86.181%

0.512% 1.348% 2.881% 5.771% 10.276% 14.271% 13.819%

Initial portfolio size: n0 = 1,000 5 10 15 20 25 30 35

95.104% 87.976% 77.124% 62.016% 46.613% 37.528% 38.409%

4.896% 12.024% 22.876% 37.984% 53.387% 62.472% 61.591%

Initial portfolio size: n0 = 10,000 5 10 15 20 25 30 35

66.013% 42.253% 25.214% 14.036% 8.030% 5.667% 5.870%

33.987% 57.747% 74.786% 85.964% 91.970% 94.333% 94.130%

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7 : The longevity risk: actuarial perspectives Table 7.15. (Unconditional) coefficient of variation () of annual outflows: CV[Bt ]

Time t

n0 = 100

n0 = 1,000

n0 = 10,000

5 10 15 20 25 30 35

6.143% 10.531% 16.084% 24.103% 37.257% 63.110% 130.126%

1.943% 3.330% 5.086% 7.622% 11.782% 19.957% 41.150%

0.614% 1.053% 1.608% 2.410% 3.726% 6.311% 13.013%

Table 7.16. Alternative probability distributions on A(τ)

Weighting system Weight ρ1 ρ2 ρ3 ρ4 ρ5

(a)

(b)

(c)

(d)

0 0 1 0 0

0.1 0.1 0.6 0.1 0.1

0.15 0.15 0.4 0.15 0.15

0.2 0.2 0.2 0.2 0.2

Example 7.3 In this example, we compare the right tail of the distribution of the present value of future payments assuming the alternative weighting systems for (7.2) that are presented in Table 7.16. System (a) is the one allowing for process risk only (see Example 7.1). System (b) is the one adopted in Example 7.2. System (c) is similar to (b), with the highest weight assigned to the best-estimate assumption; however, such weight has been reduced. System (d), finally, consists of a uniform distribution of weights. We focus on the right tail of the distribution of the present value of future payments (and not on the other risk measures considered previously, such as the risk index) due to its practical importance. Actually, reserving or capital allocation could be based on this quantity (see also Section 7.3.3). From the details presented for this example in Table 7.17, it seems that whenever parameter risk is allowed for, the magnitude of the right tail is not deeply affected by the weighting system (although, of course, the actual figure does depend on the specific weights). Indeed, an apparent difference emerges between results found under system (a), on the one hand, and systems (b)–(d), on the other. This suggests that, having poor information, the allowance for longevity risk is more important than the actual choice of the weights.

7.2 Table 7.17.

The longevity risk

Some (unconditional) percentiles of the present value of future payments, per unit

of expected value:

yt,ε [nt ] ()

E[Yt

|nt ]

, under alternative weighting systems; n0 = 1, 000

Probability Time t

ε = 0.75

ε = 0.90

ε = 0.95

ε = 0.99

0.635% 0.820% 0.898% 1.131% 1.354% 2.117% 3.638% 9.155%

1.286% 1.531% 1.923% 2.221% 2.692% 4.281% 7.355% 18.426%

1.631% 1.934% 2.423% 2.854% 3.781% 5.443% 9.765% 22.253%

2.286% 2.668% 3.386% 4.472% 6.223% 7.967% 14.334% 31.641%

1.057% 1.378% 1.878% 2.108% 2.797% 3.417% 5.100% 14.933%

5.164% 6.300% 7.501% 9.426% 11.838% 14.771% 19.791% 27.796%

7.357% 9.654% 12.471% 16.712% 22.206% 29.772% 37.774% 48.299%

8.591% 11.276% 14.889% 19.332% 25.939% 34.880% 47.734% 72.891%

2.912% 3.206% 3.615% 3.881% 4.258% 5.047% 6.192% 16.809%

6.785% 8.850% 11.893% 15.645% 20.997% 27.391% 35.431% 41.965%

7.652% 9.822% 13.119% 17.404% 23.363% 31.364% 41.423% 54.794%

8.539% 11.384% 15.178% 19.643% 26.253% 35.474% 49.504% 74.366%

3.697% 4.609% 5.079% 5.480% 5.929% 7.025% 8.782% 19.443%

7.142% 9.408% 12.195% 16.397% 21.825% 29.249% 36.966% 46.965%

7.642% 10.362% 13.497% 17.687% 23.725% 32.239% 42.059% 60.689%

8.508% 11.702% 15.037% 19.671% 26.542% 35.218% 50.565% 74.138%

System (a) 0 5 10 15 20 25 30 35 System (b) 0 5 10 15 20 25 30 35 System (c) 0 5 10 15 20 25 30 35 System (d) 0 5 10 15 20 25 30 35

We have noted that the most important aspect is to allow for parameter risk by assigning positive weights to trend assumptions alternative to the best-estimate one. However, the specific weights do affect the magnitude of quantities of interest (such as the tail of the distribution of future payments).

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A Bayesian inferential model could provide an appropriate method for updating the weights. We briefly discuss how one could structure such a procedure. We still refer to a cohort of annuitants, which is homogeneous and, conditional on a given trend, with identically distributed and independent lifetimes. The observed number of annuitants at time t is nt . As the static approach for stochastic mortality evolution, we assume that the trend of the cohort is unknown, but fixed (i.e. not subject neither to shocks nor to unanticipated shifts). The set of trend assumptions is given by (7.1). In the current context, the set of weights (7.2) will be denoted as {ρ(Ah (τ))}h=1,2,...,m

(7.21)

We let f0 (t|A(τ)) denote the probability density function (briefly: pdf) of the lifetime at birth of one individual, conditional on assumption A(τ) about the mortality trend. We then let S(t|A(τ)) denote the relevant survival function. Within the inferential procedure, the sampling pdf is defined as follows: 0 for z ≤ t ft (z|A(τ)) = f0 (z|A(τ)) (7.22) for z > t S(t|A(τ)) The multivariate sampling pdf is then given by ft (z(1) , z(2) , . . . , z(nt ) |A(τ)) =

nt !

ft (z(j) |A(τ))

(7.23)

j=1

Note that ft (z) =

m

ft (z|A(τ)) ρ(Ah (τ))

(7.24)

h=1

represents the (prior) predictive pdf restricted to the age interval [t, ω − x0 ]. Assume now the observation period [t, t ]. Let d denote the number of deaths observed in such period. With an appropriate renumbering, let x = {x(1) , x(2) , . . . , x(d) }

(7.25)

denote the array of ages at death. We note that the defined observation procedure implies a Type I-censored sampling (see, for instance, Namboodiri and Suchindran (1987)). Using the information provided by the pair (d, x), the (posterior) predictive pdf ft (z|d, x) can be constructed. With this objective in mind, we can adopt the following procedure (usual in the Bayesian context):

7.3

Managing the longevity risk

1. Update the initial opinion about the possible evolution of mortality, and hence about the probability distribution over the set of trend assumptions A(τ), by calculating the posterior pdf ρ(Ah (τ)|d, x) ∝ ρ(Ah (τ)) L(Ah (τ|d, x))

(7.26)

where L(Ah (τ|d, x)) denotes the likelihood function; 2. Calculate the (posterior) predictive pdf as ft (z|d, x) =

m

ft (z|Ah (τ)) ρ(Ah (τ)|d, x)

(7.27)

h=1

Step 1 requires the construction of the likelihood function L(Ah (τ|d, x)). We have (see, e.g. Namboodiri and Suchindran (1987)):   d ! S(t |Ah (τ)) nt −d L(Ah (τ|d, x)) ∝  ft (x(k) − t|Ah (τ)) (7.28) S(t|Ah (τ)) k=1

The inferential procedure described above could be adopted within internal solvency models, whenever alternative projected mortality tables are available. Some numerical investigations in this regard are discussed by Olivieri and Pitacco (2002a).

7.3 7.3.1

Managing the longevity risk A risk management perspective

Several tools can be developed to manage longevity risk. These tools can be placed and analysed in a risk management (RM) framework. As sketched in Fig. 7.4, the RM process consists of three basic steps, namely the identification of risks, the assessment (or measurement) of the relevant consequences, and the choice of the RM techniques. In what follows we refer to the RM process applied to life insurance, in general, and to life annuity portfolios, in particular. The identification of risks affecting an insurer can follow, for example, the guidelines provided by IAA (2004) or those provided within the Solvency 2 project (see CEIOPS, 2007 and CEIOPS, 2008). Mortality/longevity risks belong to underwriting risks; the relevant components have already been discussed (see Section 7.2.1). Obviously, for an insurer the importance of the longevity risk within the class of mortality risks is strictly related to the

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IDENTIFICATION

UNDERWRITING RISK Mortality/Longevity risk – Volatility – Level uncertainty – Trend uncertainty – Catastrophe Lapse risk ... MARKET RISK ...

ASSESSMENT

DETERMINISTIC MODELS Sensitivity testing Scenario testing STOCHASTIC MODELS Risk index, VaR, Probability of default ...

RISK MANAGEMENT TECHNIQUES LOSS CONTROL Loss prevention (frequency control) Loss reduction (severity control) RISK MITIGATION

LOSS FINANCING Hedging Transfer Retention

Figure 7.4.

PORTFOLIO STRATEGIES

PRODUCT DESIGN Pricing (life table, guarantees, options, expense loading, etc.) Participation mechanism PORTFOLIO PROTECTION Natural hedging Reinsurance, ART No advance funding Capital allocation

The risk management process.

relative weight of the life annuity portfolio with respect to the overall life business. A rigorous assessment of the longevity risk requires the use of stochastic models (i.e. approach 5 in Fig. 7.2). In Section 7.2.4 we have provided some examples of risk measurement, viz the variance, the coefficient of variation, and the right tail of liabilities – these need to be appropriately defined; in Section 7.2.4 they were stated in terms of the present value of future payments and of annual outflows. A further example is given by

7.3

Expected values

Threshold

Annual outflows

Actual outflows


Time Figure 7.5.

Annual outflows in a portfolio of immediate life annuities (one cohort).

the probability of default (or ruin probability, in the traditional language), which will be considered in Section 7.3.3 when dealing with the solvency problem. As discussed in Section 7.2.2, deterministic models (i.e. approach 4 in Fig. 7.2) can provide useful, although rough, insights into the impact of longevity risk on portfolio results. In particular, as outlined in Sections 7.2.3 and 7.2.4, deterministic models allow us to calculate the range of values that some quantities (present value of future payments, annual outflows, or others) may assume in respect of the outcome of the underlying random quantity. Risk management techniques for dealing with longevity risk include a wide set of tools, which can be interpreted, under an insurance perspective, as portfolio strategies, aimed at risk mitigation. A number of portfolio results can be taken as ‘metrics’ to assess the effectiveness of portfolio strategies. In what follows, we focus on annual outflows relating to annuity payments only, which, in any event, constitute the starting point from which other quantities (e.g. profits) may be derived. In Fig. 7.5, we present a sequence of outflows, together with a barrier (the ‘threshold’) which represents a maintainable level of benefit payment. The threshold amount is financed first by premiums via the portfolio technical provision, and then by shareholders’ capital as the result of the allocation policy (consisting of specific capital allocations as well as an accumulation of undistributed profits).

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The situation occurring in Fig. 7.5, namely, some annual outflows being above the threshold level, should be clearly avoided. To lower the probability of such critical situations, the insurer can resort to various portfolio strategies, in the framework of the RM process. Figure 7.6 illustrates a wide range of portfolio strategies which aim at risk mitigation, in terms of lowering the probability and the severity of events like the situation depicted in Fig. 7.5. In practical terms, a portfolio strategy can have as targets (i) an increase in the maintainable annual outflow, and thus a higher threshold level; (ii) lower (and smoother) annual outflows in the case of unanticipated improvements in portfolio mortality. Both loss control and loss financing techniques (according to the RM language) can be adopted to achieve targets (i) and (ii). Loss control techniques are mainly performed via the product design, that is, via an appropriate choice of the various items which constitute an insurance product. In particular, loss prevention is usually interpreted as the RM technique which aims to mitigate the loss frequency, whereas loss reduction aims at lowering the severity of the possible losses. The pricing of insurance products provides a tool for loss prevention. This portfolio strategy is represented by path (1) → (a) in Fig. 7.6. Referring to

(1) Single premiums

Reserve (a) Threshold

(2) Allocation (3) Undistributed profits

Shareholders' capital

(4) Profit partic. (5) [Reduction]

Annual benefits

Gross outflow

(b) Net outflow

(6) Reinsurance (7) Swaps (8) Longevity bonds

Figure 7.6.

Portfolio strategies for risk mitigation.

Transfers

7.3


a life annuity product, the following issues, in particular, should be taken into account. – Mortality improvements require the use of a projected life table for pricing life annuities. – Because of the uncertainty in the future mortality trend, a premium formula other than the traditional one based on the equivalence principle (see Section 1.6.1, and formula (1.57) in particular) should be adopted. It should be noted that, by adopting the equivalence principle, the longevity risk can be accounted for only via a (rough) safety loading, which is calculated by increasing the survival probabilities resulting from the projected life table. Indeed, this approach is often adopted in current actuarial practice. – The presence, in an accumulation product such as an endowment, of an option to annuitize at a fixed annuitization rate (the so-called Guaranteed Annuity Option, briefly GAO – see Section 1.6.2) requires an accurate pricing model accounting for the value of the option itself. To pursue loss reduction, it is necessary to control the annuity amounts paid out. Hence, some flexibility must be added to the life annuity product. One action could be the reduction of the annual amount as a consequence of an unanticipated mortality improvement (path (5) → (b) in Fig. 7.6). However, in this case the product would be a non-guaranteed life annuity, although possibly with a reasonable minimum amount guaranteed. A more practicable tool, consistent with the features of a guaranteed life annuity, consists of reducing the level of investment profit participation when the mortality experience is adverse to the annuity provider (path (4) → (b)). It is worth stressing that undistributed profits also increase the shareholders’ capital within the portfolio, hence increasing the maintainable threshold (path (3) → (a)). Loss financing techniques require specific strategies involving the whole portfolio, and in some cases even other portfolios of the insurer. Risk transfer can be realized via (traditional) reinsurance arrangements (path (6) → (b)), swap-like reinsurance ((7) → (b)) and securitization, that is, Alternative Risk Transfer (ART). In the case of life annuities, ART requires the use of specific financial instruments, for example, longevity bonds ((8) → (b)), whose performance is linked to some measure of longevity in a given population. A comment is required on traditional risk transfer tools. Traditional reinsurance arrangements (e.g. surplus reinsurance, XL reinsurance, and so on) at least in principle can be applied also to life annuity portfolios.

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But, it should be stressed that such risk transfer solutions mainly rely on the improved diversification of risks when these are taken by the reinsurer, thanks to a stronger pooling effect. Notably, such an improvement can be achieved in relation to process risk (i.e. random fluctuations in the number of deaths), whilst uncertainty risk (leading to systematic deviations) cannot be diversified ‘inside’ the insurance–reinsurance process. Hence, to become more effective, reinsurance transfers must be completed with a further transfer, that is, a transfer to capital markets. Such a transfer can be realized via bonds, whose yield is linked to some mortality/longevity index, so that the bonds themselves generate flows which hedge the payment of life annuity benefits. While mortality bonds (hedging the risk of a mortality higher than expected) already exist, longevity bonds (hedging the risk of a mortality lower than expected) are yet to appear in the market. To the extent that mortality/longevity risks are retained by an insurer, the impact of a poor experience falls on the insurer itself. To meet an unexpected amount of obligations, an appropriate level of advance funding may provide a substantial help. To this purpose, shareholders’ capital must be allocated to the life annuity portfolio (path (2) → (a), as well as (3) → (a) in Fig. 7.6), and the relevant amount should be determined to achieve insurer solvency. Conversely, the expression ‘no advance funding’ (see Fig. 7.4) should be referred to the situations where no specific capital allocation is provided in respect of mortality/longevity risks. In the case of adverse experience, the unexpected amount of obligations has to be met (at least partially) by the available residual assets, which are not tied up to specific liabilities. Hedging strategies in general consist of assuming the existence of a risk which offsets another risk borne by the insurer. In some cases, hedging strategies involve various portfolios or lines of business (LOBs), or even the whole insurance company, so that they cannot be placed in the portfolio framework as depicted in Fig. 7.6. In particular, natural hedging (see Fig. 7.4) consists of offsetting risks in different LOBs. For example, writing both life insurance providing death benefits and life annuities for similar groups of policyholders may help to provide a hedge against longevity risk. Such a hedge is usually named across LOBs. A natural hedge can be realized even inside a life annuity portfolio, allowing for a death benefit (possibly decreasing as the age at death increases) combined with the life annuity; see Section 1.6.4. Clearly, in the case of a higher than anticipated mortality improvement, death benefits which are lower than expected will be paid. Such a hedge is usually called across time. Clearly, mortality/longevity risks should be managed by the insurer through an appropriate mix of the tools described above. The choice of the

7.3


RM tools is also driven by various interrelationships among the tools themselves. For example, the possibility of purchasing profitable reinsurance is strictly related with the features of the insurance product and, in particular, the life tables underlying the pricing, as well as with the availability of ART for the reinsurer. The following sections are devoted to an in-depth analysis of the RM tools which currently seem to be the most practicable.

7.3.2

Natural hedging

In the context of life insurance, natural hedging refers to a diversification strategy combining ‘opposite’ benefits with respect to the duration of life. The main idea is that if mortality rates decrease then life annuity costs increase while death benefit costs decrease (and vice versa). Hence the mortality risk inherent in a life annuity business could be offset, at least partially, by taking a position also on some insurance products providing benefits in the case of death. We discuss two situations, one concerning hedging across time and one across LOBs. We first consider hedging across time. We assume that at time 0 (i.e. calendar year x0 + t0 ) an immediate life annuity is issued to a person aged x0 , with the proviso that at death (e.g. at the end of the year of death) the mathematical reserve therein set up to meet the life annuity benefit (only) is paid back to the beneficiaries. Reasonably, the reserving basis concerning the death benefit should be stated at policy issue so that the death benefit, although decreasing over time, is guaranteed. At time 0, the random present value of future (life annuity and death) benefits for an individual (generically, individual j) is defined as follows: (j)

(j)

Y0 = b(j) aK(j) + (1 + i)−(Kx0 +1) C x0

(j)

(j)

Kx0 +1

(7.29)

(j)

where Ct is the death benefit payable at time t if death occurs in (t − 1, t), defined as follows (j) Ct

=

b(j) a[A] x0 +t

(j)

=b

ω−x 0 −t

(1 + i)−h h p[A] x0 +t

(7.30)

h=1 (j)

The benefit Ct is therefore the mathematical reserve set up at time t to meet the life annuity benefit, calculated according to the mortality assumption A(τ) and the annual interest rate i. Note that the individual reserve (meeting both the life annuity and the death benefit) to be set up at time t according

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to the (traditional) equivalence principle is (j) Vt

(j)

=b

ax0 +t +

ω−x 0 −t

h/1 qx0 +t

(j)

(1 + i)−(h+1) Ct+h+1

(7.31)

h=0

(calculated according to a proper technical basis, possibly other than that (j) (j) (j) assumed in the calculation of Ct ). The sum at risk, Ct − Vt , in each year (t − 1, t) is intended to be close to 0. Intuitively, when dealing with both a life annuity and a death benefit the insurer benefits from a risk reduction, given that the longer is the annuity payment period, the lower is the amount of the death benefit. However, the risk reduction cannot be total, because of the definition of the death benefit (which is in particular guaranteed). The tricky point of this package is the cost to the annuitant. Intuitively, we expect that the death benefit (7.30) will be expensive (given that the consequence – which is the insurer’s target as well – is a strong reduction of the cross-subsidy effect); so commercial difficulties may arise. For the sake of brevity we do not give analytical details; for discussion, we only provide a numerical example. Example 7.4 We take the assumptions adopted in Examples 7.1 and 7.2. We assume that the death benefit is calculated according to the annual interest rate i = 0.03 and the mortality assumption A3 (τ). Table 7.18 quotes the risk index (i.e. the coefficient of variation of the present value of future payments), when a given mortality assumption is adopted. The reduction in the risk profile of the insurer is apparent (compare with Table 7.4). The reduction of the riskiness can be noticed also in the unconditional case; see Table 7.19, which should be compared with Table 7.11. However, the death benefit requires a 22.730% increase in the single premium at age 65 (according to a pricing basis given by i = 0.03 and the mortality assumption A3 (τ)). Actually, the mutuality effect is weaker in this case than when just a life annuity benefit is involved. For the sake of brevity, we do not investigate further risk measures.

From the point of view of the annuitant, the previous policy structure has the advantage of paying back the assets (in terms of the amount stated under policy conditions) remaining at her/his death, hence meeting bequest expectations. On the other hand, the death benefit is rather expensive. Further solutions can be studied, in order to reconcile the risk reduction purposes of the insurer with the request by the annuitant for a high level of the ratio between the annual amount and the single premium. However, the lower is the death benefit, the lower is the risk reduction gained by the insurer. To

7.3


Table 7.18. Coefficient of variation of the present value of future payments, conditional on () the best-estimate scenario: CV[Yt |A3 (τ), nt ], in the presence of death benefit (7.30)


n0 = 1

n0 = 100

n0 = 1,000

n0 = 10,000

…

n0 ∞

10.714% 13.364% 16.722% 20.925% 26.105% 32.390% 39.960% 49.174%

1.071% 1.336% 1.672% 2.093% 2.610% 3.239% 3.996% 4.917%

0.339% 0.423% 0.529% 0.662% 0.826% 1.024% 1.264% 1.555%

0.107% 0.134% 0.167% 0.209% 0.261% 0.324% 0.400% 0.492%

… … … … … … … …

0% 0% 0% 0% 0% 0% 0% 0%

Table 7.19. (Unconditional) coefficient of variation of the present value of future () payments: CV[Yt |nt ], in the presence of death benefit (7.30)


n0 = 1

n0 = 100

n0 = 1,000

n0 = 10,000

…

n0 ∞

10.804% 13.489% 16.902% 21.193% 26.511% 33.009% 40.884% 50.490%

1.764% 2.256% 2.924% 3.830% 5.043% 6.620% 8.574% 10.859%

1.442% 1.866% 2.455% 3.273% 4.390% 5.858% 7.680% 9.789%

1.405% 1.822% 2.403% 3.213% 4.319% 5.776% 7.585% 9.675%

… … … … … … … …

1.401% 1.817% 2.397% 3.206% 4.312% 5.767% 7.575% 9.663%

give an example that can be commercially practicable, we consider a death benefit defined as the difference (if positive) between the single premium S funding the life annuity benefit and the number of annual amounts paid up to death (see also Section 1.6.4); so we have (j) Ct = max S − (t − 1) b(j) , 0 (7.32) See Example 7.5. Example 7.5 With the same inputs as Example 7.4, we quote, in Tables 7.20 and 7.21, the risk index. In Table 7.20 the calculation is conditional on mortality assumption A3 (τ) and in Table 7.21 it is based on the unconditional probability distribution. The single premium has been calculated as the expected present value of future payments, conditional on assumption A3 (τ); hence, S = b(j) E[aK(j) |A3 (τ)]. When compared with Tables 7.4 and x0

7.11, we note a reduction in the risk profile to the insurer in the early policy

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7 : The longevity risk: actuarial perspectives Table 7.20. Coefficient of variation of the present value of future payments, condi() tional on the best-estimate scenario: CV[Yt |A3 (τ), nt ], in the presence of death benefit (7.32)


n0 = 1

n0 = 100

n0 = 1,000

n0 = 10,000

…

n0 ∞

18.877% 26.330% 37.817% 52.312% 61.755% 72.408% 84.929% 100.172%

1.888% 2.633% 3.782% 5.231% 6.175% 7.241% 8.493% 10.017%

0.597% 0.833% 1.196% 1.654% 1.953% 2.290% 2.686% 3.168%

0.189% 0.263% 0.378% 0.523% 0.618% 0.724% 0.849% 1.002%

… … … … … … … …

0% 0% 0% 0% 0% 0% 0% 0%

Table 7.21. (Unconditional) coefficient of variation of the present value of future () payments: CV[Yt |nt ], in the presence of death benefit (7.32)


n0 = 1

n0 = 100

n0 = 1,000

n0 = 10,000

…

n0 ∞

19.010% 26.497% 38.040% 52.659% 62.362% 73.394% 86.413% 102.214%

3.129% 4.386% 6.363% 9.063% 11.512% 14.493% 18.000% 21.929%

2.568% 3.609% 5.263% 7.594% 9.918% 12.766% 16.096% 19.756%

2.505% 3.522% 5.140% 7.431% 9.745% 12.581% 15.893% 19.525%

… … … … … … … …

2.498% 3.512% 5.126% 7.413% 9.725% 12.560% 15.870% 19.499%

years; of course, when the death benefit is zero, we find again the case of the stand-alone life annuity benefit. The risk reduction is lower than in Example 7.4, due to the lower death benefit. The increase in the single premium required at age 65 is lower as well; according to the usual pricing basis (i = 0.03, mortality assumption A3 (τ)), a 7.173% increase is required with respect to the case of the stand-alone life annuity. Death benefits like (7.32) are included in the so-called money-back annuities; see Boardman (2006). One further, very well-known, example of natural hedging across time is given by reversionary annuities (see Section 1.6.3). In this case, the longer is the payment period to the leading annuitant, the lower should be the number of payments to the reversionary annuitant. However, some increased longevity risk arises in this case, due to the fact that two (or more) lives are involved instead of just one (with a possibly correlated mortality trend).

7.3


We now address natural hedging across LOBs. A risk reduction could be pursued by properly mixing positions in life insurances and life annuities. The offset result is unlikely to be as good as those mentioned previously, given that life insurances usually concern a different range of ages than life annuities. Further, we would point out that mortality trends emerge differently within life insurance and life annuity blocks of business. Some empirical investigations have been performed (see Cox and Lin, 2007), considering a set of whole life insurances and a set of life annuities. Some interesting effects in terms of risk reduction can be gained when, at issue, the magnitude of the costs of life insurances is similar to those of life annuities. A satisfactory offsetting effect between sets of life insurances and life annuities is difficult to obtain. Only large insurance companies could be partially effective in this regard. Reinsurers, in particular, could offer proper support, also through swap-like agreements (see Section 7.3.4).

7.3.3

Solvency issues

Appropriate capital allocation policies should be undertaken to deal with the longevity risk which has been retained by the insurer. In particular, the adoption of internal models addressing longevity risk should be considered. In what follows we investigate some internal models in this regard and compare the main results with the requirements embedded in Solvency 2 for longevity risk (only). We focus mainly on longevity risk and refer to conventional immediate life annuities, so that there is no allowance for participation in financial or other profits. To make the results easier to understand, we further assume that no risk transfer (i.e. neither reinsurance nor ART) has been undertaken. Where not specified, we adopt the notation and assumptions introduced in Section 7.2.4. ()

With reference to time t, let Wt be the amount of portfolio assets and Vt the portfolio reserve (or technical provision). These quantities are random at the valuation time, because of the risks, mortality, and investment risks in particular, facing the portfolio. Let z be the valuation time (z = 0, 1, . . . ). The random path of portfolio assets is recursively described as follows: ()

Wt = Wt−1 (1 + it ) − Bt ;

t = z + 1, z + 2, . . .

(7.33)

where it is the investment yield in year (t − 1, t) and Wz is given (including both the reserve and capital in the size required according to a chosen solvency rule).

303

304


According to legislation, the portfolio reserve is normally calculated as the expected present value (using an appropriate technical basis) of future payments, increased by an appropriately defined risk margin. If the risk margin is a function of the expected present value of future payments, then (at least in principle) the mathematical reserve can be calculated by aggregating individual reserves. In this case, the reserve at time t is random because it (j) is the sum of a random number of individual reserves. If Vt denotes the individual reserve at time t, we have (j) () Vt = Vt (7.34) j:j∈t

We will adopt this assumption in the following discussion. However, we point out that if the risk margin is an appropriate risk measure assessed for the portfolio as a whole, the reserve must be calculated directly at the portfolio level, given that the number of in-force policies affects the amount of the technical provision when pooling risks are present. For example, the portfolio reserve could be defined as a given percentile (e.g., the 75th ) of the present value of future payments (see Section 1.5.3); in this case, the risk margin would be implicitly assessed as the difference between the percentile and the expected value of the distribution of the present value of future payments. The quantity ()

Mt = Wt − Vt

(7.35)

represents the assets available to meet the residual risks having allowed for those risks met by the portfolio reserve; shortly, we will refer to Wt as the total portfolio assets and to Mt as the capital assets in the portfolio (conversely, Wt − Mt represents assets backing the portfolio reserve). In line with common practice, we consider solvency to be the ability of the insurer to meet, with an assigned (high) probability, random liabilities as they are described by a realistic probabilistic structure. To implement such a concept, choices are needed in respect of the following items: 1. The quantity expressing the ability of the insurer to meet liabilities; reasonable choices are either the total portfolio assets Wt or, as it is more usual in practice, the capital assets, Mt , which (clearly) is supposed to be positive when the insurer is solvent. 2. The time span T which the above results are referred to; it may range from a short-medium term (1–5 years, say), to the residual duration of the portfolio.

7.3


3. The timing of the results, in particular annual results (e.g. the amount of portfolio assets at every integer time within T years) versus single figure results (e.g. the amount of portfolio assets at the end of the time horizon under consideration, that is, after T years). Further choices concern how to define the portfolio (just in-force policies or also future entrants). To make these choices, the point of view from which solvency is ascertained must be stated. Policyholders, investors and the supervisory authority represent possible viewpoints in respect of the insurance business. However, the perspectives of the (current or potential) policyholders and investors involve profitability requirements possibly higher than those implied by the need of just meeting current liabilities. Such requirements would lead to a concept of insurer’s solidity, rather than solvency. So, we restrict our attention to the supervisory authority’s perspective. The supervisory authority is charged to protect mainly the interests of current policyholders. So a run-off approach should be adopted (hence disregarding future entrants). Further, no profit release should be allowed for within the solvency time-horizon T, nor should any need for capital allocation be delayed. Let z be the time at which solvency is ascertained (z = 0, 1, . . . ). The capital required at time z could be assessed according to one of the following (alternative) models   z+T 5 P Mt ≥ 0 = 1 − ε1 (7.36) t=z+1



P

z+T 5

P [Mz+T ≥ 0] = 1 − ε2  ()

Wt − Y t

≥ 0 = 1 − ε3

(7.37) (7.38)

t=z+1

where εi (i = 1, 2, 3) is the accepted default probability under the chosen () requirement and Yt is defined as in (7.8). Clearly, in all the solvency models above (i.e. (7.36)–(7.38)), the relevant probability is assessed conditional on the current information at time z. With reference to requirement (7.38), first note that recursion (7.33) can be rewritten as Wt = Wz

t 1 1 () − Bh v(z, t) v(h, t) h=z+1

(7.39)

305

306


where 1 = (1 + ih+1 ) (1 + ih+2 ) . . . (1 + ik ) v(h, k)

(7.40)

is the accumulation factor based on investment returns from time h to time k, and v(h, k) = ((1 + ih+1 ) (1 + ih+2 ) . . . (1 + ik ))−1

(7.41)

is the discount factor, based on the annual investment yields, from time k to () time h. Referring to one cohort only, the quantity Yt can also be written as (see (7.10)) () Yt

=

ω−x 0

()

Bh v(t, h)

(7.42)

h=t+1

Requirement (7.38) can be rewritten as 

P

z+T 5

1 − v(z, t)

Wz

t=z+1

t

()

Bh

h=z+1

1 − v(h, t)

ω−x 0

 ()

Bh v(t, h) ≥ 0 = 1 − ε3

h=t+1

(7.43) Assume, for brevity, that the annual investment yields are constant, that is, ih = i for all h. Then we can write (7.43) as 

P

z+T 5

Wz (1 + i)t−z −

t=z+1

ω−x 0

 ()

Bh (1 + i)t−h ≥ 0 = 1 − ε3

(7.44)

h=z+1

or also as '

P

z+T 6

(1 + i)t−(ω+1−x0 ) (Wz (1 + i)ω+1−x0 −z t=z+1 ( ω−x 0 () ω+1−x −h 0 − Bh (1 + i) ) ≥ 0 = 1 − ε3

(7.45)

h=z+1

We note that Wz (1 + i)ω+1−x0 −z −

ω−x 0 h=z+1

()

Bh (1 + i)ω+1−x0 −h = Wω+1−x0

(7.46)

7.3


represents the amount of portfolio assets available when the cohort is exhausted, and so the following result can be easily justified:   z+T 5 P (1 + i)t−(ω+1−x0 ) Wω+1−x0 ≥ 0 t=z+1



= P

z+T 5

 Wω+1−x0 ≥ 0 = P[Wω+1−x0 ≥ 0] = 1 − ε3

(7.47)

t=z+1

Hence, requirement (7.38) can be replaced by the following:

P[Wω+1−x0 ≥ 0] = 1 − ε3

(7.48)

Before commenting on the above results from the perspective of solvency, it is useful to note that such results hold in particular because: (a) the portfolio is closed to new entrants; (b) the probability in requirement (7.38) (as well as in (7.36) and (7.37)) is assessed according to the natural probability distribution of assets and liabilities (so that no risk-adjustment is applied, for example, in a risk-neutral sense) and it is implicitly conditional on the information available at time z on the relevant variables (current number of survivors, investment yields, and so on). The results described in (7.44)– (7.47) could then be generalized to the case where more than one cohort is addressed and the investment yield is not constant. Turning back to the solvency requirements (7.36)–(7.38), the difference between requirement (7.37) and (7.36) is clear. The same quantity is addressed in both, but whilst under requirement (7.36) it is checked at every year within the solvency time-horizon, under (7.37) it is checked just at its end. We note that requirement (7.37) allows, in particular, for temporary shortages of money within the solvency time-horizon. In the context of a portfolio of immediate life annuities, possible deficiencies of assets may be self-financed only by healthy financial profits and, also in this case, when the participation mechanisms to such profits (when present) are under the control of the insurer (i.e. if the insurer can reduce the participation in some years to recover more easily past or future losses). In the case of immediate life annuities, therefore, the outputs of requirement (7.37) should be close to those of (7.36). Hence, in the following we will disregard requirement (7.37). The apparent difference between (7.36) and (7.38) arises from the way that the liabilities are defined. In (7.38), the liabilities are stated in terms of the random present value of future payments, whilst in (7.36) they are stated as the expected value of such a quantity (plus possibly a risk margin). So

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308


whilst in (7.38) a consistent assessment of assets and liabilities is performed, under (7.36) some intermediate step is required. To compare further (7.36) with (7.38), it is useful to note that the capital assets build up because of specific capital allocations, and also because of the annual profits which are released according to the reserve profile and, in our setting, retained within portfolio assets. On the other hand, the amount of portfolio assets at the natural maturity of the cohort represents the surplus left to the insurer at the expiry of the cohort itself. Given that, under (7.38), the maximum available time-horizon is implicitly considered (see (7.47)), we can argue that such a requirement takes care of the overall losses possibly deriving from the portfolio. Assume that a timehorizon T = ω + 1 − x0 − z is chosen in requirement (7.36); the difference between (7.36) and (7.38) lies in the fact that, under the latter, only the total amount of the surplus (and loss) is considered (see (7.48)), whilst, under the former, also the timing of their emergence is taken into consideration. According to valuation terminology, requirement (7.36) is based on a ‘deferral and matching’ logic, whilst (7.38) on an ‘asset and liability’ approach. Further, whenever a shorter time-horizon is chosen in (7.36), we note that just profits (and losses) emerging in the first T years are accounted for. Note that because of the differences among the three requirements, it is reasonable that they are implemented with different levels of the accepted default probability; in particular, we can imagine ε2 ≥ ε1 . The comparison between ε1 and ε3 is not straightforward in general, given that, in a life portfolio, short-term losses could be recovered in the long run. Referring to a portfolio of immediate life annuities, however, we can imagine that ε1 ≥ ε3 whenever T < ω + 1 − x0 − z. Should T = ω + 1 − x0 − z, then ε1 = ε3 could be a reasonable choice. Solving (7.36), through stochastic simulation, one finds the amount of capital assets required at time z; we will denote such amount by Mz[R1] (T). () Then Wz[R1] (T) = Vz + Mz[R1] (T) is the amount of total portfolio assets required at time z. Solving (7.48), again through stochastic simulation, one finds the amount of total portfolio assets required at time z, denoted as Wz[R3] ; the required amount of capital assets at time z is then: () Mz[R3] = Wz[R3] − Vz . Example 7.6 Let us adopt the inputs of Example 7.2; so, in particular, we refer to a homogeneous cohort. To focus on mortality, we disregard financial risk; so we set it = i = 0.03 for all t (i = 0.03 is adopted in the reserving basis as well). To facilitate the comparisons among the results obtained under the different requirements, we define the individual reserve as the

7.3


Table 7.22. Individual reserve (1)

Time z

Reserve Vz

0 5 10 15 20 25 30 35

15.259 12.956 10.599 8.294 6.167 4.336 2.877 1.807

expected value of future payments, under the best-estimate assumption; then (j)

(j)

Vt = E[Yt |A3 (τ)]

(7.49)

Further, the same default probability is set for all the requirements, so ε1 = ε3 = 0.005. Such a level has been chosen to be consistent with the developing Solvency 2 system (see CEIOPS (2007) and CEIOPS (2008)). We note that under Solvency 2 a risk margin should be added to (7.49), calculated according to the Cost of Capital approach; see CEIOPS (2007) and CEIOPS (2008) for details. Table 7.22 quotes the individual reserve. Clearly, at any time z the port() (1) (1) folio reserve is simply: Vz = nz Vz , where Vz is the reserve at time z for a generic annuitant. In Table 7.23, we state the amount of the capital (per unit of portfolio reserve) required according to (7.36) and (7.38) for several portfolio sizes. For (7.36), the maximum possible time-horizon has been chosen. As we would expect from the previous discussion, the two requirements lead to similar outputs, at least when mortality only is addressed. In this case, at least, the outputs suggest that requirement (7.36) is to some extent independent of the reserve when T takes the maximum possible value for the time-horizon. It should be stressed that in our investigation no risk mar() gin is included in Vz . Thus, a share of the required capital quoted in Table 7.23 should be included in the reserve and, possibly, charged to annuitants through an appropriate safety loading at the issue of the policy. When interpreting the size of the required capital per unit of the portfolio reserve, we also point out that the reserve is lower than what would be required by the supervisory authority, and so the ratios in Table 7.23 would be higher than what we would find in practice.

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310

7 : The longevity risk: actuarial perspectives Table 7.23. Required capital based on requirements (7.36) and (7.38), facing longevity risk and mortality random fluctuations

Required capital based on (7.36) Mz[R1] (ω+1−x0 −z) () Vz

Time z

n0 = 100

0 5 10 15 20 25 30 35

12.744% 16.510% 21.474% 28.097% 37.722% 53.980% 82.980% 171.782%

n0 = 1,000 n0 = 10,000 9.243% 11.938% 15.630% 20.372% 27.031% 36.129% 50.605% 79.024%

8.103% 10.525% 13.890% 18.282% 24.131% 31.832% 42.152% 56.968%

Required capital based on (7.38) Mz[R3] () Vz

n0 = 100 12.744% 16.492% 21.333% 28.007% 37.456% 53.378% 81.037% 165.842%

n0 = 1,000 n0 = 10,000 9.241% 11.938% 15.621% 20.372% 27.008% 36.113% 50.476% 77.890%

8.103% 10.525% 13.890% 18.281% 24.131% 31.832% 42.140% 56.968%

It is worthwhile to comment on the similar magnitude of the ratios () and yz,ε [nz ]/E[Yz |nz ] (see Table 7.12), when the probability ε considered for the calculation of the percentile yz,ε [nz ] is very close (or, better, the same as) the non-default probability 1 − ε3 adopted for calculating Mz[R3] . In order to deal with an example, we can compare () the ratio Mz[R3] /Vz in Table 7.23 (where 1 − ε3 = 0.995) with the () ratio yz,0.99 [nz ]/E[Yz |nz ] in Table 7.12 (thus, we are setting ε = 0.99); we can note that the two ratios have a similar magnitude at each time () z. First, we note that, as pointed out in Example 7.2, Vz (given by (1) () () nz E[Yz |A3 (τ)] = E[Yz |A3 (τ), nz ]) and E[Yz |nz ] are very close (compare in particular Tables 7.2 and 7.9). So given the similar values of the two ratios, the quantities Mz[R3] and yz,ε [nz ] are also likely to be close to one the other. Actually, under requirement (7.38) what is measured is the accumulated value of annual payments, whilst with yz,ε [nz ] the relevant present value is accounted for. Indeed in Section 7.2.4, we mentioned the practical importance of investigating the right tail of the distribution of the present value of future payments; this comes from the fact that the quantity yz,ε [nz ] may be taken as a measure of the capital required to meet liabilities under a low default probability (and according to the maximum possible solvency time horizon). () Mz[R3] /Vz

In Table 7.24, outputs from requirement (7.36) are investigated for shorter time-horizons. Comparing Tables 7.23 with 7.24, the long-term nature of longevity risk clearly emerges. We note that, both in Tables 7.23 and 7.24, at each valuation time and for each requirement, the size of the required capital decreases when a larger portfolio is considered. This is due to the fact that also random fluctuations are accounted for in the assessment.

7.3


Table 7.24. Required capital based on requirements (7.36), per unit of portfolio reserve: () [R1] (Mz (T))/Vz , facing longevity risk and mortality random fluctuations

Time-horizon T = 1

Time-horizon T = 3

Time z n0 = 100 n0 = 1, 000 n0 = 10, 000

n0 = 100 n0 = 1, 000 n0 = 10, 000

0 5 10 15 20 25 30 35

1.834% 3.358% 5.162% 8.689% 13.796% 22.727% 44.454% 124.167%

0.574% 1.058% 1.951% 3.600% 6.639% 12.246% 22.588% 41.664%

0.473% 0.743% 1.159% 1.903% 3.265% 6.070% 12.168% 26.210%

0.242% 0.397% 0.649% 1.226% 2.306% 4.465% 8.655% 16.739%

1.076% 1.711% 2.568% 4.463% 8.003% 14.314% 26.438% 51.506%

0.581% 0.983% 1.738% 3.399% 6.403% 11.790% 21.145% 36.973%

We have obtained Table 7.25 by addressing random fluctuations only. In particular, the required capital has been calculated adopting only the best-estimate mortality assumption A3 (τ). In Table 7.26, in contrast, only longevity risk has been accounted for, by assuming that whatever is the realized mortality trend, the actual number of deaths in each year coincides with what has been expected under the relevant trend assumption. We note that in the latter case the amount of the required capital per unit of portfolio reserve is independent of the size of the portfolio – this occurs because, as noted previously, longevity risk is systematic. Regarding Table 7.25, we point out that the random fluctuations accounted for there are not fully comparable to those embedded in Tables 7.23 and 7.24. Actually, in Tables 7.23 and 7.24, a mixture of the random fluctuations which can be appraised under the several mortality assumptions in A(τ) is accounted for. When comparing Table 7.25 (lower panels) with Table 7.24, we can see that, if requirement (7.36) is implemented with a short time-horizon, in practice we are mainly accounting for random fluctuations, rather than systematic deviations; this is due to the long-term nature of longevity risk. Tables 7.25 and 7.26 do provide us with some useful information. However, it must be pointed out that implementing an internal model allowing for a component only of a risk represents an improper use of the model itself. As an illustration, we note that on summing the results in Tables 7.25 and 7.26, for a given requirement and portfolio size, we do not find the correspondent results in Table 7.23 or 7.24. Thus, some aspects are missed when working with marginal distributions only (as is the case when we address either random fluctuations or systematic deviations only). Finally, it is interesting to compare the findings described by the previous Tables with some legal requirements. We refer here to the developing Solvency 2 system, which is one of the few explicitly considering longevity

311

312

7 : The longevity risk: actuarial perspectives Table 7.25. Required capital based on requirements (7.36) and (7.38), facing mortality random fluctuations only; mortality assumption A3 (τ)

Required capital based on (7.36) Mz[R1] (ω+1−x0 −z) () Vz

Required capital based on (7.38) Mz[R3] () Vz

Time z

n0 = 100 n0 = 1, 000 n0 = 10, 000

n0 = 100 n0 = 1, 000 n0 = 10, 000

0 5 10 15 20 25 30 35

7.813% 9.983% 12.144% 16.153% 22.343% 29.728% 54.183% 155.859%

7.031% 9.436% 11.543% 14.982% 21.292% 28.546% 51.253% 144.058%

2.832% 3.071% 4.040% 5.202% 6.938% 10.388% 16.871% 36.795%

0.879% 1.067% 1.217% 1.544% 2.091% 3.072% 5.547% 11.715%

Required capital based on (7.36) Mz[R1] (1) () Vz

2.698% 2.949% 3.759% 4.921% 6.554% 9.642% 16.807% 34.809%

0.800% 1.040% 1.193% 1.462% 1.936% 2.983% 5.152% 11.207%

Required capital based on (7.36) Mz[R1] (3) () Vz

Time z

n0 = 100 n0 = 1, 000 n0 = 10, 000

n0 = 100 n0 = 1, 000 n0 = 10, 000

0 5 10 15 20 25 30 35

0.574% 1.058% 1.951% 3.600% 6.639% 12.246% 22.588% 41.664%

1.834% 3.358% 5.162% 8.604% 13.304% 19.609% 41.023% 124.167%

0.473% 0.743% 0.932% 1.642% 2.458% 4.633% 7.878% 21.058%

0.171% 0.271% 0.388% 0.583% 0.806% 1.379% 2.804% 7.321%

0.983% 1.443% 1.957% 2.630% 3.775% 7.129% 13.181% 32.954%

0.378% 0.479% 0.657% 0.932% 1.329% 2.166% 4.168% 10.176%

risk. The capital required to deal with such risk is the change expected in the net asset value against a permanent reduction by 25% in the current and all future mortality rates (we do not discuss further details, such as possible reductions of this amount; see CEIOPS (2007) and CEIOPS (2008)). Under our hypotheses (we are considering just one cohort, there is no profit participation, we are disregarding risks other than those deriving from mortality, and so on), the requirement reduces to the difference between the best-estimate reserve and a reserve set up with a mortality table embedding probabilities of death 25% lower than in the best-estimate assumption. The relevant results are quoted in Table 7.27, where the required capital at time z is denoted by Mz[Solv2] . It is clear that, in relative terms, such an amount is independent of the portfolio size. We further recall that, under Solvency 2, no specific capital allocation is required for the risk of random fluctuations, since they are treated as hedgeable risks.

7.3


Table 7.26. Required capital based on requirements (7.36) and (7.38), facing longevity risk only

Required capital Time z 0 5 10 15 20 25 30 35

Mz[R1] (ω+1−x0 −z) () Vz

Mz[R3] () Vz

Mz[R1] (1) () Vz

Mz[R1] (3) () Vz

7.562% 9.895% 13.040% 17.239% 22.745% 29.762% 38.348% 48.330%

7.562% 9.895% 13.040% 17.239% 22.745% 29.762% 38.348% 48.330%

0.125% 0.205% 0.437% 0.922% 1.883% 3.727% 7.110% 12.949%

0.389% 0.651% 1.394% 2.857% 5.621% 10.564% 18.745% 30.875%

Table 7.27. Required capital according to Solvency 2 Mz[Solv2]

Time z

()

Vz

0 5 10 15 20 25 30 35

7.274% 9.080% 11.377% 14.293% 18.000% 22.767% 29.102% 38.065%

Tables 7.26 and 7.27 may suggest that a deterministic approach can be adopted for allocating capital to deal with longevity risk. In particular, the assessment of the required capital could be based on a comparison between the actual reserve and a reserve calculated under a more severe mortality trend assumption (as turns out to be the case under Solvency 2). ()[B]

Let Vz be a reserve calculated according to the same valuation prin() ciple adopted for Vz (the equivalence principle, in our implementation), but based on a worse mortality assumption, so that ()

Vz

()[B]

≤ Vz

(7.50)

The required capital would be ()[B]

Mz[R4] = Vz

()

− Vz

(7.51)

We note that requirement (7.51) would deal with longevity risk only. Further, no default probability is explicitly mentioned; however, the mortality

313

314

7 : The longevity risk: actuarial perspectives ()[B]

assumption adopted in Vz clearly implies some (not explicit) default probability. The time-horizon implicitly considered is the maximum residual duration of the portfolio, given that this is the time-horizon referred to in the calculation of the reserve. We also point out that, to simplify the assessment of the required capital and to avoid any duplication of risk margins as well, it is reasonable that reserves in (7.51) are actually based on the equivalence principle. So the required capital Mz[R4] turns out to be linear in respect of the portfolio size nz . To compare requirements (7.36) and (7.38) with (7.51), let us define the following ratios: QMz[R1] (T; nz ) = QMz[R3] (nz ) = QVz =

Mz[R1] (T) ()

Vz

Mz[R3] ()

Vz

Mz[R4] ()

Vz

(7.52) (7.53) (7.54)

Accounting also for the risk of random fluctuations, the ratios QMz[R1] (T; nz ) and QMz[R3] (nz ) depend on the size of the portfolio while, in contrast, the ratio QVz , which considers just longevity risk, is independent of portfolio size. On the other hand, requirement (7.36) and (7.38) could be implemented considering only the risk of random fluctuations or the longevity risk, as we have illustrated in the calculations in Tables 7.25 and 7.26, respectively. As noted previously, when addressing longevity risk only, the ratios QMz[R1] (T; nz ) and QMz[R3] (nz ) are independent of the size of the portfolio (as it emerges from Table 7.26). However, we have already commented on the fact that addressing just a component of the mortality risk represents an improper use of requirements (7.36) and (7.38). A further difference between ratio QMz[R1] (T; nz ) and QVz stands in the possibility to set a preferred time-horizon; indeed, time-horizons other than the maximum one may be chosen only when requirement (7.36) is adopted. It is not possible to derive general conclusions regarding the comparison between the outcoming levels of ratios QMz[R1] (T; nz ) and QMz[R3] (nz ), on one hand, and QVz , on the other. However, we comment further through an example. Example 7.7 Figure 7.7 plots the ratios (7.53) and (7.54), for several portfolio sizes, based on calculations performed at time 0. In particular:

7.3


Required capital, per unit of reserve

12%

10%

8%

6%

4%

(3)

(1)

(2)

(4)

2%

0% 0

2000

4000

6000

8000

10000

12000

Portfolio size ()[A5 (τ)] [R3] [R3] Ratios QM0 (n0 ) and QV0 . (1): QM0 (n0 ); (2): QV0 , with V0 ; (3): [R3] [R3] [R3] QM0 (n0 ), with M0 accounting for random fluctuations only; (4): QV0 + QM0 (n0 ), with [R3] M0 accounting for random fluctuations only.

Figure 7.7.

– case (1) plots the ratio QM0[R3] (nz ); – case (2) plots the ratio QV0 , obtained by choosing the mortality trend A5 (τ) as an assumption more severe than the best-estimate; – case (3) plots the ratio QM0[R3] (nz ) where, in contrast to case (1), the required capital M0[R3] has been obtained by addressing random fluctuations only (the best-estimate assumption has been used to describe mortality); – case (4) plots the required capital obtained summing the results in case (2) (accounting for longevity risk only) and in case (3) (accounting for random fluctuations only). We first note that the outputs found under case (2) are very similar to (indeed, in our example they coincide with) those found adopting requirement (7.38), as well as requirement (7.36) with T = ω + 1 − x0 (the ratio ()[A5 (τ)] QV0 , with V0 , plotted in Fig. 7.7, amounts to 7.562% for each portfolio size; compare this outcome with the ratios QM0[R1] (ω + 1 − x0 ; n0 ) = () () (M0[R1] (ω + 1 − x0 ))/V0 and QM0[R3] (n0 ) = M0[R3] /V0 in Table 7.26). This is explained by the fact that the (left) tail of the distribution of assets (addressed in (7.38) and (7.36)) is heavily affected by the worst scenario (A5 (τ), in our example) when low probabilities (of default) are addressed.

315

316


Thus, when allowing for longevity risk only, requirement (7.36) adopted with the maximum possible time-horizon and requirement (7.38) reduce to (7.51). This is why a practicable idea could be to split the capital allocation process in two steps: – one for longevity risk only, based on a comparison between reserves calculated according to different mortality assumptions (i.e. adopting requirement (7.51)); – one for random fluctuations only, adopting an internal model or some other standard formula. Case (4) in Fig. 7.7 is intended to represent such a choice. We note, however, that an unnecessary allocation may result from this procedure; as we have already commented, working separately on the components of mortality risk is improper and may lead to an inaccurate capital allocation. Undoubtedly, the advantage of requirement (7.51) is its simplicity, and we note that it seems that this requirement will be adopted by Solvency 2 in respect of many risks. Of course, it is also possible to find the reserving basis avoiding the situation plotted in Fig. 7.7 (but to be sure, one should first perform the valuation through an internal model, at least for some typical compositions of the portfolio). Another possibility supporting the separate treatment of the mortality risk components is to adopt different solvency time-horizons for the different components of mortality risk. So we could choose the maximum possible value for T for longevity risk (adopting (7.51)) and a short-medium time-horizon for random fluctuations (if requirement (7.36) is adopted, with say T = 1 to 5 years). For practical purposes, this approach could represent a good compromise, on condition that the relevant assumptions are properly disclosed. If valuation tools other than an internal model are available or are required for the risk of random fluctuations (as should be the case for Solvency 2), then requirement (7.51) is certainly able to capture properly the feature of longevity risk (only). Example 7.8 We conclude this section with a final example. So far, just homogeneous portfolios have been investigated. We now consider the case of a portfolio with some heterogeneity in annual amounts. A stronger dispersion of annual amounts usually leads to a poorer pooling effect. Also there is the danger that if the annuitants living longer are those with higher annual amounts, then the impact of longevity risk could be more severe. Even though it is reasonable to assume that, because of adverse selection, those with higher annual amounts live longer (as it is supported by some evidence), in this example we do not account for this dependence. The impact

7.3 Table 7.28.


Classes of annual amounts in five portfolios

Portf. 1 Class 1 2 3 4 5

Portf. 2

Portf. 3

Portf. 4

Portf. 5

Amount Freq. Amount Freq. Amount Freq. Amount Freq. Amount Freq. 1

100%

0.75 1 2

40% 50% 10%

0.25 0.75 1 1.25 1.75

20% 20% 20% 20% 20%

0.75 3.25

90% 0.5625 80% 10% 2 15% 5 5%

distribution of the annual amount Average value standard deviation

1

1

1

1

1

0

0.35355

0.5

0.75

1.0503

of the dispersion of annual amounts is checked through the calculation of the capital required to meet mortality risks, assuming a zero correlation between the annual amount and the lifetime of the annuitant. We test the five portfolios described in Table 7.28. We note that, to facilitate comparisons, the same average annual amount per annuitant has been assumed. The specific annual amount paid to each annuitant may, however, be different from the average value, depending on the insurance class (each class grouping people with the same annual amount). We note that the portfolios are ordered with respect to the degree of heterogeneity, as measured by the standard deviation of the distribution of the annual amounts. Adopting the inputs of Example 7.6, we have calculated the capital required based on requirement (7.38). The assessment has been performed at time 0 only, for several portfolio sizes. The outputs are plotted in Fig. 7.8. A stronger requirement emerges when portfolios with a wider dispersion of annual amounts are considered: portfolio 5 versus portfolio 1, for example. We note that the portfolio reserve at time 0 is the same in all portfolios, due to the assumption about the average annual amount. It is interesting to compare Figs. 7.8 to 7.9, where only random fluctuations have been considered. It seems that most of the change in the capital required when changing the portfolio composition is due to random fluctuations. We note, in particular, the width of the range of variation of the capital required for the several portfolios when also longevity risk is accounted for relative to what happens when only random fluctuations are accounted for. Thus, when comparing Figs. 7.8 and 7.9 in detail, we note, although the scale of the yaxis is different, that the length of the range is the same. So we can conclude that, to some extent, longevity risk is independent of the heterogeneity of the portfolio. It is important to note, again, that this result is also due to the

317


12%

Require dcapital (per unit of reserve)

318

portf 1 portf 2 portf 3 portf 4 portf 5

11%

10%

9%

8% 0

2000

4000

6000

8000

10000

12000

Portfolio size Figure 7.8.

[R3 ]

Required capital, per unit of reserve: QM0

(n0 ).

model, which does not explicitly account for any dependence of the lifetime of the individual on her/his annual amount.

7.3.4

Reinsurance arrangements

Various reinsurance arrangements can be conceived, at least in principle, to transfer longevity risk. At the time of writing, reinsurers are reluctant to accept such a transfer, due to the systematic nature of the risk of unanticipated aggregate mortality. Actually, only some slight offset (through natural hedging) can be gained by dealing with longevity risk just within the insurance-reinsurance process. Longevity-linked securities, transferring the risk to the capital market, could back the development of a longevity reinsurance market (see Section 7.4). So in the following we describe several arrangements, some of which in particular could be effective when linked to longevity securities. At the same time, we disregard any arrangement designed to deal with random fluctuations. To be consistent with the previous discussion, we refer to immediate life annuities (which in any case are the most interesting type of annuity when a transfer of longevity risk is being considered). It must be pointed out that when mortality risk is reinsured in a life annuity portfolio, one cannot be sure that just longevity risk is transferred. Indeed, random fluctuations also contribute to deviations in mortality rates,

7.3

Required capital (per unit of reserve)

4%


portf 1 portf 2 portf 3 portf 4 portf 5

3%

2%

1%

0% 0

2000

4000

6000 8000 Portfolio size [R3 ]

Figure 7.9. Required capital, per unit of reserve: QM0 tions only; mortality assumption A3 (τ).

10000

12000

(n0 ), facing mortality random fluctua-

as we have highlighted previously. If the reinsurance arrangement is meant to deal mainly with longevity risk, then before underwriting it the risk of random fluctuations has to be reduced; for example, some leveling of the annual amounts has to be achieved through a first-step surplus reinsurance. For this reason, in the following we will implicitly refer to homogeneous portfolios in respect of the amount of benefits. The more natural way to transfer longevity risk for an annuity provider is to truncate the duration of each annuity. To this purpose, an Excess-of-Loss (XL) reinsurance can be designed. Under such an arrangement, the reinsurer would pay to the cedant the ‘final’ part of the life annuity in excess of a given age xmax . Such an age should be reasonably old, but not too close to the maximum age (otherwise the transfer would be ineffective); xmax could, for example, be set equal to the Lexis point in the current projected table. Note that xmax defines the deductible of the XL arrangement. See Fig. 7.10, where x0 = 65 and xmax = 85. From the point of view of the cedant, this reinsurance treaty converts immediate life annuities payable for the whole residual lifetime into immediate temporary life annuities. From the point of view of the reinsurer, a heavy charge of risk emerges. Actually, the reinsurer takes the ‘worst part’ of each annuity, being involved at the oldest ages only. Therefore, from a practical point of view, the reinsurance treaty would be acceptable to the

319


Reinsurer's intervention Annuitants

320

n . . .

. . .

. . .

5 4 3 2 1 65 Figure 7.10.

85

Lifetime

An XL reinsurance arrangement.

reinsurer only if it were compulsory for some annuity providers. This could be the case, for example, with pension funds, which may be forced by the supervisory authority to back their liabilities through arrangements with (re-)insurers. The XL arrangement is clearly defined on a long-term basis, so implying a heavy longevity risk charged to the reinsurer. In more realistic terms, reinsurance arrangements defined on a short-medium period basis could be addressed. With this objective in mind, stop-loss arrangements could provide interesting solutions. According to the stop-loss rationale, the reinsurer’s interventions are aimed at preventing the default of the cedant, caused by (systematic) mortality deviations. The effect of mortality deviations can be identified, in particular, by comparing the total portfolio assets at a given time with the portfolio reserve required to meet the insurer’s obligations. A Stop-Loss reinsurance on assets can then be designed, according to which the reinsurer funds (at least partially) the possible deficiency in assets; Fig. 7.11 sketches this idea (in a run-off perspective). Let z be the time of issue (or revision) of the reinsurance arrangement. Adopting the notation introduced earlier, in practical terms the reinsurer’s intervention can be limited to the case ()

Wz+k < (1 − π) Vz+k ,

π≥0

(7.55)

7.3


Assets and reserve

Reinsurer's intervention

Required portfolio reserve

Assets available Time

Figure 7.11.

A Stop-Loss reinsurance arrangement on assets. ()

where the amount πVz+k represents the ‘priority’ of the stop-loss treaty and k is a given number of years. We note that setting π > 0 may contain the possibility of random fluctuations being transferred. However, thanks to the fact that the assets and the reserve of a life annuity portfolio have longterm features, the flows of the arrangement should not be heavily affected by random fluctuations, at least up to some time. In fact, close to the natural maturity of the portfolio we may expect that random fluctuations become predominant relative to systematic deviations; see also Section 7.2.4. Setting k > 1 (e.g. k = 3 or k = 5) ensures that the reinsurer intervenes in the more severe situations, and not when the lack of assets may be recovered by the subsequent flows of the portfolio. However, k should not be set too high, otherwise the funding to the cedant in the critical cases would turn out to be too delayed in time. A technical difficulty in this treaty concerns the definition of assets and reserve to be referred to for ascertaining the loss. Further, some control of the investment policy adopted by the cedant in relation to these assets could be requested by the reinsurer. For these reasons, the treaty can be conceived as an ‘internal’ arrangement, that is, within an insurance group (where the holding company takes the role of the reinsurer of affiliates) or when there is some partnership between a pension fund and an insurance company (the latter then acting as the reinsurer, the former as the cedant). A Stop-Loss reinsurance may be designed on annual outflows, instead of assets. The rationale, in this case, is that, at a given point in time, longevity risk is perceived if the amount of benefits to be currently paid to annuitants is (significantly) higher than expected. A transfer arrangement can then be

321

322


designed so that the reinsurer takes charge of such an extra amount, or ‘loss’. As in the previous case, the loss may be due to random fluctuations – here, this situation is more likely, given that annual outflows are directly referred to, instead of some accrual of outflows. By setting a trigger level for the reinsurer’s intervention higher than the expected value of the amount of benefits, we would reduce the possible transfer of such a random risk component. Reinsurance conditions should concern the following items: – Let z be the time of issue (or revision) of the arrangement. The time horizon k of the reinsurance coverage should be stated, as well as the timing of the possible reinsurer’s intervention within it. Within the time horizon k, policy conditions (i.e. premium basis, mortality assumptions, and so on) should be guaranteed. As to the timing of the intervention of the reinsurer, since reference is to annual outflows, it is reasonable to assume that a yearly timing is chosen. Hence, in the following, we will make this assumption. – The mortality assumption for calculating the expected value of the outflow, required to define the loss of the cedant. Reasonably, we will adopt the current mortality table, which will be generically denoted as A(τ) in what follows. – The minimum amount t of benefits (at time t, t = z + 1, z + 2, . . . , z + k) below which there is no payment by the reinsurer. For example, ()

t = E[Bt |A(τ), nz ] (1 + r) = b E[Nt |A(τ), nz ] (1 + r)

(7.56)

with r ≥ 0 and b the annual amount for each annuitant; thus the amount t represents the priority of the Stop-Loss arrangement. – The Stop-Loss upper limit, that is, an amount t such that t − t is the maximum amount paid by the reinsurer at time t. From the point of view of the cedant, the amount t should be set high enough so that only situations of extremely high survivorship are charged to the cedant. However, the reinsurer reasonably sets t in connection to the available hedging opportunities. We will come back to this issue in Section 7.4.3. As to the cedant, a further reinsurance arrangement may be underwritten, if available, for the residual risk, possibly with another reinsurer; in this case, the amount t − t operates as the first layer. In Fig. 7.12, a typical situation is represented. When we consider the features of this treaty, especially in relation to the Stop-Loss arrangement on assets, we note that measuring annual outflows is relatively easy, since this relies on some direct information about the

7.3

Actual outflows


Expected values

Annual outflows

Priority

Upperlimit

Reinsurer's intervention

Time Figure 7.12.

A Stop-Loss reinsurance arrangement on annual outflows.

portfolio (viz. the number of living annuitants, joint to the annual amount of their benefits). On the other hand, as already pointed out, it is more difficult to avoid the transfer of random fluctuations as well. (SL)

We now define in detail the flows paid by the reinsurer. Let Bt such flow at time t, t = z + 1, z + 2, . . . , z + k. We have  ()  if Bt ≤ t 0 (SL) () Bt = B() − t if t < Bt ≤ t t   () t − t if Bt > t

denote

(7.57)

The net outflow of the cedant at time t (gross of the reinsurance premium), (SL) denoted as OFt , is then  () ()  if Bt ≤ t Bt (SL) () (SL) () = Bt − Bt = t OFt if t < Bt ≤ t (7.58)   () () Bt − (t − t ) if Bt > t The net outflow of the cedant is clearly random but, unless some ‘extreme’ survivorship event occurs, it is protected with a cap. It is interesting (especially for comparison with the swap-like arrangement described subsequently) to comment on this outflow. First of all, it must be stressed that () Bt ≤ t represents a situation of profit or small loss to the insurer. On () the contrary, the event Bt > t corresponds to a huge loss. Whenever () t < Bt ≤ t a loss results for the insurer, whose severity may range

323

324

7 : The longevity risk: actuarial perspectives ()

()

from small (if Bt is close to t ) to high (if Bt is close to t ). So the effect of the Stop-Loss arrangement is to transfer to the reinsurer all of the loss situations, except for the lowest and the heaviest ones; any situation of profit, on the contrary, is kept by the cedant. To reduce further randomness of the annual outflow, the cedant may be willing to transfer to the reinsurer not only losses, but also profits. Thus, a reinsurance-swap arrangement on annual outflows can be designed. Let B∗t be a target value for the outflows of the insurer at time t, t = z + 1, z + 2, . . . , z + k; for example, ()

B∗t = E[Bt |A(τ), nz ]

(7.59)

where A(τ) is an appropriate mortality assumption and z is the time of issue () of the reinsurance swap. Under the swap, if Bt > B∗t the cedant receives () money from the reinsurer; otherwise, if Bt < B∗t , then the cedant gives money to the reinsurer, so that the target outflow is reached. (swap)

Let Bt follows:

be the payment from the reinsurer to the cedant, defined as (swap)

Bt

()

= Bt

− B∗t

(7.60)

The annual outflow (gross of the reinsurance premium) for the cedant at time t is (swap) (swap) () = B∗t OFt = Bt − Bt (7.61) ()

The advantage for the cedant is to convert a random flow, Bt , into a certain flow, B∗t and hence the term ‘reinsurance-swap” that we have assigned to this arrangement. Figure 7.13 depicts a possible situation. Note that, ceteris paribus, this arrangement should be less expensive than the Stop-Loss treaty on outflows, given that the reinsurer participates not only in the losses, but also in the profits. Although one advantage for the cedant of the reinsurance-swap is a possible price reduction, the cedant may be unwilling to transfer profits. On the contrary, the arrangement may be interesting for the reinsurer depending on the hedging tools available in the capital market (so that it could even be the only arrangement available on the reinsurance market); see Section 7.4.3 in this regard. The design of the reinsurance-swap can be generalized by assigning two barriers t , t (with t ≤ B∗t ≤ t ) such that  () ()  Bt − t if Bt ≤ t (swap-b) () = 0 (7.62) Bt if t < Bt ≤ t   () () Bt − t if Bt > t

7.3

Actual outflow


Target outflow

Annual outflows

bn0

Time From/to cedant Figure 7.13.

A reinsurance-swap arrangement.

Clearly, when setting t = t = B∗t in (7.62), one finds (7.60) again. The net outflow (gross of the reinsurance premium) to the cedant is then

(swap-b) OFt

=

() Bt

(swap-b) − Bt

  t () = Bt   t

()

if Bt ≤ t () if t < Bt ≤ t () if Bt > t

(7.63)

It is interesting to compare (7.63) with (7.58). We have already commented on the implications of (7.58) for the profit/loss left to the cedant. Under (7.63), large losses as well as large profits are transferred to the reinsurer; therefore, both a floor and a cap are now applied to the profits/losses of the cedant. So far we have not commented on the pricing of the reinsurance arrangements which have been examined. Actually, we will not enter into details regarding this subject, but just make some remarks. The critical issue in pricing a reinsurance arrangement involving aggregate mortality risk is the pricing of longevity risk. As already commented in Section 7.2.3, many attempts have been devoted to this issue, but no generally accepted proposal is yet available. The stochastic model used extensively in this chapter, although useful for internal purposes (such as capital allocation), is not appropriate in general for pricing, due to the wide set of items to be chosen (alternative mortality scenarios, weights attached

325

326


to such scenarios, and so on), as well as to the intrinsic static representation of stochastic mortality. As far as the XL arrangement and the Stop-Loss treaty on assets are concerned, the adoption of traditional actuarial pricing methods, such as the percentile principle, is reasonable because of the traditional structure of the arrangement. Due to the context within which they could have to be realized (such as a compulsory backing of pension fund liabilities by a life insurer), the stochastic model used so far, with the set A(τ) suggested by some independent institution, can offer an acceptable representation of stochastic mortality also for pricing purposes. The Stop-Loss arrangement on outflows and the reinsurance-swap, in contrast, have features very close to those of financial derivatives. As we have already noted, these arrangements can develop if they are properly backed by longevity-linked securities. So their pricing will depend on the pricing of the backing securities; attempts in this respect are still at an early stage. The choice of a particular reinsurance arrangement clearly depends at first on what is available in the reinsurance market. In the case that more than one solution is available, attention should not be paid just to the price, but also to the benefits obtained in terms of reduction of the required capital. For the reasons discussed above, we are not going to compare the arrangements in terms of their price but we conduct some numerical investigations concerning the capital requirements resulting from various reinsurance arrangements. Due to the practical interest that they might have, we consider just the Stop-Loss treaty on outflows and the swap-reinsurance arrangement. Given the earlier comments about solvency issues, we make use of an internal model, to account jointly (and consistently) for the risk of random fluctuations and the longevity risk. Example 7.9 We refer to the assumptions of Example 7.6. As we have highlighted in discussing this example, if one wants to deal with a model recording the overall longevity risk, the proper time-horizon is the maximum residual duration of the portfolio. So we adopt requirement (7.38). At any valuation time, we assume that the flows until the end of the reinsurance period are the net outflows OFt(·) , whilst after that time they are simply () the annual payments Bt . Therefore, when assessing the required capital, we do not assume that the reinsurance arrangement will be automatically renewed. Policy conditions have to be chosen specifically for the reinsurance arrangement; in particular, the two bounds t and t must be set differently in the Stop-Loss arrangement and in the reinsurance-swap with

7.3


barriers. The following choices are adopted: ()

t = 1.1 E[Bt |A3 (τ), nz ] ()

t = 2 E[Bt |A3 (τ), nz ]

(7.64)

for the former; ()

t = 0.75 E[Bt |A3 (τ), nz ] ()

t = 1.25 E[Bt |A3 (τ), nz ]

(7.65)

for the latter. For the reinsurance-swap we set: ()

B∗t = E[Bt |A3 (τ), nz ]

(7.66)

For all of the arrangements, a 5-year reinsurance period has been chosen. To allow for some comparisons, we have assumed that at the beginning of each reinsurance period a premium must be paid by the cedant, assessed as the (unconditional) expected present value of future reinsurance flows. We should point out that this pricing principle does not make practical sense, given that no risk margin is included; however, with this approach, we can at least take into account the magnitude of the reinsurance premium. We assume further that the reinsurer and the cedant adopt the same mortality model, with the same parameters and that the reserve must be fully set up by the cedant. The possible default of the reinsurer is disregarded when assessing the required capital. In Table 7.29, we give the required capital (per unit of reserve) for the three arrangements, for different portfolio sizes, as well as for the case of no reinsurance arrangement (these latter results are taken from Table 7.23). Because of the increased certainty of the outflows during the reinsurance period, the lowest amount of required capital is found under the reinsurance-swap (with no barriers); but clearly, in such an arrangement the premium for the risk (which we have not considered) could be higher than in other cases. As already noted, due to the different parameter values, the outflows under the alternative arrangements are not directly comparable. It is interesting to note that most of the reduction in the required capital is gained at the oldest ages, roughly after the Lexis point. Indeed, the most severe part of the longevity risk is expected to emerge after this age. So, we can argue that the need for reinsurance emerges in particular at the oldest ages; at earlier ages, the risk could be managed through other RM tools. We conclude this section by describing an arrangement which (at least in principle) could help in realizing natural hedging across LOBs.

327

328

7 : The longevity risk: actuarial perspectives Table 7.29.

[R3 ]

Required capital, per unit of reserve: Mz

No reinsurance Time z

n0 = 100

0 5 10 15 20 25 30 35

12.744% 16.492% 21.333% 28.007% 37.456% 53.378% 81.037% 165.842%

8.103% 10.525% 13.890% 18.281% 24.131% 31.832% 42.140% 56.968%

Reinsurance-swap, no barriers Time z

n0 = 100

0 5 10 15 20 25 30 35

12.451% 15.819% 20.138% 24.683% 30.776% 37.998% 45.167% 66.244%

n0 = 1,000 n0 = 10,000 9.088% 11.571% 14.731% 18.440% 22.168% 25.280% 26.260% 27.762%

, with and without reinsurance

Stop-loss on outflows

n0 = 1,000 n0 = 10,000 9.241% 11.938% 15.621% 20.372% 27.008% 36.113% 50.476% 77.890%

()

/Vz

8.002% 10.241% 13.196% 16.548% 19.918% 22.112% 21.452% 17.984%

n0 = 100 n0 = 1,000 n0 = 10,000 12.744% 16.492% 21.333% 27.603% 35.246% 44.356% 51.771% 58.926%

9.241% 11.938% 15.621% 20.372% 26.230% 31.746% 35.389% 30.841%

8.103% 10.525% 13.890% 18.281% 23.739% 28.433% 30.687% 25.540%

Reinsurance-swap, with barriers n0 = 100 n0 = 1,000 n0 = 10,000 12.744% 16.492% 21.333% 28.007% 37.299% 49.855% 62.390% 89.438%

9.241% 11.938% 15.621% 20.372% 27.008% 35.183% 41.945% 48.414%

8.103% 10.525% 13.890% 18.281% 24.131% 31.413% 36.373% 37.579%

As was mentioned in Section 7.3.2, an appropriate diversification effect between life insurance and life annuities may be difficult to obtain by an insurer on its own. Intervention of a reinsurer can help in reaching the target and, inter alia, could provide a way for reinsurers to hedge the accepted longevity risk. We sketch a simple situation, involving two insurers, labelled IA and IB respectively, and a reinsurer. Insurer IA deals with life annuities. At time 0 a (total) single premium SA is collected from the issue of immediate life annuities; the overall annual () amount paid at time t is Bt (t = 1, 2, . . . ). Insurer IB deals with whole life insurances. Let us assume that annual premiums are payable up to the time of death and the benefit is paid at the end of the year of death; the total amount of premiums collected at time t is PtB (t = 0, 1, . . . ) whilst the benefits falling due at time t (t = 1, 2, . . . ) over the portfolio are denoted as () Ct . A reinsurance arrangement is underwritten by the two insurers with the same reinsurer, according to which – at time t (t = 0, 1, . . . ) the reinsurer pays to insurer IA an amount equal to PtB and at time 0 an amount equal to SA to insurer IB ;

7.3

Annuity benefit

Death benefit Insurer IA

Annuity benefit

Annuitants Figure 7.14.

Insurer IB

Reinsurer Insurance premiums

Annuity premiums


Annuity premiums Death benefit

Insurance premiums

Insureds

Flows in the swap-like arrangement between life annuities and life insurances.

– at each time t (t = 1, 2, . . . ) the reinsurer receives from insurer IA an () amount equal to Ct and at time t (t = 1, 2, . . . ) receives from insurer () IB an amount equal to Bt . This would be a swap-like arrangement between life annuities and life insurances; Fig. 7.14 gives a graphical idea of the overall flows. Let us assume that the quantities introduced above are defined for each () time t; in particular, S0A = SA and StA = 0 for t = 1, 2, . . . whilst C0 = 0 () and B0 = 0. Then it turns out that at any time t, t = 0, 1, . . . , the net () () cashflow for both insurer IA and IB is StA + PtB − Bt − Ct , whilst for the () () reinsurer, the net cashflow is Bt + Ct − StA − PtB . Each party has both a position in life annuities and one in life insurances, and therefore gains the benefit from natural hedging. Practical difficulties inherent in such an arrangement are self-evident. Advantages may be weak, especially because of the incomplete hedging provided. It must also be pointed out that the actual duration of the life insurance covers may be shortened because of surrenders. Further, some reward has to be acknowledged to the reinsurer, which can reduce the advantages gained from the new position. However, this structure could represent a useful management framework within an insurance group, where the holding company could play the part of the reinsurer (with reduced fees charged to the counterparties). A similar swap arrangement is described by Cox and Lin (2007), however without explicit intervention of a reinsurer. Consider homogeneous port() folios, both for insurer IA and IB . Therefore: Bt = bNt and Ct = cDt , where b denotes the annual amount to each annuitant, c the death benefit to whole life policyholder, Nt the number of the annuitants at time t in the portfolio of insurer IA and Dt the number of deaths in year (t − 1, t)

329

330


in the portfolio of insurer IB . Let nt∗ and dt∗ be two given benchmarks for the number of annuitants at time t for insurer IA and the number of deaths in year (t − 1, t) for insurer IB , respectively. Insurer IA and IB agree that the flow b · max{Nt − nt∗ , 0} is paid at time t by insurer IB to insurer IA , whilst the flow c · max{Dt − dt∗ , 0} is paid at the same time by insurer IA to IB . This way, insurer IA is protected against excess survivorship, whilst insurer IB is protected in respect of excess mortality. However, insurer IA is then exposed to excess mortality, whilst insurer IB to excess survivorship. Cox and Lin (2007) show through numerical assessments that some natural hedging effects are gained by both insurers, provided that the present value of future payments for life annuities and for life insurances are the same at the time of issue.

7.4 7.4.1

Alternative risk transfers Life insurance securitization

Securitization consists in packaging a pool of assets or, more generally, a sequence of cash flows into securities traded on the market. The aims of a securitization transaction can be: – to raise liquidity by selling future flows (such as the recovery of acquisition costs or embedded profits); – to transfer risks whenever contingent payments or random cash flows are involved. We note that, since new securities are issued, a counterparty risk arises (for the investor). The organizational aspects of a securitization transaction are rather complex. Figure 7.15 sketches a simple design for a life insurance deal, focussing on the main agents involved. The transaction starts in the insurance market, where policies underwritten give rise to the cash flows which are securitized (at least in part). The insurer then sells the right to some cash flows to a special purpose vehicle (SPV), which is a financial entity that has been established to link the insurer to the capital market. Securities backed by the chosen cash flows are issued by the SPV, which raises monies from the capital market. Such funds are (at least partially) available to the insurer. According to the specific features of the transaction, further items may be added to the structure. For example, a fixed interest rate could be paid

7.4

Alternative risk transfers

Policyholders

Premiums

Benefits Price

Funding Insurer Cash flows Figure 7.15.

Special Purpose Vehicle (SPV)

Securities

Capital Market

The securitization process in life insurance: a simplified structure.

Credit Enhancement Mechanism

Policyholders

Guarantee

Premium Premiums

Benefits Funding Insurer Cash flows

Special Purpose Vehicle (SPV)

Floating interest rate

Price Securities

Capital Market

Fixed interest rate

Swap counterparty Figure 7.16.

The securitization process in life insurance: a more composite structure.

to investors, so that the intervention by a swap counterparty is required; see Fig. 7.16. As has been pointed out above, some counterparty risk is originated by the securitization transaction. This is due to the possible default of the insurer with respect to the obligations assumed against the SPV, as well as of the policyholders in respect of the insurer, in the form of surrenders and lapses (which may possibly affect the securitized cash flows). To reduce such default risks, some form of credit enhancement may be introduced, both internal (e.g. transferring to the SPV higher cashflows than those required by the actual size of the securities) and external, through the intervention of a specific entity (issuing, for example, credit insurance, letters of credit, and so on); see again Fig. 7.16. Further counterparty risk emerges from the other parties involved, similarly to any financial transaction. We note that the intervention by a third financial institution may result in an increase of the rating of the securities.

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Further details of the securitization transaction concern services for payments provided by external bodies, investment banks trading the securities on the market, and so on. Since we are only interested in the main technical aspects of the securitization process, we do not go deeper into these topics (which, nevertheless, do play an important role in the success of the overall transaction).

7.4.2

Mortality-linked securities

Mortality-linked securities are securities whose pay-off is contingent on the mortality experienced in a given population; this is obtained, in particular, by embedding some derivatives whose underlying is a mortality index assessed on the given population. These securities may serve two opposite purposes: to hedge extra-mortality or extra-survivorship. In the former case, we will refer to them as mortality bonds, in the latter case as longevity bonds. We restrict the terminology to ‘bond”, without making explicit reference (in the name) to the derivative which is included in the security (which could be option-like, swap-like, or other) because we are more interested in the hedging opportunities rather than in the organizational aspects of the deal. We are aware of the importance that such aspects play from a practical point of view, but their discussion goes beyond the aims of this book. Both for mortality and longevity bonds, a reference population is chosen, whose mortality rates are observed during the lifetime of the bond. The population may consist of a given cohort (as can be the case for longevity bonds) or a given mix of populations, possibly of different countries (typically this applies to mortality bonds). A mortality or a survivor index is defined, whose performance is assessed according to the mortality experienced in the reference population. Possible examples of an index are: the average mortality rate in one-year’s time (or a longer period) across the population, the number of survivors relative to the size of the population at the time of issue of the bond, and so on. The amount of the coupon is contingent on such index; in particular, the coupon may be higher/lower the higher is the index, depending on the specific bond design. In some cases, the principal may vary (in particular, be reduced) according to the mortality index. Specific cases are discussed below, separately for mortality and longevity bonds. We point out that, to avoid lack of confidence in the way that the pay-off of the mortality-linked security is determined, mortality data should be collected and calculated by independent analysts; so typically general population mortality data are referred to instead of insurance data (we will come back later to this aspect).

7.4


Mortality bonds are designed as catastrophe bonds. The purpose is to provide liquidity in the case of mortality being in excess of what is expected, possibly owing to epidemics or natural disasters. So, typically a short position on the bond may hedge liabilities of an insurer/reinsurer dealing with life insurances. Mortality bonds are typically short term (3-5 years) and they are linked to a mortality index expressing the frequency of mortality observed in the reference population in a given period. Some thresholds are normally set at bond issue. If the mortality index outperforms a threshold, then either the principal or the coupon are reduced. Although it is outside the scope of the discussion to deal with mortality risk in the portfolios of life insurances, we discuss in some detail possible structures for mortality bonds to give a comprehensive picture of the developing mortality-linked securities. In what follows, 0 is the time of issue of the bond and T its maturity. With It we denote the mortality index after t years from bond issue (t = 0, 1, . . . , T). Further, St denotes the principal of the bond at time t and Ct the coupon due at time t. Mortality bond – example 1. The bond is designed to protect against high mortality experienced during the lifetime of the bond itself. This is obtained by reducing the principal at maturity. Although just some ages could be considered in detecting situations of high mortality, it is reasonable to address a range of ages. Further, the index should account for mortality over the whole lifetime of the bond. So the following quantities represent possible examples of a mortality index IT = max{q(t)}t=1,2,...,T T q(t) It = t=1 T

(7.67) (7.68)

where q(t) is the annual frequency of death averaged across the chosen population in year t (we stress that although in our notation t is the time since the issue of the bond, the frequencies of death in (7.67) and (7.68) are recorded in specific calendar years, namely, in the ‘calendar year of issue + t”). It is then reasonable for I0 = q(0). At maturity the principal paid-back to investors is   if IT ≤ λ I0 1 ST = S0 × (IT ) if λ I0 < IT ≤ λ I0   0 if IT > λ I0

(7.69)

where λ , λ are two parameters (stated under bond conditions), with 1 ≤ λ < λ , and (IT ) is a decreasing function, such that (λ I0 ) = 1 and

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(λ I0 ) = 0. For example, (IT ) =

λ I0 − IT (λ − λ ) I0

(7.70)

Note that λ I0 and λ I0 represent two thresholds for the mortality index. The coupon is independent of mortality; it could be defined as follows: Ct = S0 (it + r)

(7.71)

where it is the market interest rate in year t (defined by the bond conditions) and r is an extra-yield rewarding investors for taking mortality risk. We note that for an insurer/reinsurer dealing with life insurances and taking a short position in the bond, in the case of high mortality experience, the high frequency of payment of death benefits is counterbalanced by a reduced payment to investors. An example of this security is the mortality bond issued by Swiss Re; see, for example, Blake et al. (2006a). Mortality bond – example 2. The flows of the bond described in the previous example try to match the flows in the life insurance portfolio just at the end of a period of some years. An alternative design of the mortality bond may provide a match on a yearly basis. This is obtained by letting the coupon depend on mortality. For example,   if It ≤ t it + r Ct = S0 × (it + r) φ(It ) if t < It ≤ t (7.72)   0 if It > t where t , t set two mortality thresholds. For example, t = λ E[Dt |A] t = λ E[Dt |A]

1 ≤ λ < λ

(7.73)

where Dt is the number of deaths in year (t−1, t) in the reference population and E[Dt |A] is its expected value according to the mortality assumption A. Clearly, in this structure the mortality index It should measure the number of deaths in year (t − 1, t). The function φ(·) should then be decreasing; for example, − It φ(It ) = t (7.74) t − t As in (7.71), the rate r in (7.72) is an extra-investment yield rewarding investors for the mortality risk inherent in the pay-off of the bond.

7.4


For longevity bonds the critical situation is a mortality lower than expected or, in other terms, people outliving their expected lifetime. In contrast to the situation of extra-mortality, excess survivorship is not a sudden phenomena, but rather a persistent situation. So longevity bonds are, by nature, long term. Remark It is worthwhile stressing the difference between longevity bonds and (fixed-income) long-term bonds. While the former are financial securities whose performance is linked to some longevity index (see below for details), the latter are traditional bonds with, say, a 20–25 years maturity, and (usually) a fixed annual interest (or possibly an annual interest linked to some economic or financial index, as for example an inflation index). Although not tailored to the specific needs arising from the longevity risk, long-term bonds can help in meeting obligations related to a life annuity portfolio. Actually, one of the most important problems in managing portfolios of life annuities (with a guaranteed benefit) consists in mitigating the investment risk through the availability of fixed-income long-term assets, to match the long-term liabilities. Clearly, this problem becomes more dramatic as the expected duration of the life annuities increases. Depending on its design, the longevity bond may offer hedging opportunities to an insurer/reinsurer dealing with life annuities through either a long or a short position. In the first case, the pay-off of the bond increases with decreasing mortality; vice versa in the second case. Given the longterm maturity, it is reasonable that the link is realized through the coupon, hence providing liquidity on a yearly basis. In the following, we therefore assume that the principal is fixed. The reference population should be a given cohort, possibly close to retirement, that is, with age 60–65 at bond issue. Let Lt be the number of individuals in the cohort after t years from issue, t = 0, 1, . . . ; viz, L0 = l0 is a known value. A maturity T may be chosen for the bond, with T high (e.g.: T ≥ 85−initial age). In the following, some possible designs for the coupons are examined. Longevity bond – example 1. The easiest way to link the coupon to the longevity experience in the reference population is to let it be proportional to the observed survival rate. So Ct = C ×

Lt l0

(7.75)

where C is a given amount (linking the size of the coupon to the principal of the bond). We note that in the case of unanticipated longevity the coupon increases faster than expected; so a long position should be taken by an

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insurer/reinsurer dealing with life annuities. A similar bond has been proposed by EIB/BNP Paribas, although it has not been traded on the market; see Blake, Cairns and Dowd (2006a) for details. Longevity bond – example 2. In a similar way to the mortality bond (example 1 or 2), two thresholds may be assigned, expressing survival levels. If the number of survivors in the cohort exceeds such thresholds, then the amount of the coupon is reduced, possibly to 0. The following definition can be adopted:  lt −lt  if Lt ≤ lt   l0 Ct = C × lt −Lt if lt < Lt ≤ lt (7.76) l0   0 if L > l t

t

where lt , lt are the two thresholds, expressing a given number of survivors. For example: lt = λ E[Lt |A(τ)], lt = λ E[Lt |A(τ)], where 1 ≤ λ < λ and A(τ) is a given mortality assumption for the reference cohort (assumed to be born in year τ). We note that, in this case, the lower is the mortality (i.e. the higher is Lt ), the lower is the amount of the coupon. A short position should be taken to hedge life annuity outflows. A similar bond is described by Lin and Cox (2005). Longevity bond – example 3. The coupon can be set proportional to the number of deaths observed in the reference cohort from issue. For example Ct = C ×

l 0 − Lt l0

(7.77)

where l0 − Lt is the observed number of deaths up to time t. In contrast to the previous case, no target is set for such a number. Clearly, also in this case a short position should be taken to hedge longevity risk. We will discuss in more detail how to hedge longevity risk through longevity bonds in Section 7.4.3. We now address some market issues. There are many difficulties in developing a market for longevity bonds. A first issue concerns who might be interested in issuing/investing in bonds that offer hedging opportunities to insurers/reinsurers. In general terms, one could argue that such securities may offer diversification opportunities, in particular because of their low correlation with standard financial market risk factors. Further, they may give long-term investment opportunities, which may be rarely available. From the point of view of the issuer of bonds like example 1, the possibility of building a longevity bond depends, however, on the availability of financial securities with an appropriate maturity to match the payments promised under the longevity bond.

7.4


A further issue, already mentioned, concerns the choice of mortality data. To encourage confidence in the linking mechanism, reference to insurance data should be avoided. Data recorded and analysed by an independent body should rather be adopted. This raises an issue of basis risk for hedgers (see Section 7.4.3). Conversely, there are many weak points in a mechanism linking the pay-off of the bond to insurance data; among these, we mention the following: insurance data may be affected in particular by insurers/reinsurers with large portfolios, so that some manipulation of data may be feared by investors; due to commercial reasons, the mix of the insured population may change over time, whilst reference to the general population offers more stability. A final aspect (but not least in terms of importance) concerns the pricing of the longevity risk transferred to the capital market. Also in this respect there are many difficulties. First, an overall accepted model for stochastic mortality is not yet available (see Section 7.2.3). Second, a market is not yet developed, nor are similar risks traded in the market itself. So, even if there were common agreement on a pricing model, data to estimate the relevant parameters are not yet available. Three theoretical approaches have been proposed in the literature: distortion measures, risk-neutral modelling, and incomplete markets. Researches in this respect can be considered to be at an early stage and open issues remain requiring careful investigation. See Section 7.6 for some examples, and Section 7.8 for references.

7.4.3

Hedging life annuity liabilities through longevity bonds

We refer here to an insurer or to a reinsurer dealing with immediate life annuities. In the case of an insurer, we refer to the portfolio described in Section 7.2.4; in the case of a reinsurer, we assume that support is provided to an insurer with a portfolio as the one described in Section 7.2.4. We have already noted (see Section 7.3.3) that heterogeneous annual amounts mainly impact on random fluctuations. Therefore, when managing mortality risk in a life annuity portfolio, the insurer should first underwrite some traditional surplus reinsurance to reduce the dispersion of annual amounts in its portfolio. In the following, we will assume that such action has been taken; so, unless otherwise stated, we make reference to a homogeneous life annuity portfolio, where b(j) = b for each annuitant j. We recall that in this () case Bt = b Nt . (·)

The insurer/reinsurer faces the random outflows Bt and counterbalances (·) them with random flows Ft , such that the net outflows Bt − Ft are close ∗ to some target outflows OFt . If the hedging is pursued by an insurer, then () reference is to the original outflows Bt of the life annuity portfolio. If the

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7 : The longevity risk: actuarial perspectives (SL)

hedging is realized by a reinsurer, then reference is to the outflows Bt , (swap) (swap−b) Bt , or Bt , depending on the reinsurance arrangement dealt with. In the following, we discuss how the target OFt∗ can be set and reached according to the hedging tools available in the market. For the sake of brevity, we assume that the longevity bond is issued at the same time as the life annuities; some comments will follow in this regard. Thus, unless otherwise stated, time 0 will be the time of issue of the life annuities and the bond. We first consider the case of a longevity bond with coupon (7.75). An insurer dealing with immediate life annuities should buy k units of such bond at time 0, so that Ft = k Ct > 0 at time t = 1, 2, . . . . The net outflow for the insurer at time t, t = 1, 2, . . . , is then (LB)

OFt

()

= Bt

− k Ct

(7.78)

Lt Nt − kC n0 l0

(7.79)

which can be rewritten as (LB)

OFt

= b n0

We assume that Nt /n0 = Lt /l0 for any time t; this means that mortality of annuitants is perfectly replicated by mortality in the reference population. The net outflow to the insurer then becomes (LB)

OFt

=

Lt (b n0 − k C) l0

(7.80)

Note that the net outflow is still random because of the dependence on Lt . However, if k = b n0 /C then the term b n0 − k C reduces to zero, and a situation of certainty is achieved (i.e. the hedging would be perfect); the target outflow for this situation is therefore OFt∗ = 0. In practical terms, perfect hedging is difficult to realize. Although we can rely on some positive correlation between the survival rate in the reference population, Lt /l0 , and that in the annuitants’ cohort, Nt /n0 , it is unrealistic that they coincide in each year, due to the fact that usually the annuitants are not representative of the reference population. In particular, the year of birth of the reference cohort and of annuitants may differ. This mismatching leads to basis risk in the strategy for hedging longevity risk. A second aspect concerns the lifetime of the bond. Typically the bond is not issued when the life annuity payments start. If it is issued earlier, the previous relations still hold, just with an appropriate redefinition of the quantities l0 and Lt ; the problem in this case would consist in the availability of the bond, in the required size, in the secondary market. If the bond is issued later than the life annuities, the longevity risk of the insurer would be

7.4


unhedged for some years (but in a period when annuitants are still young, and longevity risk is therefore not too severe). In both cases, the basis risk may be stronger, due to the fact that it is more likely that the years of birth of annuitants and the reference population are different. The critical aspect of the lifetime of the bond is its maturity, T. Realistically, T is a finite time, so that the hedge in (7.79) can be realized just up to time T (and not for any time t). The insurer has to plan a further purchase of longevity bonds after time T; however, the availability of bonds, in particular with the features required for the hedging, is not certain. In the case that further longevity bonds are available in the future, the basis risk may worsen in time, given that for any bond issue a cohort of new retirees is likely to be referred to. We now move to longevity bonds with coupon (7.76) and (7.77). As already mentioned in Section 7.4.2, such bonds require a short position to hedge longevity risk. This position is, however, difficult for an insurer (or other annuity provider) to realize on its own, because of the complexity of the deal. It is reasonable to assume that some form of reinsurance is purchased by the annuity provider. The reinsurer, who transacts business on a larger scale than the insurer, then hedges its position through longevity bonds, typically issued by an SPV (see Fig. 7.17). Let us assume that a reinsurer is able to issue a bond with coupon (7.76). The reinsurer should be willing, in this case, to underwrite the Stop-Loss arrangement on annual outflows, whose reinsurance flows are described

Annuitants

Premiums Annual payments

Annuity Provider

Reinsurer

Benefits

Figure 7.17.

Coupons and Principal

Premium

Premium

Capital Market

SPV

Benefits

Income from bond sale

Longevity risk transfer from the annuity provider to the capital market.

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340


by (7.57). Thus, the longevity bond should offer hedging opportunities against the liabilities of the reinsurer in respect of the insurer, as we will demonstrate. (SL)

Assume that the reinsurer matches the outflow Bt arising from the reinsurance arrangement with a short position on k units of the longevity bond with coupon (7.76). In this case, Ft = −k Ct < 0 at time t = 1, 2, . . . . If the underlying life annuity portfolio is homogeneous in respect of annual (SL) amounts, the net outflow of the reinsurer, NFt , is (SL)

NFt

(SL)

= Bt + k Ct   if b Nt ≤ t 0 = b Nt − t if t < b Nt ≤ t   t − t if b Nt > t  lt −lt  if Lt ≤ lt   l0 lt −Lt + kC × if lt < Lt ≤ lt l0   0 if L > l t

(7.81)

t

Since we are aiming at perfect hedging, the thresholds t , t in the reinsurance arrangement are reasonably chosen according to the feature of the longevity bond. So we assume that t = (lt /l0 ) b n0 and t = (lt /l0 ) b n0 . We can rewrite (replacing the relevant quantities and rearranging)  l t  ≤ lt 0 if N  n  0 0    (SL) lt lt lt Nt t NFt = b n0 × N − if < ≤ n n l0 l0 l0 0 0      lt  lt −lt t if N n0 > l0 l0  lt −lt l  if Ll t ≤ l t   l 0 0 0    l l t + k C × lt −L (7.82) if l t < Ll t ≤ lt l 0 0 0 0      l 0 if Ll t > lt 0

If Nt /n0 = Lt /l0 , this reduces to  l −l  kC tl t   0   (SL) Lt −lt = b n0 l + k C NFt 0     b n lt −lt 0 l 0

lt −Lt l0

0

if

Lt l0

≤

lt l0

if

lt l0

0

lt l0

≤

lt l0

(7.83)

7.4


Annuity outflows Upper limit

Annual outflows

Priority

Flow to investors

Flow to the insurer

Time Figure 7.18. Flows for a reinsurer dealing with a Stop-Loss arrangement on annual outflows and issuing a longevity bond – example 2.

Further, if k = b n0 /C, then

(SL)

NFt

  b n0     = b n0     b n 0 = b n0

lt −lt l0

if

Lt l0

≤

lt l0

lt −lt l0

if

lt l0

lt l0

lt − lt l0

≤

lt l0

(7.84)

which is a non-random situation. A graphical representation is provided in Fig. 7.18. The assumptions on which such a perfect hedging strategy is based are the same as those adopted for the longevity bond – example 1, that is, – the survival rate in the annuitant population, Nt /n0 , is the same as that observed in the reference population, Lt /l0 ; – the lifetime of the bond coincides with the lifetime of the life annuity portfolio; in particular, no maturity has been set. It is clear that such conditions are unrealistic, so that the reinsurer transfers just partially the longevity risk to investors. In any case, the target outflow l −l in setting the hedging strategy in this case is OFt∗ = b n0 t l t . A similar 0 strategy is described by Lin and Cox (2005), albeit without calling explicitly for a reinsurance arrangement between an insurer and a reinsurer.

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We note that the unavailability of a longevity bond which perfectly matches the reinsurer’s liability suggests that the reinsurance arrangement should be underwritten just for a finite time, as we have considered in Section 7.3.4. At any renewal time, the pricing of the arrangement, as well as the relevant conditions, can be updated to take account of the current availability of hedging tools. If the reinsurer is able to issue a bond with coupon (7.77), then the reinsurance-swap arrangement can be hedged. We assume that the reinsurer takes a short position on k units of the longevity bond with coupon (7.77) (note that in this case, similarly to the previous one, Ft = −k Ct < 0). Underwriting jointly a reinsurance-swap arrangement, the following net flow of the reinsurer is (swap)

NFt

()

= Bt

− B∗t + k Ct

(7.85)

First, we refer to a homogeneous life annuity portfolio and note that the target outflow (7.59) for the insurer under the reinsurance-swap can be restated as B∗t = b E[Nt |A(τ), nz ] = b nt∗

(7.86)

and so the net flow for the reinsurer can be rewritten as (swap)

NFt

If

Nt n0

=

Lt l0

and k =

b n0 C ,

l0 − Lt Nt − b nt∗ + k C n0 l0 Lt Nt = k C − b nt∗ + b n0 − kC n0 l0 = b n0

then

(swap)

NFt

(7.87)

= k C − b nt∗ = b (n0 − nt∗ )

(7.88)

which is again non-random. We note that the net outflow of the reinsurer is proportional to the number of deaths assumed as a target in the reinsurance-swap, namely, n0 − nt∗ . Clearly, OFt∗ = b (n0 − nt∗ ) is the target outflow for the hedging strategy. A graphical representation is provided in Fig. 7.19. Remarks on the possibility of realizing a perfect hedging are as in the previous cases. The impossibility of relying on a perfect hedging strategy suggests to adopt the reinsurance-swap arrangement with flows (7.62) instead of (7.60). The arrangement (7.62) could also be justified by a hedging strategy involving several positions on longevity-linked securities. We conclude this section by recalling that whenever longevity risk is transferred to some other entities, either to the issuer of a longevity bond or

7.5

Life annuities and longevity risk

Net outflow bn0

Annual outflow

To investors

Time From/to cedant Figure 7.19. Flows for a reinsurer dealing with a reinsurance-swap arrangement and issuing a longevity bond – example 3.

to a reinsurer, a default risk arises for the insurer. This aspect should be accounted for when allocating capital for the residual longevity risk borne by the insurer itself.

7.5 7.5.1

Life annuities and longevity risk The location of mortality risks in traditional life annuity products

So far in this chapter we have dealt with longevity risk referring to a portfolio of immediate life annuities. The need for taking into account uncertainty in future mortality trends and hence for a sound management of the impact of longevity risk has clearly emerged. However, life annuity products other than immediate life annuities are sold on a number of insurance markets and, in many products, the severity of longevity risk can be even higher than what has emerged in the previous investigations. We now introduce some remarks considering cases other than immediate life annuities. The technical features of several types of life annuities have already been examined in Chapter 1, and the relevant traditional pricing tools as well (see, in particular, Section 1.6). Unsatisfactory features of such models can

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be easily understood if one analyses the models themselves under the perspective of a dynamic mortality scenario. In this section, we develop some general comments on the pricing of life annuities allowing for longevity risk; a few examples are then mentioned in Section 7.6. In Section 1.6, we recalled that in the traditional guaranteed life annuity product the technical basis is stated when the premiums are fixed. So (a) a deferred life annuity with (level) annual premiums implies the highest longevity risk borne by the insurer, as the technical basis is stated at policy issue (hence, well before retirement); (b) a single premium immediate life annuity implies the lowest longevity risk, as the technical basis is stated at retirement time only; (c) the arrangement with single recurrent premiums represents an intermediate solution, given that the technical basis can be stated specifically for each premium. It follows that a stronger safety loading is required for solution (a) than for (b), with solution (c) at some intermediate level. Clearly, in order to calculate properly the safety loading required for the implied longevity risk, some pricing model is needed. Alternatively, policy conditions that allow for a revision of the technical basis should be included in the policy, as will be commented later. As it was recalled in Section 1.6, in case (b) the accumulation of the amount funding an immediate life annuity can be obtained through some insurance saving product, for example, an endowment insurance. A package, in particular, can be offered, in which an endowment for the accumulation period is combined with an immediate life annuity for the decumulation period. Combining an endowment insurance with a life annuity provides the policyholder with (a) an insurance cover against the risk of early death during the working period; (b) a saving instrument for accumulating a sum at retirement, to be (partly) converted into a life annuity; (c) a life annuity throughout the whole residual lifetime. It is interesting to analyse the risks involved by this product, from the point of view of the insurance company (see Fig. 7.20); we refer just to the flows given by net premiums and benefits (hence we disregard risks connected to expenses and other aspects). Consistent with the notation in

7.5

Mortality risk

Risk of surrender


Annuitization risk

0

Reserve

Investment risk

Time

n Accumulation period

Figure 7.20.

Mortality risk (Longevity risk included)

Sum at risk Reserve

Death benefit

C1

Post-retirement period

Risks in an endowment combined with a life annuity.

Section 1.6, we let 0 be the time of issue of the endowment, n the maturity of the endowment and the retirement time as well, x the age at time 0. During the accumulation period, that is, throughout the policy duration of the endowment, the insurer in particular bears: – the investment risk, related to the mathematical reserve of the endowment, if some financial guarantee operates, involving for example a minimum interest rate guarantee; – the (extra-)mortality risk, related to the sum at risk; – the risk of surrender, related to the amount of the reserve, if some guarantee on the surrender price (usually expressed as a share of the reserve) is given. During the decumulation period, as the annual amount is usually guaranteed, the insurer bears: – the investment risk, related to the mathematical reserve of the annuity, if a minimum interest rate guarantee is operating; – the (under-)mortality risk, and in particular the longevity risk. At retirement time, if some guarantee has been given on the annuitization rate, the insurer bears the risk connected to the option to annuitize. This aspect is discussed in more detail in Section 7.5.2.

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As regards the longevity risk, the time interval throughout which the insurer bears the risk itself clearly coincides with the time interval involved by the immediate life annuity, if the annuity rate 1/ax+n is stated and hence guaranteed at retirement time only. We recall that the annuity rate converts the sum at maturity S (used as a single premium) into a life annuity of annual amount b according to the relation b = S/ax+n (see (1.57)). Even if the annuity rate is stated at time n only, it is worth noting that the endowment policy contains an ‘option to annuitize’. Apart from the severity of the longevity risk implied by the guarantee on the annuity rate, the presence of this option determines the insurer’s exposure to the risk of adverse selection, as most of the policyholders annuitizing the maturity benefit will be in a good health status (see Section 1.6.5).

7.5.2

GAO and GAR

The so-called guaranteed annuity option (GAO) (see Section 1.6.2) entitles the policyholder to choose at retirement between the current annuity rate (i.e. the annuity rate applied at time n for pricing immediate life annuities) and the guaranteed one. By definition, the GAO condition implies a guaranteed annuity rate (GAR). In principle, the GAR can be stated at any time t, 0 ≤ t ≤ n. In practice, the GAR stated at policy issue, that is, at time 0, constitutes a more appealing feature of the life insurance product. If the GAR is stated at time n only, the GAO vanishes and the insurance product simply provides the policyholder with a life annuity with a guaranteed annual amount. Whatever may be the time at which the GAR is stated, the life annuity provides a guaranteed benefit, so that it can be referred to as a guaranteed annuity (see Fig. 7.21). Conversely, the expression non-guaranteed annuity denotes a life annuity product in which the technical basis (and in particular the mortality basis) can be changed during the annuity payment period; in practice, this means

GAO

GAR at time t (0 ≤ t ≤ n)

GAR at time n

Figure 7.21.

GAO, GAR and Guaranteed Annuity.

Guaranteed Annuity

7.5


that the annual amount of the annuity can be reduced, according to the mortality experience. Clearly, such an annuity is a rather poor product from the point of view of the annuitant. As a consequence of the GAR, the insurer bears the longevity risk (and the market risk, as the guarantee concerns both the mortality table and the rate of interest) from the time at which the guaranteed rate is stated on. Obviously, the longevity (and the market) risk borne by the insurer decreases as the time at which the guaranteed rate is stated increases. The importance of an appropriate pricing of a GAO, and therefore of an appropriate setting of a GAR, is witnessed by the default of Equitable Life. The unanticipated decrease in interest and mortality rates experienced during the 1990s, let the GAOs issued by Equitable during the 1980s to become deeply in the money at the end of the 1990s. As a consequence, in 2000 the Equitable was forced to close to new life and pension business. Pricing a life annuity product within the GAR framework requires the use of a projected mortality table. The more straightforward (and traditional) approach for pricing the guarantee consists of adopting a table that includes a safety loading to meet mortality improvements higher than expected. One should, however, be aware of the fact that the possibility of unanticipated mortality improvements reduces the reliability of such a safety loading (as happened to Equitable). A more appropriate approach requires a pricing model explicitly allowing for the longevity risk borne by the insurer, rather than a safety loading roughly determined; see Section 7.6.

7.5.3

Adding flexibility to GAR products

A rigorous approach to pricing a GAR product usually leads to high premium rates, which would not be attractive from the point of view of the potential clients. Conversely, lower premiums leave the insurer hardly exposed to unexpected mortality improvements. However, in both cases, adding some flexibility to the life annuity product can provide interesting solutions to the problem of pricing guaranteed life annuities. In what follows we focus on some practicable solutions. Assume that the insurer decides to set the GAR 1/a[1] x+n (h) at time h (0 ≤ h < n) for a deferred life annuity to be paid from time n. Suppose that a[1] x+n (h) is lower than the correspondent output of a rigorous approach to GAR pricing. If an amount S is paid at time n as a single premium, the

347

348


Reduction in the annual amount b[1] b'[1]

0

h

r

n

Time

A new projected table Figure 7.22.

Annual amount in a conditional GAR product.

resulting annual amount of the life annuity is given by b[1] =

S a[1] x+n (h)

(7.89)

Assume that the insurer promises to pay the annual amount b[1] from time n on, with the proviso that no dramatic improvement in the mortality experienced occurs before time n. Conversely, if such an improvement is experienced (and it results, for example, from a new projected life table available at time r, h < r ≤ n), then the insurer can reduce the annual amount to a lower level b[1] (see Fig. 7.22). So a policy condition must be added, leading to a conditional GAR product. Some constraints are usually imposed (e.g. by the supervisory authority); in particular: (a) the mortality improvement must exceed a stated threshold (e.g. in terms of the increase in the life expectancy at age 65); (b) r ≤ n − 2, say; (c) no more than one reduction can be applied in a given number of years; (d) whatever the mortality improvements may be, the reduction in the annual amount must be less than or equal to a given share ρ, that is, b[1] − b[1] ≤ρ b[1]

(7.90)

so that, combining (c) and (d), a guarantee of minimum annual amount works. Conversely, from time n the annual amount is guaranteed, irrespective of any mortality improvement which can be recorded afterwards.

7.5


Increase in the annual amount b' [2] b[2]

0

h

n

s

Time

Experienced mortality higher than expected Figure 7.23.

Annual amount in a participating GAR product.

Let us now turn to the case in which the insurer charges a rigourous (i.e. lower) annuity rate 1/a[2] x+n (h). Hence, the annuity amount is given by b[2] =

S a[2] x+n (h)

(7.91)

with b[2] < b[1] . Suppose that, at time s (s > n), statistical observations reveal that the experienced mortality is higher than expected, because of a mortality improvement lower than forecasted. Hence, a mortality profit is going to emerge from the life annuity portfolio. Then, the insurer can decide to share part of the emerging profit among the annuitants, by raising the annual amount from the (initial) guaranteed level b[2] to b[2] (see Fig. 7.23). This mechanism leads to a with-profit GAR product (or participating GAR product). Participation mechanisms work successfully in a number of life insurance and life annuity products as far as distributing the investment profits is concerned. Conversely, mortality profit participation is less common. Notwithstanding, important examples are provided by mortality profit sharing in group life insurance and, as regards the life annuity business, participation mechanisms adopted in the German annuity market. The critical point is that, in contrast to what happens for products with participation to investment profits and to mortality profits in life insurance, people participating to mortality profits in life annuity portfolios are not those who have generated such profits and, so, a tontine scheme emerges (see Section 1.4.3).

349

350


Reduction in the annual amount b[3] b[2]

0

h

n

s

Time Experienced mortality lower than expected

Figure 7.24.

Annual amount in a product with conditional GAR in the decumulation period.

It is worthwhile to note that from a technical point of view a policy condition similar to the conditional GAR may work also during the decumulation period. In this case, the amount of the benefit (possibly assessed at retirement time with an annuity rate higher than what resulting from a ring approach to GAR pricing) may be reduced in the case of strong unanticipated improvements in mortality. It would be reasonable to fix a minimum benefit level in this case. As an illustration, assume that the amount b[2] resulting from (7.91) is considered the level of benefit that is consistent with a rigorous approach to GAR pricing. However, considering that the implied safety loading could turn out to be too severe according to the actual mortality experienced, the insurer is willing to pay the annual benefit b[3] , with b[3] > b[2] . If after time n, a strong mortality improvement is recorded, then the insurer will reduce the annual amount down to b[2] (see Fig. 7.24). Constraints similar to (a) and (c) for the conditional GAR in the accumulation period should be applied. From a commercial point of view, care should be taken in making clear to the annuitant that the guaranteed benefit is b[2] and not b[3] . However, a tontine scheme emerges, given that in some sense a participation to losses is realized.

7.6

Allowing for longevity risk in pricing

As already pointed out, we are not going to discuss in details the problem of pricing long-term living benefits allowing for longevity risk. Indeed, the unsolved issues are too important and complex to allow for a complete

7.6


description in the present chapter: for example, there are different opinions on evolving mortality and hence on the appropriate stochastic model to allow for uncertain mortality trends, and the data for estimating the main parameters are unavailable. On the other hand, pricing models for longevity risk are required when dealing with life annuities and longevity bonds. Therefore, in this section, we summarize a few of the main proposals which have been described in literature. However, this is a subject which has been developing in the recent literature, and we do not aim to give a comprehensive illustration of the several proposals that have been put forward. We first address the present value of life annuities. Denuit and Dhaene (2007) and Denuit (2007) allow for randomness in the probabilities of death within a Lee–Carter framework. Due to the importance of such a framework, we briefly describe their approach. Let us adopt the standard Lee–Carter framework, where the future forces of mortality are decomposed in a log-bilinear way (see Section 4.7.2). Specifically, the death rate at age x in calendar year t is of the form exp(αx + βx κt ), where κt , in particular, is a time index, reflecting the general level of mortality. We denote as h Px0 (t0 ) the random h-year survival probability for an individual aged x0 in year t0 , that is, the conditional probability that this individual reaches age x0 + h in year t0 + h, given the κt ’s. Adopting assumptions (3.2) (from which (3.13) holds), such probability is formally defined as   h−1  mx0 +s (t0 + s) h Px0 (t0 ) = exp − s=0

 = exp−

h−1



exp αx0 +s + βx0 +s κt0 +s 

(7.92)

s=0

We refer to a basic life annuity contract paying the annual amount b = 1 at the end of each year, as long as the annuitant survives. The present value of such annuity is the expectation of the payments made to an annuitant aged x0 in year t0 , conditional on a given time index; it is calculated as ax0 (t0 ) =

ω−x 0 h=1

=

ω−x 0 h=1

h Px0 (t0 )v(0, h)

 exp−

h−1 s=0



exp αx0 +s + βx0 +s κt0 +s  v(0, h)

(7.93)

351

352


where v(0, h) is the discount factor, that is, the present value at time 0 of a unit payment made at time h. We note that ax0 (t0 ) is a random variable, since it depends on the future trajectory of the time index (i.e. on κt0 , κt0 +1 , κt0 +2 , . . .). We note also that (7.93) generalizes (1.27). The distribution function of ax (t0 ) is difficult to obtain. Useful approximations have been proposed by Denuit and Dhaene (2007) and Denuit (2007). Specifically, Denuit and Dhaene (2007) have proposed comonotonic approximations for the quantiles of the random survival probabilities h Px0 (t0 ). Since the expression for ax (t0 ) involves a weighted sum of the h Px0 (t0 ) terms, Denuit (2007) supplemented the first comonotonic approximation with a second one. This second approximation is based on the fact that the h Px0 (t0 ) terms are expected to be closely dependent for increasing values of h so that it may be reasonable to approximate the vector of random survival probabilities with its comonotonic version. Interesting information can be obtained from a further investigation of the distribution of ax0 (t0 ). We consider a homogeneous portfolio, made of n0 annuitants at time t0 . We refer now to the random variable aK(j) , where x0

(j)

(j)

Kx0 is the curtate lifetime of individual j. Given the time index, the Kx0 ’s are assumed to be independent and identically distributed, with common conditional h-year survival probability h Px0 (t0 ). We recall from Denuit et al. (2005) that a random variable X is said to precede another one Y in the convex order, denoted as X cx Y, if the inequality E[g(X)] ≤ E[g(Y)] holds for all the convex functions g for which the expectations exist. Since X cx Y ⇒ E[X] = E[Y] and Var[X] ≤ Var[Y], X cx Y intuitively means that X is ‘less variable’, or ‘less dangerous’ than Y. Now, since the aK(j) ’s are exchangeable, we have from Proposition 1.1 x0

in Denuit and Vermandele (1998) that n0 +1 ax (t0 ) = E[aK(j) |κt0 +k , k = 1, 2, . . .] cx · · · cx x0

n0

cx

j=1 aKx(j) 0

n0

.

j=1

aK(j) x0

n0 + 1 (7.94)

Increasing the size of the portfolio makes the average payment per annuity less variable (in the cx -sense), but this average remains random whatever the number of policies comprising the portfolio, being bounded from below by ax (t0 ) in the cx -sense. We note that, despite the positive dependence

7.6


existing between the Lee–Carter lifetimes, there is still some diversification effect in the portfolio. Biffis (2005) calculates the single premium of a life annuity adopting affine jump-diffusions for modelling the force of mortality and the short interest rate. In this way, one deals simultaneously with financial and mortality risks and calculates values based on no-arbitrage arguments. The setting is also applied for portfolio valuations in Biffis and Millossovich (2006a) and to the valuation of GAOs in Biffis and Millossovich (2006b). Affine mortality structures are also addressed by Dahl (2004) and Dahl and Møller (2006), where, in particular, hedging strategies for life insurance liabilities are investigated. Turning to the problem of pricing longevity bonds, Lin and Cox (2005), consider that the market is incomplete, and adopt the Wang transform (see, e.g. Wang (2002) and Wang (2004)). Given the future random flow X with cumulative probability distribution function (briefly, cdf) F(x), the one-factor Wang transform is the distorted cdf F ∗ (x) such that F ∗ (x) = (−1 (F(x)) + λ)

(7.95)

where (·) is the standard normal cdf and λ is the market price of risk (longevity risk included). The fair price of X is the present value of the expected value of X, calculated with the risk-free rate and the distorted cdf F ∗ (x). Lin and Cox (2005) take X as the lifetime of an annuitant and calibrate λ using life annuity quotations in the market (assuming that the price of a life annuity is the present value of future payments, based on the risk-free rate and the distorted cdf of the lifetime). They then apply the approach to price mortality-linked securities. The one-factor Wang transform assumes that the underlying distribution is known. However, usually F(x) is the best-estimate of the underlying unknown distribution. The two-factor Wang transform is the cdf F ∗∗ (x) such that F ∗∗ (x) = Q(−1 (F(x)) + λ)

(7.96)

where Q is the t-distribution with k degrees of freedom. Lin and Cox (2008) adopt this latter approach for pricing mortality-linked securities, with k = 6. Cairns, Blake and Dowd (2006a) assume similarities between the force of mortality and interest rates and adapt arbitrage-free pricing frameworks developed for interest-rates derivatives to price mortality-linked securities. In Cairns, Blake and Dowd (2006b) they introduce the two-factor model

353

354


described in Section 5.3 and price longevity bonds with different terms to maturity referenced to different cohorts. In particular, they develop a method for calculating the market risk-adjusted price of a longevity bond, which allows for mortality trend uncertainty and parameter risk as well. We finally address the problem of the valuation of a GAO. The GAO (see Section 7.5.2) consists of a European call option with the underlying asset the retail market value of a life annuity at retirement time and the strike the GAR set when the GAO was underwritten. The pay-off of the option by itself depends on the comparison between the guaranteed and the current annuity rate. However, the actual exercise of the option depends also on the preference that the holder expresses for a life annuity instead of self-annuitization. The intrinsic structure of the pay-off of the option is, therefore, uncertain because it depends on individual preferences, with possible adverse selection in respect of the insurer. When assessing the value of the GAO, individual preferences are usually disregarded in the current literature. The pricing problem is therefore attacked by assuming that the policyholder will decide to exercise the option just comparing the current market quotes for life annuities and the GAR. Ballotta and Haberman (2003) address this problem, assuming that the overall mortality risks (and hence also the longevity risk) are diversified. In Ballotta and Haberman (2006) the analysis is extended to the case in which mortality risk is incorporated via a stochastic model for the evolution over time of the underlying force of mortality.

7.7 7.7.1

Financing post-retirement income Comparing life annuity prices

We refer to a person buying an immediate life annuity. Let S be the capital converted into the annuity and b the annual amount. The annuity rate b/S is a function of: • the discount rate, i; • the reference mortality table, A(τ); • a safety loading (possibly explicit) for longevity risk.

The buyer may be interested in comparing the annuity rates applied by different providers, and in explaining the relevant differences. However, it may not be straightforward to understand the reasons for such differences, due to the interaction of the items building up the annuity rate and the complexity of the pricing model for longevity risk.

7.7

Financing post-retirement income

Typically, the discount rate is disclosed; this is in particular required when participation in investment profits occurs during the annuity payment. The comparison of annuity rates then concerns the incidence of mortality and the relevant interaction with the discount rate. Some equivalent parameters should be produced by the annuity provider (or by some other entity) to provide better information in this regard. It is reasonable that the comparison among annuity rates makes reference to traditional pricing models. In particular, the actuarial value of a life −t annuity, ax = ω−x t=1 (1+i) t px , and the present value of an annuity certain, k aki = t=1 (1 + i)−t = (1 − (1 + i)−k )/i, may be addressed. Given the discount rate assumed in the annuity rate bS , we can determine the equivalent number of payments of an annuity certain, that is, the number k such that aki = bS . If i > 0, we easily find k=−

ln(1 − i bS ) ln(1 + i)

(7.97)

Conversely, if i = 0 then k is simply given by bS and, according to a traditional actuarial valuation of the life annuity, it coincides with the expected lifetime assumed by the annuity provider for the annuitant. Clearly, the stronger is the cost of longevity embedded in the annuity rate bS , the lower is k. Note that if i > 0, k depends also on i. In the case where there is a prevailing mortality table referred to for the traditional actuarial valuation of life annuities, one can calculate what is the equivalent entry age x such that, according to this table and having set the discount rate i, 1/ax coincides with the annuity rate quoted by the annuity provider. Such an age should then be compared with the actual entry age, say x0 . Similarly, in the case where there is a prevailing mortality table referred to for the traditional actuarial valuation of life annuities, an alternative possibility is to refer to the actual age x0 and to calculate the equivalent discount rate, that is, the rate i such that 1/ax0 (based on the reference mortality table) coincides with the quoted annuity rate bS , as it is performed, for example, by Verrall et al. (2006). Example 7.10 With reference to the expected values quoted in Table 7.2 for time 0, we perform the comparisons discussed above. We assume that the prevailing mortality table referred to for the traditional actuarial valuation of the life annuity is given by assumption A3 (τ). All of the other assumptions are as in Example 7.1; in particular, the actual entry age is x0 = 65.

355

356

7 : The longevity risk: actuarial perspectives Table 7.30. Equivalent number of payments of an annuity certain; discount rate: i = 0.03

Mortality assumption A1 (τ) A2 (τ) A3 (τ) A4 (τ) A5 (τ)

Annuity rate 1 14.462 1 14.651 1 15.259 1 15.817 1 16.413

Equivalent number of payments

= 0.06915

19.247

= 0.06825

19.587

= 0.06554

20.707

= 0.06322

21.767

= 0.06093

22.938

Table 7.31. Equivalent number of payments of an annuity certain; mortality assumption: A3 (τ)

Discount rate i=0 i = 0.01 i = 0.02 i = 0.03 i = 0.04

Annuity rate 1 21.853 1 19.238 1 17.071 1 15.259 1 13.733

Equivalent number of payments

= 0.04576

21.853

= 0.05198

21.473

= 0.05858

21.091

= 0.06554

20.707

= 0.07282

20.321

Tables 7.30 and 7.31 give the equivalent number of payments of an annuity certain, for several quoted prices of the life annuity. In particular, in Table 7.30 the discount rate has been kept fixed, while alternative mortality assumptions have been used; in Table 7.31 the annuity rate is based on the mortality assumption A3 (τ) while alternative levels of the discount rate are chosen. Clearly, given the mortality table, the equivalent number of payments of an annuity certain is higher the lower is the discount rate. With a fixed discount rate, the equivalent number of payments is higher the stronger is the mortality improvement implied by the table. In Table 7.32 the reference mortality assumption is A3 (τ) and the reference discount rate is i = 0.03. First, the equivalent discount rate relating to different mortality assumptions is calculated (third column); then the equivalent rounded entry age is quoted (fourth column). We note that a lower equivalent discount rate and a lower equivalent entry age emerge from a stronger assumption about mortality improvements.

7.7.2

Life annuities versus income drawdown

When planning post-retirement income, some basic features of the life annuity product should be accounted for. In particular,

7.7


Table 7.32. Equivalent discount rate, equivalent entry age; reference parameters: mortality A3 (τ), discount rate: i = 0.03

Mortality assumption A1 (τ) A2 (τ) A3 (τ) A4 (τ) A5 (τ)

Annuity rate 1 14.462 1 14.651 1 15.259 1 15.817 1 16.413

Equivalent discount rate

Equivalent entry age x

= 0.06915

3.501%

67

= 0.06825

3.379%

66

= 0.06554

3%

65

= 0.06322

2.673%

64

= 0.06093

2.343%

62

(a) a life annuity provides the annuitant with an inflexible income, in the sense that, if the whole fund available to the annuitant at retirement is converted into a life annuity, the annual income is stated as defined by the annuity rate (apart from the effect of possible profit participation mechanisms); (b) a more flexible income can be obtained via a partial annuitization of the fund, or partially delaying the annuitization itself; the part of the income not provided by the life annuity is then obtained by drawdown from the non-annuitized fund; (c) the life annuity product benefits from a mortality cross-subsidy, as each life annuity in a given portfolio (or pension plan) is annually credited with ‘mortality interests’, that is, a share of the technical provisions released by the deceased annuitants, according to the mutuality principle (see Sections 1.4 and 1.4.1 in particular). Let us start with point (c). We refer to a life annuity issued at age x0 with annual amount b, whose technical provision (simply denoted by Vt ) is calculated according to rule (7.49) (adopting a mortality assumption A(τ)). Recursively, we may express the technical provision as follows: V0 = S Vt−1 (1 + i) = (Vt + b) px0 +t−1 ,

t = 1, 2, . . .

(7.98)

where i is the technical interest rate, px0 +t−1 is based on mortality assumption A(τ) and S is the single premium (see (1.28)). According to a traditional pricing structure, we may further assume S = b ax0

(7.99)

where ax0 is calculated according to the same assumptions adopted in (7.98). To be more realistic, we consider a (financial) profit participation mechanism. We denote as b0 the amount of the benefit set at policy issue (so,

357

358


b0 = b, where b comes from (7.99)). Assume that in each policy year a constant (to shorten notation) extra interest rate r is credited to the reserve. As a consequence, the annual amounts b1 , b2 , . . . , bt , . . . are paid out, at times 1, 2, . . . , t, . . . , where bt is assessed as follows bt = bt−1 (1 + r),

t = 1, 2, . . .

(7.100)

The recursion describing the behaviour of the reserve then becomes Vt−1 (1 + i) (1 + r) = (Vt + bt ) px0 +t−1 ,

t = 1, 2, . . .

(7.101)

or, defining 1 + i = (1 + i) (1 + r), so that i represents the total annual interest rate credited to the reserve Vt−1 (1 + i ) = (Vt + bt ) px0 +t−1 ,

t = 1, 2, . . .

(7.102)

Vt − Vt−1 = −bt px0 +t−1 + Vt−1 i + Vt qx0 +t−1

(7.103)

Rearranging (7.102), we obtain

which can be rewritten as Vt − Vt−1 = Vt−1 i + (Vt + bt ) qx0 +t−1 − bt

(7.104)

and, replacing Vt + bt according to (7.102), finally as Vt − Vt−1 = Vt−1 i +

qx0 +t−1 Vt−1 (1 + i ) − bt px0 +t−1

(7.105)

We note that (7.105) generalizes (1.13). Recalling that Vt − Vt−1 < 0, from (7.103) we find that the variation in the reserve is due to the following contributions: (i) a positive contribution due to the (total) amount of interest assigned to the reserve; (ii) a positive contribution due to mutuality; (iii) a negative contribution by the payment bt . The splitting of the variation of the reserve in a year is sketched in Fig. 1.4. We now address item (b) in the list at the beginning of this Section. As was discussed in Section 1.2.1, the annuitant may decide not to use S to buy a life annuity, but simply to invest it and receive the post-retirement income via a sequence of withdrawals (set at her/his choice). Suppose that the fund is credited each year with annual interest at the rate g. Further assume that the annuitant withdraws from the fund a sequence of amounts

7.7


set to be a (constant) proportion α of the annual payments she/he would have obtained under the life annuity, that is, of the sequence (7.100). Let Ft be the fund available at time t. We have F0 = S Ft−1 (1 + g) = α bt + Ft ,

t = 1, 2, . . .

(7.106)

simply generalizing (1.1). As already noted in Section 1.2.1, there is a time m such that Fm ≥ 0 and Fm+1 < 0, that is, the withdrawals b1 , b2 , . . . , bm exhaust the fund. If the lifetime of the annuitant, Tx0 , turns out to be lower than m, then the amount FTx0 is available at her/his death for bequest. However, if Tx0 > m then at time m the annuitant is unfunded. To avoid early exhaustion, the annuitant should set a low level for α or look for investments with a high yield g. In the former case, however, the annual income may then become insufficient to meet current needs; in the latter case, risky assets could be involved, so that possible losses may then emerge because of fluctuating values. Example 7.11 Let us assume that the amount S = 15.259 can be used to buy a life annuity with initial benefit b = b0 = 1, subject to profit participation. The annuity rate b/S = 1/15.259 is based on a traditional calculation of the actuarial value of the life annuity, under the mortality assumption A3 (τ) and the annual interest rate i = 0.03 (see Table 7.32, second column). We set the actual annual interest rate gained in each year on investments to be i = 0.05, so that benefits are yearly increased by the rate r = 1.05/1.03 − 1 = 0.01942. With the parameters mentioned above, we now refer to the case of drawdown, based on an annual consumption α bt , t = 1, 2, . . . . Setting g = i = 0.05, Fig. 7.25, panel (a), shows the share α as a function of the time m to fund exhaustion. Note that α becomes lower than 1 as soon as the time m is greater than the expected lifetime of the annuitant under scenario A3 (τ) (which turns out to be 21.853 years; see also Table 7.31 for i = 0). Alternatively, setting α = 1, in panel (b) of Fig. 7.25 the required annual investment yield g is quoted, again as a function of the time to exhaustion of the fund. We note that, in this case, g exceeds i = 0.05 as soon as m is greater than the expected lifetime of the annuitant.

7.7.3

The ‘mortality drag’

The absence of mutuality in an income drawdown process can be compensated (at least partially) by a higher investment yield (see Section 1.4.1). The

359

7 : The longevity risk: actuarial perspectives (a)

2.00 1.80 1.60

Share α

1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00

(b)

0

10

20 30 40 Time to exhaustion, m

50

60

0

10

20 30 40 Time to exhaustion, m

50

60

9% 8% 7% 6%

Rate g

360

5% 4% 3% 2% 1% 0%

Figure 7.25. Annual withdrawal (panel (a)) and annual investment yield (panel (b)) as a function of the time to fund exhaustion.

extra return required in each year for this purpose has been called the mortality drag. However, it is worth stressing that a fixed drawdown sequence leads in any case to wealth exhaustion in a given number of years (possibly the maximum residual lifetime), whatever the interest rate may be, as was depicted in Fig. 7.25, panel (b).

7.7


Conversely, the concept of mortality drag suggests an alternative arrangement for the post-retirement income. Assume that at time 0 no life annuity is purchased, whereas some amount will be converted into a life annuity at time k, thus with a delay of k years since the retirement time. We suppose that a traditional pricing method is adopted at time k by the insurer and that the mortality assumption for the trend of the cohort is not revised during the delay period. To facilitate a comparison, we assume that the amount to be annuitized at time k must provide the annuitant with the sequence bk+1 , bk+2 , . . . , whose items follow from (7.100) (assuming b0 = b as given by (7.99)). Hence, the amount to be converted at time k into the life annuity is bk ax+k = Vk

(7.107)

with Vk originated by (7.105). Therefore, an amount funding the reserve to be set up must be provided at time k. If the annuitant aims at getting the same income as under the life annuity also during the delay period, than the drawdown process b1 , b2 , . . . , bk must be defined. Because of the absence of mutuality, if the individual investment provides the same yield as that which the insurer is willing to recognize, then the fund available at time k, Fk , is lower then the required amount to annuitise, Vk . However, an extra return may offset the loss of ‘mutuality (or mortality) returns”, thus leading to Fk = Vk . The size of the extra investment yield required so that Fk = Vk can be obtained from (7.106), considered with α = 1. If i is the yield on the life annuity product, then intuitively g − i is an average of the annual quantities θx+t defined in Section 1.4.1. It is worthwhile stressing that given the deferment k, the extra yield g − i must be obtained in each of the k years of delay. Thus, g − i is like a yield to maturity, measuring the mortality interest in k years, whereas the quantity θx+t is the extra yield specific of year (t − 1, t) (see also (1.34) and (1.35)). Example 7.12 Under the assumptions adopted in Example 7.11 for the life annuity, Fig. 7.26 plots the extra-yield required on individual investments in each of the k years of delay to compensate the loss of mutuality. Trivially, the higher is k, the higher is the required extra yield. Given that the extra yield must be realized in each of the k years of delay, this target may be very difficult to reach when the annuitization is planned for a distant time in the future. It is worthwhile to investigate in more detail how the average mortality drag g − i is affected by the annuity rate. From (7.106), having set α = 1

361

362


9%

Extra investment yield

8%

Life annuity yield

7% 6% 5% 4% 3% 2% 1% 0% 5

Figure 7.26.

10

15

20 25 30 35 Delay period k

40

45

50

Extra investment yield required by mortality drag.

we get t

Ft = S (1 + g) −

t

bh (1 + g)t−h

(7.108)

h=1

Let gk be the rate g such that Fk = Vk for a given k. The rate gk is therefore defined by the following relation: S (1 + gk )k −

k

bh (1 + gk )k−h = Vk

(7.109)

h=1

Note that Fig. 7.26 actually plots the rate gk for several choices of k. From (7.100), we can express the annual benefit at time t as bt = b (1 + r)t

(7.110)

Replacing (7.110), (7.107), and (7.99) into (7.109), we obtain b ax0 (1 + gk )k −

k

b (1 + r)h (1 + gk )k−h = b (1 + r)k ax0 +k

(7.111)

(1 + r)k+1 1+r + = (1 + r)k ax0 +k gk − r gk − r

(7.112)

h=1

or equivalently ax0 (1 + gk )k − (1 + gk )k

7.7


which suggests that gk depends on the annuity rate applied at time k, 1/ax0 +k , but also on that applied at time 0, 1/ax0 . The rate gk obtained with r = 0 has been named the Implied Longevity Yield (ILY)1 ; see Milevsky (2005) and Milevsky (2006). The delay in the purchase of the life annuity may have some advantages. In particular: – in the case of death before time k, the fund available constitutes a bequest (which is not provided by a life annuity purchased at time 0, because of the implicit mortality cross-subsidy); – more flexibility is gained, as the annuitant may change the annual income modifying the drawdown sequence (with a possible change in the fund available at time k). Conversely, a disadvantage is due to the risk of a shift to a different mortality assumption, leading to a conversion rate at time k which is less favourable to the annuity purchaser than the one in force at time 0. Further, as already noted, in the case where k is high, it may be difficult to gain the required mortality drag.

7.7.4

Flexibility in financing post-retirement income

Combining an income drawdown with a delay in the life annuity purchase constitutes an example of a post-retirement income arrangement which is more general than the one consisting of a life annuity-based income only. We now summarize what has emerged in the previous sections, thereby defining a general framework for a discussion of post-retirement income planning. Our focus will be mostly on mortality issues, to keep the presentation in line with the main scope of the chapter. Nevertheless, important financial aspects should not be disregarded when assessing and comparing the several opportunities of meeting post-retirement income needs. We assume that an accumulation process takes place during the working period of an individual. After retirement, a decumulation process takes place and hence income requirements are met using, in some way, the accumulated fund. Figure 7.27 illustrates the process consisting of: 1. the accumulation of contributions during the working period; 1 Registered trademarks and property of CANNEX Financial Exchanges.

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364


Contributions (before retirement) Annuity purchase Non-annuitized fund

Annuitized fund

Interests

Interests Mortality Income drawdown (after retirement)

Figure 7.27.

Annuity payment (after retirement)

Accumulation process and post-retirement income.

2. a (possible) annuitization of (part of) the accumulated fund (before or after retirement); 3. receiving a post-retirement income from life annuities or through income drawdown. The annuitization of (part of) the accumulated fund consists of purchasing a deferred life annuity if annuitization takes place during the accumulation period, and an immediate life annuity otherwise. Hence, at any time, the resources available for financing post-retirement income are shared between a non-annuitized and an annuitized fund. It is reasonable to assume that a higher degree of flexibility in selecting investment opportunities is attached to the non-annuitized fund. We note that the non-annuitized fund builds up because of contributions and investment returns. Conversely, the annuitized fund builds up because of investment returns and mortality, as the fund coincides with the total mathematical reserve of the life annuities purchased, and hence it benefits from the cross-subsidy effect. Figures 7.28 and 7.29 illustrate a possible behaviour of the nonannuitized and the annuitized fund, respectively. Effects of the life annuity purchase (jumps in the processes), of the income drawdown and of the annuity payment are identified. The slope of the non-annuitized fund depends, while the fund itself is increasing, on both contributions and interest earnings, whereas it depends on the drawdown policy while the fund is decreasing. As regards the annuitized fund, as previously noted, its slope depends on interest and mortality,

7.7


Fund

Life annuity purchase

Income drawdown

Accumulation period

Post-retirement period Time

Figure 7.28.

The non-annuitized fund.

Life annuity payment

Fund

Life annuity purchase

Accumulation period


Figure 7.29.

The annuitized fund.

while it is increasing, whereas it also depends on the annuity payment while decreasing. Let us denote by Ft[NA] and Ft[A] the values of the non-annuitized and the annuitized fund, respectively, at time t. The ‘degree’ of the annuitization policy can be summarized by the annuitization ratio ar(t), defined as follows: ar(t) =

Ft[A] Ft[A] + Ft[NA]

(7.113)

Note that, obviously, 0 ≤ ar(t) ≤ 1; ar(t) = 0 means that up to time t no life annuity has been purchased, whilst ar(t) = 1 means that at time t the whole fund available consists of reserves related to purchased life annuities.

365


100% (1) Annuitization ratio

366

Deferred life annuity Income drawdown only (2)

0% Accumulation period


Figure 7.30.

Arrangements: (1) deferred life annuity; (2) income drawdown.

Example 7.13 Figures 7.30–7.33 illustrate some strategies for financing post-retirement income. In most cases, the technical tool provided by the life annuity is involved. The various strategies are described in terms of the annuitization ratio profile; thus, the value of ar(t) is plotted against time t. To improve understanding, we suppose that a specified mortality assumption is adopted when annuitizing (a part of) the accumulated fund and that the assumption itself cannot be replaced in relation to the purchased annuity, whatever the mortality trend might be (so, that a guaranteed annuity is involved). Figure 7.30 illustrates two ‘extreme’ choices. Choice (1) consists of building up a traditional deferred life annuity. In this case, each amount paid to the accumulation fund (possibly a level premium, or a single recurrent premium) is immediately converted into a deferred life annuity; this way, the accumulated fund is completely annuitized. Post-retirement income requirements are met by the life annuity (a flat annuity or, possibly, a rising profile annuity, viz an escalating annuity or an inflation-linked annuity). Choice (2) represents the opposite extreme. There is no annuitization operating, so that income requirements are fulfilled by income drawdown, which implies spreading the fund accumulated at retirement over the future life expectation, according to some spreading rule. Sometimes annuitants prefer this choice because of the high degree of freedom in selecting investment opportunities even during the post-retirement period. It should be stressed that choice (1) leads to an inflexible post-retirement income, whilst choice (2) allows the annuitant to adopt a spreading rule

7.7


Annuitization ratio

100%

Immediate life annuity

0% Accumulation period


Figure 7.31.

Immediate life annuity.

consistent with a specific income profile. Conversely, it is worth noting that arrangement (1) completely transfers the mortality risk (including its longevity component) to the insurer, whilst according to arrangement (2) the mortality risk remains completely with the annuitant (see Section 7.7.2). In more general terms, the process of transferring mortality risk depends on the annuitization profile: thus, the portion of mortality risk transferred from the annuitant to the insurer increases as the annuitization ratio increases. The following arrangements provide practical examples of how mortality risk can be transferred, as time goes by, to the insurer. The annuitization of the fund at retirement time only is illustrated in Fig. 7.31, which depicts the particular case of a complete annuitization of the fund available at retirement. This arrangement can be realized through purchasing a single-premium life annuity, and is characterized by flexibility in the investment choice during the accumulation period. Conversely, it produces an inflexible post-retirement income profile. In Fig. 7.32, the annuitization ratio increases during the accumulation period because of positive jumps corresponding to the purchase of life annuities with various deferment periods. The behaviour of the annuitization ratio between jumps obviously depends on the contributions and the interest earnings affecting the non-annuitized fund as well as on the financial and mortality experience of the annuitized fund. In contrast Fig. 7.33 illustrates the case in which no annuitization is made throughout the accumulation period, whereas the fund available after the retirement date is partially used (with delays) to purchase life annuities;

367


Annuitization ratio

100%

Combined annuities

0% Accumulation period Figure 7.32.

Time


Combined life annuities.

100%

Annuitization ratio

368

Staggered annuitization

0% Accumulation period Figure 7.33.

Time


Staggered annuitization.

such a process is sometimes called staggered annuitization or staggered vesting. The behaviour of the ratio between jumps depends on the interest earnings and income drawdown as regards the non-annuitized fund as well as financial and mortality experience of the annuitized fund. Arrangements like those illustrated by Figs. 7.32 and 7.33 are characterized by a high degree of flexibility as regards both the post-retirement income profile and the choice of investment opportunities available for the non-annuitized fund.

7.8


The framework proposed above clearly shows the wide range of choices leading to different annuitization strategies. So, convenient investment and life annuity products can be designed, to meet the different needs and preferences of the clients. An example in this regard is given by the solutions providing natural hedging across time (Section 7.3.2), such as the moneyback annuity with death benefit (7.32), which is designed so that at some future time the death benefit reduces to zero. We note that, as long as the death benefit is positive, a situation of fund just partially annuitized can be identified. As soon as the death benefit reduces to zero, the fund turns out to be fully annuitized. Thus, an annuitization strategy is embedded in the structure of money-back annuities.

7.8


In this section we summarize the main contributions on the topics dealt with in this chapter, some of which have already been mentioned while addressing specific issues. However, the purpose is to add references to those that have been previously cited. An informal and comprehensive description of longevity risk, and in particular of the relevant financial impact on life annuities, is provided by Richard and Jones (2004). See also Riemer-Hommel and Trauth (2000). A static framework for representing the longevity risk according to a probabilistic approach has been used, for example, by Olivieri (2001), Olivieri and Pitacco (2002a), Olivieri and Pitacco (2003). Olivieri and Pitacco (2002a) suggest a Bayesian-inferential procedure for updating the weighting distribution. Marocco and Pitacco (1998) adopt a continuous probability distribution for weighting the alternative scenarios. A dynamic probabilistic approach to longevity risk modelling has been proposed, among the others, by Biffis (2005), Dahl (2004), Cairns et al. (2006b). Biffis and Denuit (2006) introduce, in particular, a class of stochastic forces of mortality that generalize the Lee–Carter model. The static and the dynamic probabilistic approaches to randomness in mortality trend are addressed by Tuljapurkar and Boe (1998). The investigation in Section 7.2, and in Section 7.2.3 in particular, is based on Olivieri (2001). The analysis of the random value of future benefits is addressed also by Biffis and Olivieri (2002), where a pension scheme (or a group insurance) providing a range of life and death benefits is referred to. Following Olivieri (2001), Coppola et al. (2000) provide an investigation

369

370


also addressing financial risk for life annuity portfolios. In the Lee–Carter framework, given that the future path of the time index is unknown and modelled as a stochastic process, the policyholders’ lifetimes become dependent on each other. Consequently, systematic risk is involved. Denuit and Frostig (2007a) study this aspect of the Lee–Carter model, in particular considering solvency issues. Denuit and Frostig (2007b) further study the distribution of the present value of benefits in a run-off perspective. As the exact distribution turns out to be difficult to compute, various approximations and bounds are derived. Denuit (2008) summarizes the results obtained in this field. The literature on risk management in industry and business in general is very extensive. For an introduction to the relevant topics the reader can refer, for example, to Harrington and Niehaus (1999), and to Williams, Smith and Young (1998). Various textbooks address specific phases of the risk management process. For example, Koller (1999) focuses on the risk assessment in the risk management process for business and industry, whereas Wilkinson Tiller, Blinn and Kelly (1990) deal with the topic of risk financing. Pitacco (2007) addresses mortality and longevity risk within a risk management perspective. Several investigations have been performed with regard to natural hedging. As far as portfolio diversification effects are concerned, the reader may refer to Cox and Lin (2007), where the results of an empirical investigation concerning the US market are discussed. With regard to arrangements on a per-policy basis, some possible designs referring to pension schemes with combined benefits are discussed in Biffis and Olivieri (2002). Gründl et al. (2006) analyse natural hedging from the perspective of the maximization of shareholder value and show, under proper assumptions, that natural hedging could not be optimal in this regard. Solvency investigations in portfolio of life annuities are dealt with by Olivieri and Pitacco (2003). Solvency issues within a Lee-Carter framework are discussed by Denuit and Frostig (2007a). A review of solvency systems is provided by Sandström (2006); when the longevity risk is addressed, typically the required capital in this respect is set as a share of the technical provision. The most recent regulatory system is provided by the evolving Solvency 2 system, where the required capital is the change expected in the net asset value in case of a permanent shock in survival rates; see, for example, CEIOPS (2007) and CEIOPS (2008). The idea of assessing the required capital by comparing assets to the random value of future payments, examined in Section 7.3.3, has been put forward, for the life business in general, by Faculty of Actuaries Working Party (1986).

7.8


Reinsurance arrangements for longevity risk have not received much attention in the literature, due to the practical difficulty of transferring the systematic risk. A Stop-Loss reinsurance on the assets has been proposed by Marocco and Pitacco (1998), which the reader is referred to for some numerical examples, evaluated using both analytical and simulation methods. Olivieri (2005) deals, in a more formal setting, with both XL and Stop-Loss treaties, analysing the effectiveness of these arrangements in terms of the capital the insurer must allocate to face the residual longevity risk not covered by the reinsurer. Olivieri and Pitacco (2008) refer to a swap-like arrangement, in the context of the valuation of a life annuity portfolio. Cox and Lin (2007) also design a swap-like arrangement, based on natural hedging arguments. In contrast, considerable attention has been devoted in the recent literature to longevity bonds. Securitization of risks in general is described by Cox et al. (2000). The life insurance case is considered by Cowley and Cummins (2005). A mortality-indexed bond is described in Morgan Stanley-Equity Research Europe (2003). Various structures for longevity bonds have been proposed by Lin and Cox (2005), Lin and Cox (2007), Blake and Burrows (2001), Dowd (2003), Blake et al. (2006a), Blake et al. (2006b), Dowd et al. (2006), Olivieri and Pitacco (2008), Denuit et al. (2007). Pricing problems are also dealt with in Cairns et al. (2006b) and Denuit et al. (2007), the latter, in particular, working within the classical Lee–Carter model. The pricing of longevity risk has been addressed also in the framework of portfolio valuation. Biffis and Millossovich (2006a) consider in particular new business. Olivieri and Pitacco (2008) design a valuation setting, however without solving the problem of the appropriate stochastic mortality model to use. Friedberg and Webb (2005) analyse the pricing of the aggregate mortality risk in relation to the cost of capital of the insurance company. With reference to the problem of pricing a life annuity, Denuit and Frostig (2008) explain how to determine a conservative life table serving as first-order mortality basis, starting from a best-estimate of future mortality. Many recent papers deal with the pricing and valuation of insurance products including an option to annuitize; see, for example, Milevsky and Promislov (2001), O’Brien (2002), Wilkie et al. (2003), Boyle and Hardy (2003), Ballotta and Haberman (2003), Pelsser (2003), Ballotta and Haberman (2006), Biffis and Millossovich (2006b). Some of them, mainly deal in detail with financial aspects. Innovative ideas and proposals for structuring post-retirement benefits are presented and discussed in the reports by the Department for Work and

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Pensions (2002) in the United Kingdom, and the Retirement Choice Working Party (2001). The paper by Wadsworth et al. (2001) suggests a technical structure for a fund providing annuities. A comprehensive description of several annuities markets is provided by Cardinale et al. (2002). Piggot et al. (2005) describe Group-Self Annuitization schemes, which provide an example of flexible GAR; however, the benefit in this case is not guaranteed. Money-back annuities in the United Kingdom represent an interesting annuitization strategy; see Boardman (2006). Income drawdown issues within the context of defined contribution pension plans are discussed by Emms and Haberman (2008), Gerrard et al. (2006). An extensive presentation of issues concerning financing the post-retirement income is given by Milevsky (2006). An informal description of private solutions is provided by Swiss Re (2007). The reader interested in the impact of longevity risk on living benefits other than life annuities can refer, for example, to Olivieri and Ferri (2003), Olivieri and Pitacco (2002c), Olivieri and Pitacco (2002b). See also Pitacco (2004b), where both life insurance and other living benefits are considered.

References

Alho, J. M. (2000). Discussion of Lee (2000). North American Actuarial Journal, 4(1), 91–93. Andreev, K. F. and Vaupel, J. W. (2006). Forecasts of cohort mortality after age 50. Technical report. Ballotta, L. and Haberman, S. (2003). Valuation of guaranteed annuity conversion options. Insurance: Mathematics & Economics, 33, 87–108. Ballotta, L. and Haberman, S. (2006). The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case. Insurance: Mathematics & Economics, 38(1), 195–214. Baran, S., Gall, J., Ispany, M., and Pap, G. (2007). Forecasting hungarian mortality rates using the Lee–Carter method. Journal Acta Oeconomica, 57, 21–34. Barnett, H. A. R. (1960). The trends of population mortality and assured lives’ mortality in Great Britain. In Transactions of the 16th International Congress of Actuaries, Volume 2, Bruxelles, pp. 310–326. Beard, R. E. (1952). Some further experiments in the use of the incomplete gamma function for the calculation of actuarial functions. Journal of the Institute of Actuaries, 78, 341–353. Beard, R. E. (1959). Note on some mathematical mortality models. In CIBA Foundation Colloquia on Ageing (ed. C. E. W. Wolstenholme and M. O. Connor), Volume 5, Boston, pp. 302–311. Beard, R. E. (1971). Some aspects of theories of mortality, cause of death analysis, forecasting and stochastic processes. In Biological aspects of demography (ed. W. Brass), pp. 57–68. Taylor & Francis, London. Bell, W. R. (1997). Comparing and assessing time series methods for forecasting age-specific fertility and mortality rates. Journal of Official Statistics, 13, 279–303. Benjamin, B. and Pollard, J. H. (1993). The analysis of mortality and other actuarial statistics. The Institute of Actuaries, Oxford. Benjamin, J. and Soliman, A. S. (1993). Mortality on the move. Actuarial Education Service, Oxford. Biffis, E. (2005). Affine processes for dynamic mortality and actuarial valuations. Insurance: Mathematics & Economics, 37(3), 443–468.

374

References

Biffis, E. and Denuit, M. (2006). Lee–Carter goes risk-neutral: an application to the Italian annuity market. Giornale dell’Istituto Italiano degli Attuari, 69, 33–53. Biffis, E. and Millossovich, P. (2006a). A bidimensional approach to mortality risk. Decisions in Economics and Finance, 29, 71–94. Biffis, E. and Millossovich, P. (2006b). The fair value of guaranteed annuity options. Scandinavian Actuarial Journal, 1, 23–41. Biffis, E. and Olivieri, A. (2002). Demographic risks in pension schemes with combined benefits. Giornale dell’Istituto Italiano degli Attuari, 65(1–2), 137–174. Black, K. and Skipper, H. D. (2000). Life & health insurance. Prentice Hall, New Jersey. Blake, D., Cairns, A. J., and Dowd, K. (2007). Facing up to the uncertainty of life: the longevity fan charts. Technical Report. Blake, D. and Burrows, W. (2001). Survivor bonds: helping to hedge mortality risk. The Journal of Risk and Insurance, 68(2), 339–348. Blake, D., Cairns, A. J. G., and Dowd, K. (2006a). Living with mortality: longevity bonds and other mortality-linked securities. British Actuarial Journal, 12, 153–228. Blake, D., Cairns, A. J. G., Dowd, K., and MacMinn, R. (2006b). Longevity bonds: financial engineering, valuation, and hedging. The Journal of Risk and Insurance, 73(4), 647–672. Blake, D. and Hudson, R. (2000). Improving security and flexibility in retirement. Retirement Income Working Party, London. Blaschke, E. (1923). Sulle tavole di mortalità variabili col tempo. Giornale di Matematica Finanziaria, 5, 1–31. Boardman, T. (2006). Annuitization lessons from the UK: money-back annuities and other developments. The Journal of Risk and Insurance, 73(4), 633–646. Booth, H. (2006). Demographic forecasting: 1980 to 2005 in review. International Journal of Forecasting, 22(3), 547–581. Booth, H., Hyndman, R. J. Tickle, L., and De Jong, P. (2006). Lee– Carter mortality forecasting: a multi-country comparison of variants and extensions. Technical Report. Booth, H., Maindonald, J., and Smith, L. (2002). Applying Lee–Carter under conditions of variable mortality decline. Population Studies, 56(3), 325–336. Booth, H., Tickle, L., and Smith, L. (2005). Evaluation of the variants of the Lee–Carter method of forecasting mortality: a multi-country comparison. New Zealand Population Review, 31, 13–34.

References

Booth, P., Chadburn, R., Heberman, S., James, D., Kharasarce, Z., Plumb, R. and Rickayza, B. (2005). Modern advanced theory and practice. Boca Rator: Chapman & Hall/CRC. Bourgeois-Pichat, J. (1952). Essai sur la mortalité “biologique” de l’homme. Population, 7(3), 381–394. Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., and Nesbitt, C. J. (1997). Actuarial mathematics. The Society of Actuaries, Schaumburg, Illinois. Boyle, P. and Hardy, M. (2003). Guaranteed annuity options. ASTIN Bulletin, 33, 125–152. Brass, W. (1974). Mortality models and their uses in demography. Transactions of the Faculty of Actuaries, 33, 123–132. Brillinger, D. R. (1986). The natural variability of vital rates and associated statistics. Biometrics, 42, 693–734. Brouhns, N. and Denuit, M. (2002). Risque de longévité et rentes viagères. II. Tables de mortalité prospectives pour la population belge. Belgian Actuarial Bulletin, 2, 49–63. Brouhns, N., Denuit, M., and Keilegom, van, I. (2005). Bootstrapping the Poisson log-bilinear model for mortality forecasting. Scandinavian Actuarial Journal, (3), 212–224. Brouhns, N., Denuit, M., and Vermunt, J. K. (2002a). Measuring the longevity risk in mortality projections. Bulletin of the Swiss Association of Actuaries, 2, 105–130. Brouhns, N., Denuit, M., and Vermunt, J. K. (2002b). A Poisson logbilinear approach to the construction of projected lifetables. Insurance: Mathematics & Economics, 31(3), 373–393. Buettner, T. (2002). Approaches and experiences in projecting mortality patterns for the oldest-old. North American Actuarial Journal, 6(3), 14–25. Butt, Z. and Haberman, S. (2002). Application of frailty-based mortality models to insurance data. Actuarial Research Paper No. 142, Dept. of Actuarial Science and Statistics, City University, London. Butt, Z. and Haberman, S. (2004). Application of frailty-based mortality models using generalized linear models. ASTIN Bulletin, 34(1), 175– 197. Buus, H. (1960). Investigations on mortality variations. In Transactions of the 16th International Congress of Actuaries, Volume 2, Bruxelles, pp. 364–378. Cairns, A. J. G., Blake, D., and Dowd, K. (2006a). Pricing death: frameworks for the valuation and securitization of mortality risk. ASTIN Bulletin, 36(1), 79–120.

375

376

References

Cairns, A. J. G., Blake, D., and Dowd, K. (2006b). A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. The Journal of Risk and Insurance, 73(4), 687–718. Cairns, A., Blake, D., Dowd, K., Coughlan, G., Epstein, D., Ong, A. and Balevich, I. (2007) A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. Pensions Insitute Discussion Paper PI-0701, Cass Business School, City University. Cardinale, M., Findlater, A., and Orszag, M. (2002). Paying out pensions. A review of international annuities markets. Research report, Watson Wyatt. Carter, L. and Lee, R. D. (1992). Modelling and forecasting US sex differentials in mortality. International Journal of Forecasting, 8, 393–411. Carter, L. R. (1996). Forecasting U.S. mortality: a comparison of Box – Jenkins ARIMA and structural time series models. The Sociological Quarterly, 37(1), 127–144. Catalano, R. and Bruckner, T. (2006). Child mortality and cohort lifespan: a test of diminished entelechy. International Journal of Epidemiology, 35, 1264–1269. CEIOPS (2007). QIS3. Technical specifications. Part I: Instructions. CEIOPS (2008). QIS4. Technical specifications. Champion, R., Lenard, C. T., and Mills, T. M. (2004). Splines. In Encyclopedia of actuarial science (ed. J. L. Teugels and B. Sundt), Volume 3, pp. 1584–1586. John Wiley & Sons. CMI (2002). An interim basis for adjusting the “92” series mortality projections for cohort effects. Working Paper 1, The Faculty of Actuaries and Institute of Actuaries. CMI (2005). Projecting future mortality: towards a proposal for a stochastic methodology. Working paper 15, The Faculty of Actuaries and Institute of Actuaries. CMI (2006). Stochastic projection methodologies: Further progress and P-spline model features, example results and implications. Working Paper 20, The Faculty of Actuaries and Institute of Actuaries. CMIB (1978). Report no. 3. Continuous Mortality Investigation Bureau, Institute of Actuaries and Faculty of Actuaries. CMIB (1990). Report no. 10. Continuous Mortality Investigation Bureau, Institute of Actuaries and Faculty of Actuaries. CMIB (1999). Report no. 17. Continuous Mortality Investigation Bureau, Institute of Actuaries and Faculty of Actuaries. Coale, A. and Kisker, E. E. (1990). Defects in data on old age mortality in the United States: new procedures for calculating approximately

References

accurate mortality schedules and life tables at the highest ages. Asian and Pacific Population Forum, 4, 1–31. Congdon, P. (1993). Statistical graduation in local demographic analysis and projection. Journal of the Royal Statistical Society, A, 156, 237–270. Coppola, M., Di Lorenzo, E., and Sibillo, M. (2000). Risk sources in a life annuity portfolio: decomposition and measurement tools. Journal of Actuarial Practice, 8(1–2), 43–61. Cossette, H., Delwarde, A., Denuit, M., Guillot, F., and Marceau, E. (2007). Pension plan valuation and dynamic mortality tables. North American Actuarial Journal, 11, 1–34. Cowley, A. and Cummins, J. D. (2005). Securitization of life insurance assets and liabilities. The Journal of Risk and Insurance, 72(2), 193– 226. Cox, S. H., Fairchild, J. R., and Pedersen, H. W. (2000). Economic aspects of securitization of risk. ASTIN Bulletin, 30(1), 157–193. Cox, S. H. and Lin, Y. (2007). Natural hedging of life and annuity mortality risks. North Americal Actuarial Journal, 11, 1–15. Cramér, H. and Wold, H. (1935). Mortality variations in Sweden: a study in graduation and forecasting. Skandinavisk Aktuarietidskrift, 18, 161–241. Crimmins, E. and Finch, C. (2006). Infection, inflammation, height and longevity. Proceedings of the National Academy Sciences, 103, 498–503. Cummins, J. D., Smith, B. D., Vance, R. N., and VanDerhei, J. L. (1983). Risk classification in life insurance. Kluwer-Nijhoff Publishing, Boston, The Hague, London. Czado, C., Delwarde, A., and Denuit, M. (2005). Bayesian Poisson log-bilinear mortality projections. Insurance: Mathematics & Economics, 36(3), 260–284. Dahl, M. (2004). Stochastic mortality in life insurance. Market reserves and mortality-linked insurance contracts. Insurance: Mathematics & Economics, 35(1), 113–136. Dahl, M. and Møller, T. (2006). Valuation and hedging of life insurance liabilities with systematic mortality risk. Insurance: Mathematics & Economics, 39(2), 193–217. Davidson, A. R. and Reid, A. R. (1927). On the calculation of rates of mortality. Transactions of the Faculty of Actuaries, 11(105), 183–232. Davy Smith, G., Hart, C., Blane, D., and Hole, D. (1998) Adverse socioeconomic conditions in childhood and cause specific adult mortality: a prospective observational study. British Medical Journal, 316, 1631– 1635.

377

378

References

De Jong, P. and Tickle, L. (2006). Extending the Lee–Carter model of mortality projection. Mathematical Population Studies, 13, 1–18. Delwarde, A. and Denuit, M. (2006). Construction de tables de mortalité périodiques et prospectives. Ed. Economica, Paris. Delwarde, A., Denuit, M., and Eilers, P. (2007a). Smoothing the Lee– Carter and Poisson log-bilinear models for mortality forecasting. Statistical Modelling, 7, 29–48. Delwarde, A., Denuit, M., and Partrat, Ch. (2007b). Negative binomial version of the Lee–Carter model for mortality forecasting. Applied Stochastic Models in Business and Industry, 23, 385–401. Delwarde, A., Denuit, M., Guillen, M., and Vidiella, A. (2006). Application of the Poisson log-bilinear projection model to the G5 mortality experience. Belgian Actuarial Bulletin, 6, 54–68. Delwarde, A., Kachakhidze, D., Olié, L., and Denuit, M. (2004). Modèles linéaires et additifs généralisés, maximum de vraisemblance local et méthodes relationnelles en assurance sur la vie. Bulletin Français d’Actuariat, 6, 77–102. Denuit, M. (2007). Comonotonic approximations to quantiles of life annuity conditional expected present values. Insurance: Mathematics & Economics, 42, 831–838. Denuit, M. (2008). Life annuities with stochastic survival probability: a review. Methodology and Computing in Applied Probability, to appear. Denuit, M., Devolder, P., and Goderniaux, A.C. (2007). Securitization of longevity risk: pricing survivor bonds with Wang transform in the Lee– Carter framework. The Journal of Risk and Insurance, 74(1), 87–113. Denuit, M. and Dhaene, J. (2007). Comonotonic bounds on the survival probabilities in the Lee–Carter model for mortality projections. Computational and Applied Mathematics, 203, 169–176. Denuit, M., Dhaene, J., Goovaerts, M. J., and Kaas, R. (2005). Actuarial theory for dependent risks: measures, orders and models. Wiley, New York. Denuit, M. and Frostig, E. (2007a). Association and heterogeneity of insured lifetimes in the Lee–Carter framework. Scandinavian Actuarial Journal, 107, 1–19. Denuit, M. and Frostig, E. (2007b). Life insurance mathematics with random life tables. WP 07-07, Institut des Sciences Actuarielles, Université Catholique de Louvain, Louvain-la-Neuve, Beglium. Denuit, M. and Frostig, E. (2008). First-order mortality basis for life annuities. The Geneva Risk and Insurance Review, to appear. Denuit, M. and Goderniaux, A.-C. (2005). Closing and projecting life tables using log-linear models. Bulletin of the Swiss Association of Actuaries (1), 29–48.

References

Denuit, M. and Vermandele, C. (1998). Optimal reinsurance and stop-loss order. Insurance: Mathematics & Economics, 22, 229–233. Department for Work and Pensions (2002). Modernising annuities. Technical Report, Inland Revenue, London. Dowd, K. (2003). Survivor bonds: A comment on Blake and Burrows. The Journal of Risk and Insurance, 70(2), 339–348. Dowd, K., Blake, D., Cairns, A. J. G., and Dawson, P. (2006). Survivor swaps. The Journal of Risk and Insurance, 73(1), 1–17. Durban, I., Currie, M., and Eilers, P. (2004). Smoothing and forecasting mortality rates. Statistical Modelling, 4, 279–298. Eilers, P. H. C. and Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statistical Sciences, 11, 89–121. Emms, P. and Haberman, S. (2008). Income drawdown schemes for a defined contribution pension plan. Journal of Risk and Insurance, 75(3), 739–761. Evandrou, E. and Falkingham, J. (2002). Smoking behaviour and socioeconomic class: a cohort analysis, 1974 to 1998. Health Statistics Quarterly, 14, 30–38. Faculty of Actuaries Working Party (1986). The solvency of life assurance companies. Transactions of the Faculty of Actuaries, 39(3), 251–340. Felipe, A., Guillèn, M., and Perez-Marin, A. M. (2002). Recent mortality trends in the Spanish population. British Actuarial Journal, 8, 757–786. Finetti, de, B. (1950). Matematica attuariale. Quaderni dell’Istituto per gli Studi Assicurativi (Trieste), 5, 53–103. Finetti, de, B. (1957). Lezioni di matematica attuariale. Edizioni Ricerche, Roma. Forfar, D. O. (2004a). Life table. In Encyclopedia of Actuarial Science (ed. J. L. Teugels and B. Sundt), Volume 2, pp. 1005–1009. John Wiley & Sons. Forfar, D. O. (2004b). Mortality laws. In Encyclopedia of actuarial science (ed. J. L. Teugels and B. Sundt), Volume 2, pp. 1139–1145. John Wiley & Sons. Forfar, D. O., McCutcheon, J. J., and Wilkie, A. D. (1988). On graduation by mathematical formulae. Journal of the Institute of Actuaries, 115, 1–149. Forfar, D. O. and Smith, D. M. (1988). The changing shape of English Life Tables. Transactions of the Faculty of Actuaries, 40, 98–134. Francis, B., Green, M., and Payne, C. (1993). The GLIM system: Release 4 Manual. Clarendon Press, Oxford. Friedberg, L. and Webb, A. (2005). Life is cheap: using mortality bonds to hedge aggregate mortality risk. WP No. 2005-13, Center for Retirement Research at Boston College.

379

380

References

Gerber, H. U. (1995). Life insurance mathematics. Springer-Verlag. Gerrard, R., Haberman, S., and Vigna, E. (2006). The management of decumulation risks in a defined contribution environment. North American Actuarial Journal, 10(1), 84–110. Girosi, F. and King, G. (2007). Understanding the Lee–Carter mortality forecasting method. Technical report. Government Actuary’s Department (1995). National Population Projections 1992-based. HMSO, London. Government Actuary’s Department (2001). National Population Projections: review of methodology for projecting mortality. Government Actuary’s Department, London. Government Actuary’s Department (2002). National Population Projections 2000-based. HMSO, London. Goss, S. C., Wade, A., and Bell, F. (1998). Historical and projected mortality for Mexico, Canada and the United States. North American Actuarial Journal, 2(4), 108–126. Group Annuity Valuation Table Task Force (1995). 1994 Group annuity mortality table and 1994 Group annuity reserving table. Transactions of the Society of Actuaries, 47, 865–913. Gründl, H., Post, T., and Schulze, R. N. (2006). To hedge or not to hedge: managing demographic risk in life insurance companies. The Journal of Risk and Insurance, 73(1), 19–41. Gupta, A. K. and Varga, T. (2002). An introduction to actuarial mathematics. Kluwer Academic Publishers. Gutterman, S. and Vanderhoof, I. T. (1998). Forecasting changes in mortality: a search for a law of causes and effects. North American Actuarial Journal, 2(4), 135–138. Haberman, S. (1996). Landmarks in the history of actuarial science (up to 1919). Actuarial Research Paper No. 84, Dept. of Actuarial Science and Statistics, City University, London. Haberman, S. and Renshaw, A. (2008). On simulator-based approaches to risk measurement in mortality with specific reference to binomial Lee– Carter modelling. Presented to Living to 100. Survival to Advanced Ages international symposium. Society of Actuaries, Orlando, Florida. Haberman, S. and Renshaw, A. (2007) Discussion of “pension plan valuation and mortality projection: A case study with mortality data”, North American Actuarial Journal, 11(4), 2007, 148–150. Haberman, S. and Sibbett, T. A. (eds.) (1995). History of actuarial science, London. Pickering & Chatto. Hald, A. (1987). On the early history of life insurance mathematics. Scandinavian Actuarial Journal, (1), 4–18.

References

Hamilton, J. (1994). Time series analysis. Princeton: Princeton University Press. Harrington, S. E. and Niehaus, G. R. (1999). Risk management and insurance. Irwin/McGraw-Hill. Heligman, L. and Pollard, J. H. (1980). The age pattern of mortality. Journal of the Institute of Actuaries, 107, 49–80. Horiuchi, S. and Wilmoth, J. R. (1998). Deceleration in the age pattern of mortality at older ages. Demography, 35(4), 391–412. Hougaard, P. (1984). Life table methods for heterogeneous populations distributions describing the heterogeneity. Biometrika, 71, 75–83. Hyndman, R. J. and Ullah, Md. S. (2007). Robust forecasting of mortality and fertility rates: a functional data approach. Computational Statistics and Data Analysis, 51, 4942–4956. IAA (2004). A global framework for insurer solvency assessment. Research Report of the Insurer Solvency Assessment Working Party, International Actuarial Association. James, I. R. and Segal, M. R. (1982). On a method of mortality analysis incorporating age–year interaction, with application to prostate cancer mortality. Biometrics, 38, 433–443. Kannisto, V. J., Lauritsen, A. R. Thatcher and Vaupel, J. W. (1994). Reductions in mortality at advanced ages: several decades of evidence from 27 countries. Population and Development Review, 20, 793–810. Keyfitz, N. (1982). Choice of functions for mortality analysis: Effective forecasting depends on a minimum parameter representation. Theoretical Population Biology, 21, 329–352. Koissi, M.-C., Shapiro, A. F., and Högnäs, G. (2006). Evaluating and extending the Lee–Carter model for mortality forecasting: Bootstrap confidence interval. Insurance: Mathematics & Economics, 38(1), 1–20. Koller, G. (1999). Risk assessment and decision making in business and industry. CRC Press. Kopf, E. W. (1926). The early history of life annuity. Proceedings of the Casualty Actuarial Society, 13(27), 225–266. Kotz, S., Balakrishnan, N., and Johnson, N. L. (2000). Continuous multivariate distributions (2 edn), Volume 1: Models and applications. John Wiley & Sons. Lee, R. D. (2000). The Lee–Carter method for forecasting mortality, with various extensions and applications. North American Actuarial Journal, 4(1), 80–93.

381

382

References

Lee, R. D. (2003). Mortality forecasts and linear life expectancy trends. Technical Report. Lee, R. D. and Carter, L. R. (1992). Modelling and forecasting U.S. mortality. Journal of the American Statistical Association, 87(14), 659–675. Lee, R. and Miller, T. (2001). Evaluating the performance of the Lee–Carter approach to modelling and forecasting. Demography, 38, 537–549. Li, N. and Lee, R. D. (2005). Coherent mortality forecasts for a group of populations: an extension of the Lee–Carter method. Demography, 42, 575–594. Lin, Y. and Cox, S. H. (2005). Securitization of mortality risks in life annuities. The Journal of Risk and Insurance, 72(2), 227–252. Lin, Y. and Cox, S. H. (2008). Securitization of catastrophe mortality risks. Insurance: Mathematics & Economics, 42, 628–637. Lindbergson, M. (2001). Mortality among the elderly in Sweden 1988–97. Scandinavian Actuarial Journal (1), 79–94. Loader, C. (1999). Local regression and likelihood. Springer, New York. London, D. (1985). Graduation: the revision of estimates. ACTEX Publications. Lundström, H. and Qvist, J. (2004). Mortality forecasting and trend shifts: an application of the Lee–Carter model to swedish mortality data. International Statistical Review, 72, 37–50. Manton, K. G. and Stallard, E. (1984). Recent trend in mortality analysis. Academic Press. Marocco, P. and Pitacco, E. (1998). Longevity risk and life annuity reinsurance. In Transactions of the 26th International Congress of Actuaries, Birmingham, Volume 6, pp. 453–479. McCrory, R. T. (1986). Mortality risk in life annuities. Transactions of Society of Actuaries, 36, 309–338. McCutcheon, J. J. (1981). Some remarks on splines. Transaction of the Faculty of Actuaries, 37, 421–438. Milevsky, M. A. and Promislov, S. D. (2001). Mortality derivatives and the option to annuitise. Insurance: Mathematics & Economics, 29(3), 299–318. Milevsky, M. A. (2005). The implied longevity yield: A note on developing an index for life annuities. The Journal of Risk and Insurance, 72(2), 301–320. Milevsky, M. A. (2006). The calculus of retirement income. Financial models for pension annuities and life insurance. Cambridge University Press.

References

Miller, R. T. (2004). Graduation. In Encyclopedia of actuarial science (ed. J. L. Teugels and B. Sundt), Volume 2, John Wiley & Sons. pp. 780–784. Morgan Stanley-Equity Research Europe (2003). Swiss Re-Innovative mortality-based security. Technical report, Morgan Stanley. Namboodiri, K. and Suchindran, C. M. (1987). Life table techniques and their applications. Academic Press. National Statistics-Government Actuary’s Department (2001). National population projections: Review of methodology for projecting mortality. National Statistics Quality Review Series, Report No. 8. Nordenmark, N. V. E. (1906). Über die Bedeutung der Verlängerung der Lebensdauer für die Berechnung der Leibrenten. In Transactions of the 5th International Congress of Actuaries, Volume 1, Berlin, pp. 421–430. O’Brien, C. D. (2002). Guaranteed annuity options: five issues for resolution. British Actuarial Journal, 8, 593–629. Oeppen, J. and Vaupel, J. W. (2002). Broken limits to life expectancy. Science, 296, 1029–1031. Office of National Statistics (1997) The health of adult Britain 1841–1994. HMSO, London. Olivieri, A. (2001). Uncertainty in mortality projections: an actuarial perspective. Insurance: Mathematics & Economics, 29(2), 231–245. Olivieri, A. (2005). Designing longevity risk transfers: the point of view of the cedant. Giornale dell’Istituto Italiano degli Attuari, 68, 1–35. Reprinted on: ICFAI Journal of Financial Risk Management, 4-March 2007: 55–83. Olivieri, A. (2006). Heterogeneity in survival models. applications to pension and life annuities. Belgian Actuarial Bulletin, 6, 23–39. http://www.actuaweb.be/frameset/frameset.html. Olivieri, A. and Ferri, S. (2003). Mortality and disability risks in long term care insurance. IAAHS Online Journal. http://www.actuaries.org/ members/en/IAAHS/OnlineJournal/2003-1/2003-1.pdf. Olivieri, A. and Pitacco, E. (2002a). Inference about mortality improvements in life annuity portfolios. In Transactions of the 27th International Congress of Actuaries, Cancun (Mexico). Olivieri, A. and Pitacco, E. (2002b). Managing demographic risks in long term care insurance. Rendiconti per gli Studi Economici Quantitativi, 2, 15–37. Olivieri, A. and Pitacco, E. (2002c). Premium systems for post-retirement sickness covers. Belgian Actuarial Bulletin, 2, 15–25. Olivieri, A. and Pitacco, E. (2003). Solvency requirements for pension annuities. Journal of Pension Economics & Finance, 2, 127–157.

383

384

References

Olivieri, A. and Pitacco, E. (2008). Assessing the cost of capital for longevity risk. Insurance: Mathematics & Economics, 42, 1013–1021. Olshansky, S. J., Passaro, D., Hershaw, R., Layden, J., Carnes, B. A., Brody, J., Hayflick, L., Butler, R. N., Allison, D. B., and Ludwig, D. S. (2005). A potential decline in life expectancy in the United States in the 21st century. New England Journal of Medicine, 352, 1103–1110. Olshansky, S. J. (1988). On forecasting mortality. The Milbank Quarterly, 66(3), 482–530. Olshansky, S. J. and Carnes, B. E. (1997). Ever since Gompertz. Demography, 34, 1–15. O’Malley, P. (2007). Development of GMxB markets in Europe. In Transactions of the 1st IAA Life Colloquium, Stockholm. Pelsser, A. (2003). Pricing and hedging guaranteed annuity options via static option replication. Insurance: Mathematics & Economics, 33(2), 283–296. Petrioli, L. and Berti, M. (1979). Modelli di mortalità. Franco Angeli Editore, Milano. Piggot, J., Valdez, E. A., and Detzel, B. (2005). The simple analytics of a pooled annuity fund. The Journal of Risk and Insurance, 72(3), 497–520. Pitacco, E. (2004a). From Halley to “frailty”: a review of survival models for actuarial calculations. Giornale dell’Istituto Italiano degli Attuari, 67(1–2), 17–47. Pitacco, E. (2004b). Longevity risks in living benefits. In Developing an annuity market in Europe (ed. E. Fornero and E. Luciano), pp. 132–167. Edward Elgar, Cheltenham. Pitacco, E. (2004c). Survival models in a dynamic context: a survey. Insurance: Mathematics & Economics, 35(2), 279–298. Pitacco, E. (2007). Mortality and longevity: a risk management perspective. In Proceedings of the 1st IAA Life Colloquium, Stockholm. Pollard, A. H. (1949). Methods of forecasting mortality using Australian data. Journal of the Institute of Actuaries, 75, 151–182. Pollard, J. H. (1987). Projection of age-specific mortality rates. Population Bulletin of the UN, 21–22, 55–69. Poulin, C. (1980). Essai de mise au point d’un modèle représentatif de l’évolution de la mortalité humaine. In Transactions of the 21st International Congress of Actuaries, Volume 2, Zürich-Lausanne, pp. 205–211. Renshaw, A. E. and Haberman, S. (2000). Modelling for mortality reduction factors. Actuarial Research Paper No. 127, Dept. of Actuarial Science and Statistics, City University, London.

References

Renshaw, A. E. and Haberman, S. (2003a). Lee–Carter mortality forecasting, a parallel generalized linear modelling approach for England & Wales mortality projections. Applied Statistics, 52, 119–137. Renshaw, A. E. and Haberman, S. (2003b). Lee–Carter mortality forecasting with age specific enhancement. Insurance: Mathematics & Economics, 33(2), 255–272. Renshaw, A. E. and Haberman, S. (2003c). On the forecasting of mortality reduction factors. Insurance: Mathematics & Economics, 32(3), 379–401. Renshaw, A. E. and Haberman, S. (2005). Lee–Carter mortality forecasting incorporating bivariate time series for England and Wales mortality projections. Technical report. Renshaw, A. E. and Haberman, S. (2006). A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insurance: Mathematics & Economics, 38(3), 556–570. Renshaw, A. E. and Haberman, S. (2008). On simulation-based approaches to risk measurement in mortality with specific reference to poisson Lee–Carter modelling. Insurance: Mathematics and Economics, 42, 797–816. Renshaw, A. E., Haberman, S., and Hatzopoulos, P. (1996). The modelling of recent mortality trends in United Kingdom male assured lives. British Actuarial Journal, 2(II), 449–477. Retirement Choice Working Party (2001). Extending retirement choices. Retirement income options for modern needs. The Faculty and Institute of Actuaries. Richards, S., Ellam, J., Hubbard, J., Lu, J., Makin, S., and Miller, K. (2007). Two-dimensional mortality data: patterns and projections. Presented to the Institute of Actuaries. Richard, S. J. and Jones, G. L. (2004). Financial aspects of longevity risk. The Staple Inn Actuarial Society, London. Riemer-Hommel, P. and Trauth, T. (2000). Challenges and solutions for the management of longevity risk. In Risk management. Challenge and opportunity (ed. M. Frenkel, U. Hommel, and M. Rudolf), pp. 85–100. Springer. Rotar, V. I. (2007). Actuarial Models. The Mathematics of Insurance. Chapman & Hall/CRC. Sandström, A. (2006). Solvency. Models, assessment and regulation. Chapman & Hall, CRC. Sithole, T. Z., Haberman, S., and Verrall, R. J. (2000). An investigation into parametric models for mortality projections, with applications to immediate annuitants and life office pensioners’ data. Insurance: Mathematics & Economics, 27(3), 285–312.

385

386

References

Skwire, D. (1997). Actuarial issues in the novels of Jane Austen. North American Actuarial Journal, 1(1), 74–83. Smith, D. and Keyfitz, N. (eds.) (1977). Mathematical demography. Selected papers, Berlin. Springer Verlag. Sun, F. (2006). Pricing and risk management of variable annuities with multiple guaranteed minimum benefits. Actuarial Practice Forum. Society of Actuaries. Sverdrup, E. (1952). Basic concepts in life assurance mathematics. Skandinavisk Aktuarietidskrift, 3–4, 115–131. Swiss Re (2007). Annuities: a private solution to longevity risk. Sigma, 3. Tabeau, E., van den Berg Jeths, A., and Heathcote, C. (eds.) (2001). Forecasting mortality in developed countries. Kluwer Academic Publishers. Thatcher, A. R. (1999). The long-term pattern of adult mortality and the highest attained age. Journal of the Royal Statistical Society, A, 162, 5–43. Tuljapurkar, S., Li, N., and Boe, C. (2000). A universal pattern of mortality decline in the G7 countries. Nature, 405, 789–792. Tuljapurkar, S. and Boe, C. (1998). Mortality change and forecasting: how much and how little do we know. North American Actuarial Journal, 2, 13–47. Vaupel, J. W., Manton, K. G., and Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16(3), 439–454. Verrall, R., Haberman, S., Sithole, T., and Collinson, D. (2006, September). The price of mortality. Life and Pensions, 35–40. Wadsworth, M., Findlater, A., and Boardman, T. (2001). Reinventing annuities. The Staple Inn Actuarial Society, London. Wang, S. H. (2002). A universal framework for pricing financial and insurance risks. ASTIN Bulletin, 32(2), 213–234. Wang, S.H. (2004). Cat bond pricing using probability transforms. The Geneva Papers on Risk and Insurance: Issues and Practice, 278, 19–29. Wang, S. S. and Brown, R. L. (1998). A frailty model for projection of human mortality improvement. Journal of Actuarial Practice, 6(1–2), 221–241. Wetterstrand, W. H. (1981). Parametric models for life insurance mortality data: Gompertz’s law over time. Transactions of the Society of Actuaries, 33, 159–175. Wilkie, A. D., Waters, H. R., and Yang, S. Y. (2003). Reserving, pricing and hedging for policies with guaranteed annuity options. British Actuarial Journal, 9, 263–425. Wilkie, A. D. (1997). Mutuality and solidarity: assessing risks and sharing losses. British Actuarial Journal, 3, 985–996.

References

Wilkinson Tiller, M., Blinn, J. D., and Kelly, J. J. (1990). Essentials of risk financing. Insurance Institute of America. Willets, R. C. (2004). The cohort effect: insights and explanations. British Actuarial Journal, 10, 833–877. Williams, JR. C. A., Smith, M. L., and Young, P. C. (1998). Risk management and insurance. Irwin/McGraw-Hill. Wilmoth, J. R. (1993). Computational methods for fitting and extrapolating the Lee–Carter model of mortality change. Technical report. Wilmoth, J. R. (2000). Demography of longevity: Past, present, and future trends. Journal of Experimental Gerontology, 35, 1111–1129. Wilmoth, J. R. and Horiuchi, S. (1999). Rectangularization revisited: variability of age at death within human populations. Demography, 36(4), 475–495. Wong-Fupuy, C. and Haberman, S. (2004). Projecting mortality trends: recent developments in the United Kingdom and the United States. North American Actuarial Journal, 8, 56–83. Yaari, M. E. (1965). Uncertain lifetime, life insurance, and the theory of the consumer. Review of Economic Studies, 32(2), 137–150. Yashin, A. I. and Iachine, I. A. (1997). How frailty models can be used for evaluating longevity limits: Taking advantage of an interdisciplinary approach. Demography, 34, 31–48.

387


Index

account value 42 accumulation period 31, 32, 33–6, 344, 350, 364, 367 actuarial value 9–12 additive model 79 adverse selection 41 age at death variability 113–15 age rating models 79 age shifts 79, 127–9, 155–6 age-patterns of mortality 13–14, 159–60, 178 age-period life tables 93–5 age-period-cohort models see APC (Age-Period-Cohort) models age-specific functions 60, 139–40 aggregate table 51 alternative risk transfer (ART) 297 see also risk transfer Andreev–Vaupel life expectancy projections 235–7 annual probability of death 48 laws for 66 mortality modelling by extrapolation 141–52, 162 versus interpolation 165–6 annual survival probability 48 annuities-certain 2–8, 36 avoiding early fund exhaustion 5–6 equivalent number of payments 355 risks in 6–8 withdrawing from fund 2–5 annuitization 35, 364–9 staggered 368 annuity in advance 32–3 annuity in arrears 8, 31 APC (Age-Period-Cohort) models 173–5 application to UK mortality data 254–63 Lee–Carter APC model 246–54 error structure and model fitting 248–52 model structure 246–8 mortality rate projections 253

apportionable annuity 39 asymptotic mortality 147 autoregressive integrated moving average (ARIMA) models 221–3, 231–2, 253 B-splines 71–2, 210, 265 Balducci assumption 58 Banking, Finance, and Insurance Commission (BFIC) 92–3 Barnett law 66 Beard law 66 Belgium 130–53 Cairns–Blake–Dowd model application 207–9 Lee–Carter model application 200–3 prediction intervals 232–4 smoothing 213–14 life expectancy forecasting 237–9 optimal calibration period selection 217–18 residuals analysis 220–1 see also Federal Planning Bureau (FPB), Belgium Bernoulli model 122 binomial maximum likelihood estimation 198–9 negative 199–200 bonus rates 39 bootstrapping 229–30 application to Belgian mortality statistics 232–4 bootstrap percentiles confidence intervals 230–2 Brass logit transform 167–8 Cairns–Blake–Dowd mortality projection model 183–4, 203–9 allowing for cohort effects 263–5 application to Belgian mortality statistics 207–9

390

Index

Cairns–Blake–Dowd mortality projection model (Cont.) calibration 206–7 optimal calibration period selection 217, 218 residuals analysis 220–1 specification 203–6 time index modelling 228–9 see also mortality modelling calibration period selection 214–18 application to Belgian mortality statistics 217–18 motivation 214–16 selection procedure 216–17 capital protection 40 cash-refund annuity 40 catastrophe risk 269 central death rate 57 Coale–Kisker model 76 coefficient of variation 61 cohort effect 243–5 in Cairns-Blake-Dowd model 263–5 in P-splines model 265–6 UK 243–5 see also APC (Age-Period-Cohort) models cohort life expectancies 112–13, 153 cohort life table 46, 140 in projected table 152–3 complete expectation of life 60 complete life annuity 39 conditional GAR products 348, 349 constant-growth annuity 38 Continuous Mortality Investigation Bureau (CMIB), UK 243 software 185 cross-subsidy 14–20 mutuality 14–16 solidarity 16–18 tontine annuities 18–20 cubic spline 70 curtate expectation of life 59 curtate remaining lifetime 49 curve of deaths 54 curve squaring 105–6 death age at, variability 113–15 annual probability of 48 curve of deaths 54 death rates 96–101

central 57 observed 116–18 smoothed 118–22, 209–14 uniform distribution of deaths 57–8 see also mortality decumulation period 31, 32, 36–8, 344, 345, 350 deferred life annuity 32–3 diminished entelechy hypothesis 244–5 distribution function 53–4 dynamic mortality model 139 endowment 33–4, 344–5 endurance 61 England see United Kingdom enhanced annuities 41 enhanced pensions 41 entropy 61 Equitable Life 135 equity-indexed annuity 38 equivalence principle 12 equivalent discount rate 355 equivalent entry age 355 equivalent number of payments 355 escalating annuities 38 Esscher formula 151 excess-of-loss (XL) reinsurance 319–20, 326 exhaustion time 5 expansion 138, 161, 168, 179 expected lifetime 59, 139, 152, 170 exponential formula 145–6, 149 alternative approach 146–7 formulae used in actuarial practice 149–51 generalization 147 implementation 148 exposure-to-risk (ETR) 95–6, 97 failure rate 55 fan charts 170, 240 Federal Planning Bureau (FPB), Belgium 91–2 life expectancy projections 235 financing post-retirement income 354–69, 371–2 comparing life annuity prices 354–6 flexibility in 363–9 life annuities versus income drawdown 356–9 mortality drag 359–63

Index

first-order basis 12, 13 fixed-rate escalating annuity 38 force of mortality 55–6, 58, 82–3, 94–5 cumulative 56 laws for 64–5 frailty 80–3 models 83–5, 88 combined with mortality laws 85–7 France 130–5 fund exhaustion avoiding 5–6 exhaustion time 5 Gamma distribution 83–5, 87 Gaussian-Inverse distribution 85 Germany 130 GLB (Guaranteed Living Benefits) 43 GM (Gompertz-Makeham) models 65, 163–4 GMAB (Guaranteed Minimum Accumulation Benefit) 42, 43 GMDB (Guaranteed Minimum Death Benefit) 42 GMIB (Guaranteed Minimum Income Benefit) 42, 43 GMWB (Guaranteed Minimum Withdrawal Benefit) 42–3 GMxBs (Guarantees Minimum Benefits of type ‘x’) 41–3 Gompertz model 55–6, 64, 85–6 see also GM (Gompertz-Makeham) models graduation 67–8, 87–8 mortality graduation over age and time 163–5 see also non-parametric graduation guaranteed annuity 346 guaranteed annuity option (GAO) 35, 297, 346–7 valuation of 354 guaranteed annuity rate (GAR) 346–7 adding flexibility 347–50 conditional GAR products 348, 349 with-profit GAR products 349 Gyldén, H. 175–6 hazard function 55 cumulative 56 healthy worker effect 122 hedging 298 across LOBs 303 across time 299–302

life annuity liabilities through longevity bonds 337–43 natural hedging 298, 299–303, 370 Heligman–Pollard laws 12, 66, 75, 178, 179, 276 highest anniversary value 42 Human mortality database (HMD) 92 impaired-life annuities 41 implied longevity yield (ILY) 15, 363 inception-select mortality 51 index-linked escalating annuity 38 inflation-linked annuity 38 instalment-refund annuity 40 insurance risk 269 insured population 14 internal knots 69, 70 interquartile range 61–2 investment-linked annuities 38–9 issue-select mortality 51 Italy 130 joint-life annuity 37 K-K-K hypothesis 173 knots 69–70 Kwiatowski–Philips–Schmidt–Shin test 224 last-survivor annuity 37 Lee–Carter (LC) model 169–73, 178–80, 182–4, 186–203 age-period-cohort model 246–54 see also APC (Age-Period-Cohort) models application to Belgian mortality statistics 200–3 application to UK mortality statistics 254–63 calibration 188–200 alternative estimation procedures 198–200 identifiable constraints 188–9 least-squares estimation 189–98 optimal calibration period selection 214–18 extensions 172, 180, 192–200 life expectancy forecasting 237–9, 241–2 model tables and 173 prediction intervals 232–4 residuals analysis 218–21

391

392

Index

Lee–Carter (LC) model (Cont.) smoothing in 212–13 specification 186–8 time index modelling 221–8 random walk with drift model 225–8 stationarity 223–4 see also mortality modelling level annuities 38 Lexis diagram 94 Lexis point 60 liability 11 life annuities 2–8 accumulation period 33–6 as financial transactions 8 avoiding early fund exhaustion 5–6 cross-subsidy in 14–20 decumulation period 36–8 deterministic evaluation 8–14 actuarial value 9–12 technical bases 12–14 immediate versus deferred annuities 31–3 longevity risk and 343–50 mortality risk location 343–6 payment profile 38–40 present value of 351–2 price comparisons 354–6 risks in 6–8 stochastic evaluation 20–30 focussing on portfolio results 21–4 random present value 20–1 risk assessment 24–7 uncertainty in mortality assumptions 27–30 temporary life annuity 36 versus income drawdown 356–9 whole life annuity 36 with a guarantee period 37 withdrawing from fund 2–5 life expectancy 59–60, 89 Andreev–Vaupel projections 235–7 Belgian Federal Planning Bureau (FPB) projections 235 cohort life expectancies 112–13, 153 forecasting 234–42 application to Belgian mortality statistics 237–9 back testing 240–2 fan charts 240 heterogeneity 115–16 observed 122–3 period life expectancies 62, 111–13

life insurance market 116–29 age shifts 127–9 life expectancies 122–3 observed death rates 116–18 smoothed death rates 118–22 life insurance securitization 330–2 life tables 46–51, 93 aggregate table 51 as probabilistic models 48–9 closure 101–5 cohort life table 46, 140 in projected table 152–3 limit table 165–6 optimal 166, 177 period life table 46–7, 93, 140 age-period 93–5 population versus market tables 47–8 projected life table 47 projecting transforms of life table functions 167–9 ultimate life table 51 LifeMetrics 185 lifetime probability distribution 58 limiting age 4 linear spline 70 lines of business (LOBs) 298 natural hedging across LOBs 303 liquidation period see decumulation period liquidity risk 7 location measure 60 logit transform of the survival function 73 long-term bonds 335 longevity bonds 332, 335–7, 371 hedging life annuity liabilities through 337–43 longevity risk 1, 267, 268–93, 369 life annuities and 343–50 management 293–330 natural hedging 299–303 reinsurance arrangements 318–30, 371 risk management perspective 293–9 solvency issues 303–18, 370 see also risk management measurement in a static framework 276–93 mortality risks 268–70 pricing and 350–4, 371 representation 273–6 stochastic modelling issues 270–3 loss control techniques 296–7 loss financing techniques 297

Index

Makeham laws 64, 67, 76, 159, 176–7, 179 see also GM (Gompertz-Makeham) models market risk 7 maximum downward slope 61 median age at death 60 model risk 269 model tables 165–6, 173, 177–8 money-back annuities 302 Monte Carlo simulation 22, 230–1 mortality age-patterns 13–14, 159–60, 178 allowing for uncertainty 27–30 asymptotic 147 at very old ages 74–6, 88 best estimate 29 by causes 67, 175 force of 55–6, 58, 64–5, 82–3, 94–5 cumulative 56 forecasting see mortality modelling graduation over age and time 163–5 heterogeneity 77–87, 88 frailty models 83–7 models for differential mortality 78–80 observable heterogeneity factors 77–8 unobservable heterogeneity factors 80–3 laws 63–7, 179 combined with frailty models 85–7 projections and 156–60 risk of random fluctuation 25 select 49–51 trends see mortality trends see also death; life tables; survival mortality bonds 332, 333–4 mortality drag 15, 359–63 mortality modelling 137–9, 175–80 age-period models 181–242 age-period-cohort models 243–66 age-specific functions 139–40 cohort versus period approach 173–5 diagonal approach 157–9, 162, 177 dynamic approach 137–41 extrapolation of annual probabilities of death 141–52, 162 versus interpolation 165–6 horizontal approach 143–4, 162, 176 life expectancy forecasting 234–42 model tables 165–6, 173, 177–8 mortality by causes 175 mortality projection 221–9

projection in parametric context 156–65 prediction intervals 229–34 projected table use 152–6 projecting transforms of life table functions 167–9 relational method 178 surface approach 163 vertical approach 157, 159–60, 162, 177 see also Cairns–Blake–Dowd mortality projection model; Lee–Carter (LC) model mortality odds 49 mortality profile 138, 140 mortality risks 268–70 location in traditional life annuity products 343–6 mortality trends 93–116, 176 age-period life tables 93–5 closure of life tables 101–5 death rates 96–101 exposure-to-risk 95–6 expression via Weibull’s parameters 160–1 heterogeneity 115–16 life expectancies 111–13 life insurance market 116–29 mortality surfaces 101 rectangularization and expansion 105–11 throughout the EU 129–35 variability 113–15 see also mortality mortality-linked securities 332–7 multiplicative model 79 mutuality 6, 14–16, 17–18, 357 interest from 15 Nadaraya–Watson kernel estimate 120 natural cubic spline 70 natural hedging 298, 299–303, 370 across LOBs 303 across time 299–302 Newton–Raphson procedure 193–4 no advance funding 298 non-guaranteed annuity 346–7 non-parametric graduation 67–72 splines 69–72 Whittaker–Henderson model 68–9 non-pooling risk 285 numerical rating system 79–80

393

394

Index

option to annuitize 35, 297, 346 overdispersed Poisson and negative binomial maximum likelihood estimation 199–200 P-splines model allowing for cohort effects 265–6 smoothing approach 210–11 parameter risk 269 participating GAR products 349 payout period see decumulation period Pearson residuals 220 pension annuities 40–1 period life expectancies 62, 111–13 period life table 46–7, 140 age-period 93–5 Perks laws 65, 75–6, 86–7 Petrioli–Berti model 168–9 Poisson bootstrap 231 Poisson log-bilinear model 172 Poisson maximum likelihood estimation 196–8 overdispersed 199–200 pooling risk 285 post-retirement income financing see financing post-retirement income prediction intervals 229–34 application to Belgian mortality statistics 232–4 premium 8 return of premiums 35, 42 present value 351–2 pricing longevity risk and 350–4, 371 reinsurance arrangements 325–6 probability density function (pdf) 53–4 probability of default 295 process risk 25, 269 profit participation mechanisms 13, 39 projected life table 47 projected mortality model 139 extrapolation of annual probabilities of death 141–52, 162 versus interpolation 165–6 parametric context 156–65 see also mortality modelling; projected mortality table projected mortality table 152–6 age shifting 155–6 cohort tables in 152–3 from double-entry to single-entry projected table 153–5 prudential basis 12

R software 184–5 random present value 20–1, 24, 43 random walk with drift model 225–8 ratchet 42 rating classes 16–17 realistic basis 12 rectangularization 51, 105–11, 138, 161, 168, 179 reduction factors 124, 144–5, 179, 233, 246–8, 252 reinsurance arrangements 318–30, 371 excess-of-loss (XL) reinsurance 319–20, 326 pricing 325–6 reinsurance-swap arrangement on annual outflows 324–5 stop-loss reinsurance on annual outflows 321–4, 326 on assets 320–1, 326 swap-like arrangement between life annuities and life insurances 329–30 Renshaw–Haberman model 165 Renshaw–Haberman–Hatzopoulos model 163–4 reserve 6, 27 residuals analysis 218–21 application to Belgian mortality statistics 220–1 residuals bootstrap 231 resistance function 73–4, 178 return of premiums 35, 42 reversionary annuity 38 Richardt, T. 176 risk 6–8, 78 assessment 24–7 exposure-to-risk (ETR) 95–6, 97 management see risk management of mortality random fluctuation 25 process risk 25, 269 uncertainty risk 28–9 see also longevity risk; risk management (RM); risk transfer risk classes 16–17 risk factors 40 risk index 280 risk management (RM) 293–9, 370 natural hedging 299–303 reinsurance arrangements 318–30, 371 solvency issues 303–18, 370 risk transfer 297–8 hedging life annuity liabilities through longevity bonds 337–43

Index

life insurance securitization 330–2 mortality-linked securities 332–7 see also risk management roll-up 42 Rueff’s adjustments 127, 155 ruin probability 295

safe-side technical basis 12, 13 safety loading 13 scenario technical basis 12 second-order basis 12 securitization 330 life insurance 330–2 select mortality 49–51 select period 50 select table 51 self-selection 17, 51 single-entry projected table 153–5 Sithole–Haberman–Verrall model 164–5 smoothing 118–22, 209–14 application to Belgian mortality statistics 213–14 in Lee–Carter model 212–13 motivation 209 P-splines approach 210–12 solidarity 14, 16–18 solvency 303–18, 370 assessment 24–7 Spain 130 special-rate annuities 41 splines 69–72 B-splines 71–2, 210, 265 P-splines model allowing for cohort effects 265–6 smoothing approach 210–11 staggered annuitization 368 standard annuities 38 standardized mortality ratio (SMR) 116–18 stationarity 223–4 Statistics Belgium 91 stochastic valuation 270–3 life annuity evaluation 20–30 stop-loss reinsurance on annual outflows 321–4, 326 on assets 320–1, 326 survival, annual probability 48 see also mortality survival function 51–3

expansion 51, 138 rectangularization 51, 105–11, 138 transforms of 73–4 Sweden 130 temporary life annuity 36 Thiele law 65 time series modelling 221–3 Cairns–Blake–Dowd time indices 228–9 Lee–Carter time index 221–8 random walk with drift model 225–8 stationarity 223–4 Tonti, Lorenzo 18–19 tontine annuities 14, 18–20 ultimate life table 51 uncertainty in mortality assumptions 27–30 uncertainty risk 269, 298 uni-sex annuities 40 uniform spline 69 unit-linked life annuity 39 United Kingdom 135, 243–4 APC model application 254–63 cohort effect 243–5 value-protected annuities 40 variability measures 60–1 variable annuities 41–3 variance of the random lifetime 61 variation factor 145 voluntary annuities 40 Wales see United Kingdom Wang transform 353 Weibull law 65, 160–1, 179 Whittaker–Henderson model 68–9 whole life annuity 36 with-profit annuity 39 with-profit GAR products 349 XL (excess-of-loss) reinsurance 319–20, 326 young mortality hump 138 YourCast software 185

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