Intelligent and Other Computational Techniques in Insurance: Theory and Applications (Series on Innovative Intelligence, 6)

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Series on Innovative Intelligence - Vol. 6

10C0101110000010101000

nmmwmmmm

3101

• it*

K Intelligent and Other Computational Techniques in Insurance Theory and Applications Editors

A. F. Shapiro L. C. Jain

Intelligent and Other Computational Techniques in Insurance Theory and Applications

Series on Innovative Intelligence Editor: L. C. Jain (University of South Australia) Published: Vol. 1

Virtual Environments for Teaching and Learning (eds. L. C. Jain, R. J. Howlett, N. S. Ichalkaranje & G. Tonfoni)

Vol. 2

Advances in Intelligent Systems for Defence (eds. L. C. Jain, N. S. Ichalkaranje & G. Tonfoni)

Vol. 3

Internet-Based Intelligent Information Processing Systems (eds. R. J. Howlett, N. S. Ichalkaranje, L. C. Jain & G. Tonfoni)

Vol. 4

Neural Networks for Intelligent Signal Processing (A. Zaknich)

Vol. 5

Complex Valued Neural Networks: Theories and Applications (ed. A. Hirose)

Forthcoming Titles: Biology and Logic-Based Applied Machine Intelligence: Theory and Applications (A. Konar & L. C. Jain) Levels of Evolutionary Adaptation for Fuzzy Agents (G. Resconi & L. C. Jain)

Series on Innov.iiiw Inirlli^rnt'f - Vol. C>

Intelligent and Other Computational Techniques in Insurance Theory and Applications

Editors

A. E Shapiro Penn State University, USA

L. C. Jain University of South Australia

l j | p World Scientific NEW JERSEY • LONDON • SINGAPORE • SHANGHAI • HONGKONG • TAIPEI • BANGALOBE

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

INTELLIGENT AND OTHER COMPUTATIONAL TECHNIQUES IN INSURANCE: THEORY AND APPLICATIONS Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-718-8

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Foreword "Knowledge is Power." This maxim summarizes my attitude toward intelligence as a key driver of success,... in particular long term success of an individual or business enterprise in a competitive economy. Intelligence has a number of meanings including the intellectual capability of the individual or a group of individuals, and the ability of an organization to gather data useful for making decisions. The data itself is also often called intelligence! Intelligent systems are those automated procedures that turn intelligence, in the sense of data, into advice for decision-makers. One of the features of recently-developed intelligent systems is that they "learn" in the sense of improving decision-making through time by continually incorporating new data. With the development of high speed computers and the resulting ability to collect and store and process huge amounts of data, the insurance field is one of the industries that is in the best position to take advantage of the intelligent systems that are being developed. Insurance companies do not need to be huge in order to take advantage of intelligent systems and other data dependent systems. Indeed, an automobile or health insurer with a few hundred thousand policyholders can collect vast amounts of data on the attributes as well as the detailed experience of each of the policyholders. Some easy-to-use tools for data mining and other techniques are now already commercialized and easy to use. This book provides a series of papers that provide overviews of some of these "intelligent" techniques as well as more traditional techniques that are not necessarily described as intelligent, such as statistical methods; e.g., logistic regression. These additional tech-

V

Foreword

VI

niques do of course contribute to intelligence in the sense of the intellectual capability of the individual or enterprise. Finally, several papers focus on the process of making decisions in the presence of imperfect "fuzzy" information. Virtually all the papers include direct applications to insurance, risk management or other financial data. Actuaries form a key element of the intelligence of the insurance industry. Actuarial science is a collection of paradigms and techniques, most of which are shared by other disciplines. Some methodologies, such as bonus-malus, are used exclusively by actuaries, since they are developed for use in the areas, such as insurance, of traditional actuarial practice. This book will be successful for the insurance industry and for the actuarial profession if it stimulates interest in further more detailed reading and research. I commend it to all individuals and insurance enterprises. "Knowledge is Power.'''

Harry H. Panjer President, Society of Actuaries

Preface At the turn of this century, the premiums written by insurance companies were nearly $2.5 trillion (USD) worldwide, and growing. Many factors will influence the continued growth of the industry, including a rapidly changing risk environment and market conditions, but one of the dominant factors will be technology, and an important component of that will involve intelligent and other computational techniques. This will continue the worldwide trend to fuse these techniques into insurance-based programs. This novel book presents recent advances in the theory and implementation of intelligent and other computational techniques in the insurance industry. The paradigms covered include artificial neural networks and fuzzy systems, including clustering versions, optimization and resampling methods, algebraic and Bayesian models, decision trees and regression splines. Thus, the focus is not just on intelligent techniques, although that is a major component, the book also includes other current computational paradigms that are likely to impact the industry. The chapters are divided into two main sections: "Neural Networks, Fuzzy Systems and Genetic Algorithms" (Chapters 1-11) and "Other Computational Techniques" (Chapters 12-18). For the most part, these section headings are adhered to, but a few chapters spanned more than one of these topics and had to be arbitrarily assigned to a section. A summary of each chapter follows. The first chapter, by Shapiro, presents an overview of insurance applications of neural networks, fuzzy logic and genetic algorithms. The purposes of the chapter are to document the unique characteristics of insurance as an application area and the extent to which these technologies have been employed. The relationships between

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Vlll

Preface

previously published material and the chapters in this book are noted. Chapter 2, by Francis, walks the reader through an introduction to neural networks from a statistical perspective, in a property and casualty insurance context. One of her goals is to demystify neural networks, and, to that end, she describes them and how they work by generalizing more familiar models. Chapter 3, also by Francis, is a sequel to her first chapter. It uses fraud detection and underwriting applications to illustrate practical issues and problems associated with the implementation of neural networks. The topics addressed include goodness of fit, relevance of predictor variables, and the functional relationships between the independent and target variables. Chapter 4, by Dugas et ah, argues in favor of the use of statistical learning algorithms such as neural networks for automobile insurance ratemaking. To this end, they describe various candidate models and compare them qualitatively and numerically. Issues they address include the differences between operational and modeling objectives, the bias-variance dilemma, high order nonlinear interactions, and non-stationary data. Chapter 5, by Yeo and Smith, combines data mining and mathematical programming to establish an optimal balance between profitability and market share when determining the premium to charge automobile insurance policy holders. They demonstrate the quantitative benefits of their approach. Chapter 6, by Carretero, describes and implements two fuzzy models for adjusting third party liability insurance rates. The models, which represent the two main classes of linear programming models with imprecise information, are based on constraints or information that is supplemental to the statistical experience data. Chapter 7, by Duncan and Robb, deals with population risk management, from the perspective of reducing costs and managing risk in health insurance. In this context, the authors discuss the problem of identifying high-risk members of a population and potential prediction methodologies.

Preface

IX

Chapter 8, by Chang, applies a fuzzy set theoretical approach to asset-liability management and decision-making. It first develops fuzzy-set theoretical analogues of the classical immunization theory and the matching of assets and liabilities. Then, a fuzzy set theoretical approach is used to extend the Bayesian decision method, which accommodates inherently fuzzy new information and decision maker options. Chapter 9, by Brockett et al., gives a review of effectiveness of using neural network models for predicting market failure (insolvency or financial distress) at the firm level, with special emphasis on the US insurance industry. The early warning signals of the neural network models are contrasted with those of expert rating agencies and other statistical methods. Chapter 10, by Viaene et al., addresses the problem of determining which predictor variables are most informative to the trained neural network. The study is done in the context of automobile insurance claim fraud detection, and the neural network results are compared with those of logistic regression and decision tree learning. Chapter 11, by Shapiro, discusses the merging of neural networks, fuzzy logic and genetic algorithms within an insurance context. The topics addressed include the advantages and disadvantages of each technology, the potential merging options, and the explicit nature of the merging. Chapter 12, by Gomez-Deniz and Vazquez-Polo, illustrate notions and techniques of robust Bayesian analysis in the context of problems that arise in Bonus-Malus Systems. They suggest the use of classes of a priori distributions to reflect the a priori opinion of experts. Chapter 13, by Guillen et al, focuses on the issue of policyholder retention. The authors use logistic regression to investigate the conditional probability of a personal lines customer leaving a property and casualty insurer, given various explanatory variables. Chapter 14, by Kolyshkina et al., investigates the use of data mining techniques, such as decision trees and regression splines, in

X

Preface

workers' compensation and health insurance. They describe the methodology used and compare the result of data mining modeling to the results achievable by using more traditional techniques such as generalized linear models. Chapter 15, by Ostaszewski and Rempala, describes resampling methodology and then shows how it can be used to enhance a parametric mortality law and to develop a nonparametric model of the interest rate process associated with an insurer's asset-liability model. Chapter 16, by Craighead and Klemesrud, reports on a study of the selection and active trading of stocks by the use of a clustering algorithm and time series outlier analysis. The clustering is used to restrict the initial set of stocks, while the change in an outlier statistic is used to determine when to actively move in and out of various stocks. The authors comment on the advantages and limitations of their strategies and areas for further research. Chapter 17, by Cheung and Yang, discusses some of the recent advances in optimal portfolio selection strategies for financial and insurance portfolios. Specific topics addressed include an optimal multi-period mean-variance portfolio policy, a continuous time model, and the measures value-at-risk and capital-at-risk. The final chapter, by Hiirlimann, presents some new and interesting algebraic and probabilistic aspects of deterministic cash flow analysis, which are motivated by several concrete examples. The methodology illustrates how abstract algebraic structures can be used to resolve practical problems.

Acknowledgments The foregoing authors are leading professionals from around the world, including Asia, Australia/Oceania, Europe and North America. The editors are grateful to these authors and reviewers for their contributions and their willingness to share their knowledge and insights. We are grateful to Berend Jan van der Zwaag for his wonderful contribution.

Contents Parti Neural networks, fuzzy systems, and genetic algorithms Chapter 1.

3

Insurance applications of neural networks, fuzzy logic, and genetic algorithms 1 2

Introduction Neural network (NN) applications 2.1 An overview of NNs 2.1.1 Supervised NNs 2.1.2 Unsupervised NNs 2.2 Applications 2.2.1 Underwriting 2.2.2 Classification 2.2.3 Asset and investment models 2.2.4 Insolvency 2.2.5 Projected liabilities 2.2.6 Comment 3 Fuzzy logic (FL) applications 3.1 An overview of FL 3.1.1 Linguistic variables 3.1.2 Fuzzy numbers 3.1.3 A fuzzy inference system (FIS) 3.1.4 C-means algorithm 3.2 Applications 3.2.1 Underwriting 3.2.2 Classification 3.2.3 Pricing 3.2.4 Asset and investment models 3.2.5 Projected liabilities

xi

3 4 5 5 8 10 11 12 13 14 17 19 19 19 19 20 22 23 24 24 26 30 31 33

xii

Contents

4

Genetic algorithm (GA) applications 4.1 An overview of GAs 4.1.1 Population regeneration factors 4.2 Applications 4.2.1 Classification 4.2.2 Underwriting 4.2.3 Asset allocation 4.2.4 Competitiveness of the insurance products 5 Comment Acknowledgments References

34 35 35 36 36 37 37 38 39 39 40

Property and casualty Chapter 2. An introduction to neural networks in insurance

51

1 2

51 54 55

Introduction Background on neural networks 2.1 Structure of a feedforward neural network 3 Example 1: simple example of fitting a nonlinear function to claim severity 3.1 Severity trend models 3.2 A one node neural network 3.2.1 Fitting the curve 3.2.2 Fitting the neural network 3.2.3 The fitted curve 3.3 The logistic function revisited 4 Example 2: using neural networks to fit a complex nonlinear function 4.1 The chain ladder method 4.2 Modeling loss development using a two-variable neural network 4.3 Interactions 5 Correlated variables and dimension reduction

57 58 60 64 65 66 69 71 71 81 84 86

Contents

5.1

xiii

Factor analysis and principal components analysis 5.1.1 Factor analysis 5.1.2 Principal components analysis 5.2 Example 3: dimension reduction 6 Conclusion Acknowledgments References

86 87 89 91 98 99 100

Chapter 3 . Practical applications of neural networks in property and casualty insurance

103

1 2

104 105 105 110 117 119 122

Introduction Fraud example 2.1 The data 2.2 Testing variable importance 3 Underwriting example 3.1 Neural network analysis of simulated data 3.2 Goodness of fit 3.3 Interpreting neural networks functions: visualizing neural network results 3.4 Applying an underwriting model 4 Conclusions Acknowledgments Appendix 1 Appendix 2 References

122 128 130 130 131 132 134

Chapter 4. Statistical learning algorithms applied to automobile insurance ratemaking

137

1 2

139 142 144 146

Introduction Concepts of statistical learning theory 2.1 Hypothesis testing: an example 2.2 Parameter optimization: an example

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Contents

3

Mathematical objectives 3.1 The precision criterion 3.2 The fairness criterion 4 Methodology 5 Models 5.1 Constant model 5.2 Linear model 5.3 Table-based methods 5.4 Greedy multiplicative model 5.5 Generalized linear model 5.6 CHAID decision trees 5.7 Combination of CHAID and linear model 5.8 Ordinary neural network 5.9 How can neural networks represent nonlinear interactions? 5.10 Softplus neural network 5.11 Regression support vector machine 5.12 Mixture models 6 Experimental results 6.1 Mean-squared error comparisons 6.2 Evaluating model fairness 6.3 Comparison with current premiums 7 Application to risk sharing pool facilities 8 Conclusion Appendix: Proof of the equivalence of the fairness and precision criterions References

148 149 152 153 156 156 157 158 159 160 162 162 163

Chapter 5. An integrated data mining approach to premium pricing for the automobile insurance industry

199

1 2 3

199 200 201

Introduction A data mining approach Risk classification and prediction of claim cost

167 168 170 172 174 174 179 181 183 187 191 192

Contents

3.1

xv

Risk classification 3.1.1 K-means clustering model 3.1.2 Fuzzy c-means clustering model 3.1.3 Heuristic model 3.2 Prediction of claim cost 3.2.1 K-means clustering model 3.2.2 Fuzzy c-means clustering model 3.2.3 Heuristic model 3.3 Results 4 Prediction of retention rates and price sensitivity 4.1 Prediction of retention rate 4.1.1 Neural network model 4.1.2 Determining decision thresholds 4.1.3 Analyzing prediction accuracy 4.1.4 Generating more homogeneous models 4.1.5 Combining small clusters 4.1.6 Results 4.2 Price sensitivity analysis 4.2.1 Results 5 Determining an optimal portfolio of policy holders 5.1 Results 6 Conclusions References

201 203 204 204 205 205 205 206 207 209 209 209 210 213 213 215 217 218 219 220 222 225 226

Chapter 6. Fuzzy logic techniques in the non-life insurance industry

229

1 2

Insurance market 230 Fuzzy logic in insurance 232 2.1 Basic concepts in the fuzzy decision-making processes..234 2.2 The two fuzzy non-life insurance models 236 2.2.1 Bonus-malus system 236 2.2.2 The data 237 2.2.3 The non-life insurance model based on Zimmermann's linear approach: a symmetric model..237

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Contents

2.2.4 The non-life insurance model based on Verdegay's proach: a nonsymmetric model 2.2.5 Closing remarks about the two models 3 Some extensions 4 Classification 4.1 The fuzzy c-means algorithm 5 The future of insurance Acknowledgments References

ap247 249 250 251 253 254 256 256

Life and health Chapter 7. Population risk management: reducing costs and managing risk in health insurance

261

1

261 262 265 267 267 267 268 269 270 271 271

Background What is a high-risk member? Elements of population risk management 2 Identification (targeting) of high-risk populations 2.1 Data: available sources 2.1.1 Medical charts 2.1.2 Survey data 2.1.3 Medical claims 2.1.4 Pharmacy claims 2.1.5 Laboratory values 2.1.6 Conclusions on data 2.2 Implementation issues and how they affect prediction methodologies 2.2.1 Goals 2.2.2 Budgets 2.2.3 Staffing 2.2.4 Computing resources 2.2.5 Data warehousing 2.3 Prediction methodologies 1.1 1.2

271 272 272 272 272 273 273

Contents

xvii

2.3.1 Clinical methods 2.3.2 Statistical methods 3 Application of interventions and other risk management techniques 3.1 Targeting the right members 3.2 Effectiveness and outcomes 3.3 Results, including a methodology for optimizing inputs to and estimating return on investment from intervention programs 3.4 Using the risk management economic model 4 Summary and conclusions References

275 277 293 293 294

295 295 297 298

Asset-liability management and investments Chapter 8. A fuzzy set theoretical approach to asset and liability management and decision making

301

1 2

301 303 303 304 308 310 314 324

3 4 5 6 7 8

Introduction Fuzzy numbers and their arithmetic operations 2.1 Definitions 2.2 Fuzzy arithmetic operations Fuzzy immunization theory Fuzzy matching of assets and liabilities Bayesian decision method Fuzzy Bayesian decision method Decision making under fuzzy states and fuzzy alternatives Conclusions References

327 331 332

Contents

XV111

Industry issues Chapter 9. 337 Using neural networks to predict failure in the marketplace 1 2

Introduction 337 Overview and background 338 2.1 Models for firm failure 338 2.2 Neural network and artificial intelligence background ..340 2.2.1 The general neural network model 341 2.2.2 Network neural processing units and layers 343 2.2.3 The back-propagation algorithm 344 3 Neural network methods for life insurer insolvency prediction 345 4 Neural network methods for property-liability insurer insolvency prediction 356 5 Conclusion and further directions 360 References 362 Chapter 10. 365 Illustrating the explicative capabilities of Bayesian learning neural networks for auto claim fraud detection 1 2 3 4 5 6 7

Introduction Neural networks for classification Input relevance determination Evidence framework PIP claims data Empirical evaluation Conclusion References

365 368 376 378 384 388 393 394

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Contents

4.1

Transition rules in a robustness model Illustrations Conclusions and further works Acknowledgments Appendix A: Proof of Theorem 2 References 5 6

447 449 456 460 461 462

Chapter 13. 465 Using logistic regression models to predict and understand why customers leave an insurance company 1 2

Introduction Qualitative dependent variable models 2.1 Model specification 2.2 Estimation and inference 2.3 Stages in the modeling process 3 Customer information 4 Empirical results 5 Conclusions Acknowledgments References

466 470 470 473 475 477 481 486 488 489

Life and health Chapter 14. 493 Using data mining for modeling insurance risk and comparison of data mining and linear modeling approaches 1 2

Introduction Data mining - the new methodology for analysis of large data sets - areas of application of data mining in insurance 2.1 Origins and definition of data mining. Data mining and traditional statistical techniques 2.2 Data mining and on-line analytical processing ("OLAP")

495

497 498 499

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Contents

Chapter 11. Merging soft computing technologies in insurance-related applications

401

1 2 3

401 403 404 405 408 410 411 412 415 419 421 422 424 424

4 5 6 7 8 9

Introduction Advantages and disadvantages of NNs, FL and GAs NNs controlled by FL 3.1 Inputs and weights 3.2 Learning rates and momentum coefficients NNs generated by GAs 4.1 Implementing GAs Fuzzy inference systems (FISs) tuned by GAs FISs tuned by NNs GAs controlled by FL Neuro-fuzzy-genetic systems Conclusions Acknowledgments References

Part 2 Other computational techniques Property and

casualty

Chapter 12. Robustness in Bayesian models for bonus-malus systems

435

1 2

435 437 437 438 439 439 440 441 444

Introduction The models 2.1 The likelihood 2.2 The structure functions 2.3 Conjugate posterior structure functions 2.3.1 Negative binomial-generalizedPareto model 2.3.2 Poisson-generalized inverse Gaussian model 3 Premium calculation in a bonus-malus system 4 Technical results

Contents

2.3 3

4

5

6 7

xxi

The use of data mining within the insurance industry....499 Data mining methodologies. Decision trees (CART), MARS and hybrid models 500 3.1 Classification and regression trees (CART) 500 3.2 Multivariate adaptive regression splines (MARS) 502 3.3 Hybrid models 503 Case study 1. Predicting, at the outset of a claim, the likelihood of the claim becoming serious 504 4.1 Problem and background 505 4.2 The data, its description and preparation for the analysis 505 4.3 The analysis, its purposes and methodology 506 4.3.1 Analysis using CART 506 4.3.2 Brief discussion of the analysis using logistic regression and comparison of the two approaches ....510 4.4 Findings and results, implementation issues and client feedback 511 4.5 Conclusion and future directions 512 Case study 2. Health insurer claim cost 512 5.1 Background 512 5.2 Data 513 5.3 Overall modeling approach 514 5.4 Modeling methodology 515 5.5 Model diagnostic and evaluation 517 5.5.1 Gains chart for total expected hospital claims cost ...517 5.5.2 Actual versus predicted chart 518 5.6 Findings and results 519 5.6.1 Hospital cost model precision 519 5.6.2 Predictor importance for hospital cost 519 5.7 Implementation and client feedback 520 Neural networks 520 Conclusion 520 Acknowledgments 521 References 521

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Chapter 15. Emerging applications of the resampling methods in actuarial models

523

1

524 525 526 527 528 530 530 532

Introduction The concept Bootstrap standard error and bias estimates Bootstrap confidence intervals Dependent data 2 Modeling US mortality tables 2.1 Carriere mortality law 2.2 Fitting the mortality curve 2.3 Statistical properties of the parameter estimates in Carriere mortality model 2.4 Assessment of the model accuracy with parametric bootstrap 3 Methodology of cash-flow analysis with resampling 3.1 Interest rates process 3.2 Modeling interest rates with nonparametric bootstrap of dependent data 3.3 Model company assumptions 3.4 Interest rates process assumptions 3.5 Bootstrapping the surplus-value empirical process 3.6 Bootstrap estimates of the surplus cumulative distribution 3.7 Estimates based on the yields on the long-term treasury bonds for 1953-76 3.8 Comparison with the parametric analysis results 4 Conclusions Acknowledgments References 1.1 1.2 1.3 1.4

533 535 540 540 541 543 545 547 551 553 555 557 558 558

Contents

xxiii

Asset-liability management and investments Chapter 16. System intelligence and active stock trading

563

1 2 3 4 5 6

Introduction Determination of the asset universe Implementation Model description Results Conclusions and further research Acknowledgments Appendix A: Partitioning around mediods (PAM) Appendix B: Filter/smoother model References

564 566 568 571 574 577 579 580 583 585

Chapter 17. Asset allocation: investment strategies for financial and insurance portfolio

587

1 2 3

587 589 590 591 592 593 597 600 600 601 604 606 606 609 611 613

4 5

6

7 8

Introduction Single-period Markowitz model Multi-period mean-variance model 3.1 Model and problem formulation 3.2 Model assumption 3.3 A useful auxiliary problem A brief review of Merton's model Continuous-time VaR optimal portfolio 5.1 Value-at-risk 5.2 Model and problem formulation 5.3 Solution approach Continuous-time CaR formulation 6.1 Model and CaR 6.2 Problem formulation Optimal investment strategy for insurance portfolio Conclusions

XXIV

Contents

Acknowledgments Appendix 1: Proof of Proposition 5 Appendix 2: Proof of Proposition 6 Appendix 3: Proof of Proposition 7 References

614 615 616 618 620

Chapter 18. The algebra of cash flows: theory and application

625

1 2 3 4 5 6 7 8 9

Introduction 625 Term structure of interest rates 627 The cash flow polynomial ring 630 The economic multiplication and division of cash flows ...633 The probabilistic notions of duration and convexity 637 Convex order and immunization theory 642 Immunization of liability cash flow products 645 Examples 646 Conclusions 648 Acknowledgment 648 References 649 Appendix 1: Proof of Theorem 3 651 Appendix 2: Proof of Theorem 10 652 Index

653

List of contributors

657

Parti

Neural Networks, Fuzzy Systems, and Genetic Algorithms

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Chapter 1 Insurance Applications of Neural Networks, Fuzzy Logic, and Genetic Algorithms Arnold F. Shapiro

The insurance industry has numerous areas with potential applications for neural networks, fuzzy logic and genetic algorithms. Given this potential and the impetus on these technologies during the last decade, a number of studies have focused on insurance applications. This chapter presents an overview of these studies. The specific purposes of the chapter are twofold: first, to review the insurance applications of these technologies so as to document the unique characteristics of insurance as an application area; and second, to document the extent to which these technologies have been employed. Keywords: insurance, applications, neural networks, fuzzy logic, genetic algorithms

1

Introduction

Neural networks (NNs) are used for learning and curve fitting, fuzzy logic (FL) is used to deal with imprecision and uncertainty, and genetic algorithms (GAs) are used for search and optimization. These technologies often are linked together because they are the most commonly used components of what Zadeh (1992) called soft computing (SC), which he envisioned as being "... modes of com3

4

A. F. Shapiro

puting in which precision is traded for tractability, robustness and ease of implementation." The insurance industry has numerous areas with potential applications for these technologies. Some of these application areas, such as liability projections and mortality and morbidity studies, are unique to the insurance field; others, such as pricing, classification, underwriting, insolvency, and asset and investment models, are common to all financial intermediaries, but have features unique to the insurance area. Given this potential and the impetus on NNs, FL and GAs during the last decade, it is not surprising that a number of studies have focused on insurance applications. This chapter presents an overview of these studies. The specific purposes of the chapter are twofold: first, to review NNs, FL and GAs applications in insurance so as to document the unique characteristics of insurance as an application area; and second, to document the extent to which these technologies have been employed. The chapter has a separate section devoted to each of these technologies. Each section begins with a brief description of the technology for the reader who is unfamiliar with the topic, and then the insurance applications of the technology are reviewed. The reviews cover much of the literature and are intended to show where each technology has made inroads. The chapter ends with a comment on the merging of the technologies.

2

Neural Network (NN) Applications

NNs, first explored by Rosenblatt (1959) and Widrow and Hoff (1960), are computational structures with learning and generalization capabilities. Conceptually, they employ a distributive technique to store knowledge acquired by learning with known samples and are used for pattern classification, prediction and analysis, and control and optimization. Operationally, they are software programs that emulate the biological structure of the human brain and its associated neural complex (Bishop, 1995).

Insurance Applications of NNs, Fuzzy Logic, and Genetic Algorithms

2.1

5

An Overview of NNs

The core of a NN is the neural processing unit (neuron), a representation of which is shown in Figure 1. Inputs

Weights

Figure 1. Neural processing unit.

The inputs to the neuron, Xj, are multiplied by their respective weights, Wj, and aggregated. The weight wo serves the same function as the intercept in a regression formula. The weighted sum is then passed through an activation function, F, to produce the output of the unit. Often, the activation function takes the form of the logistic function F(z)=(l+e"z)"1, where z = Sj wj Xj, as shown in the figure. The NN can be either supervised or unsupervised. The distinguishing feature of a supervised NN is that its input and output are known and its objective is to discover a relationship between the two. The distinguishing feature of an unsupervised NN is that only the input is known and the goal is to uncover patterns in the features of the input data. The remainder of this subsection is devoted to an overview of supervised and unsupervised NNs. 2.1.1

Supervised NNs

This section gives a brief overview of supervised NNs. It covers the basic architecture of a supervised NN and its learning rules and operation.

6

A. F. Shapiro

A Three-Layer NN A supervised NN, also referred to as a multilayer perceptron (MLP) network, is composed of layers of neurons, an example of which is the three-layer NN depicted in Figure 2. Input (Layer 0)

Hidden (Layer 1) w

Output (Layer 2)

on—*£n)

w

001 / w 021

w

111

Bias

x

(nioVv»ioi-*l M 21 012 ^~s ___. \ ^ w 121

Inputs

(

Output

w

w

022

-*^«J

Figure 2. A simple representation of an FFNN.

Extending the notation associated with Figure 1, the first layer, the input layer, has three neurons (labeled noj, j=0,l,2), the second layer, the hidden processing layer, has three neurons (labeled ny, j=0,l,2), and the third layer, the output layer, has one neuron (labeled nn). There are two inputs signals, xi and X2. The neurons are connected by the weights Wijk, where the subscripts i, j , and k refer to the i-th layer, the j-th node of the i-th layer, and the k-th node of the (i+l)st layer, respectively. Thus, for example, W021 is the weight that connects node 2 of the input layer (layer 0) to node 1 of the hidden layer (layer 1). The goal is for the output of the NN to be arbitrarily close to a target output. The Learning Rules The weights of the network serve as its memory, and so the network "learns" when its weights are updated. The updating is done using a learning rule, a common example of which is the Delta rule (Shepard, 1997, p. 15), which is the product of a learning rate, which controls the speed of convergence, an error signal, and the value associated with the j-th node of the i-th layer. The choice of


1

the learning rate is critical: if its value is too large, the error term may not converge at all, and if it is too small, the weight updating process may get stuck in a local minimum and/or be extremely time intensive. The Operation of a Supervised NN A sketch of the operation of a supervised NN is shown in Figure 3.

[ Start)-

Initialize values: architecture weights learning rate (T|)

Assign Input-output values

momemtum (a)

(Stop )-

Adjust weights

-o

Compute Output Values

Compute Hidden Layer Values

Figure 3. The operation of a supervised NN.

Since supervised learning is involved, the system will attempt to match the input with a known target, such as firms that have become insolvent or claims that appear to be fraudulent. The process begins by assigning random weights to the connection between each set of neurons in the network. These weights represent the intensity of the connection between any two neurons. Given the weights, the intermediate values (a hidden layer) and the output of the system are computed. If the output is optimal, in the sense that it is sufficiently close to the target, the process is halted; if not, the weights are adjusted and the process is continued until an optimal solution is obtained or an alternate stopping rule is reached. If the flow of information through the network is from the input to the output, it is known as a feed forward network. The NN is said to involve back-propagation if inadequacies in the output are fed back through the network so that the algorithm can be im-

8

A. F. Shapiro

proved. We will refer to this network as a feedforward NN with backpropagation (FFNN with BP). An instructional tour of the FFNN methodology in a casualty actuarial context can be found in Francis (2001). The ideas in that paper are expanded upon and extended in Chapters two of this volume. 2.1.2

Unsupervised NNs

This section discusses one of the most common unsupervised NNs, the Kohonen network (Kohonen 1988), which often is referred to as a self-organizing feature map (SOFM). The purpose of the network is to emulate our understanding of how the brain uses spatial mappings to model complex data structures. Specifically, the learning algorithm develops a mapping from the input patterns to the output units that embodies the features of the input patterns. In contrast to the supervised network, where the neurons are arranged in layers, in the Kohonen network they are arranged in a planar configuration and the inputs are connected to each unit in the network. The configuration is depicted in Figure 4. Input Layer

Kohonen Layer

Figure 4. Two-dimensional Kohonen network.

As indicated, the Kohonen SOFM may be represented as a twolayered network consisting of a set of input units in the input layer


9

and a set of output units arranged in a grid called a Kohonen layer. The input and output layers are totally interconnected and there is a weight associated with each link, which is a measure of the intensity of the link. The sketch of the operation of a SOFM is shown in Figure 5.

patterns iterations

Figure 5. Operation of a Kohonen network.

The first step in the process is to initialize the parameters and organize the data. This entails setting the iteration index, t, to zero, the interconnecting weights to small positive random values, and the learning rate to a value smaller than but close to 1. Each unit has a neighborhood of units associated with it and empirical evidence suggests that the best approach is to have the neighborhoods fairly broad initially and then to have them decrease over time. Similarly, the learning rate is a decreasing function of time. Each iteration begins by randomizing the training sample, which is composed of P patterns, each of which is represented by a numerical vector. For example, the patterns may be composed of solvent and insolvent insurance companies and the input variables may be financial ratios. Until the number of patterns used (p) exceeds the number available (p > P), the patterns are presented to the units on the grid, each of which is assigned the Euclidean distance between its connecting weight to the input unit and the value of the

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input. This distance is given by [Ej ( Xj - Wy ) 2 ] 0 5 , where wy is the connecting weight between the j-th input unit and the i-th unit on the grid and Xj is the input from unit j . The unit that is the best match to the pattern, the winning unit, is used to adjust the weights of the units in its neighborhood. For this reason the SOFM is often referred to as a competitive NN. The process continues until the number of iterations exceeds some predetermined value (T). In the foregoing training process, the winning units in the Kohonen layer develop clusters of neighbors, which represent the class types found in the training patterns. As a result, patterns associated with each other in the input space will be mapped to output units that also are associated with each other. Since the class of each cluster is known, the network can be used to classify the inputs.

2.2

Applications

A survey of NN applications in insurance is provided in this section. This includes hybrid applications where the NN is the dominant technology. For the most part, these areas of application include underwriting, classification, asset and investment models, insolvency studies, and projected liabilities. Since this is a relatively new area of analysis, a number of the studies also include comparisons with rival approaches, such as discriminant analysis (DA) and logistic regression (LR).1 The overviews in this section do not include a discussion of specific steps unless the methodology of the study had some notable feature. For example, while not specifically mentioned, the methodology for each study involving a FFNN with BP generally included a feasibility stage, during which univariate and bivariate analysis was performed on the data set to determine the feasibility Discriminant analysis was the dominant methodology from the late 1960s to the 1980s, when logistic analysis was commonly used. See Jang (1997: 22, 24) for the assumptions underlying discriminant analysis and an explanation of why logistic regression is preferred.


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of the study, a training stage, a testing stage and a validation stage. Commonly, the index of discrimination for each of the NNs, as well as for the scoring system, was defined to be the area underneath their respective receiver operating characteristic (ROC) curves. 2.2.1

Underwriting

Underwriting is the process of selection through which an insurer determines which of the risks offered to it should be accepted, and the conditions and amounts of the accepted risks. The goal of underwriting is to obtain a safe, yet profitable, distribution of risks. This section reviews two NN applications in the area, mortgage insurance and bonding. One of the early applications of FFNNs to the problem of underwriting was the study by Collins et al. (1988), who used a committee of NNs to replicate the decisions made by mortgage insurance underwriters. They used an architecture of nine coupled networks in a 3x3 arrangement where each of the three networks at each of the three levels focused on a non-exclusive subset of the full feature space. Every 'expert' network rendered a decision by itself but every triad cooperated by searching for an agreement (or consensus) between networks of a level before the system yielded a classification. The system reached a high degree of agreement with human underwriters when testing on previously unseen examples and it was found that when the system and the underwriter disagreed, the system's classifications were more consistent with the guidelines than the underwriter's judgment. Bakheet (1995) used the FFNN with BP as the pattern classification tool in construction bond underwriting. Surety companies bond construction contractors based on the evaluation of a set of decision A ROC curve is a plot of the true positive rate against the false positive rate for the different possible outpoints of a diagnostic test. See Alsing et al. (1999) and Viaene et al. (2001) for an in-depth discussion of ROC curves.

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factors, such as character, capital, capacity, and continuity. The study used an independent pattern classification module for each of these four factors and an integrated pattern classification model. The conclusion was that the model was an efficient method for handling the evaluation. It is worth noting that Bakheet preferred to optimize the NN structure heuristically rather than use the structure suggested by a built-in genetic optimizer. 2.2.2

Classification

Classification is fundamental to insurance. On the one hand, classification is the prelude to the underwriting of potential coverage, while on the other hand, risks need to be properly classified and segregated for pricing purposes. This section discusses representative classification areas where NN have been employed: consumer behavior analysis, the classification of life insurance applicants, and the detection of fraudulent insurance claims. Van Wezel et al. (1996) constructed a data-mining tool based on a SOFM to investigate the consumer behavior of insurance company customers. Their goal was to determine the number of underlying dimensions influencing consumer behavior. They began with a discussion of the underlying model and problem, and the observation that they could not minimize their error measure using a gradient method, because it was not differentiable, which led them to use a competitive NN. After discussing the specifics of the NN used to solve the problem, they show the results of their method on artificially generated data and then on actual data. Vaughn et al. (1997) used a multilayer perceptron network to classify applicants for life insurance into standard and non-standard risk. They then used their knowledge discovery method (Vaughn 1996) to identify the significant, or key, inputs that the network uses to classify applicants. The ranking of these inputs enables the knowledge learned by the network during training to be presented in the form of data relationships and induced rules. These validated that the network learned sensibly and effectively when compared


13

with the training data set. Brockett et al. (1998) used a SOFM to uncover automobile bodily injury claims fraud in the insurance industry and a FFNN with BP to validate the feature map approach. In the case of the latter, a GA was used to find the optimal three-layered configuration. The focus of the study was to determine whether a claim is fraudulent and the level of suspicion of fraud associated with the claim record file. They concluded that the consistency and reliability of the fraud assessment of the SOFM exceeded that of both an insurance adjuster and an insurance investigator in assessing the suspicion level of bodily injury claims. They suggested that a similar methodology could be used for detecting fraud in Medicare claims. Viaene et al. (2001) reported on an exploratory study for benchmarking the predictive power of several prominent state-ofthe-art binary classification techniques in the context of early screening for fraud in personal injury protection automobile insurance claims. To this end, the predictive power of logistic regression, C4.5 decision tree (an algorithm for inducing decision trees from data), A;-nearest neighbor, Bayesian learning MLP, least squares support vector machine (a classification method that separates the training data into two classes), and Bayes classifiers were compared. The comparisons were made both analytically and visually. Overall, they found that logit and the Bayesian learning MLP both performed excellently in the baseline predictive performance benchmarking study, whereas none of the C4.5 variants attained a comparable predictive performance. Chapter 10 of this book, by Viaene et al., elaborates on this study and extends it. 2.2.3

Asset and Investment Models

The analysis of assets and investments is a major component in the management of an insurance enterprise. Of course, this is true of any financial intermediary, and many of the functions performed are uniform across financial companies. Thus, insurers are involved with market and individual stock price forecasting, the forecasting

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of currency futures, credit decision-making, forecasting direction and magnitude of changes in stock indexes, and so on. Numerous examples of these applications are contained in Refenes et al. (1996), which focuses on the categories of derivatives and termstructure models, foreign exchange, equities and commodities, and corporate distress and risk models. Two examples involving cash flow issues follow. Boero and Cavalli (1996) investigated a NN model for forecasting the exchange rate between the Spanish peseta and the US dollar. Their goal was to examine whether potentially highly nonlinear NN models outperform traditional methods or give at least competitive results. To this end, they compared the performance of NN with four linear models, a random walk process and three different specifications based on the purchasing power parity theory. They found mixed results. In experiments with quarterly data, they found no advantage in the use of NNs for forecasting the exchange rate, while the performance of the NNs clearly improves when they were trained on monthly data. Lokmic and Smith (2000) investigated the problem of forecasting when checks issued to retirees and beneficiaries will be presented for payment, in order to forecast daily cash flow requirements. The goal was to optimize the levels of funds kept in the company's bank account to avoid both overdraft charges and an over commitment of investment funds. In their preprocessing stage, they used SOFMs to cluster the checks into homogeneous groups. They then employed a FFNN with BP to arrive at a forecast for the date each check will be presented. They had mixed results. While they concluded that preprocessing was effective, there appeared to be no significant improvement in the cash flow forecasting over a simple heuristic approach. 2.2.4

Insolvency

A primary consideration of regulators is insurer solvency. In the past, most studies of insurance company failure prediction used sta-


15

tistical methods but recently NNs were brought to bear on the problem. This section presents an overview of five such studies and their conclusions with respect to traditional statistical methods. Three of the papers address insolvency from the perspective of a nonlife insurer while the other two articles focus on life insurers. One of the earlier studies was conducted by Park (1993), who used a NN designed by a GA to predict the bankruptcy of insurance companies, and to compare the results with those obtained using DA, LR, Classification And Regression Trees (CART), a procedure for analyzing categorical (classification) or continuous (regression) data (Breiman et al. 1984), and ID3, which is similar to CART. Using a GA to suggest the optimal structure and network parameters was found to be superior to using randomization in the design of the NN, a finding that was confirmed in a study of bank insolvencies. The major conclusion of the study was that the robust NN model outperformed the statistical models. The study also addressed the issue of conflicts in model performance. While previous studies attributed the performance of a model to the application domain, this study argued that the performance of a model is affected by the distribution of independent variables and can be compared from the viewpoint of task characteristics. A similar study was undertaken by Brockett et al. (1994), who used a three-layer FFNN with BP to develop an early warning system for U.S. property-liability insurers two years prior to insolvency. The results of the NN method were compared with those of DA, the rating of an insurer rating organization, and the ratings of an national insurance regulatory body. The conclusion was that the NN approach outperformed the DA and did far better than the rating organization and the regulators. Generally speaking, the NN results showed high predictability and generalizability, suggesting the usefulness of this method for predicting future insurer insolvency. Huang et al. (1994) used a NN optimized with a GA to forecast financial distress in life insurers, and compare the forecasts to those

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obtained using DA, k-nearest neighbor, and logit analysis. The data was limited to Insurance Regulatory Information System Ratios for the total population of insurers. They concluded that the NN dominates the other methods for both in-sample fit and out-of-sample forecasting. Jang (1997) undertook a similar comparative analysis. Here the focus was on a comparison of multiple DA and LR analysis with a FFNN with BP, learning vector quantization3 (LVQ) and SOFM. As with the Huang et al. (1994) study, the FFNN with BP outperformed the traditional statistical approaches for all data sets with a consistent superiority across the different evaluation criteria, as did the LVQ. The SOFM supported these findings by showing the distinct areas of bankrupt and non-bankrupt companies geographically, which suggests that the easier visual interpretation SOFM can be used as a management tool by both insurance regulators and the companies themselves. They also used a misclassification cost approach, like Huang et al. Kramer (1997) combined an ordered logit model with a FFNN with BP and an expert system to investigate the solvency of Dutch non-life insurers. The inputs to the study were the financial indicators of solvency ratio, profit ratio, solvency margin, investment ratios, and growth of the combined ratio. Both the logit model and the NN showed good performance for weak and strong companies, but did poorly on the classification of moderate companies. The results were markedly improved, however, when the combined output of the mathematical models were used as input to the rule-based expert system. The output of the early warning system was the priority (high, medium, low) for further investigation of a specific company.

Learning vector quantization is a two-step method of designing examples for use in nearest-neighbor procedures, which involves finding cluster centers using the k-means clustering algorithm, and then incrementally adapting the centers in order to reduce the number of misclassifications.


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Chapter 9 of this volume, by Brockett et al, gives a current perspective on solvency in the insurance industry as well as a more detailed discussion of some of the articles in this section. 2.2.5

Projected Liabilities

The evaluation of projected liabilities is fundamental to the insurance industry, so it is not surprising that we are beginning to see NNs applied in this area. The issue on the life side is mortality and morbidity rates, while on property and casualty side it is property damage and liability. The projected liability examples discussed below relate to intensive care, dental caries, in-hospital complications, insurance reserves and claim duration. Tu (1993) compared NN and LR models on the basis of their ability to predict length of stay in the intensive care unit following cardiac surgery. Structurally, the training set and test set each consisted of a large equal-sized sample of patients, and five unique risk strata were created using each model. The advantages and disadvantages of each modeling technique were explored, with the conclusion that either model could potentially be used as a risk stratification tool for counseling patients and scheduling operations. Saemundsson (1996) used a NN to address the problem of identifying individuals at risk of developing dental caries. The focus of the research was to compare the predictive capabilities of clinicians, NNs and LR. The study revealed that human prediction of dental disease was at a comparable level of performance to LR, and that the best predictive performance of any method in the study was reached by eliminating some of the uncertainty introduced by dental treatment interventions into the natural process of dental caries, and by using a NN prediction model with human prediction as one of the inputs. An example of the use of NN for a mortality study was provided by Ismael (1999), who investigated the feasibility of using NN to reliably predict in-hospital complications and 30-day mortality for acute myocardial infarction patients. A large database of American

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patients was used in this analysis and 22 history and physical variables were analyzed as predictors for 16 distinct complications. The conclusion of the study was that the NNs proved to be successful in predicting only four of the 16 specific complications: death within 30 days, shock, asystole, and congestive heart failure or pulmonary edema. The scoring system was found to have had a rather low discriminatory ability. Magee (1999) reported on a comparative analysis of FFNNs with BP and traditional actuarial methods for estimating casualty insurance reserve. Specifically, when the NNs were compared to the chain ladder, the additive method, and the least squares regression method, the NNs performed well in terms of both absolute value error and bias introduced by the methodology. A second objective of the study was to compare the simulated data to industry data. Magee found that the NNs outperformed the traditional actuarial methods insofar as these two data sets, which may be attributed to a NN's ability to model complex nonlinear patterns. His final conclusion was that NNs are more robust across data sets with different attributes. Speights et al. (1999) use a generalization of a FFNN with BP to predict claim duration in workers' compensation data in the presence of right censoring and covariate claim attributes. Right censoring occurs when only intermediate, but not final values of claim duration are known for some data points, and final values of claim duration are known for all other observations. They use a connection between least squares estimation and maximum likelihood estimation to establish the generalization, and showed that their methodology could make accurate predictions in the presence of right-censored data and many covariates. Two authors in the current volume address NNs and project liabilities. Francis investigates the chain ladder method in Chapter 2 and claim severity and loss development in Chapter 3. Duncan focuses on future high-cost members from the perspective of population risk management in Chapter 7.


2.2.6

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Comment

Notably absent from the foregoing NN articles are studies related to ratemaking and pricing. Dugas et al., Chapter 4, and Yeo and Smith, Chapter 5, focus on these issues.

3

Fuzzy Logic (FL) Applications

Fuzzy logic (FL) and fuzzy set theory4 (FST), which were formulated by Zadeh (1965), were developed as a response to the fact that most of the parameters we encounter in the real world are not precisely defined. As such, it gives a framework for approximate reasoning and allows qualitative knowledge about a problem to be translated into an executable set of rules. This reasoning and rulebased approach are then used to respond to new inputs. In this section, we review the literature where FL has been used to resolve insurance problems.5

3.1

An Overview of FL

Before proceeding to a review of the literature, this segment gives an overview of four FL concepts that are often used in articles: linguistic variables, fuzzy numbers, fuzzy inference systems and fuzzy clustering. 3.1.1

Linguistic Variables

Linguistic variables are the building blocks of FL. They may be defined (Zadeh, 1975, 1981) as variables whose values are expressed as words or sentences. Risk capacity, for example, may be viewed Following Zadeh (1994, p. 192), in this chapter the term fuzzy logic is used in the broad sense where it is essentially synonymous with fuzzy set theory. Previous reviews of this literature can be found in Yakoubov and Haberman (1998) and Derrig and Ostaszewski (1999).

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both as a numerical value ranging over the interval [0,100%], and a linguistic variable that can take on values like high, not very high, and so on. Each of these linguistic values may be interpreted as a label of a fuzzy subset of the universe of discourse X = [0,100%], whose base variable, x, is the generic numerical value risk capacity. Such a set, an example of which is shown in Figure 6, is characterized by a membership function (MF), fj-high(x), which assigns to each object a grade of membership ranging between zero and one. fuzzy

f^hiflh(x)

high

y S 0 I" 0

i 10

i 20

i 30

i 40

r —i 50 60

1 70

i 80

1 90

1 100

/ NOT high Risk Capacity in %

Figure 6. (Fuzzy) Set of clients with high risk capacity.

In this case, which represents the set of clients with a high risk capacity, individuals with a risk capacity of 50 percent, or less, are assigned a membership grade of zero and those with a risk capacity of 80 percent, or more, are assigned a grade of one. Between those risk capacities, (50%, 80%), the grade of membership is fuzzy. Fuzzy sets are implemented by extending many of the basic identities that hold for ordinary sets. Thus, for example, the union of fuzzy sets A and B is the smallest fuzzy set containing both A and B, and the intersection of A and B is the largest fuzzy set which is contained in both A and B. 3.1.2

Fuzzy Numbers

Fuzzy numbers are numbers that have fuzzy properties, examples of which are the notions of "around six percent" and "relatively high". The general characteristic of a fuzzy numbers (Zadeh, 1975 and Dubois and Prade, 1980) is represented in Figure 7.


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M*M

fi(y|M) = rri! + y( m2 - mi )

f2(y|M) = m4 - y( rru - m3 )

Figure 7. A fuzzy number.

This shape of a fuzzy number is referred to as a "flat" fuzzy number; if m2 were equal to m3, it would be referred to as a "triangular" fuzzy number. The points mj, j=l, 2, 3, 4, and the functions fj(y|M), j=l,2, M a fuzzy number, which are inverse functions mapping the membership function onto the real line, characterize the fuzzy number. These inverse functions were first used by Buckley (1987), who used fuzzy arithmetic to develop the present value for annuities-certain and other financial transactions. As indicated, a fuzzy number usually is taken to be a convex fuzzy subset of the real line. As one would anticipate, fuzzy arithmetic can be applied to the fuzzy numbers. Using the extension principle (Zadeh, 1975), the nonfuzzy arithmetic operations can be extended to incorporate fuzzy sets and fuzzy numbers. Briefly, if * is a binary operation such as addition (+) or the logical min (A) or max (v), the fuzzy number z, defined by z = x * y, is given as a fuzzy set by Uz (w) = vU;V \ix (u) A Uy (v), u,v,w e 5R,

(1)

subject to the constraint that w = u * v, where (j,x , Uy, and \iz denote the membership functions of x, y, and z, respectively and v u v denotes the supremum over u,v. Chang, in Chapter 8 of this volume, addresses fuzzy numbers and their arithmetic operations.

A. F. Shapiro

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3.1.3 A Fuzzy Inference System (FIS) The fuzzy inference system (FIS) is a popular methodology for implementing FL. FISs are also known as fuzzy rule based systems, fuzzy expert systems, fuzzy models, fuzzy associative memories (FAM), or fuzzy logic controllers when used as controllers (Jang et al. 1997 p. 73). The essence of the system can be represented as shown in Figure 8. Knowledge Base Database Rule base

Crisp Input

Fuzzification I Interface I Fuzzy Input

Inference Êngine_ I Fuzzy Output

Defuzzification I Interface I Crisp Output

- Processor -

Figure 8. A fuzzy inference system (FIS).

As indicated in the figure, the FIS can be envisioned as involving a knowledge base and a processing stage. The knowledge base provides MFs and fuzzy rules needed for the process. In the processing stage, numerical crisp variables are the input of the system. These variables are passed through a fuzzification stage where they are transformed to linguistic variables, which become the fuzzy input for the inference engine. This fuzzy input is transformed by the rules of the inference engine to fuzzy output. The linguistic results are then changed by a defuzzification stage into numerical values that become the output of the system.

The numerical input can be crisp or fuzzy. In this latter event, the input does not have to be fuzzified.


3.1.4

23

C-Means Algorithm

The foregoing fuzzy system allows us to convert and embed empirical qualitative knowledge into reasoning systems capable of performing approximate pattern matching and interpolation. However, these systems cannot adapt or learn because they are unable to extract knowledge from existing data. One approach for overcoming this limitation is to use a fuzzy clustering method such as the fuzzy c-means algorithm (Bezdek 1981). The essence of the cmeans algorithm is that it produces reasonable centers for clusters of data, in the sense that the centers capture the essential feature of the cluster, and then groups data vectors around cluster centers that are reasonably close to them. A flowchart of the c-means algorithm is depicted in Figure 9. |u ( t + 1 ) -u ( t ) U Database Calculate new partition: Q(H1)

t = ol Choose Values c = # of clusters m = exp. wt. G = pos.def (pxp) e = tolerance

Choose Initial Partition: 0(0)

Calculate Fuzzy Cluster Centers

Figure 9. The c-means algorithm.

As indicated, the database consists of the nxp matrix, xnp, where n indicates the number of patterns and p denotes the number of features. The algorithm seeks to segregate these n patterns into c clusters, 2 < c < n-1, where the within-clusters variances are minimized and the between clusters variances are maximized. To this end, the algorithm is initialized by resetting the counter, t, to zero, and choosing: c, the number of clusters; m, the exponential weight,

A. F. Shapiro

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which acts to reduce the influence of noise in the data because it limits the influence of small values of membership functions; G, a symmetric, positive-definite (all its principal minors have strictly positive determinants), pxp shaping matrix, which represents the relative importance of the elements of the data set and the correlation between them, examples of which are the identity and covariance matrixes; and e, the tolerance, which controls the stopping rule. Given the database and the initialized values, the next step is to choose the initial partition (membership matrix), t/ (0) , which may be based on a best guess or experience. Next, the fuzzy cluster centers are computed. Using these fuzzy cluster centers, a new (updated) partition, f/('+1), is calculated. The partitions are compared i,+l) {l and if the difference exceeds using the matrix norm U -U 8, the counter, t, is increased and the process continues. If the difference does not exceed s, the process stops.

3.2

Applications

This section reports on five major insurance areas where FL has been implemented: underwriting, classification, pricing, asset and investment models, and projected liabilities. Readers who are also interested in potential areas of applications will find a large number of them mentioned in Ostaszewski (1993). 3.2.1

Underwriting

The first attempt to use fuzzy theory to analyze the internal logic of the intuitive part of insurance underwriting was due to DeWit (1982). He made the case that an insurance contract is based on its (risk) premium estimated as exactly as possible, and also on a more intuitive kind of experience, which is expressed by the company's underwriting practice. His focus was on the use of basic fuzzy membership functions and their operations in an underwriting context.


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Lemaire (1990) extended DeWit (1982) to a more extensive agenda for FL in insurance theory, most notably in the financial aspects of the business. His underwriting focus was on the definition of a preferred policyholder in life insurance. In this context, he compared the classical approach of using crisp membership functions with the use of grades of membership, modified by alpha-cuts. Young (1993) used fuzzy sets to model the selection process in group health insurance. Applying the model of fuzzy multi-stage decision processes of Bellman and Zadeh (1970) to group health underwriting, Young defined fuzzy sets that characterize groups of good underwriting risks. First single-plan underwriting was considered and then the study was extended to multiple-option plans. An example of a different underwriting dimension was provided by Kieselbach (1997) who used FL for systematic failure analysis. The role of the research was to analyze structure failures in order to prevent similar cases in the future, clarify responsibilities, add to experience, and contribute to design theory. The possible advantages of using fault tree analysis as a tool for systematic failure analysis was also discussed. The failure of the railing of a scaffold, which led to an accident, was discussed as an example of the feasibility of this kind of systematic approach and the application of FL to failure analysis. Most reported studies of FL applications in insurance underwriting were written by academicians. Two exception were the studies of Erbach and Seah (1993) and Horgby et al. (1997). One of the first practical insurance application of FL took place in Canada in 1987, and involved the development of Zeno, a prototype life automated underwriter using a mixture of fuzzy and other techniques. According to Erbach and Seah (1993), it was carried as far as the development of an underwriter (coded in C++) that ran on portable computers. With the aid of electronic links to the home office, it was intended to make turnaround a matter of minutes for cases complicated enough to require human intervention. Unfortunately, while Zeno was carried far enough to show that it could

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work, when it was turned over to the system people, they "promptly abandoned it." The other underwriting application was provided by Horgby et al. (1997), who applied FL to medical underwriting of life insurance applicants. In a step-by-step fashion, the authors show how expert knowledge about underwriting diabetes mellitus in life insurance can be processed for a fuzzy inference system. The article was based on one of the first computer-based fuzzy underwriting system implemented in the industry and, given the success of the application, the authors conclude that techniques of fuzzy underwriting will become standard tools for underwriters in the future. 3.2.2

Classification

One of the earliest classification studies was Ebanks et al. (1992), who discussed how measures of fuzziness can be used to classify life insurance risks. They envisioned a two-stage process. In the first stage, a risk was assigned a vector, whose cell values represented the degree to which the risk satisfies the preferred risk requirement associated with that cell. In the second stage, the overall degree of membership in the preferred risk category was computed. This could be done using the fuzzy intersection operator of Lemaire (1990) or the fuzzy inference of Ostaszewski (1993). Measures of fuzziness were compared and discussed within the context of risk classification, including an entropy measure, which measures of the average information content of a set of datapoints, the distance measure of Yager (1979), and the axiomatic product measure of Ebanks (1983). Ostaszewski (1993, Ch. 6) observed that lack of actuarially fair classification is economically equivalent to price discrimination in favor of high risk individuals and suggested "... a possible precaution against [discrimination] is to create classification methods with no assumptions, but rather methods which discover patterns used in classification." To his end, he was the first to suggest the use of the c-means algorithm for classification in an insurance context.


27

By way of introducing the topic to actuaries, he discussed an insightful example involving the classification of four prospective insureds, two males and two females, based on the features height, gender, weight, and resting pulse. Age also was a factor, but it was irrelevant to the analysis since the applicants were all the same age. Moreover, the other feature values were intentionally exaggerated for illustrative purposes. The two initial clusters were on the basis of gender, and, in a step-by-step fashion through three iterations, Ostaszewski showed the existence of a more efficient classification based on all the features. Hellman (1995) used a fuzzy expert system to identify Finnish municipalities that were of average size and well managed, but whose insurance coverage was inadequate. The steps taken included: identify and classify the economic and insurance factors, have an expert subjectively evaluate each factor, preprocessing the conclusions of the expert, and incorporate this knowledge base into an expert system. The economic factors included population size, gross margin rating (based on funds available for capital expenditures), solidity rating, potential for growth, and whether the municipality was in a crisis situation. The insurance factors were nonlife insurance premiums written with the company and the claims ratio for three years. Hellman concluded that important features of the fuzzy expert systems were that they were easily modified, the smooth functions give continuity to the values, and new fuzzy features could easily be added. Cox (1995) reported on a fraud and abuse detection system for managed healthcare that integrated NNs and fuzzy models. The essence of the system was that it detected anomalous behavior by comparing individual medical providers to a peer group. To this end, the underlying behavior patterns for the peer population were developed using the experience of a fraud-detection department, an unsupervised NN that learnt the relationships inherent in the claim data, and a supervised approach that automatically generated the fuzzy model from a knowledge of the decision variables. Cox con-

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eluded that the system was capable of detecting anomalous behaviors equal to or better than the best fraud-detection departments. Derrig and Ostaszewski (1995) extended the previous works by showing how the c-means clustering algorithm can provide an alternative way to view risk classification. Their focus was on applying fuzzy clustering algorithms to problems of auto rating territories and fraud detection and on the classification of insurance claims according to their suspected level of fraud. Conceptually, when the fuzzy cluster algorithm is applied to the classification of rating territories, the clusters are the risk classes, and the degree of belief that each territory belongs to a given cluster (risk class) is quantified as a real-valued number between zero and one. McCauley-Bell and Badiru (1996a,b) discussed a two-phase research project to develop a fuzzy-linguistic expert system for quantifying and predicting the risk of occupational injury of the forearm and hand. The first phase of the research developed linguistic variables to qualify risk levels. The second phase used a multi-criteria decision making technique to assign relative weights to the identified risk factors. Using the linguistic variables obtained in the first part of the research, a fuzzy rule base was constructed with all of the potential combinations for the given factors. This study was particularly interesting because it used processed expert opinion, unlike most previous studies, which rely on unprocessed expert opinion. McCauley-Bell et al. (1999) extended the study by incorporating a fuzzy linear regression model to predict the relationship of known risk factors to the onset of occupational injury. Jablonowski (1996, 1997) investigated the use of FST to represent uncertainty in both the definition and measurement of risk, from a risk manager's perspective. He conceptualized exposure in terms of three fuzzy concepts: risk, which he viewed as a contoured function of frequency and severity; the probability of loss, point estimates of which he assigned membership functions; and the risk profile, which was the intersection of the first two. He concluded that FST provides a realistic approach to the formal analysis of risk.


29

Chen and He (1997) presented a methodology for deriving an Overall Disability Index (ODI) for measuring an individual's disability. Their approach involved the transformation of the ODI derivation problem into a multiple-criteria decision-making problem. Essentially, they used the analytic hierarchy process, a multicriteria decision making technique that uses pairwise comparisons to estimate the relative importance of each risk factors (Saaty 1980), along with entropy theory and fuzzy set theory, to elicit the weights among the attributes and to aggregate the multiple attributes into a single ODI measurement. Verrall and Yakoubov (1999) showed how the fuzzy c-means algorithm could be used to specify a data-based procedure for investigating age groupings in general insurance. Starting with the assumption that distortion effects have already been removed and policyholder age was the only significant factor, they pre-processed the data by grouping the ages where the data was sparse. Then they used a model based on six clusters and an oc-cut of 20 percent to analyze policyholder age groupings associated with the coverages of automobile material damage and bodily injury. They concluded that the flexibility of the fuzzy approach makes it most suitable for grouping policyholder age. Bentley (2000) used an evolutionary-fuzzy approach to investigate suspicious home insurance claims, where genetic programming was employed to evolve FL rules that classified claims into "suspicious" and "non-suspicious" classes. Notable features of his methodology were that it used clustering to develop membership functions and committee decisions to identify the best-evolved rules. With respect to the former, the features of the claims were clustered into low, medium, and high groups, and the minimum and maximum value in each cluster was used to define the domains of the membership functions. The committee decisions were based on different versions of the system that were run in parallel on the same data set and weighted for intelligibility, which was defined as inversely proportional to the number of rules, and accuracy. Bentley

A. F. Shapiro

30

reported that the results of his model when applied to actual data agreed with the results of previous analysis. 3.2.3

Pricing

In the conclusion of DeWit's (1982) article, he recognized the limitations of his analysis and he speculated that "eventually we may arrive, for branches where risk theory offers insufficient possibilities, at fuzzy premium calculation." He was right, of course, and eight years later Lemaire (1990) discussed the computation of a fuzzy premium for an endowment policy, using fuzzy arithmetic. His methodology followed closely that of Buckley (1987). Young (1996) described how FL can be used to make pricing decisions in group health insurance that consistently consider supplementary data, including vague or linguistic objectives of the insurer, which are ancillary to statistical experience data. Using group health insurance data from an insurance company, an illustrative competitive rate changing model was built that employed fuzzy constraints exclusively to adjust insurance rates. Young did not necessarily advocate using such a model without considering experience studies but presented the simplified model to demonstrate more clearly how to represent linguistic rules. The foregoing analysis was extended to include claim experience data in Young (1997). In this case, the author described, stepby-step, how an actuary/decision maker could use FL to adjust workers compensation insurance rates. The supplementary data may be financial or marketing data or statements that reflect the philosophy of the insurance company or client. Cummins and Derrig (1997) used FL to address the financial pricing of property-liability insurance contracts. Both probabilistic and nonprobabilistic types of uncertainty in the pricing and underwriting accept/reject context were incorporated. The authors focused primarily on FL aspects needed to solve the insurance-pricing problem, and in the process fuzzified a well-known insurance financial pricing model, provided numerical examples of fuzzy pric-


31

ing, and proposed fuzzy rules for project decision-making. They concluded that FL can lead to significantly different pricing decisions than the conventional approach. Carretero and Viejo (2000) investigated the use of fuzzy mathematical programming for insurance pricing decisions related to a bonus-malus rating system7 in automobile insurance. Their assumed objective was "attractive income from premiums" while the constraints involved the spread of policies among the risk classes, the weighted sum of the absolute variation of the insured's premium, and the deviation from perfect elasticity of the policyholder's payments with respect to their claim frequency. The system was tested on a large database of third-party personal liability claims of a Spanish insurer and they concluded that their fuzzy linear programming approach avoids unrealistic modeling and may reduce information costs. Carretero, Chapter 6, provides further commentary on this approach and FL techniques in general. A novel extension of the pricing issue was provided by Lu et al. (2001), who investigated the use of game theory to decide on a product line in a competitive situation, where the decision factors are fuzzy. To accommodate the fuzzy aspect of their study they focus on the linguistic operators of importance and superiority. Their model is simplistic, in the sense that it assumed complete information and a static game, as well as only two companies with two potential product lines, but it did walk the reader through to the development of a Nash equilibrium. 3.2.4 Asset and Investment Models As mentioned in the section on NNs, a good deal of the research on allocation of assets and investment analysis can be drawn from research with respect to other financial intermediaries. Two interestA bonus-malus rating system rewards claim-free policyholders by awarding them bonuses or discounts and penalizes policyholders responsible for accidents by assessing them maluses or premium surcharges.

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ing sources in this regard are Chorafas (1994, chapters 8-10), which describes several applications of FL in finance, and Siegel et al. (1995), which describes many applications of fuzzy set from an accounting perspective. Four interesting applications that are relevant to insurance involve cash flow matching, immunization, optimal asset allocation, and insurance company taxation. An overview of these studies follows. Buehlmann and Berliner (1992) used fuzzy arithmetic and the fuzzy numbers associated with cash-flow matching of assets and liabilities due at different times and in different amounts to arrive at a cash-flow matching approximation. The essence of their approach, which they labeled "fuzzy zooming," is that a set of cash flows can be replaced by a single payment, with an associated triangular fuzzy value. In this way, insecure future interest rates can be modeled by fuzzy numbers and used for the calculation of present values. Berliner and Buehlmann (1993) generalized the fuzzy zooming concept and showed how it can be used to materially reduce the complexity of long-term cash flows. Chang and Wang (1995) developed fuzzy mathematical analogues of the classical immunization theory and the matching of assets and liabilities. Essentially, they reformulated concepts about immunization and the matching of assets and liabilities into fuzzy mathematics. This approach offers the advantages of flexibility, improved conformity with situations encountered in practice and the extension of solutions. Chang, in Chapter 8 of this volume, presents further commentary on this application and extends the study to include a fuzzy set theoretical view of Bayesian decision theory. Guo and Huang (1996) used a possibilistic linear programming method8 for optimal asset allocation based on simultaneously maximizing the portfolio return, minimizing the portfolio risk and maximizing the possibility of reaching higher returns. This is analoBrockett and Xia (1995: 34-38) contains an informative discussion of fuzzy linear programming and how it can be transformed to its crisp counterpart.


33

analogous to maximizing mean return, minimizing variance and maximizing skewness for a random rate of return. The authors concluded that their algorithm provides maximal flexibility for decision makers to effectively balance a portfolio's return and risk. Derrig and Ostaszewski (1997) viewed the insurance liabilities, properly priced, as a management tool of the short position in the government tax option. In this context, they illustrated how FL can be used to estimate the effective tax rate and after-tax rate of return on the asset and liability portfolio of a property-liability insurance company. To accomplish this, they modeled critical parameters of underwriting and investment as fuzzy numbers, which lead to a model of uncertainty in the tax rate, rate of return, and the assetliability mix. 3.2.5

Projected Liabilities

Boissonnade (1984) used pattern recognition and FL in the evaluation of seismic intensity and damage forecasting, and for the development of models to estimate earthquake insurance premium rates and insurance strategies. The influences on the performance of structures include quantifiable factors, which can be captured by probability models, and nonquantifiable factors, such as construction quality and architectural details, which are best formulated using fuzzy set models. Accordingly, two methods of identifying earthquake intensity were presented and compared. The first method was based on the theory of pattern recognition where a discriminative function is developed based on Bayes' criterion and the second method applied FL. Cummins and Derrig (1993 p. 434) studied fuzzy trends in property-liability insurance claim costs as a follow-up to their assertion that "the actuarial approach to forecasting is rudimentary." The essence of the study was that they emphasized the selection of a "good" forecast, where goodness was defined using multiple criteria that may be vague or fuzzy, rather than a good forecasting model. They began by calculating several possible trends using ac-

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cepted statistical procedures and for each trend they determined the degree to which the estimate was good by intersecting several fuzzy goals. They suggested that one may choose the trend that has the highest degree of goodness and proposed that a trend that accounts for all the trends can be calculated by forming a weighted average using the membership degrees as weights. They concluded that FL provides an effective method for combining statistical and judgmental criteria in insurance decision-making. A final perspective was provided by Zhao (1996), who addressed the issues of maritime collision prevention and liability. The essence of the analytical portion of the study was the use of fuzzy programming methods to build decision making simulation models and the development of an automatic collision avoidance decision-making system using FL and NNs. In this case, the offline FFNN was used to map input data to membership functions.

4

Genetic Algorithm (GA) Applications

Genetic algorithms9 (GAs) were proposed by Holland (1975) as a way to perform a randomized global search in a solution space. In this space, a population of candidate solutions, each with an associated fitness value, are evaluated by a fitness function on the basis of their performance. Then, using genetic operations, the best candidates are used to evolve a new population that not only has more of the good solutions but better solutions as well. This section reviews GA applications in insurance.

GAs are a subset of the broader category of Evolutionary Computing (EC), which is comprised of the evolutionary optimization methods that work by simulating evolution on a computer. The three main subcategories of EC are Genetic Algorithms (GAs), Evolutionary Programming (EP) and Evolution Strategy (ES) (Thomas (1996). GAs are the most commonly used.


4.1

35

An Overview of GAs

Before proceeding, let us briefly review the GA process. This process, which can be described as an automated, intelligent approach to trial and error, based on principles of natural selection, is depicted in Figure 10. Initialize values: population size (M) (Start |-»- regeneration factors termination criterion

Create new individual(s) reproduction crossover mutation

AP(g+1) = new individual^)

Figure 10. The GA process.

As indicated, the first step in the process is initialization, which involves choosing a population size (M), population regeneration factors, and a termination criterion. The next step is to randomly generate an initial population of solutions, P(g=0), where g is the generation. If this population satisfies the termination criterion, the process stops. Otherwise, the fitness of each individual in the population is evaluated and the best solutions are "bred" with each other to form a new population, P(g+1); the poorer solutions are discarded. If the new population does not satisfy the termination criterion, the process continues. 4.1.1

Population Regeneration Factors

There are three ways to develop a new generation of solutions: reproduction, crossover, and mutation. Reproduction adds a copy of a

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fit individual to the next generation. Crossover emulates the process of creating children, and involves the creation of new individuals (children) from the two fit parents by a recombination of their genes (parameters). Under mutation, there is a small probability that some of the gene values in the population will be replaced with randomly generated values. This has the potential effect of introducing good gene values that may not have occurred in the initial population or which were eliminated during the iterations. In Figure 10, the process is repeated until the new generation has the same number of individuals (M) as the previous one.

4.2

Applications

GAs have become a popular alternative to linear programming and other optimization routines because of their power and ease of use. Insurance applications have included classification, underwriting, asset allocation, and optimizing the competitiveness of an insurance product. This section provides a brief overview of examples of these studies. The discussion excludes the previous mentioned studies where GAs were used to determine the architecture and parameters of NNs. 4.2.1

Classification

Lee and Kim (1999) used GAs to refine the classification system of Korean private passenger automobile insurance. The study was based on a large sample of cases randomly selected from automobile policies in force at the largest auto insurer in Korea, which was divided into two sets for training and testing purposes. The research employed a hybrid learning methodology that integrated GAs and the decision tree induction algorithm, C4.5. First, the GA was used to explore the space of all possible subsets of a large set of candidate discriminatory variables; then the candidate variables subsets were evaluated using the decision tree induction algorithm to produce a classifier from the given variables and the training data. As a


37

benchmark, the classification performance of this technique was compared with that obtained using a Logit model. It was concluded that the hybrid GA-C4.5 outperforms the Logit model both in training and in testing but is quite time consuming. 4.2.2

Underwriting

Nikolopoulos and Duvendack (1994) described the application of GAs and classification tree induction to insurance underwriting. The basis of their study was that underwriter rules were to be extracted from a set of policyholder records, each of which had 19 attributes describing the policyholder, so as to determine when a policy should be cancelled. When the results of the GAs and classification tree techniques were reviewed by an underwriting domain expert, semantically incorrect rules were found in each, which, when eliminated, drastically reduced the accuracy of their results. To overcome this problem, a hybrid model was constructed whereby the rule set generated by the classification tree algorithm was used to construct the initial population of the GA. The authors concluded that the hybrid model produces far superior results than either of the techniques alone. 4.2.3 Asset Allocation Wendt (1995) used a GA to build a portfolio efficient frontier (a set of portfolios with optimal combinations of risk and returns). The underlying data consisted of 250 scenarios of annual returns for eight asset classes. To evaluate the GA process, the final GA output was compared to the efficient frontier created by a sophisticated nonlinear optimizer. After about 50 cycles, the GA found portfolios very close to the efficient frontier generated by the nonlinear optimizer. While not insurance motivated, an interesting study in this area was conducted by Frick et al. (1996), who investigated price-based heuristic trading rules for buying and selling shares. Their method-

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ology involved transforming the time series of share prices using a heuristic charting method that gave buy and sell signals and was based on price change and reversals. Based on a binary representation of those charts, they used GAs to generate trade strategies from the classification of different price formations. They used two different evaluation methods: one compared the return of a trading strategy with the corresponding riskless interest rate and the average stock market return; the other used its risk-adjusted expected return as a benchmark instead of the average stock market return. Their analysis of over one million intra-day (during a single trading day) stock prices from the Frankfurt Stock Exchange (FSE) showed the extent to which different price formations could be classified by their system and the nature of the rules, but left for future research an analysis of the performance of the resulting trading strategies. Jackson (1997) used the asset allocation problem to investigate the performance of a GA when confronted first with a wellconditioned situation and then a more complex case. As a benchmark he used Newton's method of optimization. In the first situation, he assumed a quadratic utility function and found that the GA and Newton method produced essentially the same asset allocation, but the GA took much longer to converge. In the second case, he used a step utility function and found that the GA, while still slower than Newton's method, was more robust. 4.2.4

Competitiveness of the Insurance Products

The profitability, risk, and competitiveness of insurance products was investigated by Tan (1997). He began the analysis with a Monte Carlo simulation, which was used to develop a flexible measurement framework. A GA was then used to seek the optimum asset allocations that form the profitability-risk-competitiveness frontier and to examine the trade-offs. The paper showed how to select the appropriate asset allocation and crediting strategy in order to position the product at the desired location on the frontier.


5

39

Comment

The purpose of this chapter has been to provide the reader with an overview of where NNs, FL and GAs have been implemented in insurance. While it is clear that these technologies have made inroads into many facets of the business, in most instances the applications focused on each technology separately, and did not capitalized on potential synergies between the technologies. However, a natural evolution in soft computing has been the emergence of hybrid systems, where the technologies are used simultaneously (Shapiro and Gorman 2000). FL based technologies can be used to design NNs or GAs, with the effect of increasing their capability to display good performance across a wide range of complex problems with imprecise data. Thus, for example, a fuzzy NN can be constructed where the NN possesses fuzzy signals and/or has fuzzy weights. Conversely, FL can use technologies from other fields, like NNs or GAs, to deduce or to tune, from observed data, the membership functions in fuzzy rules, and may also structure or learn the rules themselves. Chapter 11 explores the merging of these soft computing technologies.

Acknowledgments This work was supported in part by the Robert G. Schwartz Faculty. The assistance of Asheesh Choudhary, Angela T. Koh, Travis J. Miller, and Laura E. Campbell is gratefully acknowledged.

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vol. 29, no. 1, February. Nikolopoulos, C. and Duvendack, S. (1994), "A hybrid machine learning system and its application to insurance underwriting," Proceedings of the IEEE conference on Evolutionary Computation 2, 27-29 June, pp. 692-695. Ostaszewski, K. (1993), Fuzzy Set Methods in Actuarial Science, Society of Actuaries, Schaumburg, IL. Park, J. (1993), "Bankruptcy Prediction of Banks and Insurance Companies: an Approach Using Inductive Methods," Ph.D. dissertation, University of Texas at Austin. Refenes, A.-P.N., Abu-Mostafa, Y., Moody, J., and Weigend, A. (eds) (1996), Neural Networks in Financial Engineering. Proceedings of the Third International Conf. on Neural Networks in the Capital Markets, World Scientific Pub. Co., London. Rosenblatt, F. (1959), "Two theorems of statistical separability in the perceptron," Mechanization of Thought Processes, Symposium held at the National Physical Laboratory, HM Stationary Office, pp. 421-456. Saaty, T.L. (1990), "How to make a decision: the analytic hierarchy process," European J. Oper. Res. 48, pp. 9-26. Saemundsson, S.R. (1996), "Dental Caries Prediction by Clinicians and Neural Networks," Ph.D. dissertation, University of North Carolina at Chapel Hill. Shapiro, A.F. and Gorman, R.P. (2000), "Implementing adaptive nonlinear models," Insurance: Mathematics and Economics 26, pp. 289-307. Shepherd, A.J. (1997), Second-Order Method for Neural Networks. Springer. Siegel, P.H., de Korvin, A. and Omer, K. (eds) (1995), Applications of Fuzzy Sets and the Theory of Evidence to Accounting, JAI Press, Greenwich, Connecticut. Speights, D.B., Brodsky, J.B., and Chudova, D.I. (1999), "Using neural networks to predict claim duration in the presence of right censoring and covariates," Casualty Actuarial Society Forum,

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Winter, pp. 256-278. Tan, R. (1997), "Seeking the profitability-risk-competitiveness frontier using a genetic algorithm," J Actuarial Practice 5(1), pp. 49-77. Thomas, B. (1996), Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms, Oxford University Press. Tu, J.V. (1993), "A Comparison of Neural Network and Logistic Regression Models for Predicting Length of Stay in the Intensive Care Unit Following Cardiac Surgery," Ph.D. dissertation, University of Toronto. van Wezel, M.C., Kok, J.N., and Sere, K. (1996), "Determining the number of dimensions underlying customer-choices with a competitive neural network," IEEE International Conference on Neural Networks 1, pp. 484-489. Vaughn, M.L. (1996), "Interpretation and knowledge discovery from a multilayer perceptron network: opening the black box," Neural Computing and Applications 4(2), pp. 72-82. Vaughn, M.L., Ong, E., and Cavill, S.J. (1997), "Interpretation and knowledge discovery from a multilayer perceptron network that performs whole life assurance risk assessment," Neural Computing and Applications 6, pp. 201-213. Verrall, R.J. and Yakoubov, Y.H. (1999), "A fuzzy approach to grouping by policyholder age in general insurance," Journal of Actuarial Practice, Vol. 7, pp. 181-203. Viaene, S., Derrig, R., Baesens, B., and Dedene, G. (2001), "A comparison of state-of-the-art classification techniques for expert automobile insurance fraud detection," The Journal of Risk & Insurance 69 (3), pp. 373-421. Wendt, R.Q. (1995), "Build your own GA efficient frontier," Risks and Rewards, December 1, pp. 4-5. Widrow, B. and Hoff, M.E. (1960), "Adaptive switching circuits" IRE Western Electric Show and Convention Record, Part 4, August, pp. 96-104.


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Yager, R.R., Ovchinnikov, S., Tong, R.M., and Ngugen, H.T. (1987), Fuzzy Sets and Applications: Collected Papers of Lotfi A. Zadeh, John Wiley & Sons, Inc., New York. Yakoubov, Y.H. and Haberman, S. (1998), "Review of actuarial applications of fuzzy set theory," Actuarial Research Report, City University, London, England. Young, V.R. (1993), "The application of fuzzy sets to group health underwriting," Transactions of the Society of Actuaries 45, pp. 551-590. Young, V.R. (1996), "Insurance rate changing: a fuzzy logic approach," J Risk and Insurance 63, pp. 461-483. Young, V.R. (1997), "Adjusting indicated insurance rates: fuzzy rules that consider both experience and auxiliary data," Proceedings of the Casualty Actuarial Society 84, pp. 734-765. Zadeh, L.A. (1965), "Fuzzy sets," Information and Control, vol. 8, pp.338-353. Zedeh, L.A. (1975), "The concept of a linguistic variable and its application to approximate reasoning," Part I, Information Sciences 8, pp. 199-249; Part II, Information Sciences 8, pp. 30157. Zedeh, L.A. (1981), "Fuzzy systems theory: a framework for the analysis of humanistic systems," in Cavallo, R.E. (ed.), Recent Developments in Systems Methodology in Social Science Research, Kluwer, Boston, pp. 25-41. Zadeh, L.A. (1992), Foreword of the Proceedings of the Second International Conference on Fuzzy Logic and Neural Networks, pp. xiii-xiv, Iizuka, Japan. Zadeh, L.A. (1994), "The role of fuzzy logic in modeling, identification and control," Modeling Identification and Control 15(3), p. 191. Zhao, J. (1996), "Maritime Collision and Liability," Ph.D. Dissertation, University of Southampton.


Property and Casualty


Chapter 2 An Introduction to Neural Networks in Insurance Louise A. Francis

The chapter will provide a basic introduction to the use of neural networks as an analytical modeling tool in insurance, taking the view that the neural network technique is a generalization of more familiar models such as linear regression. The reader will also be introduced to the traditional characterization of a neural network as a system modeled on the functioning of neurons in the brain. The chapter will explain how neural networks are particularly helpful in addressing the data challenges encountered in real insurance applications, where complex data structures are the rule rather than the exception. The chapter will introduce and illustrate three specific modeling challenges: 1) nonlinear functions, 2) correlated data, and 3) interactions. The chapter will explain why neural networks are effective in dealing with these challenges. The issues and concepts will be illustrated with applications that address common Property and Casualty insurance problems. Applications will include trend analysis and reserving.

1

Introduction

Artificial neural networks are the intriguing new high tech tool for finding hidden gems in data. Neural networks belong to a category of techniques for analyzing data that is known as data mining. Other widely used data mining tools include decision trees, genetic

51

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L. A. Francis

algorithms, regression splines and clustering. Data mining techniques are used to find patterns in data not easily identified by other means. Typically the data sets used in data mining are large, i.e., have many records and many predictor variables. The number of records is typically in the tens of thousands, or more, and the number of independent variables is often in the hundreds. Data mining techniques, including neural networks, have been applied to portfolio selection, credit scoring, fraud detection and market research. When data mining techniques are applied to data sets containing complex relationships, the techniques can identify the relationships and provide the analyst with insights into those relationships. An advantage data mining techniques have over classical statistical models used to analyze data such as regression and ANOVA is that they can fit data where the relation between independent and dependent variables is nonlinear and where the specific form of the nonlinear relationship is unknown. Artificial neural networks (hereafter referred to as neural networks) share the advantages just described with the many other data mining tools. However, neural networks have a longer history of research and application. As a result, their value in modeling data has been extensively studied and is established in the literature (Potts 2000). Moreover, they sometimes have advantages over other data mining tools. For instance, decisions trees, a method of splitting data into homogenous clusters with similar expected values for the dependent variable, are often less effective when the predictor variables are continuous than when they are categorical.1 Neural networks work well with both categorical and continuous variables. Neural networks are among the more glamorous of the data mining techniques. They originated in the artificial intelligence discipline where they are often portrayed as a brain in a computer. Neural networks are designed to incorporate and mimic key features of the neurons in the brain and to process data in a manner analogous Salford System's course on Advanced CART, October 15, 1999.

An Introduction to Neural Networks in Insurance

53

to that of the human brain. As a result, much of the terminology used to describe and explain neural networks is borrowed from biology. Many other data mining techniques such as decision trees and regression splines were developed by statisticians and are described in the literature as computationally intensive generalizations of classical linear models. Classical linear models assume that the functional relationship between the independent variables and the dependent variable is linear. However, they allow some nonlinear relationships to be approximated by linear functions by utilizing a transformation of dependent or independent variables. Neural networks and other data mining techniques do not require that the relationships between predictor and dependent variables be linear (whether or not the variables are transformed). The various data mining tools differ in their approach to approximating nonlinear functions and complex data structures. Neural networks use a series of "neurons" in what is known as the hidden layer that apply nonlinear activation functions to approximate complex functions in the data. The details are discussed in the body of this chapter. As the focus of this chapter is neural networks, the other data mining techniques will not be discussed in any further detail. Despite the analytic advantages, many statisticians and actuaries are reluctant to embrace neural networks. One reason is that neural networks appear to be a "black box." Because of the complexity of the functions used in neural network approximations, neural network software typically does not supply the user with information about the nature of the relationship between predictor and target variables. The output of a neural network is a predicted value and some goodness of fit statistics. However, the functional form of the relationship between independent and dependent variables is not made explicit. In addition, the strength of the relationship between dependent and independent variables, i.e., the importance of each variable, is also often not revealed. Classical models as well as other popular data mining techniques, such as decision trees, sup-

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ply the user with a functional description or map of the relationships. This chapter seeks to open that "black box" and show what is happening inside the neural network. While a description of neural networks will be presented and some of the artificial intelligence terminology covered, this chapter's approach is predominantly from the statistical perspective. The similarity between neural networks and regression will be shown. This chapter will compare and contrast how neural networks and classical modeling techniques deal with three specific modeling challenges: 1) nonlinear functions, 2) interactions, and 3) correlated data. The examples in this chapter are drawn from property and casualty insurance.

2

Background on Neural Networks

A number of different kinds of neural networks exist. This chapter will discuss feedforward neural networks with one hidden layer, the most popular kinds of neural networks. A feedforward neural network is a network where the signal (i.e., information from an external source) is passed from an input layer of neurons through a hidden layer to an output layer of neurons. The function of the hidden layer is to process the information from the input layer. The hidden layer is denoted as hidden because it contains neither input nor output data and the output of the hidden layer generally remains unknown to the user. A feedforward neural network can have more than one hidden layer; however, such networks are not common. The feedforward network with one hidden layer is one of the older neural network techniques. As a result, its effectiveness has been established and software for applying it is widely available. The feedforward neural network discussed in this chapter is known as a Multilayer Perceptron (MLP). The MLP is a feedforward network that uses supervised learning. A network that is trained using supervised learning is presented with a target variable and fits a function that can be


55

used to predict the target variable. Alternatively, it may classify records into levels of the target variable when the target variable is categorical. This is analogous to the use of such statistical procedures as regression and logistic regression for prediction and classification. Other popular kinds of feedforward networks often incorporate unsupervised learning into the training. A network trained using unsupervised learning does not have a target variable. The network finds characteristics in the data that can be used to group similar records together. This is analogous to cluster analysis in classical statistics. This chapter will discuss only supervised learning using neural networks and, further, the discussion will be limited to a feedforward MLP neural network with one hidden layer. This chapter will primarily present applications of this model to continuous rather than discrete data.

2.1

Structure of a Feedforward Neural Network

Figure 1 displays the structure of a feedforward neural network with one hidden layer. The first layer contains the input nodes. Each node is a separate independent variable and represents the actual data used to fit a model to the dependent variable. These are connected to another layer of neurons called the hidden layer or hidden nodes, which modifies the data. The nodes in the hidden layer connect to the output layer. The output layer represents the target or dependent variable(s). It is common for networks to have only one target variable, or output node, but there can be more. An example would be a classification problem where the target variable can fall into one of a number of categories. Sometimes each of the categories is represented as a separate output node. As can be seen from Figure 1, each node in the input layer connects to each node in the hidden layer and each node in the hidden layer connects to each node in the output layer.

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Figure 1. Diagram of three-layer feed forward neural network.

This structure is viewed in the artificial intelligence literature as analogous to that of the biological neurons contained in the human brain. The arrows leading to a node correspond to the dendrites leading to a neuron, and, like the dendrites, they carry a signal to the neuron or node. The arrows leading away from a node correspond to the axons of a neuron, and they carry a signal (i.e., the activation level) away from a neuron or node. The neurons of a brain have far more complex interactions than those displayed in the diagram; however developers view neural networks as abstracting the most relevant features of the neurons in the human brain. Neural networks "learn" by adjusting the strength of the signal coming from nodes in the previous connecting layer. As the neural network better learns how to predict the target value from the input pattern, each of the connections between the input neurons and the hidden or intermediate neurons and between the intermediate neu-


57

rons and the output neurons increases or decreases in strength. A function called a threshold or activation function modifies the signal coming into the hidden layer nodes. In the early days of neural networks, this function produced a value of 1 or 0, depending on whether the signal from the prior layer exceeded a threshold value. Thus, the node or neuron would only "fire" (i.e., become active) if the signal exceeded the threshold, a process thought to be similar to the behavior of a neuron. It is now known that biological neurons are more complicated than previously believed and they do not follow a simple all or none rule. Currently, activation functions are typically sigmoid in shape and can take on any value between 0.0 and 1.0 or between -1.0 and 1.0, depending on the particular function chosen. The modified signal is then output to the output layer nodes, which also apply activation functions. Thus, the information about the pattern being learned is encoded in the signals carried to and from the nodes. These signals map a relationship between the input nodes or the data and the output nodes or dependent variable.

3

Example 1: Simple Example of Fitting a Nonlinear Function to Claim Severity

A simple example will be used to illustrate how neural networks perform nonlinear function approximations. This example will provide details about the activation functions in the hidden and output layers to facilitate an understanding of how a neural network works. When actuaries project future losses on insurance policies using historic data containing losses for past years, they must consider the impact of inflationary factors on the future loss experience. To estimate the impact of these inflationary factors, actuaries use regression techniques to estimate "trends" or the rate of change in insurance losses across successive accident or policy periods. The

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L. A. Francis

trends are often fit separately to frequencies and severities, where frequency is defined as the number of claims per policy or per other unit of exposure, and severity is defined as the average dollars of loss per claim. This example will involve estimating severity trends.

3.1

Severity Trend Models

When fitting trends to loss data, whether severity data or frequency data, an exponential regression is one of the most common approaches used. For the exponential regression, the underlying model for loss severity is: Severityt

= Severityt

f'ê

(1)

That is, the severity at time t is equal to the severity at some starting point to times an inflation factor I (which may be thought of as one plus an inflation rate) raised to the power t - to. This is multiplied by a random error term, e, as actual observed severities contain a random component. If the logs are taken of both sides of the equation, the function is linear in the dependent variable, t—to: \n(Severityt)

= ]n(Severityt

) + (t-t0)ln(7)

+e

(2)

As a variation on the exponential trend model, we will model trend as a power function that is close to exponential in form, but increases more rapidly than an exponential trend. In this example, a curve will be fit to average severities where the "true" relationship is of the form: Severityt = SeveritytQ I{t~to)P e

(3)

or, for this particular example, using t as the input variable (with to equal to 0) and Y as the output variable:

59


7 = 10,000* I'1'2

(4)

/~N(1.05,.004) where Y is the dependent variable, average claim severity, t is the time period from the first observation, / is a random inflation or trend factor, and N (u, a) is understood to denote the normal probability distribution, with parameters u, the mean of the distribution, and a, the standard deviation of the distribution. Because this is a simulated example developed as a simple illustration intended primarily to illustrate some of the key features of neural network analysis, the values for this example are likely to be significantly less variable than the severities encountered in actual practice. A sample of forty quarters (10 years) of observations of t and Y was simulated. A scatterplot of the observations is shown in Figure 2. The scatterplot in Figure 3 displays the "true" curve for Fas well as the random Y (severity) values.

• 22000 -

• • »

•C 18000"

>

••

•

•

• •

14000"

10000-

1

3

5

7

9

t (Time in quarters)

Figure 2. Scatterplot of Severity vs. Time.

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L. A. Francis

• 22000 -

• •* • /^

18000-

• .,—• .

14000 "

•

^ ^ ™ "True" Expected Severity • Severity 10000 -

—

•**** "* 1

3

5 7 t {Time in Quarters)

9

Figure 3. Scatterplot of Severity and Expected Severity vs. Time.

3.2

A One Node Neural Network

A simple neural network with one hidden layer was fit to the simulated data. In order to compare neural networks to classical models, a regression curve was also fit. The result of that fit will be discussed after the presentation of the neural network results. The structure of this neural network is shown in Figure 4. As neural networks go, this is a relatively simple network with one input node. In biological neurons, electrochemical signals pass between neurons with the strength of the signal measured by the electrical current. In neural network analysis, the signal between neurons is simulated by software, which applies weights to the input nodes (data) and then applies an activation function to the weights: Neuron signal of the biological neuron system -> Node weights of neural networks.


61

One Hidden Node

•

ti •1

—

Input Layer (Input Data)

Hidden Layer (Process Data)

fci

*l Output Layer (Predicted Value)

Figure 4. One-node neural network.

The weights are used to compute a linear sum of the independent variables. Let Y denote the weighted sum: Y = WQ + w]*Xl+w2X2...

+ wnXn

(5)

The activation function is applied to the weighted sum and is typically a sigmoid function. The most common of the sigmoid functions is the logistic function: f(X)

=— ^

(6)

The logistic function takes on values in the range 0 to 1. Figure 5 displays a typical logistic curve. This curve is centered at an X value of 0, (i.e., the constant wo is 0). Note that this function has an inflection point at an X value of 0 andJ{X) value of 0.5, where it shifts from a convex to a concave curve. Also, note that the slope is steepest at the inflection point where small changes in the value of Xcan produce large changes in the value of the function. The curve becomes relatively flat as X approaches both 1.0 and -1.0.

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Figure 5. Logistic function.

Another sigmoid function often used in neural networks is the hyperbolic tangent function that takes on values between -1.0 and 1.0: ex-e -x (7) f(X) = ex +e~ x In this chapter, the logistic function will be used as the activation function. The Multilayer Perceptron is a multilayer feedforward neural network that uses a sigmoid activation function. The logistic function is applied to the weighted inputs. In this example, there is only one input, therefore the activation function is: 1 h = f(t;w0wl) = f(w0 + wlt) (8) l -(wo+w\0 +

e

This gives the value or activation level of the node in the hidden layer. Weights are then applied to the hidden node (i.e., w2 + w3 h). An activation function is then applied to this "signal" coming from the hidden layer:

63


Q=/(/

*;M^3)=1+e_(i2+W0

(9)

The weights wo and W2 are similar to the constants in a regression and the weights wi and W3 are similar to the coefficients in a regression. The output function o for this particular neural network, with one input node and one hidden node, can be represented as a double application of the logistic function: f(f(t;wo,wl);w2,w3)

=

j

(10)

It will be shown later in this chapter that the use of sigmoid activation functions on the weighted input variables, along with the second application of a sigmoid function by the output node, is what gives the MLP the ability to approximate nonlinear functions. One other operation is applied to the data when fitting the curve: the dependent variable X is normalized. Normalization is used in statistics to minimize the impact of the scale of the independent variables on the fitted model. Thus, a variable with values ranging from 0 to 500,000 does not prevail over variables with values ranging from 0 to 10 merely because the former variable has a much larger scale. Various software products will perform different normalization procedures. The software used to fit the networks in this chapter normalizes the data to have values in the range 0.0 to 1.0. This is accomplished by subtracting a constant from each observation and dividing by a scale factor. It is common for the constant to equal the minimum observed value for X in the data and for the scale factor to equal the range of the observed values (the maximum minus the minimum). Note also that the output function takes on values between 0 and 1 while the dependent variable takes on values between -00 and +00 (although the probability of negative The second function may not be a sigmoid function. A linear function, rather than a sigmoid function, is often applied to the output of the hidden node.

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values for the data in this particular example is nil). When producing predicted values, (i.e., the output, o), the information must be renormalized by multiplying by the target variable's scale factor (the range of Y in our example) and adding back the constant (the minimum observed Y in this example). 3.2.1 Fitting the Curve The process of finding the best set of weights for the neural network is referred to as training or learning. The approach used by most commercial software to estimate the weights is called backpropagation. Backpropagation is performed as follows: • Each time the network cycles through the training data, it produces a predicted value for the target variable. This value is compared to the actual value for the target variable and an error is computed for each observation; • The errors are "fed back" through the network and new weights are computed to reduce the overall error. Despite the neural network terminology, the training process is actually a statistical optimization procedure. Typically, the procedure minimizes the sum of the squared residuals: Min(X(Y-Y)2)

(11)

Warner and Misra (1996) point out that neural network analysis is in many ways like linear regression, which can be used to fit a curve to data. Regression coefficients are solved for by minimizing the squared deviations between actual observations on a target variable and the fitted value. In the case of linear regression, the curve is a straight line. Unlike linear regression, the relationship between the predicted and target variable in a neural network is nonlinear; therefore, a closed form solution to the minimization problem may not exist. In order to minimize the loss function, a numerical technique such as gradient descent (which is similar to backpropagation) is used. Traditional statistical procedures such as


65

nonlinear regression or the solver in Microsoft Excel use an approach similar to that used by neural networks to estimate the parameters of nonlinear functions. A brief description of the procedure is as follows: 1. Initialize the neural network model using an initial set of weights (usually randomly chosen). Use the initialized model to compute a fitted value for an observation. 2. Use the difference between the fitted and actual value on the target variable to compute the error. 3. Change the weights by a small amount that will move them in the direction of a smaller error • This involves multiplying the error by the partial derivative of the function that is being minimized with respect to the weights, because the partial derivative gives the rate of change with respect to the weights. The result is then multiplied by a factor representing the "learning rate," which controls how quickly the weights change. Since the function being approximated involves logistic functions of the weighted input and hidden layers, multiple applications of the chain rule are needed. While the derivatives are a little messy to compute, it is straightforward to incorporate them into software for fitting neural networks. 4. Continue the process until no further significant reduction in the squared error can be obtained. Further details are beyond the scope of this chapter. However, more detailed information is supplied by some authors (Warner and Misra 1996, Smith 1996). The manuals for a number of statistical packages (for example, SAS Institute 1988) provide an excellent introduction to several numerical methods used to fit nonlinear functions. 3.2.2 Fitting the Neural Network For the more ambitious readers who wish to create their own program for fitting neural networks, Smith (1996) provides an Appen-

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dix with computer code for constructing a backpropagation neural network. Chapter 3 in this volume provides the derivatives, mentioned above, which are incorporated into the computer code. However, the assumption, for the purposes of this chapter, is that the overwhelming majority of readers will use a commercial software package when fitting neural networks. Many hours of development by advanced specialists underlie these tools. 3.2.3 The Fitted Curve The parameters fitted by the neural network are shown in Table 1. Table 1. Neural network weights.

Input Node to Hidden Node 2.48 -3.95 Hidden Node to Output Node 2.16 -4.73 To produce the fitted curve from these coefficients, the following procedure must be used: 1. Normalize each input by subtracting the minimum observed value and dividing by the scale coefficient equal to the maximum observed t minus the minimum observed t. The normalized values will be denoted t*. 2. Determine the minimum observed value for Y and the scale coefficient for Y. 3. For each normalized observation t*t compute "(f

"

=

j

-(2.48-3.95**,)

^ >

4. For each h(t*,) compute o(/z(x*,)) = l + g _ ( 2 , 1 6 _4. 7 3 ^ | ) )

(13)

5. Compute the estimated value for each y; by multiplying the


67

normalized value from the output layer in step 4 by the Y scale coefficient and adding the Y constant. This value is the neural network's predicted value for yj. Figure 6 provides a look under the hood at the neural network's fitted functions. The graph shows the output of the hidden layer node and the output layer node after application of the logistic function. The outputs of the hidden node are a declining function, while the output of the output node is an increasing curve with an exponential-like shape. Figure 7 displays the final result of the neural network fitting exercise: a graph of the fitted and "true" values of the dependent variables versus the input variable.

1

3

5

7

9

t (Time in Quarters)

Figure 6. Hidden and output nodes. It is natural to compare this fitted value to that obtained from fitting a linear regression to the data. Two scenarios were used in fitting the linear regression. Since Y is an exponential-like function of t, the log transformation is a natural transformation for Y, as discussed above. However, because the relationship between the independent and dependent variable is not strictly exponential, but is close to an exponential relationship, applying the log transformation to Y produces a regression equation that is not strictly linear in both the independent variable and the error term. That is, in this

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example, the inflation factor is raised to the power t1'2, so the relationship cannot be made strictly linear. Nonetheless, the log transformation should provide a better approximation to the true curve than fitting a straight line to the data. It should be noted that the nonlinear relationship in this example could be fit using a nonlinear regression procedure that would address the concern about the log transform not producing a relationship that is linear in the independent variable. The purpose here, however, is to keep the exposition simple and use techniques that the reader is familiar with.

22000 "

/ ^^—

20000 -

Expected Severity Neural Fitted Severity

sr

18000-

•?

16000"

14000 -

12000 "

10000

1

3

5

7

9

t (Time in Quarters)

Figure 7. Neural network fitted and "true" severity.

The table below presents the goodness of fit results for both regression and the neural network. In this example, the "true" value for severity is known, because the data is simulated. Thus, the average squared difference between the fitted and "true" values can be computed. In addition, it is possible to compute the percentage of the variance in the "true" severity that can be explained by the fitted model. Table 2 indicates that the fitted severity for both models explain was close to the "true" severity. Also note that both the neural network and regression had an R2 (that is the variance of actual data around the fitted values) of about 0.86.


69

Table 2. Results of fit to simulated severity.

average % of variance explained „2 squared error fitted vs. true severity Neural network 47,107 99.6% 86.0% Regression 46,260 99.6% 86.0% The results of this simple example suggest that the exponential regression and the neural network with one hidden node are fairly similar in their predictive accuracy. In general, one might not use a neural network for this simple situation where there is only one predictor variable, and a simple transformation of one of the variables produces a curve that is a reasonably good approximation to the actual data. In addition, if the analyst knew the true function for the curve, a nonlinear regression technique would probably provide the best fit to the data. However, in actual applications, the functional form of the relationship between the independent and dependent variable is often not known.

3.3

The Logistic Function Revisited

The two parameters of the logistic function give it a lot of flexibility in approximating nonlinear curves. Figure 8 presents logistic curves for various values of the coefficient w\. The coefficient controls the steepness of the curve and how quickly it approached its maximum and minimum values. Coefficients with absolute values less than or equal to 1.0 produce curves that are straight lines. Figure 9 presents the effect of varying wo on logistic curves. Varying the values of wo while holding w\ constant shifts the curve right or left. A great variety of shapes can be obtained by varying the constant and coefficients of the logistic functions. A sample of some of the shapes is shown in Figure 10. Note that the lvalues on the graph are limited to the range of 0 to 1, since this is what the neural networks use. In the previous example the combination of shifting the curve and adjusting the steepness coefficient

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L. A. Francis

was used to define a simple nonlinear curve that is close to exponential in shape in the region between 0 and 1.

1.0 -

"""•vTN\

••'"

—

o.e - - - 0.6 -

0.4 -

=-10 = -5 « -1 = 1 - 5 = 10

Z1 V*

0.2 -

w1 w1 w1 W1 w1 w1

*

^y /

\

i

«-

•-.. s

v! ! > — -

Figure 8. Logistic function for various values of wx.

1.0 -

v 0.8 "

0.6 "

0.4 '

N

\

X

\

•nnm—

W0 = -1

^——

W0 = 0 wO = 1

\ \ \ N \X \ \ \ \ \ \ \ \ > \ \ \ \ \ \

0.0 "

Figure 9. Logistic curve with varying constants.


71

Figure 10. Examples of shapes from logistic curve.

4

Example 2: Using Neural Networks to Fit a Complex Nonlinear Function

To facilitate a clear introduction to neural networks and how they work, the first example in this chapter was intentionally simple. The next example is a somewhat more complicated curve.

4.1

The Chain Ladder Method

Table 3 3 presents a development triangle for automobile bodily injury paid losses. This data has been used by property and casualty actuaries to illustrate actuarial approaches to estimating loss reserves and measuring the variability around loss reserves (Francis 1998, Hayne 2002). The data is grouped by accident year and development age. An accident year is the year in which the claim occurred. The development age is the time period after the beginning In order to fit the table on the page, only a subset of the data is shown.

L. A. Francis

72

of the accident year in which claim data are being evaluated. Thus development age 12 displays cumulative paid losses 12 months (or one year) after the beginning of the accident year and development age 24 displays cumulative paid losses 24 months (or two years) after the beginning of the accident year. Table 3. Cumulative paid loss triangle. Accident Year

12

1990 1991 1992 1993 1994 1995 1996 1997

2164 1922 1962 2329 3343 3847 6090 5451

Months of Development 60 36 48 11538 34440 21549 29167 32982 10939 21357 28488 13053 27869 38560 44461 38099 51953 58029 18086 52054 66203 24806 59232 34171 33392

24

72 36528 35330 45988

Table 4. Age-to-age factors. Months of Development

Accident Year

12

24

36

48

60

1990 1991 1992 1993 1994 1995 1996 All yr Avg 5 Yr Avg Selected

5.332 5.691 6.653 7.766 7.420 8.883 5.483 6.747 7.241 7.241

1.868 1.952 2.135 2.107 2.098 1.733

1.354 1.334 1.384 1.364 1.272

1.181 1.158 1.153 1.117

1.061 1.071 1.034

1.982 2.005 2.005

1.341 1.341 1.341

1.152 1.155 1.155

1.055 1.061 1.061

Figure 11 presents a graph of the data on the triangle. It is apparent that as time passes and as development age increases, cumu-


73

lative paid losses tend to increase. This "development" is due to several factors including; 1) late reporting of claims, 2) investigation and settlement of claims by claims adjusters, and 3) the resolution of claims litigated in the courts. After the passage of many years, all losses for a given accident year have been paid and losses for the year reach an "ultimate" value. While a discussion of the use of loss development factors to evaluate future unpaid losses and estimate loss reserves is beyond the scope of this chapter (see Berquist and Sherman 1977), some introduction to this topic is necessary to understand the next application of neural networks.

Figure 11. Scatterplot of cumulative paid losses vs. development age.

The following notation will be used to describe the loss development process. This notation follows that of England and Verrall (2002). Cjj denotes incremental claims, where i denotes the row, (in this case, accident year4) andy denotes the column (development age): The rows of triangles can also represent policy years or report years.

74

L. A. Francis

Dij=icik

(14)

*=1

where Ay is the cumulative paid losses. The amount of development that occurs between subsequent evaluations of the loss data is measured with development factors: A ,+1

(15)

*v=-JTij

Expected or average factors are typically computed using one of the following formulas: n-j

n-j

h-^r—

or

IA,

Xj^f-r

(16)

J

Table 3 displays a triangle of paid loss development factors. These factors measure the increase in paid losses as accident years mature and losses are paid. Thus the factor 5.332 for the 1990 accident year is the result of dividing $11,538 by $2,264. It measures the development of the accident year losses from 12 months after the beginning of the accident year to 24 months after the beginning of the accident year. The all year average of the 12-month to 24month factors is 6.747 and the average for the latest 5 years' factors is 7.241. If 7.241 is selected as the "expected" future development between ages 12 months and 24 months for losses aged 12 months, we can use this factor to estimate future payments for accident years at 12 months of maturity by multiplying losses at development age 12 by the factor C 2 4 = DnXn

- Dn

or $39,471 = $5,451 * 7.241

(17)

That is, the payment from 12 months to 24 months equals cumulative payments as of 12 months multiplied by the 12-month devel-


75

opment factor minus payments as of 12 months. Similarly the all year average for development between 24 months and 36 months is 2.174. Thus, on average, cumulative paid losses aged 24 months will more than double in the next 12 months. The historic data can be used to measure development from 12 to 24 months, 24 to 36 months, 36 to 48 months, etc. The measure is typically an average, such as an all year weighted average, an all year simple average, an average of the last 5 years, etc. However, judgment is often applied, and the actual factors used may reflect a review of trends in the factors, as well as knowledge of the claims to which the factors are applied. The product of these "age-to-age" factors can then be computed and can be used to estimate "ultimate" losses.

A M =A/fU*

(18)

k=j

Table 4 (bottom) displays selected age-to-age factors and the ultimate factors for the automobile bodily injury data. The approach just illustrated for estimating future development on claims that have been incurred by an insurance company is also known as the "chain ladder" approach. The approach is also referred to as the "deterministic chain ladder," as a statistical model does not underlie the calculation. It has been shown by England and Verrall (2002) that the chain ladder method can be reformulated to a generalized linear model (GLM) allowing the machinery of statistical analysis to be used to estimate parameters and assess the quality of the fit. One of the GLM formulations introduced by England and Verrall (2002) is the over-dispersed Poisson. If the chain ladder is modeled as an over-dispersed Poisson:

L. A. Francis

76

E(Cij) = mij=xiyj VariCij^fcyj

(19)

n

k=\

In this formulation, the x's are viewed as accident year ultimate losses, and the y's are the proportion of losses paid in each development year j and, therefore, sum to 1. The variance for the overdispersed Poisson is proportional to the mean. GLM software, that is widely available, can be used to estimate the parameters of this model. Another model that can be fit with GLM methods is the negative binomial model:

Var{Cij) = cpXj{Xj-X)DUj_,

(20)

The X parameter in this model is analogous to the loss development factors in the deterministic chain ladder. The Poisson and negative binomial models are two of many statistical models for the chain ladder that have appeared in the literature. See England and Verrall (2002) for a review of chain ladder models. An alternative to the chain ladder is to fit a curve to the payment data. It is an application that is well suited to the use of neural networks, with their ability to approximate complex nonlinear functions. It is common to fit a curve to incremental as opposed to cumulative data (England and Verrall 2002, Zenwirth 1994, Halliwell 1996). Figure 12 displays a scatterplot of the incremental paid losses. The plot indicates that the amount of paid losses tends to increase until about age three years and then decrease.

11


0

30000 "

a

e

• a o

20000 "

Y

X3 Input Variables

Factor

Dependent Variable

Figure 21. Using factor analysis to create an index for prediction.

Prediction with a Neu ral Network

•

^

+% A

Input Layer (Input Data)

^0^^

*

Hidden Layer (Process Data)

Output Layer (Predicted Value)

Figure 22. Diagram showing analogy between factor analysis and neural networks.


93

On average this factor produces a 5% inflation rate. To make this example concrete, Factor \ will represent the economic factor driving the inflationary results in a line of business, say Workers Compensation. Factor1 drives the observed values of three simulated economic variables, Wage Inflation, Medical Inflation and Benefit Level Inflation. Although unrealistic, in order to keep this example simple it was assumed that no factor other than the economic factor contributes to the value of these variables and the relationship of the factors to the variables is approximately linear. Also, to keep the example simple it was assumed that one economic factor drives Workers Compensation results. A more realistic scenario would separately model the indemnity and medical components of Workers Compensation claim severity. The economic variables are modeled as follows10: \n(J¥ageInflation) = .7 * \n(Factorl) + e e ~ JV(0,.005) \n(MedicalInflation) = 1.3* \n(Factor\) + e e~N(0,.Ql) \n(Benefit _ level _ trend) = .5 * \n(Factor\) + e e ~ 7V(0,.005)

(25) (26) (27)

Two hundred fifty records of the unobserved economic inflation factor and observed inflation variables were simulated. Each record represented one of 50 states for one of five years. As a result, in the simulation, inflation varied by state and by year. The annual inflation rate variables were used to compute cumulative inflationary factors and indices. For each state, the cumulative product of the prior year's index and that year's observed inflation measures (the Note that the according to Taylor's theorem the natural log of a variable whose value is close to one is approximately equal to 1 minus the variable's value, i.e., ln(l+x) « x. Thus, the economic variables are, to a close approximation, linear functions of the factor.

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L. A. Francis

random observed independent variables) were computed. For example the cumulative unobserved economic factor is computed as: Cumfactor\t = f] Factor\k

(28)

k=\

A base severity, intended to represent the average severity over all claims for the line of business for each state for each of the five years was generated from a lognormal distribution.11 To incorporate inflation into the simulation, the severity for a given state for a given year was computed as the product of the simulated base severity and the cumulative value for the simulated (unobserved) inflation factor for its state. Thus, in this simplified scenario, only one factor, an economic factor, is responsible for the variation over time and between states in expected severity. The parameters for these variables were selected to make a solution using Factor Analysis or Principal Components Analysis straightforward and are not based on an analysis of real insurance data. This data, therefore, had significantly less variance than would be observed in actual insurance data. Note that the correlations between the variables is very high. All correlations between the variables are at least 0.90. This means that the problem of multi-collinearity exists in this data set. That is, each variable is nearly identical to the others, adjusting for a constant multiplier, so typical regression procedures have difficulty estimating the parameters of the relationship between the independent variables and severity. Dimension reduction methods such as Factor Analysis and Principal Components Analysis address this problem by reducing the three inflation variables to one, the estimated factor or index. Factor Analysis was performed on variables that were standardized. Most Factor Analysis software standardizes the variables used

This distribution has an average of $5,000 the first year (after application of the inflationary factor for year 1). Also \n(Severity) ~ N(8.47,.05)

95


in the analysis by subtracting the mean and dividing by the standard deviation of each series. The coefficients linking the variables to the factor are called loadings. That is: XI = bi FactorX X2 = b 2 FactorX

(29)

X3 = b3 Factor I where XI, X2 and X3 are the three observed variables, FactorX is the single underlying factor and bi, b 2 and b3 are the loadings. In the case of Factor Analysis, the loadings are the coefficients linking a standardized factor to the standardized dependent variables, not the variables in their original scale. Also, when there is only one factor, the loadings also represent the estimated correlations between the factor and each variable. The loadings produced by the Factor Analysis procedure are shown in Table 7. Table 7. Factor analysis results.

Variable

Loading

Weights

Wage Inflation Index

0.985

0.395

Medical Inflation Index

0.988

0.498

Benefit Level Inflation Index

0.947

0.113

Table 7 indicates that all the variables have a high loading on the factor, and, thus, all are likely to be important in the estimation of an economic index. An index value was estimated for each record using a weighted sum of the three economic variables. The weights used by the Factor Analysis procedure to compute the index are shown in Table 7. Note that these weights (within rounding error) sum to 1.0. The resulting index was then used as a dependent variable to predict each state's severity for each year. The regression model was of the form: Index = 0.395 (Wage Inflation) + .498 (Medical Inflation) + 0.113 (Benefit Level Inflation)

L. A. Francis

96

Severity = a + b * Index + e

(30)

where Severity is the simulated severity, Index is the estimated inflation Index from the Factor Analysis procedure, and e is a random error term. The results of the regression will be discussed below where they are compared to those of the neural network. The simple neural network diagrammed in Figure 22 with three inputs and one hidden node was used to predict a severity for each state and year. The relationship between the neural network's predicted value and the independent variables is shown in Figure 23. This relationship is linear and positively sloped. The relationship between the unobserved inflation factor driving the observed variables and the predicted values is shown in Figure 24. This relationship is positively sloped and nearly linear. Thus, the neural network has produced a curve that is approximately the same form as the "true" underlying relationship. I P r e d i c t o r Variable vs N e u r a l N e t w o r k P r e d i c t e d

BenRate by NeuraINotworkPredicted

1.6 1.2 "

mE>**i**wc»ipi> -«*»

«s»R«s,S*» «• * " n _

D

°

M e d C P I by N auralN elworkPredicted

- 1.6 - 1.2

^ * ^ W a o e C P I by N euraIN etwork Predicted

1.6 1.2 '

5000

5500 6000 NeuralNetworkPredicted

6500

Figure 23. Plot of each of the three inflation indices vs. the neural network predicted.

97


I N e u r a l Network P r e d i c t e d vs Inflation F a c t i

1 1

8oS^°

twork

SoOÎ^S § S*ô

0

ra

£ 6200

°J%T o^l6

•

a

z

o

y*%°2°° 5200 -

_ o8*tf8°

1 1.2 1.3 U n o b s e r v e d Inflation Factor

Figure 24. Plot of neural network predicted vs. unobserved inflation factor driving observed inflation in severity.

Figure 25 shows the actual and fitted values for the neural network and Factor Analysis predicted models. This figure displays the fitted values compared to actual randomly generated severities (on the left) and to "true" expected severities on the right. The xaxis of the graph is the "true" cumulative inflation factor used to generate the random severities in the simulation. However, it should be noted that when working with real data, information on an unobserved variable would not be available. The predicted neural network values appear to be more jagged than the Factor Analysis predicted values. This jaggedness may reflect a weakness of neural networks: over fitting. Sometimes neural networks do not generalize as well as classical linear models, and fit some of the noise or randomness in the data rather than the actual patterns. Looking at the graph on the right showing both predicted values as well as the "true" value, the Factor Analysis model appears to be a better fit as it has less dispersion around the "true" value. Although the neural network fit an approximately linear model to the data, the Factor Analysis model performed better on the data used in this example. Since the relationships between the

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L, A. Francis

independent and dependent variables in this example are approximately linear, this is an instance of a situation where a classical linear model would be preferred over a more complicated neural network procedure. However, if the relationships had been nonlinear, neural networks would probably provide a better approximation. Neural Network and Factor Predicted Values rt

7000

ft

7000 a,_d

6500"

if

6000

6000

Q OAM

AjtjS$tn

'^^'V Jfflr*1 — •

*>Jjff^

^ —

NeuralNetworkPredicte H 5500 Actual Severity Factor Predicted

5000

o NeuralNetworkPredicted ^ Factor Predicted - o " True Severity

if

5000

J? 4500 4000

1.0

1.2 1.4 Cumulative Factor

1.0

1.2

1.4

Cumulative Factor

Figure 25. Left panel displays neural network predicted, factor analysis predicted and actual severities vs. underlying factor, right panel displays neural network predicted and factor analysis predicted vs. the "true" expected severity.

6

Conclusion

This chapter has gone into some detail in describing neural networks and how they work. The chapter has attempted to remove some of the mystery from the neural network "black box". The author has described neural networks as a statistical tool that mini-

An Introduction

to Neural Networks in

Insurance

99

mizes the squared deviation between target and fitted values, much like more traditional statistical procedures do. Examples were provided which showed that neural networks: 1) are universal function approximators, 2) can model complicated data patterns such as interactions between variables, and 3) perform dimension reduction on correlated predictor variables. Classical techniques can be expected to outperform neural network models when data is well behaved and the relationships are linear or variables can be transformed into variables with linear relationships. However neural networks seem to have an advantage over linear models when they are applied to complex nonlinear data. This is an advantage neural networks share with other data mining tools not discussed in detail in this chapter. Practical examples of applications of neural networks to complex data are presented in Chapter 3 of this volume, "Practical Applications of Neural Networks in Property and Casualty Insurance." Note that this chapter does not advocate abandoning classical statistical tools, but rather adding a new tool to the actuarial toolkit. Classical regression performed well in many of the examples in this chapter. Some classical statistical tools such as Generalized Linear Models have been applied successfully to problems similar to those in this chapter.

Acknowledgments The author wishes to acknowledge Virginia Lambert and Jane Taylor for their encouragement and helpful comments on this project.

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References Berquist, J.R. and Sherman, R.E. (1977), "Loss reserve adequacy testing; a comprehensive, systematic approach," Proceedings of the Casualty Actuarial Society, pp. 10-33. Berry, M.J.A. and Linoff, G. (1997), Data Mining Techniques, John Wiley and Sons Brockett, P.L., Xiaohua, X. and Derrig, R.A. (1998), "Using Kohonen's self-organizing feature map to uncover automobile bodily injury claims fraud," Journal of Risk and Insurance, June, 65:2. Dhar, V. and Stein, R. (1997), Seven Methods for Transforming Corporate Data Into Business Intelligence, Princeton Hall. Dunteman, G.H. (1989), Principal Components Analysis, SAGE Publications. Derrig, R.A. (1999), "Patterns, fighting fraud with data," Contingencies, pp. 40-49. England, P.D. and Verrall, R.J. (2002), "Stochastic claims reserving in general insurance," Presented at the Institute of Actuaries, January. Francis, L. (1998) "Regression methods and loss reserving," presented at Casualty Actuarial Society Loss Reserve Seminar. Francis, L. (2001), "Neural networks demystified," Casualty Actuarial Society Forum, Winter, pp. 253-320. Freedman, R.S., Klein, R.A., and Lederman, J. (1995), Artificial Intelligence in the Capital Markets, Probus Publishers. Halliwell, L.J. (1996), "Loss prediction by generalized least squares," Proceeding of the Casualty Actuarial Society, pp. 436489. Hastie, T., Tibshirani, R., and Freidman, J. (2001), The Elements of Statistical Learning: Data Mining, Inference and Prediction, Springer. Hayne, R. (2002), "Determining reserve ranges and variability of loss reserves," presented at Casualty Actuarial Society Loss Reserve Seminar


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Hatcher, L. (1996), A Step by Step Approach to Using the SAS System for Factor Analysis, SAS Institute. Heckman, P.E. and Meyers, G.G. (1986), "The calculation of aggregate loss distributions from claim severity and claim cost distributions," Proceedings of the Casualty Actuarial Society, pp. 22-61. Holler, K., Somner, D., and Trahair, G. (1999), "Something old, something new in classification ratemaking with a new use of GLMs for credit insurance," Casualty Actuarial Society Forum, Winter, pp. 31-84. Hosmer, D.W. and Lemshow, S. (1989), Applied Logistic Regression, John Wiley and Sons. Keefer, J. (2000), "Finding causal relationships by combining knowledge and data in data mining applications," presented at Seminar on Data Mining, University of Delaware. Kim, J. and Mueler, C.W. (1978), Factor Analysis: Statistical Methods and Practical Issues, SAGE Publications. Lawrence, J. (1994), Introduction to Neural Networks: Design, Theory and Applications, California Scientific Software. Martin, E.B. and Morris, A.J. (1999), "Artificial neural networks and multivariate statistics," in Statistics and Neural Networks: Advances at the Interface, Oxford University Press, pp. 195292. Masterson, N. E. (1968), "Economic factors in liability and property insurance claims cost: 1935 - 1967," Proceedings of the Casualty Actuarial Society, pp. 61-89. Monaghan, J.E. (2000), "The impact of personal credit history on loss performance in personal lines," Casualty Actuarial Society Forum, Winter, pp. 79-105. Plate, T.A., Bert, J., and Band, P. (2000), "Visualizing the function computed by a feedforward neural network," Neural Computation, June, pp. 1337-1353. Potts, W.J.E. (2000), Neural Network Modeling: Course Notes, SAS Institute. SAS Institute (1988), SAS/STAT Users Guide: Release 6.03.

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Smith, M. (1996), Neural Networks for Statistical Modeling, International Thompson Computer Press. Speights, D.B, Brodsky, J.B., Chudova, D.I. (1999), "Using neural networks to predict claim duration in the presence of right censoring and covariates," Casualty Actuarial Society Forum, Winter, pp. 255-278. Venebles, W.N. and Ripley, B.D. (1999), Modern Applied Statistics with S-PLUS, 3 r ed., Springer. Viaene, S., Derrig, R., Baesens, B., and Dedene, G. (2002), "A comparison of state-of-the-art classification techniques for expert automobile insurance fraud detection," Journal of Risk and Insurance. Warner, B. and Misra, M. (1996), "Understanding neural networks as statistical tools," American Statistician, November, pp. 284293. Zehnwirth, B. (1994) "Probabilistic development factor models with applications to loss reserve variability, prediction intervals and risk based capital," Casualty Actuarial Society, Spring Forum, pp. 447-606. Zehnwirth, B. and Barnett, G. (1998), "Best estimate for loss reserves" Casualty Actuarial Society Forum, Fall 1998, pp. 55102.

Chapter 3 Practical Applications of Neural Networks in Property and Casualty Insurance Louise A. Francis

This chapter will use two property and casualty insurance applications to illustrate practical issues and problems encountered when using neural networks. The two applications are fraud detection and underwriting. Neural networks seem to show promise as a tool for finding patterns in data with complex data structures. Many statisticians, actuaries and other insurance analysts are reluctant to embrace neural networks because of concern about the pitfalls associated with neural networks. One criticism is that neural networks operate as a "black box" - data goes in, answers come out, but what happens inside the "box" remains a mystery. The output of neural networks reveals little about relationship between predictor and target variables. Thus, it can be difficult to present the results of neural network modeling to management. This chapter will address the "black box" issue that arises when using neural networks. It will show how to derive meaningful business intelligence from the fitted model. It will also explain how to avoid overparameterization.

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104

1

L. A. Francis

Introduction

Neural networks are one of the more popular data mining techniques. Chapter 2 of this volume, "An Introduction to Neural Networks in Property and Casualty Insurance," showed that neural networks can be viewed as computationally intensive extensions of better-known statistical techniques. Some of the strengths of neural networks were presented, i.e., their ability to approximate complex patterns such as those found in Property and Casualty insurance data. Some of these complexities include nonlinear relationships between the independent and dependent variables, interactions and correlated independent variables. Perhaps the greatest disadvantage of neural networks is the inability of users to understand or explain them. Because the neural network is a very complex function, there is no accepted way to illustrate the relationships between independent and dependent variables with functions that can be interpreted by data analysts or management. Thus, users must accept on faith the relationships between the independent and dependent variables that give rise to the predictions they get, and trust that the neural network has produced a good prediction. In the words of Berry and Linoff (1997), "Neural networks are best approached as black boxes with mysterious inner workings, as mysterious as the origins of our own consciousness." More conventional techniques such as linear regression result in simple mathematical functions where the relationship between predictor and target variables is clearly described and can be understood by audiences with modest mathematical expertise. The "black box" aspect of neural networks is a serious impediment to more widespread use. A number of other practical issues arise when using neural networks. For instance, the analyst must choose the number of hidden nodes to include in the neural network. This number should be high enough so that the fitted model approximates the data well, but small enough so that the model is not overparameterized. In addi-

Practical Applications of NNs in Property and Casualty Insurance

105

tion, the analyst may wish to eliminate variables from the model that do not make a significant contribution to predicting the dependent variable. In this chapter, methods for evaluating and interpreting neural networks will be presented. These will include techniques for assessing goodness of fit of the model, techniques for assessing the relevance of variables in the model, and techniques for visualizing the functional relationships between the independent and dependant variables. Two applications will be presented in this chapter and will be used to illustrate the evaluation and interpretation techniques. The first application will be the fraud model. This application will be used to illustrate methods of assessing goodness of fit and methods of determining the importance of predictor variables in explaining the target variable. The second example, underwriting, will be used to illustrate techniques for understanding the functional relationships between the independent and target variables.

2

Fraud Example

2.1

The Data

The data for the fraud application was supplied by the Automobile Insurers Bureau of Massachusetts (AIB). The data consists of information on 1400 closed claims from personal automobile claimants. The data are a random sample of Massachusetts PIP claims that occurred in 1993. The database was assembled with the cooperation of ten large insurers. This data has been used by the AIB, the Insurance Fraud Bureau of Massachusetts (IFB), and other researchers to investigate fraudulent claims or probable fraudulent claims (Derrig and Ostaszewski 1995, Weisberg and Derrig 1995, Viaene et al. 2002). Most data mining applications would use a much larger database. However, the AIB PIP data is well suited to illustration of the use of data mining techniques in insurance. Viaene et al. (2002) used the AIB data to compare the performance of

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a number of data mining and conventional classification techniques. A measure of fraud collected in the study was an overall assessment (ASSESS) of the likelihood that the claim was fraudulent or abusive. Each record in the data was assigned a value by an expert. The value indicated the expert's subjective assessment as to whether the claim was legitimate or whether fraud or abuse was suspected. Experts were asked to classify suspected fraud or abuse claims into the following categories: exaggerated damages, opportunistic fraud, or planned fraud. As shown in Table 1, the assessment variable can take on 5 possible values. Overall, about one third of the claims were coded by adjusters as probable abuse or fraud claims. In this chapter, all categories of the assessment variable except "probably legitimate" will be treated as suspected fraud. However, it should be noted that "soft fraud" such as exaggerating the extent of the injury, often does not meet the legal definition of fraud. This section will present an example where neural networks are used for classification, that is, in a model developed to predict in which of two categories or classes each claim belongs. The dependent variable for the model is ASSESS, the adjustor's assessment of the likelihood that the claim was fraudulent or abusive. The values ranged from 1 (probably legitimate) through 2 to 5 (the various kinds of suspected fraud or abuse). This variable was then converted to a binary dependent variable. Thus, if a claim was other than probably legitimate, it was treated as a suspected fraud. Table 1. Assessment variable. Value 1 2 3 4 5

Assessment Probably legitimate Excessive treatment only Suspected opportunistic fraud, no injury Suspected opportunistic fraud, exaggerated injury Suspected planned fraud


107

When the dependent variable is binary, it can take on one of two possible values. For the purposes of this analysis the values are 0 (legitimate) or a 1 (suspected fraud or abuse). Ordinary least squares regression can be performed by regressing a binary variable on the predictor variables. A more common procedure when the dependent variable is binary is to use logistic regression. Suppose that the true target variable is the probability that a given claim is abusive, and this probability is denoted by p(x). The model relating p(x) to the a vector of independent variables x is: H-7£-;x)

=

B0+B{xl+...+Bnxn

1-/7

where the quantity ln(p(x)/(l-p(x))) is known as the logit function. Logistic regression can be used to produce scores that are between zero and one, consistent with viewing the model's score as a probability. If one does not use a logistic regression approach, but leaves the dependent variable at its original binary values when fitting a model, predicted values which are less than zero and greater than one can result. One solution to this issue is to truncate the predicted values at zero and one. Another solution is to add the extra step of fitting a logistic regression to the data using the neural network predicted value as the independent variable and the binary ASSESS variable as the dependent variable. The fitted probabilities from the logistic regression can then be assigned as a score for the claim. In this example, logistic regression was used after applying the neural network model to convert the predicted values into probabilities. There are two kinds of predictor variables that were used in the analysis. The first category is red flag variables. These are subjective variables that are intended to capture features of the accident, injury or claimant that are believed to be predictive of fraud or abuse. Many red flag variables represent accumulated industry wisdom about which indicators are likely to be associated with fraud. The data on these variables represent an adjustor's subjective as-

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L. A. Francis

sessment of a red flag indication of fraud, such as the "claimant appeared to be claim wise." These variables are binary, that is, they are either true or false. Such red flag variables are often used to target certain claims for further investigation. The data for these red flag variables is not part of the claim file; it was collected as part of the special effort undertaken in assembling the AIB database for fraud research. The red flag variables were supplemented with claims file variables deemed to be available early in the life of a claim and were, therefore, of practical value in predicting fraud. The variables selected for this example are the same as those used by Viaene et al. (2002) in their comparison of statistical and data mining methods. These same variables were also used by Francis (2003) to compare two data mining techniques. While a much larger number of predictor variables is available in the AIB data for modeling fraud, the red flag and objective claim variables selected for incorporation into their models by Viaene et al. were chosen because of early availability. Therefore, they are likely to be useful in predicting fraud soon enough in the claim's lifespan for action to mitigate the cost of the claim to be effective. Tables 2 and 3 present the claim file variables and red flag. Note that one of the claim file variables, treatment lag, was missing values on a significant number of records. For this reason, an additional dummy variable was created. This variable served as an indicator variable for the presence or absence of a value for the treatment lag. One of the objectives of this research is to investigate which variables are likely to be of greatest value in predicting fraud. To do this, procedures were needed for evaluating the importance of independent variables in predicting the target variable.


109

Table 2. Objective claim file variables. Value AGE POLLAG RPTLAG TREATLAG AMBUL PARTDIS TOTDIS LEGALREP TRTMIS

Description Age of claimant Lag from inception date of policy to date reported Lag from date of accident to date reported Lag from date of accident to earliest treatment by service provider Ambulance charges Is the claimant partially disabled? Is the claimant totally disabled? Is the claimant represented by an attorney? Is treatment lag missing? Table 3. Red flag variables.

Subject Accident

Indicator variable Description No report by police officer at scene ACC01 Single vehicle accident ACC04 No plausible explanation for accident ACC09 Claimant in old, low valued vehicle ACC10 Rental vehicle involved in accident ACC11 Property damage was inconsistent with accident ACC14 Very minor impact collision ACC15 Claimant vehicle stopped short ACC16 Insured felt set up, denied fault ACC19 Claimant CLT02 Had a history of previous claims Was an out of state accident CLT04 Was one of three or more claimants in vehicle CLT07 Injury consisted of strain or sprain only Injury INJ01 No objective evidence of injury INJ02 Police report showed no injury or pain INJ03 No emergency treatment was given INJ05 Non-emergency treatment was delayed INJ06 Unusual injury for auto accident INJ11 Insured INS01 Had history of previous claims INS03 Readily accepted fault for accident INS06 Was difficult to contact/uncooperative INS07 Accident occurred soon after effective date Lost wages LW01 Claimant worked for self or a family member LW03 Claimant recently started employment

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2.2

L. A. Francis

Testing Variable Importance

Because of the complicated functions involved in neural network analysis, interpretation of the variables is more challenging than for classical statistical models. One approach (Potts 1999) is to examine the weights connecting the input variables to the hidden layer nodes. Those weights that are closest to zero are viewed as least important. A variable is deemed unimportant only if all of these connections are near zero. Figure 1 displays the absolute values for the weights connecting the input layer (independent variables) to the hidden layer. A table displaying the variables that correspond to the numbers displayed on the graph is provided in Appendix 1. Since there are five nodes in the hidden layer, there are five weights for each variable. Using this procedure, several variables appear to have low weights on all hidden layer nodes and might be deemed "unimportant." This procedure is typically used to eliminate variables from a model, not to quantify their impact on the outcome. Potts points out that this procedure has a number of limitations. Large weights on variables do not necessarily mean that the variables are important and small weights do not guarantee that they are unimportant. Figure 1 indicates that all weights for the 3 rd and 4th variables (TRTLAG and POLLAG) are "small," thus these variables could be pruned from the model. In the next procedure described, these two variables rank in the top 11 (out of 33) in importance. Another limitation of examining weights is that the weights do not provide a means for evaluating the relative importance of variables. There is no established mechanism for combining the weights into a summary statistic that can meaningfully be used to evaluate the variables' importance. In other words, the sum of the weights (which is shown in Figure 1) has no statistical meaning. The next procedure introduced provides a means for assessing variable importance. The measure is referred to as sensitivity and is described by Potts (1999). The sensitivity is a measure of how


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much the predicted value's error increases when the variables are excluded from the model one at a time. However, instead of actually excluding variables and refitting the model, which could be very time consuming, they are fixed at a constant value. The sensitivity is computed as follows: 1. Hold one of the variables constant; say at its mean or median value. 2. Apply the fitted neural network to the data with the selected variable held constant. 3. Compute the squared errors for each observation produced by these modified fitted values. 4. Compute the average of the squared errors and compare it to the average squared error of the full model. 5. Repeat this procedure for each variable used by the neural network. The sensitivity is the percentage reduction in the error of the full model, compared to the model excluding the variable in question. 6. If desired, the variables can be ranked based on their sensitivities.

Figure 1. Plot of weights to hidden layer nodes.

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Since the same set of parameters is used to compute the sensitivities for all variables, this procedure does not require the user to refit the model each time a variable's importance is being evaluated. Table 4 presents the sensitivities of the neural network model fitted to the fraud data. The sensitivity statistic was computed as a ratio of the total squared error for the model, excluding the variable, to the total squared error for the model, including the variable. A higher ratio indicates a more significant impact. In the neural network model, the sensitivity statistic indicates that the involvement of a lawyer is the most important variable in predicting fraud. This variable has a sensitivity ratio well above that of any of the others. Table 5 provides further insight into the relationship between LEGALREP and the dependent variable. The table shows that 61% of claims with legal representation are probably fraudulent while 89% of claims with no representation are probably legitimate. Table 4. Neural network variable ranking and sensitivity. Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Variable LEGALREP TRTMIS AMBUL AGE PARTDIS RPTLAG ACC04 POLLAG CLT02 INJ01 TRTLAG ACC01 ACC14 INJ02 TOTDIS INJ06 ACC15

Ratio 1.4514 1.0905 1.0872 1.0666 1.0605 1.0389 1.0354 1.0329 1.0304 1.0291 1.0286 1.0206 1.0162 1.0160 1.0143 1.0137 1.0107

Rank 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Variable INS06 ACC10 INS01 CLT04 INJ11 INJ05 INJ03 CLT07 ACC09 INS03 ACC19 ACC16 LW01 INS07 ACC11 LW03

Ratio 1.0091 1.0076 1.0068 1.0061 1.0054 1.0053 1.0042 1.0041 1.0029 1.0023 1.0020 1.0019 1.0019 1.0006 1.0003 1.0000


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Table 5. Crosstabulation of ASSESS and LEGALREP. T FfiAl RFP

No Yes

ASSESS Probably Legitimate Probably Fraud 89% 11% 39% 61%

The variable TRTMIS, treatment lag missing, ranks second in importance. The value on this variable is missing when the claimant has not been to an outpatient health care provider, although in over 95% of these cases the claimant has visited an emergency room. Note that both medical paid and total paid for this group are less than one third of the medical paid and total paid for claimants who have visited a provider. Table 6 is a cross-tabulation of the values of TRTMIS versus ASSESS. It indicates that 97% of claims with treatment lag missing are probably legitimate while 51% of claims not missing on this variable are probably fraudulent. Note that the actual lag in obtaining treatment ranks far lower in importance. Table 6. Crosstabulation of ASSESS and TRTMIS. TRTMIS No Yes

ASSESS Probably Legitimate Probably Fraud 49% 51% 97% 3%

Table 7 presents a cross-tabulation for AMBUL, the ambulance cost variable. The table suggests that a higher proportion of claims with no ambulance costs are probably fraudulent, but the effect does not appear to be as strong as LEGALREP or TRTMIS. Note that the variable may be related to the dependent variable in a complicated way, i.e., the function may be nonlinear and may involve interactions with other variables, and a simple cross-tabulation will not uncover the more complicated relationship. These simple descriptive statistics for several of the highest ranking variables indicate that the neural network analysis uncov-

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ered relationships between independent and dependent variables that help us understand the fitted model. Table 7. Crosstabulation of ASSESS and AMBUL. AMBUL 0 1-200 201-300 301-400 >400

ASSESS Probably Legitimate Probably Fraud 60% 40% 82% 18% 71% 29% 69% 31% 81% 19%

A task that is often performed when using conventional statistical procedures is determining which variables are important enough to keep in the model. To develop a parsimonious model it is not sufficient to know the variable's importance ranking. Rather, a test of the variable's significance to the goodness of fit of the model is needed. One approach is to create training and testing samples. Most neural network software allows the user to hold out a portion of the sample for testing. This is because most modeling procedures fit the sample data better than they fit new observations that were not in the sample. One can typically improve the fit on the sample by adding variables, even when the added variables do not improve the fit on out -of -sample data. Data is separated into training and test data.1 The model is fitted using the training data and tested using the test data. The testing helps to determine how well the dependent variable was predicted on data not used for fitting. The test data can be used to determine if the goodness of fit improves or deteriorates as variables are added to or subtracted from the model. As stated previously, this fraud application involves a classification problem. That is, the model can be used to classify claims into 1

A more rigorous testing procedure is to create three samples; one for training, one for validation or fine tuning of the model (such as deciding how many variables to retain in the model), and one for testing the goodness of fit of the model.


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the categories "probably legitimate" and "probably fraudulent." The classification in this example is based on the predicted values for the neural network. Whenever the predicted value exceeds 50%, claims are classified as probably legitimate; otherwise they are classified as probably fraudulent. These classifications are then compared to the actual values in the data for the ASSESS variable to compute the percent correctly classified. This statistic was computed for the test data while varying the numbers of variables in the model. Variables were eliminated from the model based on their rank in the sensitivity test. In addition, the squared correlation coefficient between the neural network's predicted values for the claims and the actual values was computed. Table 8 presents the results, ranked by number of variables in the model. The table indicates that the model achieves its maximum goodness of fit at around 9 or 10 variables. Thus, it seems unnecessary to include more than 10 variables in the model, although the model's performance on out of sample claims does not appear to degrade when additional variables are added. The table indicates that a good model for predicting fraud could be constructed from about 10 variables. Of the top 10 variables, based on the sensitivity test, three are red flag variables. They are CLT02 (claimant has a prior history of claims), ACC04 (single vehicle accident), and INJ01 (injury was a sprain or strain only). According to this test, seven of the top ten predictors are claim file variables. This may indicate that even when a company does not have the resources to collect subjective fraud-related information on claims, there may be value in using only claim file information to build a fraud model. Another procedure that can be used to test the fit of a model is cross-validation. This procedure might be preferred when applying neural networks to a small database. Cross-validation involves iteratively holding out part of the sample, fitting the model to the remainder of the sample, and testing the accuracy of the fitted model on the held out portion. For instance, the sample may be di-

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vided into ten groups. Nine of the groups are used to fit the model and one is used for testing. The process is repeated ten times, and the goodness of fit statistics for the ten test samples are averaged. Though the reader should be aware of cross-validation, it was not used in this example. Table 8. Goodness of fit on test data by number of variables. Number of .. . . . . Variables in -. , , Model 1 2 3 4 5 6 7 8 9 10 11 12 13 15 20 26

„2 ,-, ~- . t , R : Coefficient of _. . . ,. Determination 0.195 0.253 0.267 0.270 0.290 0.282 0.307 0.274 0.328 0.331 0.332 0.365 0.326 0.338 0.340 0.362

D

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Percentage _ f Correct 72.6% 74.1% 74.3% 75.2% 76.0% 76.4% 76.2% 74.9% 77.7% 78.2% 78.4% 78.2% 77.9% 78.2% 77.7% 78.3%

A more comprehensive introduction to the techniques and uses of fraud modeling can be found in the many references on that subject found at the end of this chapter. Derrig (2002) provides an excellent overview of some of the literature on fraud modeling and discusses how analytical models can be used to sort claims for further processing by adjusters.


3

117

Underwriting Example

In this example additional methods for understanding the relationship between the predictor variables and target variables when using neural networks will be introduced. The example makes use of underwriting and ratemaking variables to model an underwriting outcome variable: claim severity.2 The data in this example were produced by simulation and will have nonlinearities, interactions, correlated variables, and missing values, i.e., the data contains complexities that are often encountered in actual practice. The predictor variables in the simulated data are intended to represent those that would be available when underwriting or rating a policy. A random sample of 5,000 claims was simulated. The sample represents 6 years of claims history. (A multiyear period was chosen, so that inflation could be incorporated into the example.) Each claim represents a personal automobile claim severity developed to ultimate settlement.3 As an alternative to using claims developed to ultimate, an analyst might use a database of claims that are all at the same development age.4 Random claim values were generated from a lognormal distribution. In the simulation, the scale parameter, |a, of the lognormal, varies with the characteristics of the policyholder and with factors, such as inflation, that impact the entire line of business simultaneously. The policyholder and line of business characteristics in the simulation were generated by eight variables. The (J, parameter itself has a probability distribution. A graph of the distribution of the parameter in the simulated sample is shown in Figure 2. The parameter had a standard deviation of approximately 0.47. The objective of the analysis is to dis2

Severity is the cost of the claim, given that a claim has occurred. The analyst may want to use neural networks or other data mining techniques to develop the data. 4 For readers who are not familiar with Property and Casualty insurance terminology, a discussion of ultimate losses and loss development is provided in chapter 2 of this volume. 3

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tinguish high severity policyholders from low severity policyholders. This translates into an estimate of (i which is as close to the "true" î as possible.

Figure 2. Histogram of severity |x parameter.

Table 9 lists the eight variables used to generate the data (by affecting the u. parameter) in this example. These variables are not intended to serve as an exhaustive list of factors that are associated with loss exposure for the personal automobile line. Rather, they are examples of the kinds of variables one could incorporate into an underwriting or ratemaking analysis. A ninth variable (labeled Bogus) has no causal relationship to average severity. It is included as a noise variable to test the effectiveness of the neural network procedure. An effective prediction model should be able to distinguish between meaningful variables and variables that have no relationship to the dependent variable. Note that in the analysis of the data, two of the variables used to create the data are unavailable to the analyst as they represent unobserved inflation factors, i.e., the auto bodily injury and auto prop-


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erty damage/physical damage underlying inflation factors (see the discussion of Factor Analysis in the previous chapter, "Introduction to Neural Networks in Property and Casualty Insurance"). Instead, six inflation indices that are correlated with the unobserved factors are available to the analyst for modeling. The inflation indices include a hospital cost index and a medical services index that measure medical cost inflation, and a wage cost index that measures economic damages associated with wage loss. The inflation indices are the actual variables that the neural network uses as predictor variables to capture the inflation effect on claim severity. The variables are further described in Appendix 2 at the end of this chapter. Table 9. Variables and factors used to simulate severity data. Variable Variable Type Number of Categories Missing Data Age of Driver Continuous No Territory Categorical 45 No Age of Car Continuous No No Car Type Categorical 4 Credit Rating Continuous Yes Auto BI Inflation Factor Continuous No Auto PD Inflation Factor Continuous No No Law Change Categorical 2 Bogus Continuous No

3.1

Neural Network Analysis of Simulated Data

The dependent variable for the model was the log of severity. A general rule in statistics is that variables that show significant skewness should be transformed to approximate normality before fitting is done. The log transform is a common transform for accomplishing this. In general, Property and Casualty severities are positively skewed. The data in this example have a skewness of 10, a relatively high skewness. Figure 3, a graph of the distribution of the log of severity indicates that approximate normality is attained after the data is logged.

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The data was separated into a training database of 4,000 claims and a test database of 1,000 claims. A neural network with 6 nodes in the hidden layer was run on the 4,000 claims in the training database. As will be discussed later, this network was larger than the final fitted network. This network was used to rank variables in importance and to eliminate variables. Because the amount of variance explained by the model is relatively small (13%), the sensitivities were relatively small. Table 10 displays the results of the sensitivity test for each of the variables. These rankings indicate that the Bogus variable had no effect on the dependent variable, therefore it was eliminated from the model. Despite their low sensitivities, the inflation variables were not removed. The low sensitivities were probably a result of the high correlations of the variables with each other. The inflation variables, when tested as a group, had a sensitivity of 1.014, significantly higher than their individual sensitivities. In addition, it was deemed necessary to include a measure of inflation in the


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model. Since the neural network's hidden layer performs dimension reduction on the inflation variables, in a manner analogous to Factor or Principal Components Analysis, it seemed appropriate to retain these variables. Table 10. Results of sensitivity test. Rank 1 2 3 4 5 6 7 8 9 10 11 12 13

Variable Sensitivity Ratio 1.098 Credit 1.073 Car Age 1.057 Car Type 1.038 Age Territory 1.023 1.022 Law Effect 1.002 Hospital Cost Index 1.002 Bogus 1.002 Medical Services Index Car Part Cost Index 1.000 Other Services Cost Index 1.000 1.000 Wage Cost Index 1.000 Car Body Cost Index

One peril that is present with neural network models is overfitting. As more hidden layer nodes are added to the model, the fit to the data improves and the R of the model increases. However, the model may simply be fitting the features of the training data, therefore its results may not generalize well to a new database. A rule of thumb for the number of intermediate nodes to include in a neural network is to use one half of the number of variables in the model. After eliminating the Bogus variable, 12 variables remained in the model. The rule of thumb would indicate that 6 nodes should be used. The test data was used to determine how well networks fit with 3, 4, 5, 6 and 7 hidden nodes performed when presented with new data. The fitted model was then used to predict values of claims in the test data. Application of the fitted model to the test data indicated that a 4 node neural network provided the best model. (It produced the highest R2 in the test data.)

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Goodness of Fit

The fitted model had an R2 of 13%. This is a relatively low R2 , but not out of line with what one would expect with the highly random data in this example. The "true" \i (true expected log (severity)) has a variance equal to 13% of the variance of the log of severity. (While these percentages happen to be equal, note that the model did not perfectly predict \i and the measured R2 is almost certainly an optimistic estimate of the goodness of fit.) Thus, if one had perfect knowledge of \i, one could predict individual log(severities) with only 13% accuracy. However, if one had perfect knowledge of the true mean value for severity for each policyholder, along with knowledge of the true mean frequency for each policyholder, one could charge the appropriate rate for the policy, given the particular characteristics of the policyholder. In the aggregate, with a large number of policyholders, the insurance company's actual experience should come close to the experience predicted from the expected severities and frequencies. With simulated data, the "true" \i for each record is known. Thus, the model's accuracy in predicting the true parameter can be assessed. For this example, the correlation between the neural network's predicted values and the parameter [x (mean of the logs of severity) is 0.87.

3.3

Interpreting Neural Networks Functions: Visualizing Neural Network Results

When a model is used to fit a complex function, one can attempt to understand the fitted function by plotting the fitted value against a dependent variable of interest. However, because the fitted values for a given value of a predictor variable are influenced by the values of many other independent variables, such graphs may not provide much insight into the nature of the relationships between independent and dependent variables. Figure 4 displays the relationship


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between the neural network predicted value and the policyholder's age.

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It is difficult to discern the relationship between age and the fitted value for claim severity from this graph. There is a great deal of dispersion of predicted values at any given age due to other effects of variables in the model that disguise the fitted relationship between age and the dependent variable. Researchers have been exploring methods for understanding the function fit by a neural network. Recently, a procedure for visualizing neural network fitted functions was published by Plate at al. (2000). Plate et al. describe their plots as Generalized Additive Model style plots. Rather than attempting to describe Generalized Additive Models, an algorithm for producing the plots is presented below. Hastie et al. (2002), Venables and Ripley (1999), and Plate et al. provide descriptions of Generalized Additive Models. The procedure is implemented as follows: 1. Set all the variables except the one being visualized to a constant value. Means and medians are logical choices for the constants. 2. Apply the neural network function to this dataset to produce a predicted value for each value of the independent variable. Al-

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ternatively, one could choose to apply the neural network to a range of values for the independent variable selected to represent a reasonable set of values of the variable. The other variables remain at the selected constant values. 3. Plot the relationship between the neural network predicted value and the variable. 4. Plate et al. recommend scaling all the variables onto a common scale, such as 0 to 1. This is the scale of the outputs of the logistic functions in the neural network. In this chapter, variables remain in their original scale. The result of applying the above procedure is a plot of the relationship between the dependent variable and one of the independent variables. Multiple applications of this procedure to different variables in the model provides the analyst with a tool for understanding the functional form of the relationships between the independent and dependent variables. As an illustration, the visualization method was applied to the data with all variables set to constants except for driver age. The result is shown in Figure 5. From this graph, we can conclude that the fitted function increases at first and then declines with driver age. Figure 6 shows a similar plot for car age. This function declines with car age, but then increases at older ages.


Figure 5. Neural network age function.

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One of the independent variables in the model is a dummy variable for law change. In the simulation, legislative reform affecting automobile liability costs was passed and implemented after 12 quarters of experience. The dummy variable, intended to capture an intervention effect for the law change, was set to 1 after 12 quarters and was 0 otherwise. The visualization procedure can be employed if the user wishes to obtain an estimated value for the law change. In Figure 7, the line for claims subject to the law change (a value of 1 on the graph) is about 0.27 units below the line of claims not subject to the law change. This suggests that the neural network estimates the law effect at 0.27 on a log scale or about 27%. A 0.27 impact on a log scale corresponds approximately to a multiplicative factor of 1.27, or 0.73 in the case of a negative effect.5 The "true" impact of the law change is a 22% reduction in claim severity, therefore the neural network overestimates the impact.

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The visualization procedure can also be used to evaluate the impact of inflation on the predicted value. All variables except the six economic inflation factors were fixed at a constant value, while the inflation index variables entered the model at their actual values. This case differs from the previous visualization examples in that six variables, rather than one, were allowed to vary. The predicted values are plotted against time. Figure 8 shows that the neural network estimated that inflation increased by about 30% during the six year time period of the sample data. This corresponds roughly to an annual inflation rate of about 4.8%. The "true" inflation underlying the model was approximately 5%.

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One way to visualize two-way interactions is to allow two variables to take on their actual values in the fitting function while keeping the others constant. Figure 9 displays a panel graph for the age and car type interaction. The graph suggests that the function relating policyholder age to severity varies with the value of car type.

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These few examples indicate that visualization can be a powerful tool for understanding the relationships between the independent and dependent variables when using neural networks to find patterns in the data.

3.4

Applying an Underwriting Model

Many Property and Casualty insurance applications of neural networks can utilize predictions of claim severity. A company may want to devise an early warning system to screen newly reported claims for those with a high probability of developing into large settlements. A severity model utilizing only information available early in the life of a claim could be used in an early warning system. A fraud detection system could also be based on claim severity. One approach to fraud detection that was not presented in this chapter is to produce a severity prediction for each claim. The actual value of the claim is compared to the predicted value. Those


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with a large positive deviation from the predicted are candidates for further investigation. However, many of the potential underwriting applications of neural networks require both a frequency and a severity estimate. A company may wish to prune unprofitable risks from its portfolio, pursue profitable risks, or actually use models to establish rates. For such applications, either the loss ratio or pure premium6 will be the target variable of interest. There are two approaches to estimating the needed variable. One can develop models to separately estimate frequency and severity and combine the two estimates. An illustration of fitting neural network models to frequencies is provided in Francis (2001). Alternatively, one can estimate a pure premium or loss ratio model directly. One difficulty of modeling pure premiums or loss ratios is that in some lines of business, such as personal lines auto, most of the policyholders will have no losses, since the expected frequency is relatively low. Because loss ratios and pure premiums are restricted to the range [0, <x>], it is desirable to transform the data onto a scale that does not allow for negative values. The log transformation accomplishes this. However, since the natural log is not defined for a value of zero, it may be necessary to add a very small constant to the data in order to apply the log transform. Once a pure premium is computed, it can be converted into a rate by loading for expenses and profit. Alternatively, the pure premium could be divided by the premium at current rate levels to produce an expected loss ratio. A decision could be made as to whether the predicted loss ratio is acceptable before writing a risk. Alternatively, the loss ratio prediction for a company's portfolio of risks for a line of business can be loaded for expenses and profit and the insurance company can determine if a rate increase is needed. 6

The pure premium is the losses divided by the exposure base for the line of business (in personal auto this might be the number of vehicles insured).

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Conclusions

This chapter is a sequel to the previous chapter, which introduced the neural network technique. Whereas Chapter 2 sought to remove some of the "black box" stigma from neural networks by explaining how they work, this chapter addresses another "black box" challenge: understanding the relationship between independent and target variables uncovered by the neural network. Because of the complexity of the functions used in the neural network approximations, neural network software typically does not supply the user with information about the nature of the relationship between predictor and target variables. Two key tools for understanding this relationship, the sensitivity test and a visualization technique, were presented in this chapter. The sensitivity test helps the user rank the variables in importance. The visualization technique helps the user understand the functional relationships fit by the model. Of these techniques, the visualization technique is not commonly available in commercial data mining software. Incorporating the procedures into neural network software should help address the "black box" criticism of neural networks.

Acknowledgments The author wishes to acknowledge the helpful comments of Patricia Francis-Lyon.


Appendix 1 Table Al. Variable order for Figure 1. Order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Variable AGE RPTLAG TRTLAG POLLAG AMBUL TRTMIS LEGALREP PARTDIS TOTDIS ACC01 ACC04 ACC09 ACC10 ACC11 ACC14 ACC15 ACC16 ACC19 CLT02 CLT04 CLT07 INJ01 INJ02 INJ03 INJ05 INJ06 INJ11 INS01 INS03 INS06 INS07 LW01 LW03

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Appendix 2 This appendix is provided for readers wishing a little more detail on the structure of the data in Example 2. The predictor variables are: Driver age: Age of the driver in years. Car type: This is intended to represent classifications like compact, midsize, sports utility vehicle and luxury car. There are 4 categories. Car age: Age of the car in years. Representative parameters for the Driver age, Car type and Car age and their interactions variables were determined from the Baxter automobile claims database.7 Territory: Intended to represent all the territories for 1 state. There are 45 categories. Credit: A variable using information from the insured's credit history was included in the model. Some recent research has suggested credit information may be useful in predicting personal lines loss ratios. Monaghan (2000) shows that credit history has a significant impact on personal automobile and homeowners' loss ratios. Some insurance companies develop a credit score (perhaps using neural networks) that use the information from a number of credit history variables. For the purposes of illustrating this technique, it was assumed that the entire impact of the credit variable is on severity, although this is unlikely in practice. Automobile Bodily Injury (ABI) inflation factor and Automobile Property Damage and Physical Damage (APD) inflation factors: These factors drive quarterly increases in the bodily injury, property damage, and physical damage components of average 7

This database of Automobile claims is available as an example database in SPLUS. Venables and Ripley supply the S-PLUS data for claim severities in a SPLUS library. See Venables and Ripley (1999), p.467.


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severity. They are unobserved factors. The ABI factor is correlated with three observed variables: the producer price index for hospitals, the medical services component of the consumer price index, and an index of average hourly earnings. The APD factor is correlated with three observed variables: the produce price index for automobile bodies, the producer price index for automobile parts, and the other services component of the consumer price index. The ABI factor was given a 60% weight and the APD factor was given a 40% weight in computing each claim's expected severity. Law Change: A change in the law is enacted which causes average severities to decline by 22% after the third year. Interactions: Table A2 shows the variables with interactions. Three of the variables have interactions. In addition, some of the interactions are nonlinear (or piecewise linear). An example is the interactions between age and car age. This is a curve that has a negative slope at older car ages and younger driver ages, but is flat for older driver ages and younger car ages. In addition to these interactions, other relationships exist in the data, which affect the mix of values for the predictor variables in the data. Young drivers (55) drivers are more likely to have older cars. Table A2. Interactions Driver Age and Car Type Driver Age and Car Age Driver Age and Car Age and Car Type

Nonlinearities: A number of nonlinear relationships were built into the data. The relationship between age and severity for certain car types follows an exponential decay. The relationships be-

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tween some of the inflation indices and the factors generating actual claim inflation are nonlinear. The relationship between car age and severity is piecewise linear. That is, there is no effect below a threshold age, then effect increases linearly up to a maximum effect and remains at that level at higher ages. Missing Data: In our real life experience with insurance data, values are often missing on variables that have a significant impact on the dependent variable. To make the simulated data in this example more realistic, data is missing on one of the independent variables, the credit variable. For records with missing data, two dummy variables were created with a value of 0 for most of the observations, but a value of 1 for records with a missing value on car age and/or credit information. In addition, a value of -1 was recorded for car age and credit leverage where data was missing. These values were used in the neural network analysis.

References Berry, M.J.A. and Linoff, G. (1997), Data Mining Techniques, John Wiley and Sons. Brockett, P.L., Xiaohua, X., and Derrig, R.A. (1998), "Using Kohonen's self-organizing feature map to uncover automobile bodily injury claims fraud," Journal of Risk and Insurance, June, 65:2. Brockett, P.L., Derrig, R.A., Golden, L.L., Levine, A., and Alpert, M. (2002), Fraud classification using principal component analysis of RIDITs, Journal of Risk and Insurance, (to be published). Dhar, V. and Stein, R. (1997), Seven Methods for Transforming Corporate Data Into Business Intelligence, Prentice Hall. Dunteman, G.H. (1989), Principal Components Analysis, SAGE Publications.


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Derrig, R.A. (1999), "Patterns, fighting fraud with data," Contingencies, pp. 40-49. Derrig, R. (2002), "Insurance fraud," Journal of Risk and Insurance. Derrig, R.A., and Ostaszewski, K.M. (1995), Fuzzy techniques of pattern recognition in risk and claim classification, Journal of Risk and Insurance, September, 62:3: 447-482. Derrig, R.A., Weisberg, H., and Chen, X. (1994), Behavioral factors and lotteries under no-fault with a monetary threshold: a study of Massachusetts automobile claims, Journal of Risk and Insurance, June, 61:2: 245-275. Derrig, R.A. and Zicko, V. (2002), Prosecuting insurance fraud: a case study of the Massachusetts experience in the 1990s, Risk Management and Insurance Research. Francis, L. (2001), "Neural networks demystified," Casualty Actuarial Society Forum, Winter, pp. 253-320. Francis, L. (2003), "Martian chronicles: is MARS better than neural networks," Casualty Actuarial Society Forum, Winter. Freedman, R.S., Klein, R.A., and Lederman, J. (1995), Artificial Intelligence in the Capital Markets, Probus Publishers. Hastie, T., Tibshirani, R., and Freidman, J. (2001), The Elements of Statistical Learning: Data Mining, Inference and Prediction, Springer. Hatcher, L. (1996) A Step by Step Approach to Using the SAS System for Factor Analysis, SAS Institute. Holler, K., Somner, D., and Trahair, G. (1999) "Something old, something new in classification ratemaking with a new use of GLMs for credit insurance," Casualty Actuarial Society Forum, Winter, pp. 31-84. Hosmer, D.W. and Lemshow, S. (1989), Applied Logistic Regression, John Wiley and Sons. Keefer, J. (2000), "Finding causal relationships by combining knowledge and data in data mining applications," presented at Seminar on Data Mining, University of Delaware.

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Kim, J. and Mueler, C.W. (1978), Factor Analysis: Statistical Methods and Practical Issues, SAGE Publications. Lawrence, J. (1994), Introduction to Neural Networks: Design, Theory and Applications, California Scientific Software. Martin, E.B. and Morris, A.J. (1999), "Artificial neural networks and multivariate statistics," in Statistics and Neural Networks: Advances at the Interface, Oxford University Press, pp. 195-292. Masterson, N.E. (1968), "Economic factors in liability and property insurance claims cost: 1935 - 1967," Proceedings of the Casualty Actuarial Society, pp. 61-89. Monaghan, J.E. (2000) "The impact of personal credit history on loss performance in personal lines," Casualty Actuarial Society Forum, Winter, pp. 79-105. Plate, T.A., Bert, J., and Band, P. (2000), "Visualizing the function computed by a feedforward neural network," Neural Computation, June, pp. 1337-1353. Potts, W.J.E. (2000), Neural Network Modeling: Course Notes, SAS Institute. SAS Institute (1988), SAS/STAT Users Guide: Release 6.03 Smith, M. (1996), Neural Networks for Statistical Modeling, International Thompson Computer Press. Speights, D.B, Brodsky, J.B., and Chudova, D.I. (1999), "Using neural networks to predict claim duration in the presence of right censoring and covariates," Casualty Actuarial Society Forum, Winter, pp. 255-278. Venebles, W.N. and Ripley, B.D. (1999), Modern Applied Statistics with S-PLUS, 3rd ed., Springer. Viaene, S., Derrig, R., Baesens, B., and Dedene, G. (2002), "A comparison of state-of-the-art classification techniques for expert automobile insurance fraud detection, Journal of Risk and Insurance. Warner, B. and Misra, M., (1996), "Understanding neural networks as statistical tools," American Statistician, November, pp. 284293.

Chapter 4 Statistical Learning Algorithms Applied to Automobile Insurance Ratemaking C. Dugas, Y. Bengio, N. Chapados, P. Vincent, G. Denoncourt, and C. Fournier The chapter will start from a description of the fundamentals of statistical learning algorithms and highlight how its basic tenets and methodologies differ from those generally followed by actuaries and econometricians. The main premise is that reality is too complex to be captured with a single unifying model, although some aspects may be well approximated by models. Therefore the statistical learning approach does not presume that reality is perfectly captured by a model, or at least tries to minimize the assumptions about the true generating distribution of the data. The approach is empirical: good models will be distinguished from poor models by comparing their predictive power and explanatory power on new data. At this point it is interesting to consider that choosing among models may be guided by two different objectives, which sometimes lead to different answers: an operational objective (which model will make the best decisions/predictions on new data), or a "modeling" objective (which model better describes the true underlying nature of the data). We will show an example in which the two approaches lead to different statistical tests and the operational approach makes more conservative decisions (chooses simpler models). Another example of the difference between the two approaches is illustrated by the case of ridge regression: there is a regularized (biased) regression that brings better out-of-sample expected predictions than the maximum likelihood (unbiased) estimator. This example will be used to illustrate the famous bias-variance dilemma that is so pervasive in statistical 137

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learning algorithms. The above discussion and introduction to the principles of statistical learning will naturally bring up the issue of methodology. We will describe and justify the main methodological tools of the statistical learning approach for selecting and comparing models, either based on theoretical bounds or on resampling techniques (such as the cross-validation and bootstrap techniques). A special section on the particular (and rarely discussed) issue of non-stationary data will explain how the above resampling methods can be generalized to data whose distribution varies over time, which is the case with insurance data. In order to evaluate and compare models, one needs to build statistical tools to evaluate the uncertainty in the measurements of out-of-sample performance (due to finite data and non-stationarity). We applied the principles and methodology described above in a research contract we recently conducted for a large North American automobile insurer. This study was the most exhaustive ever undertaken by this particular insurer and lasted over an entire year. We analyzed the discriminative power of each variable used for ratemaking. We analyzed the performance of several statistical learning algorithms within five broad categories: Linear Regressions, GLMs, Decision Trees, Neural Networks and Support Vector Machines. We present the main results of this study. We qualitatively compare models and show how Neural Networks can represent high order nonlinear dependencies with a small number of parameters, each of which is estimated on a large proportion of the data thus, has low variance. We thoroughly explain the purpose of the nonlinear sigmoidal transforms which are at the very heart of Neural Networks' performances. The main numerical result is a statistically significant reduction in the out-of-sample mean-squared error using the Neural Network model. In some provinces and states, better risk discrimination, if not directly implemented because of market share concerns or legislative constraints, can also be used for the purpose of choosing the risks to be sent to "risk-sharing pools." According to these plans, insur-

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ers choose a portion of their book of business which they ceed to the pool. Losses (seldom gains) are shared by participants and/or insurers doing business in the province or state of the plan. Since the selection of risks to be sent to the pool bears no effect on market share (the insured is unaware of the process) and legislation is generally looser than that of ratemaking, highly discriminative statistical learning algorithms such as Neural Networks can be very profitably used to identify those most underpriced risks that should be ceded to the pool. We compare Generalized Linear Models to our Neural Network based model with respect to their risk-sharing pool performance.

1

Introduction

Ratemaking is one of the main mathematical problems faced by actuaries. They must first estimate how much each insurance contract is expected to cost. This conditional expected claim amount is called the pure premium and it is the basis of the gross premium charged to the insured. This expected value is conditioned on information available about the insured and about the contract, which we call the input profile. Automobile insurance ratemaking is a complex task for many reasons. First of all, many factors are relevant. Taking account of each of them individually, i.e., making independence assumptions, can be hurtful (Bailey and Simon 1960). Taking account of all interactions is intractable and is sometimes referred to as the curse of dimensionality (Bellman 1957). In practice, actuarial judgment is used to discover the most relevant of these interactions and feed them explicitly to the model. Neural networks, on the other hand, are well-known for their ability to represent high order nonlinear interactions with a small number of parameters, i.e., they can automatically detect those most relevant interactions between variables (Rumelhart, Hinton and Williams 1986). We explain how and why in section 5. A second difficulty comes from the distribution of claims: asym-

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metric with fat tails with a large majority of zeros and a few unreliable and very large values, i.e., an asymmetric heavy tail extending out toward high positive values. Modeling data with such a distribution is essentially difficult because outliers, which are sampled from the tail of the distribution, have a strong influence on parameter estimation. When the distribution is symmetric around the mean, the problems caused by outliers can be reduced using robust estimation techniques (Huber 1982, Hampel, Ronchetti, Rousseeuw and Stahel 1986, Rousseeuw and Leroy 1987) which basically intend to ignore or down-weight outliers. Note that these techniques do not work for an asymmetric distribution: most outliers are on the same side of the mean, so down-weighting them introduces a strong bias on its estimation: the conditional expectation would be systematically underestimated. Recent developments for dealing with asymmetric heavytail distributions have been made (Takeuchi, Bengio and Kanamori 2002). The third difficulty is due to the non-stationary nature of the relationship between explanatory variables and the expected claim amount. This has an important effect on the methodology to use, in particular with respect to the task of model selection. We describe our methodology in section 4. Fourth, from year to year, the general level of claims may fluctuate heavily, in particular in states and provinces where winter plays an important role in the frequency and severity of accidents. The growth of the economy and the price of gas can also affect these figures. Fifth, one needs sufficient computational power to develop models: we had access to a large database of « 8 x 106 records, and the training effort and numerical stability of some algorithms can be burdensome for such a large number of training examples. Sixth, the data may be of poor quality. In particular, there may be missing fields for many records. An actuary could systematically discard incomplete records but this leads to loss of information. Also, this strategy could induce a bias if the absence of a data is not random but rather correlated to some particular feature which affects


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the level of risk. Alternatively one could choose among known techniques for dealing with missing values (Dempster, Laird and Rubin 1977, Ghahramani and Jordan 1994, Bengio and Gingras 1996). Seventh, once the pure premiums have been established the actuary must properly allocate expenses and a reserve for profit among the different contracts in order to obtain the gross premium level that will be charged to the insureds. Finally, an actuary must account for competitive concerns: his company's strategic goals, other insurers' rate changes, projected renewal rates and market elasticity. In this chapter, we address the task of setting an appropriate pure premium level for each contract, i.e., difficulties one through four as described above. Our goal is to compare different models with respect to their performance in that regard, i.e., how well they are able to forecast the claim level associated to each contract. We chose several models within five broad categories: Linear Regressions, Generalized Linear Models (McCullagh and Nelder 1989), Decision Trees (Kass 1980), Neural Networks and Support Vector Machines (Vapnik 1998). The rest of the chapter is organized as follows: in section 2 we introduce the reader to some of the fundamental principles underlying statistical machine learning, compare them to those that govern more traditional statistical approaches and give some examples. Then, we describe usual candidate mathematical criteria that lead to insurance premium estimation in section 3. Statistical learning methodologies are described in section 4, with an emphasis on the one that was used within the course of the study. This is followed in section 5 by a review of the statistical learning algorithms that we considered, including our best-performing mixture of positive-output Neural Networks. We describe experimental results with respect to ratemaking in section 6. In section 7, we compare two models on the task of identifying the risks to be sent to a risk sharing pool facility. In view of these results we conclude with an examination of the prospects for applying statistical learning algorithms to insurance modeling in section 8.

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Concepts of Statistical Learning Theory

Statistical inference is concerned with the following: Given a collection of empirical data originating from some functional dependency, provide answers to questions that could be answered if that dependency were known. Although elements of statistical inference have existed for more that 200 years (Gauss, Laplace), it is within the last century that the development of methods and their formal analysis began. Fisher (Fisher 1912, Fisher 1915, Fisher 1922, Fisher 1925, Aldrich 1995) developed the framework of parametric statistics and suggested one method of approximating the unknown parameter values of a particular model: maximum likelihood. Other researchers (Glivenko 1933, Cantelli 1933, Kolmogorov 1933) used a more general approach as they proved that the empirical distribution function converges exponentially to the actual distribution function. Most importantly, this result is independent of the unknown actual distribution function. These two fundamental results can be seen as the seeds of two philosophically diverging frameworks of statistical inference. The goal of the first approach is to identify the data generating process. For the purpose of this modeling goal, one must have sufficient knowledge of the physical laws that govern the process in order to build a corresponding model. The essence of that branch of statistical inference is therefore to estimate the unknown parameter values of a (presumably) known model, using the available collection of empirical data, then to devise statistical tests that lead to rejection or not of the model (or of some of its parameters). For the purpose of parameter estimation, one often adopts the maximum likelihood method, which enjoys attractive asymptotic properties. On the other hand, according to the second approach, one merely attempts to predict properties of future data, based on the already given observed data. The belief is that the reality of the process is too complex to be identified and captured in a single unifying model.


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In particular, multivariate processes are faced with the problem of the curse of dimensionality (Bellman 1957), i.e., the number of combinations of variable values increases exponentially with the dimensionality (the number of explanatory variables) of the problem. In real-life problems, such as automobile insurance where one considers dozens of variables, the belief that one can truly identify the generating process looks naive. The goal of the second approach is therefore less ambitious: given a collection of data and a set of candidate functions, find an approximation to the observed unknown process (or a function that can answer the desired questions about the data, such as future conditional expectation) in order to obtain the best performance on predictive tasks, on new data. In the face of this operational goal, statistical inference had to evolve. The robust approach to parametric statistics appeared in the 1960's (Huber 1964, Huber 1965, Huber 1966, Huber 1968, Huber 1981, Huber and Rieder 1996). In the 1970's, Generalized Linear Models were developed in order to widen the sets of functions to choose from (Nelder and Wedderburn 1972). These models have lately become increasingly popular in the actuarial community. The availability of such wider sets of functions lead to the problem of model selection. In the 1980's, many researchers (Breiman, Friedman, Olshen and Stone 1984, Huber 1985, Rumelhart et al. 1986) started to consider special types of function, nonlinear in their parameters and with fewer distributional assumptions (in particular Decision Trees and Neural Networks) and also developed the regularization method (Tikhonov and Arsenin 1977) as an alternative to the maximum likelihood method, one better suited to the operational goal of achieving the best out-of-sample predictive performance. A branch of statistical learning (or machine learning) is mainly concerned with the development of proper refinements of the regularization and model selection methods in order to improve the predictive ability of algorithms. This ability is often referred to as generalization, since the algorithms are allowed to generalize from the observed training data to new data. One crucial element of the

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evaluation of the generalization ability of a particular model is the measurement of the predictive performance results on out-of-sample data, i.e., using a collection of data, disjoint from the in-sample data that has already been used for model parameter estimation. In the case of automobile insurance, where data is not iid. but rather bears a sequential structure with potential non-stationarities (changes in the underlying generating process with respect to its explanatory variables), this requirement leads to the particular methodology of sequential validation which we shall explain in detail in section 4.

2.1

Hypothesis Testing: an Example

Let us illustrate these concepts with a simple example where the two approaches yield different statistical tests, thus possibly different conclusions. Consider the classical statistical linear regression tests for deciding to keep a coefficient. Let the relation i£|Y|:r] = a + j3x hold, with Var"[Y|:E] = a2 the output noise. The classical statistical test for rejecting the input X (i.e., setting the (3 coefficient to 0 in the model) is based on the null hypothesis (3 = 0. In this context, however, one should distinguish two questions: (1) is (3 really equal to zero? (this is what the above classical test tries to determine), or (2) would choosing (3 — 0 give better or worse out-of-sample expected generalization than choosing (3 that minimizes the in-sample error? Let us define the generalization error as the expected out-of-sample error: ESE

= E[(Y - (a + J3X))2].

(1)

If one is more interested in generalization error, then one should not use the classical test, but rather choose an unbiased out-of-sample test (Gingras, Bengio and Nadeau 2000, Nadeau and Bengio 2000). In particular one can show that if the true (3 is such that the signal to noise ratio ^ is less than (greater than) some positive threshold value 8, then setting (3 to zero (the in-sample estimator) will generalize better. When the data set has input values (Xx,X2,... ,Xn), and


writing the input average X = -^ YA=I p [

XU

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the threshold value is

(x-xy

£[(X-X)2] ' where X is an out-of-sample example (not one of the training set Xi). Thus the out-of-sample tests tell us to reject a parameter when the signal-to-noise for that parameter is too small, even if the "true" value of that parameter is non-zero. This is because by trying to estimate that parameter from too little data, one is bound to worsen expected generalization error. If one is really interested in knowing whether (3 = 0 in the above example, then one should really use the classical statistical test rather than a generalization error statistic, because for that null hypothesis the former has greater power than the latter (Gingras et al. 2000). On the other hand, if one is really interested in out-of-sample predictive performance, one should use a generalization error test, because the classical test is liberal (i.e., it keeps a non-zero (3 too often), which can be very dangerous in applications. Finally, although the difference between the two types of statistics becomes small as n goes to infinity (6 above goes to zero), it should be noted that in insurance applications with many input variables, the "small-sample" effect is not negligible, for two reasons: 1. when the number of discrete variables is large, and we want to take into account their joint effects, the number of possible combinations of their values is very large; thus there is really very little data to decide whether a parameter associated to a particular combination is useful or not, 2. when the claims distribution is highly asymmetric (i.e., the mean is far from the median), the rare large claims can have a very strong effect on the parameter estimation (i.e., the noise is strong), which increases the discrepancy between the conclusions reached with the in-sample and out-of-sample statistics.

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In the above analysis, there is another reason for preferring the operational approach in practical applications: the out-of-sample statistical tests do not require any assumption on the form of the underlying distribution.1 In other words, when performing a classical parametric test, the conclusion of the test could generally be invalidated if strictly speaking the data was not generated from the presumed class of parametric distributions. When the livelihood of a corporation is at stake in these choices, it might be wiser to avoid relying on such assumptions.

2.2

Parameter Optimization: an Example

Consider the same problem of linear regression as described above but, let us now turn the task of parameter estimation. Our objective is to minimize the expected out-of-sample squared error which does not mean that we should minimize the in-sample mean squared error: MSE

1 N = -^(y.-ia

+ Px,))2

(2)

i=i

Minimizing the MSE is what the maximum likelihood calls for in the classical framework. The reason for that apparent discrepancy has to do with the statistical learning principles defined above. Instead, in order to obtain better generalization, we turn to the regularization framework and accordingly choose to minimize a penalized criterion leading to what is often referred to as ridge regression: 1

N

MSEX = ^ 5 > i - ( a + /^)) 2 + A/32

(3)

2=1

'the only assumption, in ordinary tests, is that the data points are generated i.i.d., independently from the same distribution. Even this assumption can be relaxed in order to deal with sequentially dependent data (Newey and West 1987, Diebold and Mariano 1995, Campbell, Lo and MacKinlay 1997, Chapados and Bengio 2003).


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with A > 0 and a minimum achieved at (3X. Thus /30 is the Ordinary Least Squares estimator. This minimum is always achieved with shrinked solutions, i.e., \\(3\\\ < \\Po\\ for A > 0. Note that this solution is generally biased, unlike /30, in the sense that if the data is generated from a multivariate normal distribution, the expected value of ||P\11 is smaller than the true value \\(3\\ from the underlying distribution. In the case where linear regression is the proper model it is easy to show that the optimal fixed value of A is

x

*" NW-

(4)

Note therefore that the optimal model is biased (optimal in the sense of minimizing out-of-sample error). Considering the case of automobile insurance, with noisy observations (large a2) and the small sample effect (small N, as argued above), we obtain large optimal values for A. In other words, for this particular case, regularization significantly differs from maximum likelihood. This example illustrates the more general principle of biasvariance trade-off (Geman, Bienenstock and Doursat 1992) in generalization error. Increasing A corresponds to "smoothing more" in non-parametric statistics (choosing a simpler function) or to the choice of a smaller capacity ("smaller" class of functions) in Vapnik's VC-theory (Vapnik 1998). A too large value of A corresponds to underfitting (too simple model, too much bias), whereas a too small value corresponds to overfitting (too complex model, too much variance). Which value of A should be chosen? (The above formula is not practical because it requires the true j3, and is only applicable if the data is really Gaussian). It should be the one that strikes the optimal balance between bias and variance. This is the question that model selection algorithms address. Fortunately, the expected outof-sample error has a unique minimum as a function of A (or more generally of the capacity, or complexity of the class of functions). Concerning the above formula, note that unfortunately the data is

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generally not normal, and a2 and (3 are both unknown, so the above formula can't be used directly to choose A. However, using a separate held-out data set (also called a validation set, here), and taking advantage of that unique minimum property (which is true for any data distribution), we can quickly select a good value of A (essentially by searching), which approximately minimizes the estimated out-of-sample error on that validation set. Note that we arrive at the conclusion that a biased model is preferable because we set as our goal to minimize out-of-sample error. If our goal was to discover the underlying "truth," and if we could make very strong assumptions about the true nature of the data distribution, then the more classical statistical approach based on minimum variance unbiased estimators would be more appropriate. However, in the context of practical insurance premia estimation, we don't really know the form of the true data distribution, and we really care about how the model is going to perform in the future (at least for ratemaking).

3

Mathematical Objectives

The first goal of insurance premium modeling is to estimate the expected claim amount for a given insurance contract for a future period (usually one-year). Here we consider that the amount is 0 when no claim is filed. Let X € R" denote the customer and contract input profile, a vector representing all the information known about the customer and the proposed insurance policy before the beginning of the contract. Let A e R + denote the amount that the customer claims during the contract period; we shall assume that A is non-negative. Our objective is to estimate this claim amount, which is the pure


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premium ppure of a given contract x:2 Ppure(x) = EA[A\X

= X\.

(5)

where EA[] denotes expectation, i.e., the average over an infinite population, and £^[A|X = x] is a conditional expectation, i.e., the average over a subset of an infinite population, comprising only the cases satisfying the condition X = x.

3.1

The Precision Criterion

In practice, of course, we have no direct access to the quantity (5), which we must estimate. One possible criterion is to seek the most precise predictor, which minimizes the expected squared error (ESE) over the unknown distribution: EAJCMX)

- A)2},

(6)

where p(X) is a pure premium predictor and the expectation is taken over the random variables X (input profile) and A (total claim amount). Since P(A, X), the true joint distribution of A and X, is unknown, we can unbiasedly estimate the ESE performance of an estimator p on a data set Dtest = {(XJ, a j ) } ^ , as long as this data set is not used to choose p. We do so by using the mean squared error on that data set:

1

£

(p(x^)-a,)2,

(7)

(Xi,ai)eDtest

where 9 is the vector of parameters of the model used to compute the premiums. The vector Xi represents the ith input profile of dataset Dtest and a* is the claim amount associated to that input profile. 2

The pure premium is distinguished from the premium actually charged to the customer, which must account for the underwriting costs (marketing, commissions, premium tax), administrative overhead, risk and profit loadings and other costs.

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Thus, Dtest is a set of N insurance policies. For each policy, Dtest holds the input profile and associated incurred amount. We will call the data set Dtest a test set. It is used only to independently assess the performance of a predictor p. To choose p from a (usually infinite) set of possible predictors, one uses an estimator L, which obtains a predictor p from a given training set D. Such an estimator is really a statistical learning algorithm (Hastie, Tibshirani and Friedman 2001), generating a predictor p = LD for a given data set D. What we call the squared bias of such an estimator is EX[{EA[A\X] - ED[LD(X)})2}, where ED[LD{X)\ is the average predictor obtained by considering all possible training sets D (sampled from P(A, X)). It represents how far the average estimated predictor deviates from the ideal pure premium. What we call the variance of such an estimator is EXtD[(LD(X) — E[LD(X)])2]. It represents how the particular predictor obtained with some data set D deviates from the average of predictors over all data sets, i.e., it represents the sensitivity of the estimator to the variations in the training data and is related to the classical measure of credibility. Is the mean squared error (MSE) on a test set an appropriate criterion to evaluate the predictive power of a predictor pi First one should note that if pi andp 2 are two predictors of £^[/4|^], then the MSE criterion is a good indication of how close they are to £U|yl|X], since by the law of iterated expectations, EAX[(Pl(X) - A)2} - EAX[(P2(X) - A)2} = Ex[(Pl(X) - EA[A\X])2}Ex[(p2(X) - EA[A\X])% and of course the expected MSE is minimized when p(X) = EA[A\X\. For the more mathematically-minded readers, we show that minimizing the expected squared error optimizes simultaneously both the precision (low bias) and the variance of the estimator. The expected

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squared error of an estimator LD decomposes as follows: EAXiD[(A-LD(X))2} = EA<XtD[((A - EA[A\X}) + [EA[A\X] = EAX[(A - EA[A\X]f] + EX,D[(EA[A\X] s

v noise

LD(X)))2} LD{X)f)

'

+2EA<X,D[(A - EA[A\X])(EA[A\X} >

-

LD(X))} '

v

zero

= =

EX,D[({EA[A\X] - ED[LD(X)}) + (ED[LD(X)] - LD(X)))2} +noise +EX[(EA[A\X]-ED[LD(X)])2} +EX n ^ + E &•>•' i=l

^

j=jn+l

where (3itj is the relativity for the jth category of the ith variable and Po is the standard premium. We consider the case where the first m factors are multiplicative and the last n — m factors are additive. The formula above assumes that all variables have been analyzed individually and independently. A great deal of effort is often put in trying to capture dependencies (or interactions) between some variables and to encode them into the premium model. An extension of the above is to multiplicatively combine multiple tables associated to different subsets of variables. This is in effect a particular form of Generalized Linear Model (see below), where each table represents the interdependence effects between some variables.

5.4

Greedy Multiplicative Model

Greedy learning algorithms "grow" a model by gradually adding one "piece" at a time to the model, but keeping the already chosen pieces

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fixed. At each step, the "piece" that is most helpful to minimize the training criterion is "added" to the model. This is how decision trees are typically built. Using the validation set performance we can decide when to stop adding pieces (when the estimated out-of-sample performance starts degrading). The GLM described in the next section is a multiplicative model because the final premium function can be seen as a product of coefficients associated with each input variable. The basic idea of the Greedy Multiplicative Model is to add one of these multiplicative coefficients at a time. At each step, we have to choose one among the input variables. We choose the variable which would reduce most the training MSE. The coefficient for that component is easily obtained analytically by minimizing the MSE when all the previously obtained coefficients are kept fixed. In the tables we use the short-hand name "CondMean" for this model because it estimates and combines many conditional means. Note that like the GLM, this model provides positive premiums.

5.5

Generalized Linear Model

The Generalized Linear Models (GLM) were introduced to the actuarial community a while ago (Bailey and Simon 1960). More recently, (Brown 1988, Holler, Sommer and Trahair 1999, Murphy, Brockman and Lee 2000) some experiments have been conducted using such models. GLMs, at their roots, are simple linear models that are composed with a fixed nonlinearity (the so-called link function); a commonly-used link function is simply the exponential function ex. GLMs (with the exponential link) are sometimes used in actuarial modeling since they naturally represent multiplicative effects, for example, risk factors whose effects should combine multiplicatively rather than additively. They are attractive since they incorporate problem-specific knowledge directly into the model. These


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models can be used to obtain a pure premium: p(x)

= explpo

+ ^PiXij,

(13)

where, the exponentiation ensures that the resulting premiums are all positive. In figure 5, we can see that the model generates an exponential function in terms of the input variable.

Figure 5. The Generalized Linear Model fits an exponential of a linear transformation of the variables.

In their favor, GLMs are quite easy to estimate3, have interpretable parameters, can be associated to parametric noise models, and are not so affected when the number of explanatory variables increases, as long as the number of observations used in the estimation remains sufficient. Unfortunately, they are fairly restricted in the shape of the functions they can estimate. The capacity of a GLM model can be controlled using the same techniques as those mentioned above (5.2) in the context of linear 3

We have estimated the parameters to minimize the mean-squared error, but other training criteria have also been proposed in the GLM literature and this could be the subject of further studies.

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models. Again, note that the GLM always provides a positive premium.

5.6

CHAID Decision Trees

Decision trees split the variable space in smaller subspaces. Any input profile x fits into one and only one of those subspaces called leaves. To each leaf is associated a different premium level,

1=1

where Iêz*} is an indicator function equal to 1 if and only if x belongs to the zth leaf. In that case, I{xeh} = 1 a n d p(x) = Pi. Otherwise, I{xek}1S ec l ua l to zero, meaning x belongs to another leaf. The number of leaves is rt[. The premium level Pi is set equal to the average incurred amount of the policies for which the profile x belongs to the zth leaf. In figure 6, the decision tree is viewed as generating a piecewise constant function. The task of the decision tree is to choose the "best" possible partition of the input variable space. The basic way in which capacity is controlled is through several hyper-parameters: minimum population in each leaf, minimum population to consider splitting a node, maximum height of the decision tree and Chi-square statistic threshold value.

5.7

Combination of CHAID and Linear Model

This model is similar to the previous one except that, in each leaf, we have replaced the associated constant premium value with a linear regression. Each leaf has its own set of regression coefficients. There are thus ni different linear regressions of n + 1 coefficients each. nt X

P( ) =

/

n

^hx&i}

[Pifl + ^PijXj].

i=l

\

j=l

\

(15) /


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P(x)

Figure 6. The CHAID model fits constants to partitions of the variables. The dashed lines in the figure delimit the partitions, and are found automatically by the CHAID algorithm.

Each linear regression was fit to minimize the mean-squared error on the training cases that belong to its leaf. For reasons that are clear in the light of learning theory, a tree used in such a combination should have less leaves than an ordinary CHAID tree. In our experiments we have chosen the size of the tree based on the validation set MSE. In these models, capacity is controlled with the same hyperparameters as CHAID, and there is also the question of finding the right weight decay for the linear regression. Again, the validation set is used for this purpose.

5.8

Ordinary Neural Network

Ordinary Neural Networks consist of the clever combination and simultaneous training of a group of units or neurons that are individually quite simple. Figure 8 illustrates a typical multi-layer feedforward architecture such as the ones that were used for the current project.

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p(x)

Figure 7. The CHAID+Linear model fits a straight line within each of the CHAID partitions of the variable space. "1.0

Input .... Variable 1 -"'

Input Variable 2

/

^ Final fTlOutput

Input Variable n

Input Layer

Hidden Layer (and hidden weights)

jîSoftplus Output Layer

P(X)

where p(x) is the premium computed by the Neural Network, nh is the number of hidden units and @0 and Pi are the slope intercept and the weights of the final linear combination. Put all together in a single equation, we obtain: nh

/

1

P(x) = Po + Y^P 2=1

tanh

n

ai

( >° \

+

\

ai x

5Z J i j=l

)• J

(19)

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Figure 9 depicts a smooth non-linear function which could be generated by a Neural Network.

Figure 9. The Neural Network model learns a smooth non-linear function of the variables.

The number of hidden units (n^ above) plays a crucial role in our desire to control the capacity of the Neural Network. If we choose a value for nh that is too large, we obtain overfitting: the number of parameters of the model increases and it becomes possible, during the parameter optimization phase, for the Neural Network to model noise or spurious dependencies. These dependencies, present in the training dataset used for optimization, might not apply to other datasets. Conversely, setting nh to a value that is too low corresponds to underfitting: the number of parameters becomes too small and the Neural Network can not capture all of the relevant interactions in order to properly compute the premiums. Thus, choosing the optimal number of hidden units is an important part of modeling with Neural Networks. Another technique for controlling the capacity of a Neural Network is to use weight decay, i.e., a penalized training criterion as described in subsection 2.2 that limits the size of the parameters of the Neural Network.


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Choosing the optimal values for the parameters is a complex task and out of the scope of this chapter. Many different optimization algorithms and refinements have been suggested (Bishop 1995, Orr and Miiller 1998) but in practice, the simple stochastic gradient descent algorithm is still very popular and usually gives good performance. Note that like the linear regression, this model can potentially yield negative premiums in some cases. We have observed much fewer such cases than with the linear regression.

5.9

How Can Neural Networks Represent Nonlinear Interactions?

For the more mathematically-minded readers, we present a simple explanation of why Neural Networks are able to represent nonlinear interactions between the input variables. To simplify, suppose that we have only two input variables, x\ and x2- In classical linear regression, a common trick is to include fixed nonlinear combinations among the regressors, such as x\, x\, x\X2, x\x2, • • • However, it is obvious that this approach adds exponentially many terms to the regression, as one seeks higher powers of the input variables. In contrast, consider a single hidden unit of a Neural Network, connected to two inputs. The adjustable network parameters are named, for simplicity, a0, a\ and a2. A typical function computed by this unit is given by t a n h ( a 0 + OL\XI + 012X2).

Here comes the central part of the argument: performing a Taylor series expansion of tanh(y + a0) in powers of y, and letting a\X\ +

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a2x2 stand for y, we obtain (where /5 = tanh a0), tanh(o;o + cx.\Xi + 0.2X2) = P + (l- P2){alXl + a2x2) + (-P + P3)(axXi + a2x2)2+ - \ + ^L-pi){aixl + a2x2f+ f - ^ f + /35) (ctixi + a2x2f

+ 0{alxl

+ a2x2f.

In fact the number of terms is infinite: the nonlinear function computed by this single hidden unit includes all powers of the input variables, but they cannot all be independently controlled. The terms that will ultimately stand out depend on the coefficients a0, a.\, and a2. Adding more hidden units increases the flexibility of the overall function computed by the network: each unit is connected to the input variables with its own set of coefficients, thereby allowing the network to capture as many (nonlinear) relationships between the variables as the number of units allows. The coefficients linking the input variables to the hidden units can also be interpreted in terms of projections of the input variables. Each set of coefficients for one unit represents a direction of interest in input space. The values of the coefficients are found during the network training phase using iterative nonlinear optimization algorithms.

5.10 Softplus Neural Network This new type of model was introduced precisely to make sure that positive premiums are obtained. The softplus function was recently introduced (Dugas, Bengio, Belisle, Nadeau and Garcia 2001) as a means to model a convex relationship between an output and one of its inputs. We modified the Neural Network architecture and included a softplus unit as a final transfer function. Figure 10 illustrates this new architecture we have introduced for the purpose of computing


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'I."

Input Variable 1

Input Variable 2

«1,1

g

"*J'•*#"

"" ^'"hT*-*" V ***• ~ _ 2* r

V

—'

/ _^Final ^"Output

Input Variable n 'liiH.t>

afcjLayer a . . Hidden Layer |^r'.f{iSi[ah4hidden weightet

Softplus Output Lavor

(.ind oulpul iM-it-lil*)

Figure 10. Topology of a one-hidden-layer softplus Neural Network. The hidden layer applies a non-linear transformation of the variables, whose results are linearly combined by the output layer. The softplus output function forces the function to be positive. To avoid cluttering, some weights linking the variables to the hidden layer are omitted on the figure.

insurance premiums. The corresponding formula is as such: p(x)

= F f A> + £

A tanh f aifi + J ] atjXj I I , (20)

where F(-) is the softplus function which is actually simply the primitive (integral) function of the "sigmoid" function. Thus F(y)

= log(l + e").

(21)

The softplus function is convex and monotone increasing with respect to its input and always strictly positive. Thus, as can be seen in Figure 11, this proposed architecture leads to strictly positive premiums. In preliminary experiments we have also tried to use the exponential function (rather than the softplus function) as the final transfer

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function. However we obtained poor results due to difficulties in the optimization (probably due to the very large gradients obtained when the argument of the exponential is large).

Figure 11. The softplus Neural Network model learns a smooth non-linear positive function of the variables. This positivity is desirable for estimating insurance premiums.

The capacity of the softplus Neural Network is tuned just like that of an ordinary Neural Network. Note that this kind of Neural Network architecture is not available in commercial Neural Network packages.

5.11 Regression Support Vector Machine Support Vector Machines (SVM) have recently been introduced as a very powerful set of non-parametric statistical learning algorithms (Vapnik 1998, Scholkopf, Smola and Muller 1998). They have been very successful in classification tasks, but the framework has also been extended to perform regression. Like other kernel methods the


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class of functions has the following form: P(x)

=

ÔLiK(x,Xi)

(22)

i

where Xj is the input profile associated with one of the training records, and cti is a scalar coefficient that is learned by the algorithm and K is a kernel function that satisfies the Mercer condition (Cristianini and Shawe-Taylor 2000): J K{x,y)g{x)g(y)dxdy

> 0

(23)

for any square integrable function g(x) and compact subset C of R n . This Mercer condition ensures that the kernel function can be represented as a simple dot product: K(x,y)

= cf>{x) • (t>{y)

(24)

where () is a function that projects the input profile vector into a (usually very) high-dimensional "feature" space, usually in a nonlinear fashion. This leads us, to a simple expression for the premium function: i

=

w{x).

(25)

Thus, in order to compute the premium, one needs to project input profile x in its feature space and compute a dot product with vector w. This vector w depends only on a certain number of input profiles from the training dataset and their associated coefficients. These input profiles are referred to as the support vectors and have been selected, along with their associated coefficients by the optimization algorithm.

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SVMs have several very attractive theoretical properties, including the fact that an exact solution to the optimization problem of minimizing the training criterion can be found, and the capacity of the model is automatically determined from the training data. In many applications, we also find that most of the c^ coefficients are zero. However, in the case of insurance data, an important characteristic of regression SVMs is that they are NOT trained to minimize the training MSE. Instead they minimize the following criterion: J = \\\w\\2 + \J2\at

~P(xi)\e

(26)

i

where |e| e = max(0, |e| — e), A and e trade-off accuracy with complexity, di is the observed incurred claim amount for record i, Xj is the input profile for record i, and the vector w is defined in terms of the a, coefficients above. It can therefore be seen that this algorithm minimizes something close to the absolute value of the error rather than the squared error. As a consequence, the SVM tends to find a solution that is close to the conditional median rather than the conditional expectation, the latter being what we want to evaluate in order to set the proper value for a premium. Furthermore, note that the insurance data display a highly asymmetric distribution, so the median and the mean are very different. In fact, the conditional median is often exactly zero. Capacity is controlled through the e and A coefficients.

5.12 Mixture Models The mixture of experts has been proposed (Jacobs, Jordan, Nowlan and Hinton 1991) in the statistical learning literature in order to decompose the learning problem, and it can be applied to regression as well as classification. The conditional expectation is expressed as a linear combination of the predictions of expert models, with weights determined by a gater model. The experts are specialized predictors


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C iislmiii'i Input I'mlile «-W-BSB9W«g|

Ixpert 1 Output 1

Expert 2

• • •

Output 2

Expert n

Galer Model

Output rt

Probability (1)

Prediction Combination

Probability (.2)

*--»—*—.

Probability (n)

Final Mixture Output

Figure 12. Schematic representation of the mixture model. The first-stage models each make an independent decision, which are linearly combined by a second-stage gater.

that each estimate the pure premium for insureds that belong to a certain class. The gater attempts to predict to which class each insured belongs, with an estimator of the conditional probability of the class given the insured's input profile. For a mixture model, the premium can be expressed as P(x)

=

^2P(C\X)PC{X)

(27)

where p{c\x) is the probability that an insured with input profile x belongs to class c. This value is determined by the gater model. Also, pc{x) is the premium, as computed by the expert model of class c, associated to input profile x. A trivial case occurs when the class c is deterministically found for any particular input profile x. In that case, we simply split the training database and train each expert model on a subset of the data.

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Table 1. Comparison between the main models, with MSE on the training set, validation set, and test sets. The MSE is with respect to claim amounts and premiums expressed in thousands of dollars.

Model Constant Linear GLM NN Softplus CHAID CondMean Mixture

Train MSE 56.1108 56.0780 56.0762 56.0706 56.0704 56.0917 56.0827 56.0743

Valid MSE Test MSE 67.1192 56.5744 67.0909 56.5463 67.0926 56.5498 67.0903 56.5468 67.0918 56.5480 67.1078 56.5657 67.0964 56.5508 67.0851 56.5416

The gater then simply assigns a value of pc{x) = 1 if c is the appropriate class for input profile x and zero otherwise. This is in fact fundamentally equivalent to other techniques such as decision trees or table-based methods. A more general and powerful approach is to have the learning algorithm discover a relevant decomposition of the data into different regions of the input space which then become the classes and are encoded in the gater model. In that case, both the gater and the experts are trained together. In this study both the experts and the gater are softplus Neural Networks, but any other model can be used. In Figure 12, we schematically illustrate a mixture model as the one that was used in the framework of this project.

6

Experimental Results

6.1

Mean-Squared Error Comparisons

Table 1 summarizes the main results concerning the comparison between different types of statistical machine learning algorithms. All the models have been trained using the same input profile variables.

Statistical Learning Algorithms

in Automobile

Insurance

175

Ratemaking

Mean-Squared Error

67.1192 Test 67.0851

56.5744 Validation

56.5416 56.1108

Training

56.0743

mW

^

0„ea
c 0 ( 1 6^ a °

C^°

sVa< * Co«

Models

Figure 13. MSE results (from table 1) for eight models . Models have been sorted in ascending order of test results. The training, validation and test curves have been shifted closer together for visualization purposes. The out-of-sample test performance of the mixture model is significantly better than any of the other. Validation based model selection is confirmed on test results.

For each insurance policy, a total of 33 input variables were used and the total claims for an accident came from five main coverages: bodily injury, accident benefit, property damage, collision and comprehensive. Two other minor coverages were also included: death benefit and loss of use. In the table, NN stands for Neural Network, GLM for Generalized Linear Model, and CondMean for the Greedy Multiplicative Model. The MSE on the training set, validation set and test set are shown for all models. The MSE is with respect to claim amounts and premiums expressed in thousands of dollars. The model with the lowest MSE is the "Mixture model," and it is the

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model that has been selected for the comparisons with the insurer's current rules for determining insurance premiums to which we shall refer as the Rule-Based Model. One may wonder from the previous table why the MSE values are so similar across various models for each dataset and much different across the datasets. In particular, all models perform much worse on the testset (in terms of their MSE). There is a very simple explanation. The maximum incurred amount on the test set and on the validation set is around 3 million dollars. If there was one more such large claim in the test set than in the validation set, one would expect the test MSE (calculated for premiums and amounts in thousands of dollars) to be larger by about 7 (these are in units of squared thousand dollars). Thus a difference of 11 can easily be explained by a couple of large claims. This is a reflection of the very thick right-hand tail of the incurred amount distribution (whose standard deviation is only of about 8 thousand dollars). Conversely, this also explains why all MSE are very similar across models for one particular dataset. The MSE values are all mainly driven by very large claims which no model could reliably forecast (no model could lead the insurer to charge a million dollars to a particular insured!). Consequently, truly significant differences between model performances are shadowed by the effect of very large claims on the MSE values. Although the differences between model performance are relatively small, we shall see next that careful statistical analysis allows us to discover that some of them are significant. Figure 13 illustrates graphically the results of the table, with the models ordered according to the validation set MSE. One should note that within each class of models the capacity is tuned according to the performance on the validation set. On the test and validation sets, the Mixture model dominates all the others. Then come the ordinary Neural Network, linear model, and softplus Neural Network. Only slightly worse are the GLM and CondMean (the Greedy Multiplicative model). CHAID fared poorly on this dataset. Note that the CHAID + linear model described in section 5.7 performed worse


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Table 2. Statistical comparison between different learning models and the mixture model. The p-value is for the null hypothesis of no difference between Model #1 and the best mixture model. Symbols \x and a stand for sample mean and standard deviation. Note that ALL differences are statistically significant.

Model #1 Constant Linear GLM NN Softplus CHAID

Model #2 Mixture Mixture Mixture Mixture Mixture Mixture

A 3.41e-02 5.82e-03 7.54e-03 5.24e-03 6.71e-03 2.36e-02

a 3.33e-03 1.32e-03 1.15e-03 1.41e-03 1.09e-03 2.58e-03

Z 10.24 4.41 6.56 3.71 6.14 9.15

p-value 0 5.30e-06 2.77e-ll 1.03e-04 4.21e-10 0

than ordinary CHAID. Finally, the constant model is shown as a baseline (since it corresponds to assigning the same premium to every 1-year policy). It is also interesting to note from the figure that the model with the lowest training MSE is not necessarily the best out-of-sample (on the validation or test sets). The SVM performance was appalling and is not shown here; it did much worse than the constant model, because it is aiming for the conditional median rather the conditional expectation, which are very different for this kind of data. Table 2 shows a statistical analysis to determine whether the differences in MSE between the Mixture model and each of the other models are significant. The Mean column shows the difference in MSE with the Mixture model. The next column shows the Standard Error of that mean. Dividing the mean by the standard error gives Z in the next column. The last column gives the p-value of the null hypothesis according to which the true expected squared errors for both models are the same. Conventionally, a value below 5% or 1% is interpreted as indicating a significant difference between the two models. The p-values and Z corresponding to significant differences are highlighted. Therefore the differences in performance between the mixture and the other models are all statistically significant. As

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Table 3. Statistical comparison between pairs of learning models. Models are ordered from worst to best. Symbols ft and a stand for sample mean and standard deviation. The test is for comparing the sum of MSEs. The p-value is for the null hypothesis of no difference between Model #1 and Model #2.

Model #1 Model #2 Constant CHAID CHAID GLM GLM Softplus Softplus Linear Linear NN NN Mixture

p, 1.05e-02 1.60e-02 8.29e-04 8.87e-04 5.85e-04 5.23e-03

a 2.62e-03 2.15e-03 8.95e-04 1.09e-03 1.33e-03 1.41e-03

Z 3.99 7.46 0.93 0.82 0.44 3.71

p-value 3.24e-05 4.23e-14 1.77e-01 2.07e-01 3.30e-01 1.03e-04

mentioned above, the MSE values are very much affected by large claims. Does such a sensitivity to very large claims make statistical comparisons between models incorrect? No. Fortunately all the comparisons are performed on paired data (the squared error for each individual policy), which cancel out the effect of these very large claims (since, for these special cases, the squared error will be huge for all models and of very close magnitude) Table 3 has similar columns, but it provides a comparison of pairs of models, where the pairs are consecutive models in the order of validation set MSE. What can be seen is that the ordinary Neural Network (NN) is significantly better than the linear model, but the latter, the softplus Neural Network and GLM are not statistically distinguishable. Finally GLM is significantly better than CHAID, which is significantly better than the constant model. Note that although the softplus Neural Network alone is not doing very well here, it is doing very well within the Mixture model (it is the most successful one as a component of the mixture). The reason may be that within the mixture, the parameter estimation for model of the low incurred amounts is not polluted by the very large incurred amounts (which are learned in a separate model).


6.2

179

Evaluating Model Fairness

Although measuring the predictive accuracy—as done with the MSE in the previous section—is a useful first step in comparing models, it tells only part of the story. A given model could appear significantly better than its competitors when averaging over all customers, and yet perform miserably when restricting attention to a subset of customers. We consider a model to be fair if different cross-sections of the population are not significantly biased against, compared with the overall population. Model fairness implies that the average premiums within each sub-group should be statistically close to the average incurred amount within that sub-group. Obviously, it is nearly impossible to correct for any imaginable bias since there are many different criteria to choose from in order to divide the population into subgroups; for instance, we could split according to any single variable (e.g., premium charged, gender, rate group, territory) but also combinations of variables (e.g., all combinations of gender and territory, etc.). Ultimately, by combining enough variables, we end up identifying individual customers, and give up any hope of statistical reliability. As a first step towards validating models and ensuring fairness, we choose the subgroups corresponding to the location of the deciles of the premium distribution. The i-th decile of a distribution is the point immediately above 10i% of the individuals of the population. For example, the 9-th decile is the point such that 90% of the population come below it. In other words, the first subgroup contains the 10% of the customers who are given the lowest premiums by the model, the second subgroup contains the range 10%-20%, and so on. The subgroups corresponding to the Mixture Model (the proposed model) differ slightly from those in the Rule-Based Model (the insurer's current rules for determining insurance premiums). Since the premium distribution for both models is not the same. The subgroups used for evaluating each model are given in Table 4. Since they cor-

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Table 4. Subgroups used for evaluating model fairness, for the Mixture and Rule-Based Models. The lowest and highest premiums in the subgroups are given. Each subgroup contains the same number of observations, w 28,000.

Rule-Based Model Mixture Model High Low High Low 245.0145 166.24 139.27 50.81 Subgroup 1 297.0435 166.24 214.10 245.01 Subgroup 2 336.7524 259.74 297.04 Subgroup 3 214.10 378.4123 336.75 259.74 306.26 Subgroup 4 417.5794 357.18 378.41 306.27 Subgroup 5 460.2658 415.93 417.58 357.18 Subgroup 6 507.0753 490.34 460.26 Subgroup 7 415.93 554.2909 597.14 507.07 Subgroup 8 490.35 617.1175 783.90 554.29 597.14 Subgroup 9 Subgroup 10 783.90 4296.78 617.14 3095.7861 respond to the deciles of a distribution, all the subgroups contain approximately the same number of observations (£s 28,000 on the 1998 test set). The bias within each subgroup appears in Figure 14. It shows the average difference between the premiums and the incurred amounts, within each subgroup (recall that the subgroups are divided according to the premiums charged by each model, as per Table 4). A positive difference implies that the average premium within a subgroup is higher than the average incurred amount within the same subgroup. 95% confidence intervals on the mean difference are also given, to assess the statistical significance of the results. Since subgroups for the two models do not exactly represent the same customers, we shall refrain from directly comparing the two models on a given subgroup. We note the following points: • For most subgroups, the two models are being fair: the bias is usually not statistically significantly different from zero.


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• More rarely, the bias is significantly positive (the models overcharge), but never significantly negative (models undercharge). • The only subgroup for which both models undercharge is that of the highest-paying customers, the 10-th subgroup. This can be understood, as these customers represent the highest risk; a high degree of uncertainty is associated with them. This uncertainty is reflected in the huge confidence intervals on the mean difference, wide enough not to make the bias significantly different from zero in both cases. (The bias for the Rule-Based Model is nearly significant.) From these results, we conclude that both models are usually fair to customers in all premium subgroups. A different type of analysis could also be pursued, asking a different question: "In which cases do the Mixture and the Rule-Based Models differ the most?" We address this issue in next section.

6.3

Comparison with Current Premiums

For this comparison, we used the best (on the validation set) Mixture model and compare it on the test data of 1998 against the insurer's Rule-Based Model. Note that for legislative reasons, the Rule-Based Model did not use the same variables as the proposed Mixture Model. Histograms comparing the distribution of the premiums between the Rule-Based and the Mixture models appear in Figure 15. We observe that the premiums from the Mixture model is smoother and exhibits fatter tails (more probability mass in the right-hand side of the distribution, far from the mean). The Mixture model is better able to recognize risky customers and impose an appropriately-priced premium. This observation is confirmed by looking at the distribution of the premium difference between the Rule-Based and Mixture models, as shown in Figure 16. We note that this distribution is extremely skewed to the left. This

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C. Dugas et al. Difference with incurred claims (sum of all KOL-groups) 200

E a o n £ 3 O

c -200

-400 -600

- © - Mixture Model (normalized premia) -fc - Rule-Based Model (normalized premia)

10 Decile

Figure 14. Average difference between premiums and incurred amounts (on the sum over all coverage groups), for the Mixture and Rule-Based models, for each decile of the models' respective premium distribution. We observe that both models are being fair to most customers, except those in the last decile, the highest-risk customers, where they appear to undercharge. The error bars represent 95% confidence intervals. (Each decile contains « 28,000 observations.)

means that for some customers, the Rule-Based model considerably under-charges with respect to the Mixture model. Yet, the median of the distribution is above zero, meaning that the typical customer pays more under the Rule-Based model than under the Mixture model. At the same time, the Mixture model achieves better prediction accuracy, as measured by the Mean-Squared Error (MSE) of the respective models, all the while remaining fair to customers in all categories. Our overriding conclusion can be stated plainly: the Mixture model correctly charges less for typical customers, and correctly charges more for the "risky" ones. This may be due in part to the

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5

!

4

i

i

i

i

i

Rule-Based Model

o

i

Mean =298.5184 • — Median = 286.499 • • • Stddev = 78.7417 -

a) 2

:

i

1

Mixture Model

j

Mean =298.5184 • - Median = 246.5857 • • • Stddev = 189.5272

i

I

I

I

I

1500 2000 Premium amount {$)

Figure 15. Comparison of the premium distribution for the current RuleBased model and the Mixture model. The distributions are normalized to the same mean. The Mixture model distribution has fatter tails and is much smoother.

use of more variables, and in part to the use of a statistical learning algorithm which is better suited to capturing the dependencies between many variables.

7

Application to Risk Sharing Pool Facilities

In some provinces and states, improved discrimination between good and bad risks can be used for the purpose of choosing the insureds to be ceded to risk-sharing pool facilities. In this section, we illustrate the performance of some algorithms when applied to this feat according to the rules that apply in Quebec Plan de Repartition des Risques (PRR). According to this facility, an insurer can choose to cede up to 10% of its book of business to the pool by paying 75% of the gross premium that was charged to the insured. Then, in case an accident occurs, the PRR assumes all claim payments. The losses

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Rule-Based minus Mixture 1

-

1

i

i

i

-

. "-

Mean =-1.5993e-10 • — Median = 37.5455 • • • Stddev= 154.65

j>.

-

"i

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P I m

1

-3000

-2500

1

-2000

i i .-.-w»s^l^fllPiliillllll& -1500 -1000 -500 0 Difference between premia ($)

i 500

1000

Figure 16. Distribution of the premium difference between the Rule-Based and Mixture models, for the sum of the first three coverage groups. The distribution is negatively skewed: the Rule-Based model severely undercharges for some customers.

(gains) in the pool are then shared among the insurers in proportion of their market share. Since automobile insurance is mandatory in Quebec, the PRR was initially created in order to compensate insurers that were forced by the legislator to insure some risks that had been previously turned down by multiple insurers. The idea was that the insurer could then send these risks to the pool and the losses would be spread among all insurers operating in the province. Of course, such extreme risks represent far less than the allowed 10% of an insurer's volume. The difference can then be used for other profitable purposes. One possibility is for an insurer to cede the risks that bring most volatility in the book of business and the pool therefore becomes a means of obtaining reinsurance. In this section, we take a different view: our interest is to use highly discriminative models to identify "wholes" in the ratemaking model, i.e., to identify the risks that have been underpriced the most. Mathematically, this correspond to identifying risks for which the expected value of the claims is higher than 75% of the gross premium, i.e., those risks with an expected loss ratio of at least


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75%, a figure above the industry's average performance. For a particular insurer, the lower the loss ratio, the more difficult it becomes to identify risks that can be (statistically) profitably ceded. Still, there are a few reasons why important underpricings can be identified: 1. legislation related to ratemaking is more restrictive than the one that pertains to the risk-sharing pool, 2. strategic marketing concerns may have forced the insurer to underprice a certain part of its book of business and, 3. other concerns may not allow the insurer to use highly discriminative models for the purpose of ratemaking. The last two items can possibly be handled by rule-based systems if the insurer clearly knows which segments of its book of business are underpriced. The legislative context is of more interest to us: stringent legislators refrain insurers from using highly explanatory variables such as sex or age for the purpose of ratemaking. If the pool facility ruling is silent in that regard, then underpricings can easily be identified. But this can be done with traditional models. The interest in highly discriminative models such as Neural Networks comes from the necessity of filing ratemaking plans in a clear fashion. Often, this filing operation limits an actuary in his desire to exploit relevant dependencies between explanatory variables. A lot of insurers still analyze variables independently, in silos, in order to compute individual parameters for each one of them. In that case, no dependency can be captured unless a "home-brewed" variable, resulting from the combination of many, is added. But this is a highly informal procedure which relies on the actuary's thorough knowledge of the problem at hand and technical assistance such as visualization tools. As we have shown in subsection 5.9, Neural Networks are able to automate this procedure and capture the most relevant of these dependencies w.r.t ratemaking. This is where comes in the most important difference between Neural Networks and Generalized Linear Models: automating the detection of dependencies.

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The superiority of the Neural Network Model is illustrated in Figure 17, where we have simulated the profits that can be generated by an insurer with 100M$ book of business operating at a global loss ratio of 65%. We compare Neural Networks and Generalized Linear Models as they take turns as ratemaking model and facility model (the model used to identify underpricings in the ratemaking and to choose the risks to be ceded to the facility). We measured profits as such: for a particular insured risk, let Pr and Pf be the premiums computed according to the ratemaking and facility models, respectively. Let C be the level of claims that occurred for that risk (usually zero). The premiums Pr and Pf are pure premiums. Since we have assumed a loss ratio of 65%, we can compute the gross premium as Pr/65% for the ratemaking model. Then, when a risk is ceded, the facility keeps 75% of that premium. So the actual profit of ceeding a particular risk is Actual Profit = C - 75% x P r /65%. Similarly, the facility premium Pf corresponds to the expected level of claims, so the projected profit of ceding a risk is: Projected Profit =

Pf - 75% x P r /65%.

Accordingly, the facility premium must be 15.4% higher than the corresponding ratemaking premium in order to profitably (statistically) cede a risk. The top graphic of Figure 17 shows that the Neural Network, used as a facility model can help generate substantial profits (between 1.25M$ and 1.5M$ for the 100M$ book of business insurer) when a GLM is used for ratemaking. It profitably identifies underpricings on more than 10% of the insurer's book of business. Also observe that the difference between the actual and relative profits is relatively small. Since Actual Profit - Projected Profit =

C -

Pf,

we conclude that the Neural Network is very precise at estimating the expected claims level for high risks.


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According to the graphic, the insurer has been able to cede 1.25MS + 75% x 10% x 100M$ =

8.75M$

in claims to the pool. Thus, the ceded risks had an average loss ratio of 87.5% up from the global figure of 65%. On the other hand the second graphic of Figure 17 shows that the GLM model, when used as the facility model mistakingly identifies underpricings in the ratemaking Neural Network model that appear in the projected profit but do not translate in real, actual profit.

8

Conclusion

Neural networks have been known to perform well in tasks where discrimination is an important aspect of the task at hand and this has lead to many commercially successful applications of these modeling tools (Keller 1997). We have shown that, when applied properly while taking into account the particulars of insurance data, that ability to discriminate is also revealed with insurance data. When applied to automobile insurance ratemaking, they allow us to identify more precisely the true risk associated to each insured. We have argued in favor of the use of statistical learning algorithms such as Neural networks for automobile insurance ratemaking. We have described various candidate models and compared them qualitatively and numerically. We have found that the selected model has significantly outperformed all other models, including the current premium structure. We believe that their superior performance is mainly due to their ability to capture high-order dependencies between variables and to cope with the fat tail distribution of the claims. Other industries have adopted statistical learning algorithms in the last decade and we have shown them to be suited for the automobile insurance industry as well. Completely changing the rate structure of an insurer can be a costly enterprize, in particular when it involves significant changes

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C. Dugas et al. PRR Profit (Volume: 100M$, Loss Ratio: 65%) 1

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Figure 17. Profit from the PRR facility as a function of the ceding percentage. Both, the projected profit (dashed) and the actual profit (solid) are shown. These illustrations apply to an insurer with a volume of business of 100MS and a global loss ratio of 65%. In the top (bottom) figure, the benchmark model is the Neural Network model (GLM model) and the model used to identify the underpricings is the GLM model (Neural Networks model).


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in the computer systems handling transactions, or the relations with brokers. We have shown that substantial profit can be obtained from the use of Neural Networks in the context of risk-sharing pools. There are still many other applications where better discrimination of risks can be used profitably, in particular target marketing, fraud detection and elasticity modeling. Target Marketing When an insurer sends out mail solicitation, only a portion (5%10%) of the whole population will be contacted. The goal here is, given this fixed portion, to reach the maximum number of people who will respond positively to the solicitation. Another possibility would be for the insurer to develop a "customer lifetime value" model that would predict, given an insured's profile, what is the expected present value of the future profits that will be generated by acquiring this particular insured's business. Then, by using the customer lifetime value model in conjunction with a model for the probability of positive response, an insurer could attempt to maximize the profit of its solicitation campaign instead of simply maximizing the number of new insureds. Fraud Detection Fraud represents 10%-15% of all claims. Usually only a portion of the claims will be looked at by an insurer's investigators. The goal of fraud detection is to develop a model that will help an insurer increase the effectiveness of its investigators by referring them cases that are more likely to be fraudulent. In order to do so, one needs a database of previous successful and unsuccessful investigations. Neural Networks have been applied with great success to credit card fraud detection. Elasticity Modeling The greatest benefit from an improved estimation of pure premium derives by considering its application to ratemaking. The main rea-

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son for these benefits is that a more discriminant predictor will identify a group of insureds that are significantly undercharged and a (much larger) group that is significantly overcharged. Identifying the undercharged will increase profits: increasing their premiums will either directly increase revenues (if they stay) or reduce underwriting losses (if they switch to another insurer). The advantage of identifying the insured profiles which correspond to overcharged premiums can be coupled with a marketing strategy in order attract new customers and increase market share, a very powerful engine for increased profitability of the insurer (because of the fixed costs being shared by a larger number of insureds). To decide on the appropriate change in premium, one also needs to consider market effects. An elasticity model can be independently developed in order to characterize the relation between premium change and the probability of loosing current customers or acquiring new customers. A pure premium model such as the one described in this chapter can then be combined with the elasticity model, as well as pricing constraints (e.g., to prevent too much rate dislocation in premiums, or to satisfy some jurisdiction's regulations), in order to obtain a function that "optimally" chooses for each insured profile an appropriate change in gross premium, in order to maximize a financial criterion. Clearly, the insurance industry is filled with analytical challenges where better discrimination between good and bad risks can be used profitably. We hope this chapter goes a long way in convincing actuaries to include Neural networks within their set of modeling tools for ratemaking and other analytical tasks.


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Appendix Proof of the Equivalence of the Fairness and Precision Criterions In this section, we show that, when all subpopulations are considered to evaluate fairness, the precision criterion and the fairness criterion, as they were defined in section 3, both yield the same premium function. Theorem 1 The premium function which maximizes precision (in the sense of equation 6) also maximizes fairness (in the sense of equation 9, when all subpopulations are considered), and it is the only one that does maximize it. Proof: Let P be a subset of the domain of input profiles. Let q be a premium predictor function. The bias in P is defined by

(xi,a,i)eP 2

Let Fq = —E[J2p bq(P) ] be the expected "fairness" criterion using premium function q, to be maximized (by choosing q appropriately). Let p(x) = l£[a|:r] be the optimal solution to the precision criterion, i.e., the minimizer of

E[{p{X)-Af\. Consider a particular population P. Let q(P) denote the average premium for that population using the premium function q(x),

(xi,a.i)eP

and similarly, define a(P) the average claim amount for that population, (xj,ai)eP

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Then the expected squared bias for that population, using the premium function q, is E[bq{Pf] = E[(q(P) -

a(P)f]

which is minimized for any q such that q(P) = E[a(P)}. Note in particular that the optimal ESE solution, p, is such a minimizer of Fq, since p

^

=

jp-\ E

£feisi] = £[j4 E

(xi,a.i)eP

*i = w ) ]

{xitcn)eP

We know therefore that g = p is a minimizer of Fq, i.e., Vg, F p < Are there other minimizers? Consider a function q ^ p, that is a minimizer for a particular population Pi. Since q ^ p,3x s.t. g(x) 7^ p(x). Consider the particular singleton population Px = {x}. On singleton populations, the expected squared bias is the same as the expected squared error. In fact, there is a component of F which contains only the squared biases for the singleton populations, and it is equal to the expected squared error. Therefore on that population (and any other singleton population for which q ^ p) there is only one minimizer of the expected squared bias, and it is the conditional expectation p{x). So E[(q(x) - A)2\X = x] > E[(p(x) - A)2\X = x] and therefore E[bq(Px)} > E[bp(Px)]. Since p is a maximizer of fairness for all populations, it is enough to prove that q is sub-optimal on one population to prove that the overall fairness of q is less than that of p, which is the main statement of our theorem:

\/q^p,Fq>Fp.

References Aldrich, J. 1995, R.A. Fisher and the making of maximum likelihood 1912-22, Technical Report 9504, University of Southampton, Department of Economics.


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Bailey, R. and Simon, L. 1960, 'Two studies in automobile insurance ratemaking', ASTIN Bulletin 1(4), 192-217. Bellman, R. 1957, Dynamic Programming, Princeton University Press, NJ. Bengio, Y. and Gingras, F. 1996, Recurrent neural networks for missing or asynchronous data, in M. Mozer, D. Touretzky and M. Perrone, eds, 'Advances in Neural Information Processing System', Vol. 8, MIT Press, Cambridge, MA, pp. 395^101. Bishop, C. 1995, Neural Networks for Pattern Recognition, Oxford University Press. Breiman, L., Friedman, J., Olshen, R. and Stone, C. 1984, Classification and Regression Trees, Wadsworth Int. Group. Brown, R. 1988, Minimum bias with generalized linear models, in 'Proceedings of the Casualty Actuarial Society'. Campbell, J., Lo, A. W. and MacKinlay, A. 1997, The Econometrics of Financial Markets, Princeton University Press, Princeton. Cantelli, F. 1933, 'Sulla probabilita come limita della frequenza', Rend. Accad. Lincei 26(1), 39. Chapados, N. and Bengio, Y. 2003, 'Extensions to metricbased model selection', Journal of Machine Learning Research 3, 1209-1227. Special Issue on Feature Selection. Cristianini, N. and Shawe-Taylor, J. 2000, An Introduction to Support Vector Machines, Cambridge Press. Dempster, A. P., Laird, N. M. and Rubin, D. B. 1977, 'Maximumlikelihood from incomplete data via the EM algorithm', Journal of Royal Statistical Society B 39, 1-38.

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Diebold, F. X. and Mariano, R. S. 1995, 'Comparing predictive accuracy', Journal of Business and Economic Statistics 13(3), 253263. Dugas, C , Bengio, Y., Belisle, R, Nadeau, C. and Garcia, R. 2001, A universal approximator of convex functions applied to option pricing, in 'Advances in Neural Information Processing Systems', Vol. 13, Denver, CO. Fisher, R. A. 1912, 'On an absolute citerion for frequency curves', Messenger of Mathematics 41, 155-160. Fisher, R. A. 1915, 'Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population', Biometrika 10, 507-521. Fisher, R. A. 1922, 'On the mathematical foundations of theoretical statistics', Philosophical Transactions of the Royal Society of London A222, 309-368. Fisher, R. A. 1925, 'Theory of statistical estimation', Proceedings of the Cambridge Philosophical Society 22, 700-725. Geman, S., Bienenstock, E. and Doursat, R. 1992, 'Neural networks and the bias/variance dilemma', Neural Computation 4(1), 1-58. Ghahramani, Z. and Jordan, M. I. 1994, Supervised learning from incomplete data via an EM approach, in J. Cowan, G. Tesauro and J. Alspector, eds, 'Advances in Neural Information Processing Systems', Vol. 6, Morgan Kaufmann, San Mateo, CA. Gingras, R, Bengio, Y. and Nadeau, C. 2000, On out-of-sample statistics for time-series, in 'Computational Finance 2000'. Glivenko, V. 1933, 'Sulla determinazione empirica delle leggi di probabilita', Giornale dell'Istituta Italiano degli Attuari 4, 92.


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Hampel, R, Ronchetti, E., Rousseeuw, P. and Stahel, W. 1986, Robust Statistics, The Approach based on Influence Functions, John Wiley & Sons. Hastie, T., Tibshirani, R. and Friedman, J. 2001, Data Mining, Inference and Prediction, Springer. Holler, K., Sommer, D. and Trahair, G. 1999, 'Something old, something new in classification ratemaking with a novel use of glms for credit insurance', Casualty Actuarial Society Forum pp. 3 1 84. Huber, P. 1982, Robust Statistics, John Wiley & Sons Inc. Huber, P. J. 1964, 'Robust estimation of a location parameter', Annals of Mathematical Statistics 35, 73-101. Huber, P. J. 1965, 'A robust version of the probability ratio', Annals of Mathematical Statistics 36, 17-53. Huber, P. J. 1966, 'Strict efficiency excludes superefficiency', Annals of Mathematical Statistics 37, 14-25. Huber, P. J. 1968, 'Robust confidence limits', Zeitschrift fur Wahrscheinlichkeits Theorie 10, 269-278. Huber, P. J. 1981, Robust Statistics, John Wiley & Sons, New York. Huber, P. J. 1985, 'Projection pursuit', Annals of Statistics 13, 435525. Huber, P. J. and Rieder, H. 1996, Robust statistics, data analysis, and computer intensive methods, number 109 in 'Lecture Notes in Statistics', Springer, New York. Jacobs, R. A., Jordan, M. I., Nowlan, S. J. and Hinton, G. E. 1991, 'Adaptive mixture of local experts', Neural Computation 3, 7987.

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Kass, G. 1980, 'An exploratory technique for investigating large quantities of categorical data', Applied Statistics 29(2), 119-127. Keller, P. E. 1997, 'Neural networks: Commercial applications'. http://www.emsl.pnl.gov:2080/proj/neuron/neural/products. Kolmogorov, A. 1933, 'Sulla determinazione empirica di una leggi di distribuzione', Giornale dell'Istituta Italiano degli Attuari 4, 33. McCullagh, P. and Nelder, J. 1989, Generalized Linear Models, Chapman and Hall, London. Murphy, K., Brockman, M. and Lee, P. 2000, 'Using generalized linear models to build dynamic pricing systems', Casualty Actuarial Society Forum pp. 107-139. Nadeau, C. and Bengio, Y. 2000, Inference for the generalization error, in S. Solla, T. Leen and K.-R. Mller, eds, 'Advances in Neural Information Processing Systems', Vol. 12, MIT Press, pp. 307313. Nelder, J. and Wedderburn, R. 1972, 'Generalized linear models', Journal of the Royal Statistical Society 135, 370-384. Newey, W. and West, K. 1987, 'A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix', Econometrica 55, 703-708. Orr, G. and Miiller, K.-R. 1998, Neural Networks: Tricks of the Trade, Springer. Rousseeuw, P. and Leroy, A. 1987, Robust Regression and Outlier Detection, John Wiley & Sons Inc. Rumelhart, D., Hinton, G. and Williams, R. 1986, 'Learning representations by back-propagating errors', Nature 323, 533-536.


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Scholkopf, B., Smola, A. and Miiller, K.-R. 1998, 'Nonlinear component analysis as a kernel eigenvalue problem', Neural Computation 10, 1299-1319. Takeuchi, I., Bengio, Y. and Kanamori, T. 2002, 'Robust regression with asymmetric heavy-tail noise distributions', Neural Computation 14, 2469-2496. Tikhonov, A. and Arsenin, V. 1977, Solutions of Ill-posed Problems, W.H. Winston, Washington D.C. Vapnik, V. 1998, Statistical Learning Theory, John Wiley, Lecture Notes in Economics and Mathematical Systems, volume 454.


Chapter 5 An Integrated Data Mining Approach to Premium Pricing for the Automobile Insurance Industry A.C. Yeo and K.A. Smith

In setting the optimal combination of premiums, the insurance company has to find a balance between two conflicting objectives of market share and profitability. The premiums must cover the cost of expected claims and also ensure that a certain level of profitability is achieved. However, the premiums must not be set so high that market share is jeopardized as consumers exercise their rights to choose their insurers in a competitive market place. We used a data mining approach, which endeavours to find a balance between the two conflicting objectives in determining optimal premium, and we were able to demonstrate the quantitative benefits of the approach.

1

Introduction

To succeed in a highly competitive environment, insurance companies strive for a combination of market growth and profitability, and these two goals are at times conflicting. Premiums play a critical role in enabling insurance companies to find a balance between these two goals. The challenge is to set the premium so that expected claim costs are covered and a certain level of profitability is achieved; yet not to set premiums so high that market share is jeopardized as consumers exercise their rights to choose their insurers. 199

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A. C. Yeo and K. A. Smith

Insurance companies have traditionally determined premiums by assigning policy holders to pre-defined groups and observing the average claim behavior of each group. The groups are formed based on industry experience about the perceived risk of different groups of policy holders. With the advent of data warehouses and data mining however comes an opportunity to consider a different approach to assessing risk; one based on data-driven methods. By using data mining techniques, the aim is to determine optimal premiums that more closely reflect the genuine risk of individual policy holders as indicated by behaviors recorded in the data warehouse.

2

A Data Mining Approach

Figure 1 presents a data mining approach for determining the appropriate pricing of policies. The approach consists of three main components. The first component involves identifying risk classifications and predicting claim costs using k-means clustering. The total premiums charged must be sufficient to cover all claims made against the policies and return an acceptable cost ratio. The second component involves price sensitivity analysis using neural networks. Premiums cannot be set at too high a level as customers may terminate their policies, thus affecting market share. The third component combines the results of the first two components to provide information on the impact of premiums on profitability and market share. The optimal mix of policy holders for a given termination rate can be determined by non-linear integer programming. An Australian motor insurance company supplied the data for this study. Two data sets (training set and test set), each consisting of 12-months of comprehensive motor insurance policies and claim information were extracted. The training set consisted of 146,326 policies with due dates from 1 January to 31 December 1998 while the test set consisted of 186,658 policies with due dates from 1 July 1998 to 30 June 1999. The data was selected to enable comparison

201

An Integrated Data Mining Approach to Premium Pricing

of exposure and retention rates over a one-year period and to ensure that sample sizes are sufficiently large within the constraints of the data available at the time of collection. Forty percent of the policies in the test set were new policies. The training set was used to train the models while the test set was used to evaluate the results. (

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Risk Classification and Prediction of Claim Cost 3.1

Risk Classification

Insurance companies classify policy holders into various risk groups based on factors such as territory, demographic variables (such as age, gender and marital status) and other variables (such as use of vehicle, driving record and years of driving experience). These factors are considered predictors of claim costs (Dionne and

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Vanasse 1992, Samson 1986, Tryfos 1980). Risk classification has traditionally been achieved using heuristic methods, both within the industry and in academic studies. For example, Samson and Thomas (1987) selected four variables: age of policy holder, area in which the policy holder lives, group rating of the insured automobile and level of no-claim discount; and categorized each variable into three levels of activity. Each policy was placed into one ofthe81(3 4 )risk groups. However, the number of factors that can be included is limited for the heuristic method. This is because the number of exposure units in each risk group must be sufficiently large to make claim costs reasonably predictable (Vaughan and Vaughan 1996). To ensure that there is a large number of exposure units in each risk group, the number of risk groups has to be kept small, which in turn affects the number of factors that can be considered. For example adding an additional factor in Samson and Thomas' study would increase the number of risk groups to 243 (35), which would significantly reduce the number of exposure units in each risk group. The classification structure is normally designed to achieve maximum homogeneity within groups and maximum heterogeneity between groups. This can be achieved through clustering, whether crisp or fuzzy. Clustering places objects into groups or clusters based on distances computed on the attributes of the data. The objects in each cluster tend to be similar to each other in some sense, and objects in different clusters tend to be dissimilar. (Anderberg 1973, Everitt 1993, Johnson 1998, Kaufman and Rousseeuw 1990). Clustering allows more factors to be included in risk classification without compromising the size of each risk group compared to the heuristic method. Several researchers have used clustering techniques for the risk classification stage of the claim cost prediction problem. Williams and Huang (1997) used k-means clustering to identify high claiming policy holders in a motor vehicle insurance portfolio. Derrig and Ostaszewski (1995) used fuzzy c-means clustering for territo-


203

rial rating of Massachusetts towns. Smith, Willis and Brooks extended the use of clustering by proposing the use of k-means clustering to predict claim costs (Smith et al. 2000). Our work extends this approach by evaluating the k-means clustering model and the fuzzy c-means clustering model as techniques for predicting claims cost, and comparing the results obtained with a heuristic model (Samson and Thomas 1987) to determine the advantages of a datadriven approach. 3.1.1 K-Means Clustering Model The k-means clustering model used to classify policies performs disjoint cluster analysis on the basis of Euclidean distances computed from variables and seeds that are generated and updated by the k-means algorithm (Anderberg 1973, MacQueen 1967). Using the least squares clustering criterion, the sum of the squared distances of observations to the cluster means is minimized. Thirteen variables were used for clustering. They were: (1) policy holder's age, (2) policy holder's gender, (3) area in which the vehicle was garaged, (4) rating of policy holder, (5) years on current rating, (6) years on rating one, (7) number of years policy held, (8) category of vehicle, (9) sum insured, (10)total excess, (11) vehicle use, (12) vehicle age, (13) whether or not the vehicle is under finance. A minimum cluster size of 1,000 was specified to satisfy the insurability requisite of mass. The initial clustering yielded 6 clusters, with cluster sizes ranging from 1,600 to 58,000. Two more rounds of clustering were done to reduce the cluster sizes to no more than

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20,000. A total of 30 risk groups were generated through the three rounds of clustering, each containing between 1,000 and 20,000 policy holders. This process ensures the clustering algorithm finds a balance between the requisite mass and homogeneity criteria. 3.1.2 Fuzzy C-Means Clustering Model The fuzzy c-means clustering algorithm assigns each policy holder to different clusters to varying degrees specified by a membership grade. The algorithm minimizes an objective function that represents the distance from any given data point to a cluster center weighted by that data point's membership grade (Bezdek 1981). To ensure comparability with the k-means clustering, the number of clusters specified was 30. The same thirteen variables used in the kmeans clustering were used for the fuzzy c-means clustering. The training data was clustered using MATLAB Fuzzy Logic Toolbox. A neural network was then trained to learn the fuzzy inference system using the thirteen variables as input and the membership grades of the 30 clusters as output. The number of hidden neurons was fifty, and the hyperbolic tangent activation function was used. Thirty percent of the data was reserved for a validation set, and the R Squared obtained on this set was 0.9608 (with an R squared of 0.97 for the total data set) giving us the confidence to apply the network to the test set. 3.1.3 Heuristic Model To determine how well the two clustering models group policy holders into various risk groups, a heuristic model based on the approach of Samson and Thomas (1987) was used for comparison. Three factors were used in the heuristic model: age of the policy holders, area in which the vehicles were garaged and the category of the vehicle. Rating area and category of vehicle were sorted by average claim cost per policy holder and age of policy holder was sorted by age. The three variables were then split into 5 classes each. An attempt was made to minimise the difference in average


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claim cost per policy between the classes and to ensure that each class had at least 10,000 policies, in other words, to comply with the two requirements of mass and homogeneity. A total of 125 (53) groups were created.

3.2

Prediction of Claim Cost

Having grouped the policy holders into various risk groups using one of the three methods, the next stage is to predict claim costs for each of the risk groups. For the heuristic method, we have chosen regression, firstly so that we can compare our results to Samson and Thomas (1987), and secondly because regression has been shown in the literature to be preferable to other models that have been used (Chang and Fairley 1979, Sant 1980). As for the two clustering methods, the average claim cost per policy holder within each cluster of the training set was used as the basis for prediction of the test set. The two clustering approaches (k-means and fuzzy c-means) to the risk classification problem utilize all of the available information except claim behavior to find groups of policy holders exhibiting similar characteristics (demographic and historical). The heuristic method of Samson and Thomas (1987) classifies policy holders according to a set of three pre-defined factors. These methods are described in more detail below. 3.2.1

K-Means Clustering Model

The actual average claim cost per policy for each cluster, found by clustering the training set, was used as the basis for predicting average claim cost per policy for the test set. 3.2.2

Fuzzy C-Means Clustering Model

Claim cost of a policy holder is apportioned to the clusters he belongs to according to his/her calculated membership grade. For example, if the claim cost of a policy holder was $ 1,000 and his/her

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membership grade of Cluster 1 is 0.8 and that of Cluster 2 is 0.2, $800 will be apportioned to Cluster 1 and $200 will be apportioned to Cluster 2. The claim cost per policy holder in each cluster will be the total cost apportioned to that cluster divided by the total membership value for that cluster. An illustrative example is shown in Table 1. Table 1. Illustrative example of computing claims cost per policy holder (fuzzy clustering). Apportioning of Claim Cost ($) Membership Grade [Claim Cost * Membership Grade Policy Holder Claim Cost ($) Clusterl Cluster2 Clusterl Cluster2 1 0 0.98 0.02 0 0 2 300 1.00 0.00 300 0 0.24 3 0 0.76 0 0 4 500 0.99 0.01 497 3 5 0 0.00 1.00 0 0 6 0 0.02 0.98 0 0 7 1,000 0.98 0.02 981 19 8 0 0.03 0.97 0 0 9 900 0.16 0.84 144 756 10 0 0.01 0.99 0 0 Total 2,700 4.93 5.07 1921 779 154 389 Claim Cost Per Policy Holder (Total Claim Cost/Total Membership Value)

3.2.3 Heuristic Model A linear regression model, similar to Samson's (Samson and Thomas 1987) was used to predict average claim cost per policy for the risk groups found by the heuristic model. The linear model we used is shown in the following equation: y = 123.2ai + 13.1459a2 - 33.6505a3 - 29.7609a4 - 116bi - 82.6731b2 + 7.9831b3 - 51.317b4 - 77.8226ci - 65.8959c2 - 61.8716c3 - 3.5125c4 + 402.4 where y = claim cost per policy holder in a cell, and each risk factor is represented by two binary variables as shown below:


ai = 1 for age group 1 = 0 otherwise a3 = 1 for age group 3 = 0 otherwise bj = 1 for rating area 1 = 0 otherwise b 3 = 1 for rating area 3 = 0 otherwise Ci = 1 for vehicle category 1 = 0 otherwise C3 = 1 for vehicle category 3 = 0 otherwise

3.3

207

a2 = 1 for age group 2 = 0 otherwise a4 = 1 for age group 4 = 0 otherwise b2 = 1 for rating area 2 = 0 otherwise b4 = 1 for rating area 4 = 0 otherwise C2 = 1 for vehicle category 2 = 0 otherwise C4 = 1 for vehicle category 4 = 0 otherwise

Results

Figures 2, 3 and 4 show the plots of predicted claim cost per policy against the actual claim cost per policy for the k-means clustering model, fuzzy c-means clustering model and heuristic model respectively. From the graphs, it can be seen that the predicted claim cost of the two clustering models are closer to the actual claim cost compared to the heuristic model. However the fuzzy c-means does not appear to be able to discriminate policy holders with high claim cost. Average Claim Cost($)

Figure 2. Prediction of claim cost (k-means clustering).

208


Average Claim Cost ($)

800.00

•Actual — - Predicted

Figure 3. Prediction of claim cost (fuzzy c-means clustering). Average Claim Cost ($) 1,400 1,200

1,000 800

Risk Group -Actual ——Predicted

Figure 4. Prediction of claim cost (heuristic).

Table 2 shows the various measures of performance of the three models. The weighted mean absolute deviation of predicted claim cost per policy from the actual claim cost per policy for the k-


209

means clustering model was 8.3% which was significantly lower than the 15.63% for the fuzzy c-means clustering model and the 13.3%) for the regression model. The k-means clustering model provided a more accurate prediction than the fuzzy c-means clustering and heuristic models. Table 2. Measurement of performance of models. K-Means Fuzzy C- Heuristic Clustering Model Means Model Model 13.30% 15.63% Weighted mean absolute deviation 8.30% $403 $433 $111 Maximum deviation 129.47% 93% Maximum % deviation 23% 39% Deviation within 10% 43% 57% 72% 77% 90% Deviation within 20% Measurements

4

Prediction of Retention Rates and Price Sensitivity

Having classified the policy holders into 30 risk groups using kmeans clustering, we can now examine the price sensitivity within each cluster. This is the second component of the data mining framework. Within each cluster a neural network is used to predict retention rates given demographic and policy information, including the premium change from one year to the next. Sensitivity analysis of the neural networks was then performed to determine the effect of changes in premium on retention rate.

4.1

Prediction of Retention Rate

4.1.1

Neural Network Model

A multilayered feedforward neural network was constructed for each of the clusters with 25 inputs, 20 hidden neurons and 1 output neuron (whether the policy holder renews or terminates the con-

210


tract). The inputs consist of the thirteen variables used for risk classification and the following premium and sum insured variables: (1) "old" premium (premium paid in the previous period), (2) "new" premium (premium indicated in renewal notice), (3) "new" sum insured (sum insured indicated in the renewal notice), (4) change in premium ("new" premium - "old" premium), (5) change in sum insured ("new" sum insured - "old" sum insured), (6) percentage change in premium, (7) percentage change in sum insured, (8) ratio of "old" premium to "old" sum insured, (9) ratio of "new" premium to "new" sum insured. (10) whether there is a change in rating (11) whether there is a change in postcode (12) whether there is a change in vehicle Several experiments were carried out on a few clusters to determine the most appropriate number of hidden neurons and the activation function. Twenty hidden neurons and the hyperbolic tangent activation function were used for the neural networks for all the clusters. A uniform approach is preferred to enable the straightforward application of the methodology to all clusters, without the need for extensive experimentation by the company in the future. Input variables which were skewed were log transformed. 4.1.2

Determining Decision Thresholds

The neural network produces output between zero and one, which is the probability that a policy holder will terminate his/her policy. Figure 5 shows the probability of termination of Cluster 11 based on the training data. A threshold value is used to decide how to categorize the output data. For example a threshold of 0.5 means that if the probability of termination is more than 0.5, then the policy will be classified as terminated. Usually a decision threshold is selected based on the classification accuracy using a confusion ma-

211


trix. Table 3 shows a confusion matrix for Cluster 11 with a decision threshold of 0.5. The overall classification accuracy is 88.8% (of the 11,463 policies, 569 are correctly classified as terminated and 9,615 are correctly classified as renewed), while the classification accuracy for terminated policies is 33.8% and renewed policies is 98.3%. Frequency

200 Termination Rate = 14.7%

100

minim mn-n urtmi nffl iH iniffli H w imlTi ilHli n m u n

0.00 0.16 0.32 0.48 0.64 0.80 0.96 Probability of Termination Figure 5. Determining the threshold value of the neural network output (cluster 11).

Table 3. Confusion matrix for cluster 11 with decision threshold (training set).

0.5

Classified as Renewed Total Terminated Terminated 569 (33.8%) 1,114 (66.2%) 1,683 Actual Renewed 165 (1.7%) 9,615 (98.3%) 9,780 Total 734 11,463 10,729 Overall Accuracy 88.8%

The decision threshold is usually chosen to maximize the classification accuracy. However in our case we are more concerned with achieving a predicted termination rate that is equal to the actual termination rate. This is because we are more concerned with the performance of the portfolio (balancing market share with prof-

212


itability) rather than whether an individual will renew or terminate his/her policy. The actual termination rate for cluster 11 is 14.7%. A threshold of 0.5 yields a predicted termination rate of 6.4%. To obtain a predicted termination rate of 14.7%, the threshold has to be reduced to 0.204 (See Figure 5). The confusion matrix for a threshold of 0.204 is shown in Table 4. The overall classification accuracy has decreased from 88.8% to 85.3% and that of renewed policies from 98.3% to 91.4%. However, the classification accuracy for terminated policies has improved from 33.8%) to 50.0%. The confusion matrices for the test set with threshold of 0.5 and 0.204 are shown in Tables 5 and 6 respectively. Table 4. Confusion matrix for cluster 11 with decision threshold = 0.204 (training set). Classified as Renewed Total Terminated Terminated 842 (50.0%) 1,683 841 (50.0%) Actual 845 (8.6%) 8,935 (91.4%) 9,780 Renewed 11,463 9,777 Total 1,686 Overall Accuracy 85.3% Table 5. Confusion matrix for cluster 11 with decision threshold = 0.5 (test set). Classified as Renewed Total Terminated 284 (10.2%) 2,510 (89.8%) 2,794 Terminatec Actual Renewed 350 (2.6%) 13,234 (97.4%) 13,584 Total 634 15,744 16,378 Overall Accuracy 82.5% Table 6. Confusion matrix for cluster 11 with decision threshold = 0.204 (test set).

Actual

Classified as Renewed Total Terminated Terminatec 948 (33.9%) 1.846 (66.1%) 2,794 Renewed 1,778 (13.1%) 11,806 (86.9%) 13,584 Total 13,652 16,378 2,726 Overall Accuracy 77.9%


4.1.3

213

Analyzing Prediction Accuracy

The confusion matrix provides the prediction accuracy of the whole cluster. It does not tell us how a given percentage change in premium will impact termination rate, however. To determine how well the neural networks are able to predict termination rates for varying amounts of premium changes, the policies in each cluster are grouped into various bands of premium changes as shown in Table 7. Table 7. Dividing clusters into bands of premium change. Band = 50%

Actual >= -22.5% >= -17.5% >= -12.5% >= -7.5% >= -2.5% >= 2.5% >= 7.5% >= 12.5% >= 17.5% >= 22.5% >= 27.5% >= 32.5% >= 37.5% >= 42.5% >= 47.5%

& & & & & & & & & & & & & &

< -22.5% < -17.5% < -12.5% < -7.5% < -2.5% < 2.5% < 7.5% < 12.5% < 17.5% < 22.5% < 27.5% < 32.5% < 37.5% < 42.5% < 47.5%

The predicted termination rates of each band of policies are then compared to the actual termination rates for that band. For all the clusters the prediction accuracy of the neural networks starts to deteriorate when premium increases are between 10% and 20%. Figure 6 shows the actual and predicted termination rates for one of the clusters (Cluster 24). 4.1.4

Generating More Homogeneous Models

In order to improve the prediction accuracy, the cluster was then split at the point when prediction accuracy starts to deteriorate. Two separate neural networks were trained for each sub-cluster.

214


The prediction accuracy improved significantly with two neural networks as can be seen from Figure 7. The average absolute deviation decreased from 10.3% to 2.4%. The neural network was then applied to the test set. The neural network performed reasonably well on the test set with an average absolute deviation of 4.3% (Figure 8). TRAINING SET 60%

=50%

Premium Change - Predicted •

-Actual

Figure 6. Prediction accuracy of one neural network model for cluster 24 (training set). TRAINING SET

60% 50%

Neural Network 1

Neural Network 2

40% 30% 20% 10%

pj|=-*^

^^>^r"^

V

0% , * y ) i > ( X A,*, - C , ) '•=1

7=1

(1)

7=1

There are also some constraints affecting the system. They are:

5>„=1V/

(2)

7=1

x,e{0,l}

(3)

X^(l-«,)>(l-« ( )i;^ i=l

(4)

i=l

Equation 1 maximizes the total expected revenue associated with the portfolio. Equations 2 and 3 ensure that each cluster is assigned a termination rate and hence a calculated premium. Equation 4 ensures that the portfolio has a termination rate no greater than the desired termination rate of at. The mathematical formulation was solved using Lingo (Schrage 1999). Since our model contains integer constraints, Lingo uses the branch-and-bound method, which is the most commonly used enumerative approach. The "branching" refers to the enumeration part of the solution technique and "bounding" refers to the fathoming of possible solutions by comparison to a known upper or lower bound on the solution value (Land and Doig 1960).

5.1

Results

In our case study we have used termination rates of 9%, 11%, 15%, 20%, 25%o and 30% as inputs for the above mathematical formulation. Suppose the insurance company wants to maintain its current portfolio termination rate of 11%. Can adjusting the premiums of different clusters and thereby changing the termination rate of different clusters while maintaining the same termination rate for the

223


portfolio improve the expected revenue? Solving the above formulation for a desired termination rate of 11%, we obtain values of xy as shown in Table 11 and an optimal portfolio as shown in Table 12. The expected revenue derived from the optimal portfolio was $39.11 million compared to $35.54 million for the current portfolio. This represents an increase of 10% in revenue, without affecting the market share. Table 11. Values of xtj for optimal solution for overall portfolio termination rate of 11%. Termination Rate 25% Cluster 30% 20% 15% 11% 9% 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 1 0 1 1 1 0 0 0 0 1 1 0 1

0 0 0 0 0 0 0 1 1 1 0 0 1 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

The optimization problem was solved for varying values of the termination rate at: 15%, 20% and 25%. The results are shown in Figure 13. The curve is similar to the efficient frontier of portfolio optimization. It is a smooth non-decreasing curve that gives the best possible tradeoff between revenue and termination rate. The insurance company selects an acceptable termination rate and the model will determine a portfolio that maximizes expected revenue. We can also add other constraints to our model to better reflect practical insurance portfolio optimization. For instance the insurance company may decide that under no circumstances should their


224

market share in any cluster fall below a certain level. Such a constraint could be encoded by stipulating the minimum number of policy holders in a cluster, for example, each cluster should have at least 6,000 policy holders. The following constraint could be added: P, > 6,000 Table 12. Current portfolio versus optimal portfolio. Current Portfolio Termination Rate 11.2% 570 8.8%

Risk Class 4

Premium 424

9

Optimal Portfolio Termination Premium Rate 403 9.0% 598 9.0%

11

764

14.7%

786

19

377

7.5%

390

15.0% 9.0%

20

463

6.8%

493

9.0%

24

533

9.5%

533

9.0%

27

445

11.6%

1,090

25.0%

29

570

605 510

2,12,30

499 685

9.5% 10.9%

11.0%

1,6,15,16

13.5%

660

11.0%

3,7,8,10,18

508

11.7%

483

9.0%

5,23,26

561

14.7%

502

13,14,28

688

15.1%

338 532 $35.54m

8.7% 11.1%

630 345 490 $39.11m

17,21,22,25 ALL Objective Function Value (Revenue)

11.0%

9.0% 11.0% 9.0% 11.0%

$60,000,000 -i -

.

-

•

'

-

*

'

§ $50,000,000 c a > £ $40,000,000

• $30,000,000 9%

i

14%

i

i

19%

24%

Termination Rate * Optimal • Current

Figure 13. Optimal revenue for varying termination rates.

Such an approach may help when considering concepts such as life-time value of customers and finds analogy in portfolio theory


225

where cardinality constraints are included to impose limits on the proportion of the portfolio held in a given asset (Chang et al. 2000). The insurance company has two conflicting objectives of profitability and market share. How does the company decide on the tradeoff between the two objectives or in other words, how does it pick a point on the efficient frontier? This answer to this question depends on the elected strategy of the company, whether this be to maximize profit regardless of the impact on market share, or to accept less profit for the sake of retaining a certain percentage of market share. Another common strategy would be to maximize customer life-time value (Berger and Nasr, 1998). Lifetime value is the net present value of profit that a company can earn from a typical customer over a specified number of years. This value depends on a number of factors such as retention rate, profit margin and referral rate. The chosen strategy will determine where the company prefers to lie on the efficient frontier, and the proposed mathematical programming approach is able to find the optimal mix of premiums across the portfolio in order to implement this strategy.

6

Conclusions

We have provided evidence of the benefits of an approach which combines data mining and mathematical programming in determining the premium to charge automobile insurance policy holders in order to arrive at an optimal portfolio. The approach has addressed the insurer's need to find a balance between profitability and market share when determining optimal premiums. Using our approach we have been able to increase revenue without affecting the predicted market share. This kind of quantitative evidence is essential before we can expect the insurance industry to embrace this new approach to premium pricing.

226


References Anderberg, M. (1973), Cluster Analysis for Applications, Academic Press. Berger, P. and Nasr, N. (1998), "Customer lifetime value: marketing models and applications," Journal of Interactive Marketing, 18(1), 17-30. Bezdek, J.C. (1981), Pattern Recognition with Fuzzy Objective Function Algorithms, New York: Plenum Press. Chang, L. Fairley, W.B. (1979), "Pricing automobile insurance under multivariate classification of risks: additive versus multiplicative," Journal of Risk and Insurance, 46(1), 75-98. Chang, T., Meade, N., Beasley, J., and Sharaiha, Y. (2000), "Heuristics for cardinality constrained portfolio optimisation," Computers & Operations Research, 27, 1271-1302. Constantinides, G. and Malliaris, A. (1995), "Portfolio theory," in Jarrow, R., Maksimovic, V., and Ziemba, W. (eds.), Handbooks in Operations Research and Management Science, vol. 9, pp. 130, Elsevier Science. Derrig, R.A. and Ostaszewski, K.M. (1995), "Fuzzy techniques of pattern recognition in risk and claim classification," The Journal of Risk and Insurance, 62(3), 447-482. Dionne, G. and Vanasse, C. (1992), "Automobile insurance ratemaking in the presence of asymmetrical information," Journal of Applied Econometrics, 7(2), 149-165. Everitt, B.S. (1993), Cluster Analysis, 3rd ed., London: E. Arnold. Johnson, D.E. (1998), Applied Multivariate Methods for Data Analysts, Pacific Grove, CA: Duxbury Press. Kaufman, L. and Rousseeuw, P.J. (1990), Finding Groups in Data: an Introduction to Cluster Analysis, New York: Wiley. Land, A.H. and Doig, A.G. (1960), "An automatic method for solving discrete programming problems," Econometrica, 28, 497520. MacQueen, J. (1967), "Some methods for classification and analysis of multivariate observations," Proceedings of the 5th Berke-


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ley Symposium on Mathematical Statistics and Probability. Markowitz, H. (1952), "Portfolio selection," Journal of Finance, 7, 77-91. Samson, D. (1986), "Designing an automobile insurance classification system," European Journal of Operational Research, 27, 235-241. Samson, D. and Thomas, H. (1987), "Linear models as aids in insurance decision making: the estimation of automobile insurance claims," Journal of Business Research, 15, 247-256. Sant, D.T. (1980), "Estimating expected losses in auto insurance," The Journal of Risk and Insurance, 47(1), 133-151. Schrage, L. (1999), Optimization Modeling with Lingo, Lindo Systems Inc. Smith, K.A., Willis, R.J., and Brooks, M. (2000), "An analysis of customer retention and insurance claim patterns using data mining: a case study," Journal of the Operational Research Society, 51(5), 532-541. Tryfos, P. (1980), "On classification in automobile insurance," Journal of Risk and Insurance, 47(2), 331-337. Vaughan, E.J. and Vaughan, T.M. (1996), Fundamentals of Risk and Insurance, 7th ed., New York: Wiley. Williams, G.J. and Huang, Z. (1997), "Mining the knowledge mine: the hot spots methodology for mining large real world databases," Lecture Notes in Artificial Intelligence, 1342, 340348.


Chapter 6 Fuzzy Logic Techniques in the Non-Life Insurance Industry Raquel Caro Carretero

This chapter attempts to bring research on the insurance market and Fuzzy Logic together. It examines the current situation in the insurance industry and how fuzzy models may provide another alternative in the developing of mathematical models. At the beginning of this century, non-life insurers reported higher growth rates than in the nineties. However, we have been seeing the start of a "hard" market cycle. As a result, underwriting losses are more pervasive and stress in the insurance industry has become more prevalent. Insurers are, therefore, under pressure to find ways to return value to their policyholders. This represents an opportunity to companies that can provide key solutions to help the insurer mitigate this market effect. After an introduction to the insurance industry in this chapter, an overview of non-life insurers' pricing decision making processes is given from the perspective of possible applications of fuzzy settheoretic methods. The implementation of two "non-precise" fuzzy reasoning techniques is proposed. The Fuzzy Set Theory is developed to improve the classical bonus-malus rating system used in the automobile insurance industry. The second part proposes a fuzzy method of clustering. Classical clustering algorithms generate patterns such that each object is assigned to exactly one cluster. Often, however, objects can not appropriately be assigned to strictly one cluster, because they are, in some way, "between" clusters. Fuzzy clustering methods have

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proved themselves to be a powerful tool for representing data structure in such situations. The chapter concludes with some observations about the future of the insurance industry.

1

Insurance Market

This section details the current state of the insurance market and explores its capability to embrace the modernization movement and e-business initiatives that are changing the insurance industry. The insurance industry is currently facing a period of unprecedented change. Legal regulatory changes, the need to globalize, and shifting economic conditions are just a few of the market challenges facing today's insurance leaders. Insurance companies are looking for ways to mitigate these market conditions in order to maintain and increase their market share. Exerting additional pressure is the entrance of non-traditional players such as banks and brokerages (financial intermediaries). In fact, banks are now one of the fastest-growing distribution systems for annuity sales and some banks have shown interest in life insurance and personal propertycasualty lines and, to this end, have purchased insurance agencies. At the same time, insurers are experimenting with distributing insurance products through banks. The insurance market typically follows economic cycles characterized as either "hard" or "soft" market cycles, that is, high or low premium rates.1 Underwriters offer a broad range of coverage and typically charge low rates, because the investment of the premium 1

The phenomenon of the underwriting cycle has been analyzed extensively in the economic and financial literature. For details, see, e.g., Swiss Re (2001b). The cycle is best explained by a combination of two hypotheses. The first hypothesis of cycles is that the market is basically competitive and that participants make rational decisions. The second hypothesis, capital constraint, is that the market is susceptible to the influence of external events and inherent industry features.

Fuzzy Logic Techniques in the Non-Life Insurance Industry

231

revenue provides the greatest value. This typically leads to greater losses, since there are more policies in effect for greater risks. Eventually, market conditions change and the insurers' profitability decreases. This leads to a hardening of the market in which insurers look to increase prices and maintain profit goals despite the diminishing returns on their investments and their losses from claims. According to Swiss Re (2001a), worldwide, non-life insurance markets are currently in a period of declining profitability. The decline reflects the soft underwriting cycle, weakening investment performance, and the high level of capital funds. Falling interest rates and robust stock markets have increased insurers' capital funds. Insurance capacity reached historically high levels in 1999, which led to downward pressure on premium rates. As a result, underwriting results declined in all mayor markets. Nevertheless, nonlife insurance markets are currently in a transition period between a severe soft market and a hard market. During the soft market, rising insurance capacity and fierce price and markets share competition led to widely available coverage and low premium rates. Underwriting results declined in all major markets. Then in 2000, commercial lines and reinsurance premium rates started to rise, and have gained momentum in 2001. The need is now for improved underwriting results. As a result of the start of the hardening in the property and casualty market, according to METAgroup (2001), rates are predicted to reach bottom in a number of lines as underwriting losses become more pervasive and stress in the industry is more prevalent. Insurance industry analysts predict that rates will continue to decline. Concurrently, incidences of reserve charges are on the increase. All of this adds up to dramatic restrictions in coverage, fewer available limits, and significant increases in premiums/price. These conditions impact all insurance companies, including the reinsurance companies that provide support to primary insurers. After a prolonged soft market and with investment prospects diminished, premium rates and underwriting results need to improve significantly to achieve widespread profitability.

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R. C. Carretero

Although the insurance industry is expected to grow, insurers are not spending their information technology budgets in accordance with increasing Internet initiatives. Historically, technology spending has not been a priority for insurance carriers. Most companies use technology, but adoption has generally been slow and internally focused, with little standardization or integration of architecture. Cost often determines development, not usefulness or need. As a result, current insurance technology is generally archaic with limited functionality and information exchange between systems. Most insurance carriers utilize legacy systems, which require batch or other time-consuming processing needs. Data tend to be either too discrete or too aggregated to get meaningful feedback from. Additionally, integration can be limited when systems operations exit in silos. As a consequence, technology is gaining in importance within the insurance industry and is being reflected in increased annual technology-related budgets. Insurers are under pressure to find ways to return value to their policyholders and shareholders. This represents an opportunity to companies that can provide cost cutting solutions to help insurers mitigate this market effect.

2

Fuzzy Logic in Insurance

Most actuaries rely on probabilistic models. This kind of approach assumes that the outcome of a random event can be observed and identified with precision. Any vagueness of observation is not considered significant to the construction of the theoretical model. For this reason, it is desirable to include models of a form of uncertainty or vagueness other than randomness. Fuzzy sets are, therefore, useful to describe uncertain statements where the uncertainty is due to the nature of the phenomenon, its perception by humans or when it arises from complexity. According to Ostaszewski (1993), there may be several reasons


233

for wanting to apply Fuzzy Set Theory in modeling uncertainty in insurance science. First and foremost is that vagueness is unavoidable. If the degree of vagueness is so small that one can ignore it in the model, disregarding it is justifiable. On the other hand, if the area studied is permanently "tainted" with imprecision or the limitations of human perception or natural language, vagueness can become a major factor in any attempt to model an event. In addition, when the phenomena observed become so complex that exact measurement would take more than a lifetime, or longer than economically feasible for the study, mathematical precision is often abandoned in favor of more workable and simple, "common sense" models. Real situations are very often not crisp,2 and they can not be described precisely. Therefore, classical mathematical language is not enough to tackle all practical problems. In Fuzzy Logic "either-or" classification does not exist. Rather, the idea is that an item may be part of a class to a greater or lesser extent. The Fuzzy Set Theory is a theory of graded concepts, and Fuzzy or Multivalued Set Theory develops the basic insight that categories are not absolutely clear cut. This concept is not unique, it also found in every-day speech. One can be young, middle-aged or old. Age could be given in years instead of using the terms "young" and "old," but such precise delineators will often be too rigid. There is much that only applies to "young" and much that only applies to "old" people, but there is little that applies only to twenty-five-year-olds and not to twenty-six-years-olds. Statements with vague terms are not only more flexible, but also more "durable." Most of the relevant distinctions we make in our life are of this kind. They can retain validity for a long time, even if their concrete significance varies with time. The Fuzzy Set Theory can help to improve oversimplified (crisp) models and provide more robust and flexible models for 2

By crisp we mean dichotomous, that is, yes-or-no-type rather than more-orless-type. Classical Logic stipulates that an object either is a member of a class or not. There is no room for gradual transitions. Most of our traditional tools for formal modeling and reasoning are crisp and precise.

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R. C. Carretero

real-world complex systems. The methodology developed in fuzzy sets can be successfully applied in several areas of actuarial science, based on classical probabilistic models, such as rating systems used to provide the public with a sound insurance system. This chapter describes how Fuzzy Logic can be used to make insurance pricing decisions.

2.1

Basic Concepts in the Fuzzy DecisionMaking Processes

Fuzzy Sets were introduced by Zadeh (1965) with a view to reconciling mathematical modeling and human knowledge in the engineering sciences. Since its inception more than decades ago, the theory has matured into a wide-ranging collection of concepts and techniques for dealing with complex phenomena that do not lend themselves to analysis by classical methods using probability theory and bivalent logic. The Fuzzy Set Theory was first developed for solving the inexact and vague problems in the field of artificial intelligence/expert systems, especially for imprecise reasoning and modeling linguistic terms. In solving decision making problems, the pioneer work came from Bellman and Zadeh (1970) and Zimmermann (1987). In classical normative decision theory the components of the basic model of decision making under certainty are taken to be crisp sets or functions. Vagueness only enters the picture when considering decisions under risk or uncertainty and then this uncertainty concerns the happening of a state or event and not the event itself. This uncertainty is generally modeled by means of probability, no matter whether the decision is to be made once only or frequently. In descriptive decision theory this precision is no longer assumed, but ambiguity and vagueness is very often modeled only verbally, which usually does not permit the use of powerful mathematical methods for the purposes of analysis and computation. The first publication in Fuzzy Set Theory by Zadeh (1965) shows the intention of the author to generalize the classical notion of a set and a


235

proposition to accommodate uncertainty in the non-stochastic sense. The basic traditional structure underlying any decision-making problem can be summarized as follows: the existence of limited resources generates the constraints of the problem. In other words, the value of the decision variables that satisfy the constraints defines what is known as the feasible set. Once the feasible set is established, the objective function that suitably reflects the preferences of the decision maker is defined. Finally, by utilizing to more or less sophisticated mathematical techniques, the "best" or "optimal" solution among all possible choices is obtained. In a fuzzy environment, the fuzzy objective function and the fuzzy constraints are defined by their corresponding membership functions. Decision-making problems can be addressed by mathematical programming approaches. Next, a function is developed in which values assigned to the elements of a set fall within a specified range. This function indicates the membership grade of the elements in the set in question. Larger values denote higher degrees of set membership. Such a function is called a membership function (u) and the set it defines is a fuzzy set. In this way, the membership function indicates a subjective degree, within given tolerances, to which an element belongs to set. The value of a membership function is 0 if the constraints (including the objective function) are "strongly violated," is 1 if the constraints are very "well satisfied," or can lie on a continuum from 0 to 1. We argue that the membership function is equivalent to the utility function when the membership function(s) are based on a preference concept such as the utility theory. We can classify linear programming models with imprecise information into two main classes: symmetric and nonsymmetric. The symmetric models are based on the definition of fuzzy decision proposed by Bellman and Zadeh (1970). They assumed that the objective^) and constraints in an imprecise situation can be represented by fuzzy sets. A decision, then, may be stated as the confluence of the fuzzy objective(s) and the constraints, and may be de-

R. C. Carretero

236

fined by a Max-Min operator. The nonsymmetric models are based on the determination of a crisp maximizing decision by aggregating the objective, after appropriate transformations, with the constraints. This interpretation is similar to the well known Lagrangian relaxation approach within the classic techniques of optimization. An application making use of these linear programming models will be given in the non-life insurance section below.

2.2

The Two Fuzzy Non-Life Insurance Models

This section covers the implementation of two "non-precise" fuzzy reasoning methods in a non-life insurance line: automobile insurance. These fuzzy models may provide alternative business models that may be very helpful for insurance carriers. Probability and statistics are basic tools used by an actuary to develop models that describe the nature of both claim frequency and claim severity. Fuzzy models may provide an alternative when creating mathematical models of phenomena that can not be adequately described solely as stochastic. The Fuzzy Set Theory is developed to improve the classical bonus-malus rating system in automobile insurance. 2.2.1

Bonus-Malus System

In most developed countries, rating systems are now in force that penalize the insured responsible for one or more accidents through premium surcharges or maluses, and that reward claim-free policyholders by awarding them discounts or bonuses. Their main purpose, besides encouraging policyholders to drive carefully, is to better assess individual risks, so that everyone will pay, in the long run, a premium corresponding to his/her own claim frequency. The adopted terminology for this system in many European and Asian countries is bonus-malus system.


2.2.2

111

The Data

The sample upon which this study is based includes a set of data on 48,666 policyholders from a Spanish insurance company. The database contains four types of information on each driver: the personal characteristics appearing on the driver's license itself as of the date of the sampling (age, time transpired since the license was issued); the same information for the three year period from January 1, 1992, to December 31, 1994; the number of accidents per year; and the size or severity of each claim. This data is limited to automobile third party liability insurance. This chapter analyzes only the bonus-malus system data applied to "regular" passenger cars. Additional information came from interviews with experts, supervisory authorities and insurance companies. Classical cluster analysis has served to group policyholders into five clusters based on the characteristics they possess, with the claim frequency as the most significant variable. That is to say, as is done almost everywhere in the world, we work with a bonusmalus system exclusively based on the number of accidents reported to the company. It is supposed that every cluster is distributed by a Poisson model with claim frequency A,j (j= 1,..., 5). 2.2.3

The Non-Life Insurance Model Based on Zimmermann's Linear Approach: a Symmetric Model

If one wants to satisfy (optimize) the objective function as well as the constraints, a decision in a fuzzy environment is defined, by analogy to a non-fuzzy environment, as the selection of activities that simultaneously satisfy the objective function "and" the constraints. The logical "and" corresponds to the intersection of the set-theoretic and the intersection of fuzzy sets is defined by the Min-operator. The decision in a fuzzy environment can, therefore, be viewed as the intersection of the fuzzy objective function O and the fuzzy constraints Rj. The relationship between constraints and 3

For the results of this multivariable statistical analysis, see Caro (1999).

238

R. C. Carretew

objective function is fully symmetric; that is to say, there is no longer a difference between the former and the latter. If the decision maker wants to have a "crisp" decision proposal, it seems appropriate to choose the dividend with the highest possible degree of membership in the fuzzy set decision. Then a "maximizing decision" can be defined as |iD(x*) = Max uc(x);

x>0

(1)

X

where the fuzzy set "decision" is characterized by its membership function UD (X) = Min{//b(x), /fc,(x)} ;

i=l,...,n

(2)

If uo(x) has a unique maximum at x*, then the "maximizing decision" is a uniquely defined crisp decision that can be interpreted as the action that belongs to all fuzzy sets representing either the objective function or the constraints and which has the highest possible degree of membership. This becomes a fuzzy rating bonus-malus system based on the Max-Min operator following Zimmermann's linear approach. When the decision-making process is used in a fuzzy environment, Bellman and Zadeh (1970) proposed that the symmetry between goal and constraints is the most important feature. The symmetry eliminates the differences between them. In this context, the authors considered the classical model of a decision under certainty, and suggested a model for decision making in which the objective function as well as the constraint(s) are fuzzy. The fuzzy objective function and the constraints are defined by their corresponding membership functions //o(x) and //R.(X) respectively. This model served as a point of departure for most of the authors in the area of fuzzy decision theory, such as Zimmermann (1987), and many have developed their own methodology. The first non-life rating system described here aims to satisfy an objective function by maximizing the income from premiums. This objective function is subject to the conditions that (1) the system is


239

fair, that is to say, every insured has to pay, at each renewal, a premium proportional to the estimate of his/her claim frequency, and (2) the condition that the system is financially balanced, that is to say, at each stage of this sequential process, the mean of the individual claim frequencies is equal to the overall mean. In addition, the objective function ought to satisfy the optimal conditions of three measures that reflect the severity or "toughness" of a bonusmalus system: the relative stationary average level, the coefficient of absolute variation of the insured's premiums, and the elasticity of the policyholder's payments with respect to their claim frequency.5 The financial balance coefficient is defined as 5

ZniaJ FBC = -tl A

(3) v

'

where nj represents the number of policyholders in the risk group Gj, Oj is the coefficient applied to the premium for a new policyholder (Po), so that the premium that might pay the holder that belongs to Gj is Pj = ctj Po, 0 = 1> •••> 5), and A is the total number of policyholders in the portfolio. The FBC is related to the income from premiums (I) given that 5

I=

n

Z JaJP°

(4)

The three measures that are to be used in the model are defined as follows: the relative stationary average level is a measure of the degree of clustering of the policies in the lowest classes of a bonus4

It is supposed that the premiums are assigned correctly to every risk group, that is to say, the system is fair, after a non-fuzzy approach with the sample available. For details, see Caro (1999). 5 Lemaire (1995) uses similar tools in order to compare and rank thirty different bonus-malus systems from twenty-two countries. However, some variations in the definitions of this non-life rating system are introduced. For further details, see Caro (1999).

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R. C. Carretero

malus system; the coefficient of absolute variation of the insured's premiums is a measure of the severity of a bonus-malus system; and the elasticity of the policyholder's payments with respect to their claim frequency measures, as is well known, the financial response of the system to a change in the claim frequency. The relative stationary average level is defined as FBC-a, l

RSAL =

(5) a5 - a , where, after a non-fuzzy approach with the sample available,6 is considered, that 0:5 - cci = 6.5

(6)

A low value of RSAL indicates a high clustering of policies in the high-discount bonus-malus system classes. A high RSAL suggests a better spread of policies among these classes. The coefficient of absolute variation of the insured's premiums, if the system is financially balanced, has been defined as tz—jV lJIc r J - F B C CAV = -^ (7) A Given the spread of admissible values for the coefficients Oj, we consider that 0Cj - F B C

20

0, and Z, = 0 if Oj-FBCO. 6

For further details, see Caro (1999).

241


Also, two non-negative variables are defined for every coefficient j , Vj and Uj, so that c i j - F B C ^ V j ^ a j - F B C + O-Zj) FBC-aj ^ U j ^ F B C - a j + Z j

; ;

VjP k -(l+Cj k )P J = 0 where k * j , k = 1,2,..., 5, and Cjk * ckj. The values A,j and Xk are estimated with the sample data available.7 As a general rule, however, there will not be perfect elasticity: the change in premium is less than the change in claim frequency. Then, a variable is defined as

Ejk = P k - ( l + c j k ) P j

(15)

and sometimes will be negative and other times positive. If Ejk < 0, when the polyholder goes from Gj to Gk, he/she pays a premium lower than the corresponding to this polyholder and vice versa for E j k >0. 7

For details, see Caro (1999).

242

R. C. Carretero

The insurer seeks to find the optimal incomes, which for financial reasons, ought to be "attractive." If we assume that the objective "attractive income from premiums" or "good FBC" and constraints related to RSAL, CAV and Ejk in a imprecise situation, which depends on the decision-makers, can be represented by fuzzy sets with the assumption of the symmetry, we can make decisions that satisfy both the constraints "and" the goal. The objective function and the constraints are connected to each other by the operator "and" which corresponds to the intersection of fuzzy sets. In such a situation, if the decision-maker is interested not in a fuzzy set but in a crisp decision proposal, it seems appropriate to suggest to him/her the income that has the highest degree of membership in the fuzzy set "decision." Let us call this the "maximizing solution." The fuzzy set of the objective function "good FBC' could, for instance, be defined by H(FBC) =

0 5C -19 1

FBC < 0.95 0.95 < FBC < 1 FBC>1

(16)

and represented graphically as shown in Figure 1. In order to define this membership function (MF) in terms of linear constraints, which has been called MFFBC, three binary variables8 FBC1, FBC2 and FBC3 are defined so that FBC1 = 1 if FBC < 0.95 and 0 else; FBC2 = 1 if 0.95 < FBC < 1 and 0 else; and FBC3 = 1 if FBC > 1 and 0 else. In this sense, FBC1+FBC2 + FBC3 = 1 FBC3 + 0,95FBC2 < FBC FBC < 0, 95FBC1 + FBC2 + 2FBC3 FBC3 < MFFBC < 1 MFFBC < 1 - F B C 1 As an usual practice in lineal programming, we utilize binary variables when the function is defined in sections.


20FBC - 19 - 21 + 21FBC2 < MFFBC MFFBC < 20FBC - 19 + 21 - 21FBC2

243

(17)

Figure 1. Membership function FBC.

The fuzzy set related to the constraint "good RSAL" could be represented, for instance, by 0 RSAL-0.01 H(RSAL) =

0.49 0.8 - RSAL 0.3 0

if RSAL < 0.01 if 0.01 < RSAL < 0.5 (18) if 0.5 < RSAL < 0.8 if RSAL > 0.8

and represented graphically as shown in Figure 2. In order to model this membership function in terms of linear constraints, called MFRSAL, four binary variables RSAL1, RSAL2, RSAL3 and RSAL4 are defined in similar way that the above coefficient so that RSAL1 + RSAL2 + RSAL3 + RSAL4 = 1 0.01RSAL2 + 0.5RSAL3 + 0.8RSAL4 < RSAL RSAL < 0.01RSAL1 + 0.5RSAL2 + 0.8RSAL3 + 6RSAL4

244

R. C. Carretero

MFRSAL < 1 - R S A L 1 MFRSAL < 1 - RSAL4 2.04RSAL - 0.0204 - 100 + 100RSAL2 < MFRSAL MFRSAL < 2.04RSAL - 0.0204 + 100 - 100RSAL2 2.66 - 3.33RSAL- 100 + 100RSAL3 < MFRSAL MFRSAL < 2.66 - 3.33RSAL + 100 - 100RSAL3

(19)

Figure 2. Membership function RSAL.

Ideally, this measure takes value 0.5, that is, 50% of the polyholders pay more than Po and the other 50% pay less than Po. The fuzzy set related to the constraint "good CA V" is characterized, for instance, by 0 if CAV < 0.10 CAV-0.10 if 0.10 < CAV < 0.20 0.10 (20) H(CAV) if 0.20 < CAV < 0.45 1 0.50-CAV if 0.45 < CAV < 0.50 0.05 0 if CAV > 0.50 9

As Lemaire (1995) indicates, in bonus-malus systems that are in force the coefficient of variation of the insured's premiums varies around 0.30.


245

and is represented graphically as shown in Figure 3.

MCAV

A

0.10 0.20

0.45 0.50

> CAV

Figure 3. Membership function CAV.

In order to model this membership function in terms of linear constraints, called MFCAV, five binary variables CAV1, CAV2, CAV3, CAV4 and CAV5 are defined in similar way that the two above coefficients so that CAV1 + CAV2 + CAV3 + CAV4 + CAV5 = 1 0.5CAV5 + 0.1CAV2 + 0.2CAV3 + 0.45CAV4 < CAVAP CAVAP X).

Let kh i = 1,2,3,4, be real numbers and k\ < k2 < kj < k4. A fuzzy number B is a fuzzy subset of the real line R whose membership function //5(a) = //g(a; k\, k2, £3, £4) and is defined as follows: 1. //g: R—»[0,1] is continuous, 2. //s(a) = 0 for a e (-QO, k\\, 3. //fi(a) is strictly increasing in [k\, ki] and is denoted //si(a) and called left wing, 4. juB(a) = 1 for a e [k2, k3], 5. jUsia) is strictly decreasing in [£3, £4] and is denoted //52(a) and called right wing, 6. //^(a) = 0 for a e [£4, °°)Since //si and fisi are continuous and strictly monotonic, their inverse functions exist and are denoted respectively as Vgiy) = VB\(y) = Vmiy) and Vsiy) = Vsi(y) = jumiy)- It follows from the definition of B&, clearly we have Bx = \yB(X), VB(A)] for any A, 0 0 and b2>b\ > 0, we have that [a\, a2\[b\, b2] = [a\b\, a2b2] and [«i, a2\/[bu b2] = [ f l i4 2 ' flî]- F o r any A e [0, 1], we have (A*B)X = AX*BX. ' Following from the definition, we have Theorem 1. If A and B are triangular fuzzy numbers, then we have ki(A + B) = k(A) + kiB),

1= 1,2,3,4

UA - B) = kiA) - kUE),

(1)

i = 1,2,3,4

(2)

IfJfci(4)>0,Jfci(fl)>0,then k£A-E) = UA)-kiB),

/= 1,2,3,4

(3)

'" = ! ' 2 ' 3 ' 4

(4)

If*i04)>0, **(£)>(), then ^ / 5 ) = ^)/£ 5 _,(£>

Proof: It follows from above that (A + B)x = AX + BX. Since (A + B)x = [VA+B(X), VA+B(X)1 for any X > 0, Ax + Bx=

[VA(A), = in(X)

VXX)\

+ WB(X),

+ VB(A), Vl{X)

Vfo)]

+ V&)]

and

= VA(X) + VB(A), Vt+B(A) = Vl{X) + V$(X). Thus UA + B) = k(A) + k(B), i = 1,2,3,4. Similarly, we can prove that VA+B(X)

VA.B{X)=VA{X)-VB'{X)

VtB(X)=VA(X)-VB(X) kt{A -B) = kiA) - ks-iB), i =1,2,3,4. The proofs for the remaining parts are similar.

306

C.-C. Chang

Note that the multiplication and the division of two triangular fuzzy numbers do not necessarily result in a triangular fuzzy number. However, given two TFNs, A and B, we will treat A B and A/B as a TFN simply as a matter of convenience and approximation. This kind of approximation should be acceptable since the very idea of using fuzzy numbers themselves is an attempt to approximate real world problems. While the advantages of treating A-B and A/B this way are obvious, we like especially the fact that fuzzy linear programming (FLP) can then be transferred into ordinary linear programming when multiplication and/or division of fuzzy numbers appear in the FLP. We will see this further in Section 4. If a fuzzy number A degenerates to a positive real number a > 0, i.e., /4A(X) = MA(X', a, «> a, a), then we have V-aB(y) = a- VE(y), Vwa{y) = vB(y)/a,

Ksiy) = a- V£(y), V^/a(y) = rtiy)/a

Similarly, for any sequence of positive real numbers {a,}, / = 1,2,.. .,n and fuzzy numbers B\, Bi,..., Bn, we have

Definition 1. The set of solutions to the fuzzy inequality fix) >b{0) is a fuzzy subset and is defined by

\/sup{y\f{x)>b-0y}\ where 0 < 6< 1 is called tolerance. Note that the concept of tolerance used here was introduced by Zimmermann (1980). A fuzzy mapping/: X -> TFN(Y) is a function that maps x in its domain io fix) in TFN(Y), the set of fuzzy numbers on Y.

307

A Fuzzy Set Approach to Asset and Liability Management

Definition 2. The set of solutions to the inequality fix) > b involving a fuzzy mapping is a fuzzy subset defined by suV\y\Vnx)(\-y)>b\ Definition 3. The set of solutions to the fuzzy inequality fix) > b(0) involving a fuzzy mapping is a fuzzy subset and is defined by

/«

(\-y)>b-0y

Note that we can define fix) < b(0), fix) < b, and fix) < b{0) analogously. Definition 4. The set of solutions to the inequality fix) > g(x) involving two fuzzy mappings is a fuzzy subset and is defined by

[/sup{y\Vl{x)(\-y)>V;jx)(l-y)}\Definition 5. Two fuzzy numbers A and B are said to be s-equal, denoted by A = B(z) if \ki(A) - kt(B)\ < s, for i = 1,2,3,4. Definition 6. Let A„, n = 1,2,..., be a sequence of triangular fuzzy numbers. The fuzzy number A is said to be the limit of An, denoted by \m\A„ = A if lim kj(An) = kt{A) for all / = 1,2,3,4. Theorem 2. Y\mAn = A'\f and only if, for any y, 0 < y < 1, lim F7„(y) = V~A(y) andlim Vl(y) = V}(y). Definition 7. Let F : X -> TFN(Y) be a fuzzy mapping where X and Y are the set of real numbers. Then dx

ô

^

is the derivative of F at x = XQ if the limit as Ax —> 0 exists.

C.-C. Chang

308

Theorem 3. Let At be fuzzy numbers and ft : X —» Y non-fuzzy mappings from X to Y, where X and Y are the set of real numbers. Then iiYAJiix)) = IA^RX).

3

Fuzzy Immunization Theory

For the first example, to show how fuzzy set theory may be applied to the matching of assets and liabilities, we will develop the fuzzy set theoretical analogues of Redington's classic theory of immunization (1952). Let Lt be the net liability-outgo and A, the asset-proceeds, both at time t years. We assume that both At and Lt are fuzzy numbers. Let SQ be the current force of interest. Let VA(5Q) and VL{8Q) be the fuzzy values at this force of interest of the asset-proceeds and the net liability-outgo respectively. The conditions for fuzzy immunization against small changes in the rate of interest are as follows:

vA(s0) = (yL(s0) + T)(£l)

(5)

where ris a small positive number, VAS0) = VAS0)(e2)

(6)

vl'{S0) > v"(50)

(7)

If these conditions are satisfied, it follows that the function7(<S) = (YA{S) - VL(S) - T) (si) equals zero when 8= 8$ and has a relative minimum value there. Thus, there is a neighborhood of 8Q such that if 8\\QS therein but is not equal to 8Q, then VA(8) > Vi{8) + r. An investor whose investments are such that the above three conditions hold is thus said to be immunized against small changes in the rate of interest. The immunization exists because any immediate small change in the rate of interest will lead to a surplus, since the present value of the assets will exceed that of the net liabilities plus r.

309


Now we proceed to discuss how the above conditions may be interpreted in practice. Since Lt = St- Ph where St is the liabilities and Pt is the receipts at time t, we can rewrite equations (5) and (6) as follows:

*,Ete+4y)-*,(S<S/)-r I < ex, i=l,2,3,4 and kfet(Pt

+A,y)-k,fetStv')

| < s2,

i= 1,2,3,4,

both evaluated at force of interest So. Since r Y,KPl+A,y}_ k,fct(pl+Aty) and

TJts,v')_ kfosy) .5>'J k^syy we have

1(c/r))]^ r

A Fuzzy Set Approach

to Asset and Liability

Management

313

K(y)= (

W-t+l

i=2 V nTi

N-k+l

\k-\

'=1

/ '=1

\k-\

r

I i- lMlk(rt,k(cFr)+(^(^x4(cFr)-^(7W>3(cFr))]^ *=2 V

i'=l

J '=1

0,m > k-\ where kiiyhK) = \,m = k-\ k-\

k-l-\

Vi+^,0';) I" ZO */0'teU-i>'"î l=m+\

Let S be a specified set of interest rate patterns /. Let C = (Ci, C2, ..., C„) be the current market price of investment instruments. Assume C„ /' = l,2,...,n, are fuzzy numbers. An objective of fuzzy matching of assets and liabilities is to maximize / = C\P\ + C2P2 + ... + CnPn, where P, > 0, XJ=1 Pj = 1 and AÛ, r,P)>0,l= l,2,...,w. We have thus expressed the objective in terms of a fuzzy linear programming (FLP) problem. Since the fuzzy number AN is treated as triangular, the wings of AN, i.e., VA„ and VA„, are linear functions and so there are ways to transform FLP problems to ordinary linear programming problems and solve accordingly. For the above fuzzy linear programming problem, the equivalent ordinary linear programming problem can be described as follows: Given fi> 0, solve the linear programming problem: Maximize IM = wVJ(ji) + (1 - w)Vl{ju) such that Pj > 0,1,]=^= 1 and VAn(ihr,p){\ - ju) + judi > 0, / = 1,2,...,m, where Oi is a given set of tolerances. For further examples, see Chang and Lee (1993) and Zimmermann(1980). The above development of Tilley's conclusions on the matching of assets and liabilities into fuzzy set theoretical analogues can be

314

C.-C. Chang

similarly extended to recent results of Shiu (1988), Kocherlakota (1988, 1990) and Reitano (1991, 1993). However, the extensions are quite tedious and the resulting formulas become very cumbersome. We thus stop our fuzzy set theoretical approach to asset and liability management here and turn our attention to fuzzy decision making.

5

Bayesian Decision Method

Classical statistical decision making involves the notion that uncertainty in the future can be characterized probabilistically. When one wants to make a decision among various alternatives, one's choice is predicated on information about the future that is normally classified into various states of nature. If one knew with certainty the future states of nature (crisp states), one would not need an analytic method to assess the likelihood of a given outcome. Unfortunately, we do not know what the future will entail, under most circumstances, so we have devised methods to make the best choices given an uncertain environment. Classical Bayesian decision methods presume that future states of nature can be characterized as probability events. For example, consider the condition of cloudiness in tomorrow's weather by classifying the state space into, say, three levels (very cloudy, cloudy, and sunny) and assessing each level probabilistically. The problem with the Bayesian method is that the events are vague and ambiguous. How can one clearly distinguish between cloudy and very cloudy? If there is one small cloud in the sky, does one classify it sunny or cloudy? In what follows, we shall first present Bayesian decision making and then start to consider ambiguity in the value of new information, in the states of nature, and in the alternatives in the decision process. Let S = {s\, S2, ..-, sn} be a set of possible states of nature and let the probabilities that these states will occur be listed in a vector as

315


follows:

?={p{s]),p(s2),-,p(sn)},

(8)

where Z"=1 p(s,) = 1. The probabilities shown in (8) are called "prior probabilities" in the Bayesian decision method, because they express prior knowledge about the true states of nature. Assume now that the decision maker can choose among m alternatives A = {a\, a-i, ..., am). For a given alternative a,- we assign a utility value, Ujh if the future state of nature turns out to be state st. These utility values should be determined by the decision maker since they express value, or cost, for each alternative-state pair, i.e., for each a, - s, combination. The utility values are usually arranged in a matrix as shown in Table 1. Table 1. Utility matrix.

^\States Alternatives^'^ a, • a

m

*1

s2

uu

un

.

. U ml i

"ml

s„ . . .

U

\n

* . . .

u mn

The expected utility associated with, say, the y'th alternative would be

4*,) = i X , / 4 )

(9)

(=i

The most common decision criterion is the maximum expected utility among all the alternatives, i.e., E\uj=

max E{Uj)

(10)

which leads to the selection of alternative a^ if E(u*) = E(uk). A simple example will illustrate the above. Suppose you are an

C.-C. Chang

316

engineer who is asked by the CEO of a large oil company to help make a decision about whether to drill for natural gas in a particular geographic region of a foreign country. You determine that there are only two states of nature regarding the existence of natural gas in the region: si = there is natural gas; S2 - there is no natural gas, and you are able to find from previous drilling data that the prior probabilities for each of these states is p(si) = 0.5; p{s2) = 0.5. Assume that there are two alternatives in this decision: a\ = drill for gas; «2 = do not drill for gas. By way of helping you assemble a utility matrix, the CEO tells you that the best situation for him is to decide to drill for gas, and subsequently find that gas is indeed in the region. He assesses this value Mn as 5. However, he thinks that the worst possible situation would be to drill for gas and subsequently find that there is no gas at all in the area. He determines that the value for this would be «i2 = -10. The other two utilities are assessed by the CEO to be W21 = -2 and «22 = 4. Hence, the utility matrix is given by U=

r 5 [-2

-io" 4_

The situation may further be shown as a decision tree in the following:

317


utility 0.5

s2 \

H„

=5

0.5 un = - 1 0

Decision flX

S{

y °-5

2i = - 2

M

0.5 u22 = 4 The expected utility for each alternative a\ and a% is: E(u{) = (0.5)-(5) + (0.5H-10) = -2.5; E(u2) = (0.5)-(-2) + (0.5>(4) = 1.0. The maximum utility is 1.0, which comes from alternative a2. Hence, based on prior probabilities only, the CEO decides against drilling for natural gas. In many decision situations an intermediate issue arises: Should you get more information about the true state of nature prior to making the decision? Suppose some new information regarding the true states of nature S is available from r experiments or observations and is collected in a vector, X= {x\, x2, ..., xr). This information can be used in the Bayesian approach to update the prior probabilities, p(s,), in the following manner. First, the new information is expressed in the form of conditional probabilities, where the probability of each piece of data, Xk, k = l,2,...,r, is assessed according to whether the true state of nature, st, is known. This means that, given that we know the true state of nature is Sj, the probability that the piece of new information Xk confirms that the true state is s.

318

C.-C. Chang

is p(x/c\si). These conditional probabilities p(xk\s,) are also called likelihood values. The likelihood values are then used as weights on the prior probabilities p(s,) to find updated probabilities called posterior probabilities and denoted p(sj\xk). The posterior probabilities are equivalent to this statement: Given that the piece of new information Xk is true, the probability that the true state of nature is St is p{s,\xk). These updated probabilities are determined by Bayes' rule as follows: P\si\xk)=

(

\ P\si)

(11)

PM where the term in the denominator, p(xk), is the marginal probability of the data Xk and is determined using the total probability theorem P(xk) = 1[dp(xk\s,)-P(si)

(12)

Now the expected utility for they'th alternative, given the data Xk, is determined from the posterior probabilities instead of the prior probabilities, E u

( j\xk)=^jiP(si\xk)

(13)

1=1

and the maximum expected utility, given the new data x^ is now given by ii(w*.xt) = max E\Uj\xk) (14) To determine the unconditional maximum expected utility we need to weigh each of the r conditional expected utilities given by (14) by the respective marginal probabilities for each datum Xk, i.e., by p(xk) as shown below: Eu =

( l)

T,Eiu*\xk)-PM k=\

(15)


319

We can now introduce the concept of value of information denoted V(x). In the case where there is some uncertainty about the new information X — {x\, xi, ••-, xr), we call the information imperfect information. The value of this imperfect information V(x), can be assessed by taking the difference between the maximum expected utility without any new information and the maximum expected utility with the new information, i.e., V(x)=

E(U*X)-E{U)

(16)

We now introduce the concept of perfect information. Perfect information can never be achieved in reality, but can be used to assess the value of imperfect information. If information is considered to be perfect, then each new piece of information or data predicts one and only one state of nature and there is no ambivalence about what state is predicted by the data. However, if there is more than one piece of information, the probabilities for a particular state of nature have to be shared by all the data. Mathematically, perfect information is represented by posterior probabilities of 0 or 1, i.e.,

We call this perfect information xp. For perfect information, the maximum expected utility becomes E(u;p)=f,E(u;p\xk)-p(xk)

(18)

and the value of perfect information becomes V(XP)=E(U;P)-E(U')

(19)

Let us continue the natural gas example where we had two states of nature — s\ with gas and S2 without gas — and two alternatives — a\ to drill and 02 not to drill. The prior probabilities were uniform: p(si) = 0.5, p(s2) = 0.5.

320

C.-C. Chang

Suppose that the CEO reconsiders his utility values and provides you with a new utility matrix as follows: Table 2. New utility matrix. \

s2

4 -1

-2 2

s

U]i a

\

a2

Suppose further that the CEO has asked you to collect new information by taking eight geological testing samples from the region being considered for drilling. Assume that the results of these eight tests can be expressed as conditional probabilities in the form of a matrix as shown in Table 3. Table 3. Conditional probabilities for imperfect information. X,

p(xk

h)

p(xk

\sl)

0

6

x7

xg

0.2

0.4

0.1

0.5 y"Vow==i

0.1

0.1 0.05

xA

x5

0.05 0.1

0.1

0.4

0.2

x2

0.05 0.1

*3

X

0

yyow==i

For comparison purpose, we assume that the eight tests were capable of providing perfect information and the conditional probabilities are thereby changed as follows: Table 4. Conditional probabilities for perfect information.

p(xk

h)

p(xk

\sl)

X,

x2

x3

x4

x5

*6

x7

xg

0

0

0

0

0.2

0.5

0.2

0.1 ^row- = 1

0.1

0.2

0.5

0.2

0

0

0

0

y^row -= 1

Since the CEO changed his utility values, you have to recalculate the expected utility of making the decision on the basis of just the prior probabilities, before any new information is acquired. The


321

decision tree for this situation now looks as follows: 4 «„

-2 un Decision -1 u2X

The expected utilities and maximum expected utility, based on prior probabilities only, are £(a,) = (4>(0.5) + (-2H0.5)=1.0; E(a2) = (-1X0.5) + (2)-(0.5) = 0.5. E(u*) = 1 therefore, the choice is alternative a\\ drill for natural gas. We are now ready to assess the changes in this decision process by considering additional information, both imperfect and perfect. Let us first calculate marginal probabilities for the new imperfect information. Using (12), the conditional probabilities for imperfect information and the prior probabilities, we obtain, for example, p(Xl) = (0)-(0.5) + (0.05)-(0.5) = 0.025 p(x4) = (0.1)-(0.5) + (0.2)-(0.5) = 0.15 Now we can calculate the posterior probabilities. Using (11), the conditional probabilities for imperfect information, the prior probabilities and the marginal probabilities just obtained, we have, for example,

322

C.-C. Chang

I | \ 0.05-(0.5) 1 / | \ 0.4-(0.5) 4 p[s, \x2) = -—- = - , pis, \x6) = -—'- = —, v ' ; 0.075 3 V ' ' 0.25 5 / , \ 0.1-(0.5) 2 / , \ 0.1(0.5) 1 p\sV2\x2) = -—- = —,P\s -—- = -,•••• 2\X(,) = ' ' 0.075 3 V21 6 / 0.25 5 The conditional expected utilities, E(u*\xk), are calculated using first (13), then (14); for example,

£("i|x3) = (4>(4) + (4).(-2) = - 4 ; ^ 2 | X 3 ) = ( 4 K - 1 ) + ( 4 H 2 ) = 4.

Hence, E{u*\x^) = max(--j, 3) = -| (choose alternative ai). £(«I|X 8 ) = ( 1 H 4 ) + ( 0 K - 2 ) = 4

£(«2|x8) = (l>(-l) + (0H2) = - l Hence, £(«*|x8) = max(4, -1) = 4 (choose alternative a\). All the above results are summarized in Table 5. Table 5. Posterior probabilities based on imperfect information. x8 x7 x5 x3 X, x2 x4 *6 p(>\ ) p\s\ \

•*

x

k)

l

p(s2

\xk)

E\u* a

j \

h)

x

k

0.025 0.075 0.25 1 1 0 3 5 2 4 — — 1 3 5 7 2 1 5 a2 a2 a2

0.15 1

0.15 2

0.25 4

3 2

3 1

5 1

—

—

—

—

—

3

3

1

2

5 14 5

a2

"1

«i

0.075 0.025 2 — 1 3 1 — 0 3 2

4

a,

a,

We can now use (15) to calculate the overall unconditional expected utility for imperfect information, which is actually the sum of pairwise products of the values in rows 1 and 4 of the above table, e.g., E(ux) = (0.025)-(2) + (0.075X1) + - + (0.025)-(4) = 1.875

323


and the value of the new imperfect information is V(x) = E{u*x) E(u)= 1.875 - 1 = 0 . 8 7 5 . To decide which alternative to choose, notice from Table 5 that the total utility favoring a\ is 10.8 = (2 + -y +2 + 4) and the total utility favoring a2 is 5.4 = (2 + 1 + 3 + 1). Therefore, the CEO chooses alternative a\ to drill for gas. We can now use conditional probabilities for perfect information to replace those for imperfect information and perform all the calculations leading to p(xk), p{s]\xk), pfalxk), E(u*\xk), and a}\xk- The results are summarized in Table 6. Table 6. Posterior probabilities based on perfect information. 6

x7

xg

0.1

0.25

0.1

0.05

0

1

1

1

1

1

1

0

0

0

0

2

2

2

4

4

4

4

a2

a2

a2

a

a,

a

a,

X,

x2

x3

x4

x5

PM

0.05

0.1

0.25

0.1

p(sl\*k)

0

0

0

p(S2^k)

1

1

E(u'\xk)

2

°j\xk

a2

\

X

\

Equation (18) can be used to calculate the overall unconditional expected utility for perfect information, which is actually the sum of pairwise products of the values in rows 1 and 4 of the above table. For example, E(u*Xp) = (0.05>(2) + (0.1>(2) +...+ (0.05>(4) = 3.0 and the value of the new perfect information is v(xp)=E(u;p)-E(u*)

= 3- 1=2.0.

Alternative a\ is still the choice here. Note that the hypothetical perfect information has a value of 2 and the imperfect information has a value of 0.875. This difference can be used to assess the value of the imperfect information compared to both no information (1) and perfect information (3).

324

6

C.-C. Chang

Fuzzy Bayesian Decision Method

We now consider the situation where the new information may be inherently fuzzy. Let X = {x\, X2, ..., xr} be the new information. Then we can define fuzzy events, M, on this information, such as "good" information, "moderate" information, and "poor" information. The fuzzy event will have membership function, juyixk), k = l,2,...,r. We can now define the concept of a "probability of a fuzzy event", i.e., the probability of Mas

P(M) = YJMMPM

(20)

If the fuzzy event is, in fact, crisp, i.e., M=M, then the probability reduces to

P{M)= £ > ( * * ) / < * = { \

êM

(21

>

X$M

[ 0 otherwise where (20) describes the probability of a crisp event simply as the sum of the marginal probabilities of those data points Xk that are defined to be in the event M. Based on this, the posterior probability of st, given fuzzy information Mis P(v

I w\ = £M**hW(**M j ,-) =

{ , m )

P(M\S,)P(S,)

p(M)

p(M)

(

n~ ]

where

P(M\S, ) = X p(xk \s, )MK (xk )

(23)

k=\

We can now define the collection of all the fuzzy events describing fuzzy information as an orthogonal fuzzy information system, Y= {Mi, M2, ..., Mz} where by orthogonal we mean that the sum of the membership values for each fuzzy event Mj, for every data point Xk, equals 1 (Tanaka 1976). That is

325


I>M ; ta) = lfora11 xk eX

(24)

/=i

If the fuzzy events on the new information are orthogonal, we can extend the Bayesian approach to consider fuzzy information. The fuzzy equivalents of (13), (14), and (15) become, for a fuzzy event, Mt, E[UJ\MJ=YJUMP{S}ML)

(25)

E\u* \Mt) = max£(w J M , J

(26) (27)

E{UY)=YSE(U*M^-P{M)

The value of fuzzy information can now be determined analogously as V(Y)

= E(u'y )-E(U')

(28)

We can continue our gas example to illustrate the above by assuming that test samples are inherently fuzzy and by defining an orthogonal fuzzy information system F a s 7 = {M\, Mj, M$} ~ {poor data, moderate data, good data} with membership functions as shown in Table 7. Table 7. Orthojjonal membership functions for orthogonal fuzzy events. X,

x2

x3

x4

x5

*6

x7

xg

/"M,(*J

1

1

0.5

0

0

0

0

0

MM^M

0

0

0.5

1

1

0.5

0

0

MuMk)

0

0

0

0

0

0.5

1

1

0.15

0.15

0.25

P(xk)

0.025 0.075 0.25

0.075 0.025

Note that the fourth row is the same as that from posterior prob-

326

C.-C. Chang

abilities based on imperfect information and the sum of the membership values in each column (the first three rows) equals 1 as required for orthogonality. We can now use (20) to determine the marginal probabilities for each fuzzy event, P ( M j = 0.0225 P(M2) = 0.55 P ( M J = 0.225 and (23) to determine the fuzzy conditional probabilities, P{M^SX)

= 0.\

P{M2\SX)

=

0.55 p(M3\s1) = 0.35;

P[Mj\s2) = 0.35 P(M^s2) = 0.55 P(M 3 \S 2 ) = 0.1 and (22) to determine the fuzzy posterior probabilities,

^ U M J = 0.222 p(sx M J = 0.5 p(sx AfJ = 0.778; p[s2 Mx) = 0.778 p(s2 M2) = 0.5 p(s2 M3) = 0.222. The conditional fuzzy expected utilities can now be determined by using (25),

^ U M J = (4X0.222) + (-2)-(0.778) = -0.668; ^(WJMJ

= (-1X0.222) + (2)-(0.778) = 1.334;

E(U\M^)

= (4)-(0.5) + (-2)-(0.5) = 1.0;

£(W 2 |MJ

= (-l)-(0.5) + (2)-(0.5) = 0.5;

EUMJ

= (4)(2.668) = 1.45 and the value of the fuzzy information from (28), V(Y) = 1.45 - 1 = 0 . 4 5 . We can see here that the value of the fuzzy information is less than the value of the perfect information (2.0), and less than the value of the imperfect information (0.875). However, it may turn out that fuzzy information is far less costly than either the imperfect or perfect information.

7

Decision Making under Fuzzy States and Fuzzy Alternatives

We will now extend the Bayesian decision method to include the possibility that the states of nature are fuzzy and the decision makers' alternatives are also fuzzy (Tanaka 1976). As an example, suppose your company wants to expand and you are considering three fuzzy alternatives in terms of the size of a new facility: A\ = small-scale facility Ai = middle-scale facility A3 = large-scale facility Suppose further that the economic climate in the future is very fuzzy and you pose the following three possible fuzzy states of nature: F\ = low rate of economic growth F2 = medium rate of economic growth F\ = high rate of economic growth all of which are defined on a universe of numerical rates of economic growth, say S, where S = {s\, S2, ..., sn} is a discrete universe of economic growth rates. The fuzzy states Fs. will be required to be orthogonal and this orthogonal condition on the fuzzy states will be

C.-C. Chang

328

the same constraint as shown in (24), i.e.,

5 X M = 1, i = l,2,...,n

(29)

Since we need utility values to express the relationship between crisp alternative-state pairs, we still need a utility matrix to express the value of all the fuzzy alternative-state pairings. Such a matrix will have the form shown in Table 8. Table 8. Utility values for fuzzy states and fuzzy alternatives.

5_

5L

^

A

un

"12

«13

4

"21

W

22

«23

"31

W

32

W33

A

Now with fuzzy states of nature, the expected utility of fuzzy alternative Aj is E(UJ)=Y.UJSP{F),

(30)

s=\

where

pfe) = i>£,W/>fo)

(31)

.5=1

and the maximum utility is E[u*) = max E\Uj).

(32)

We can have crisp or fuzzy information on a universe of information X= {xi, X2, ..., xr}. Our fuzzy information will again reside on a collection of orthogonal fuzzy sets o n J , Y= {M\, M2,..., Mz}, that are defined on X. Given probabilistic information xr and fuzzy information Mj, we can derive the posterior probabilities of fuzzy states Fs_ as follows:


329

n

M F ,k = n

p^Mj

v(xp)>V{x)>V{Y)>0

(46)

These inequalities are consistent with our intuition. The inequality V(x) > V(Y) is due to the fact that information Y is characterized by fuzziness and randomness. The inequality V(xp) > V(x) is true because xp is better information than x; it is perfect. The inequality V(YP) > V(xp) holds because the uncertainty expressed by the probability P(Fj) still remains, even if we know the true state st. Hence,


331

our interest is not in the crisp states of nature S but rather in the fuzzy states F which are defined on S.

8

Conclusions

This chapter first develops fuzzy-set theoretical analogues of the Redington theory of immunization and the matching of assets and liabilities. By translating the existing knowledge about immunization and the matching of assets and liabilities into fuzzy set theory, we are able to view the problems and solutions on a broader horizon. Moreover, this approach offers the advantages of using fuzzy set theory such as flexibility and closer to real-world situations. It is believed that successful applications of fuzzy set theory to many other areas of finance and economics are highly likely. Based on Bayesian decision method, this chapter then uses a fuzzy set theoretical approach to develop a fuzzy Bayesian decision method, and finally extends the Bayesian decision method to include the possibility that the states of nature are fuzzy and the decision makers' alternatives are also fuzzy. The attributes of a typical decision making problem are that there are many states of nature, feasible policy alternatives and available information. Usually, the utilities for all the states and all the alternatives cannot be formulated because of insufficient data, the high cost of obtaining the data, or time constraints. On the other hand, top managers mostly want to decide roughly what alternatives to select simply as indicators of policy directions. Thus, an approach that is based on fuzzy states and fuzzy alternatives and that can accommodate fuzzy information is a very powerful tool for making such preliminary policy decisions.

332

C.-C. Chang

References Buehlmann, N. and Berlinder, B., "The fuzzy zooming of cash flows," (unpublished working paper). Chang, P.T. and Lee, E.S. (1993), "Fuzzy decision making: a survey," in Wang, P.Z. and Lee, K.F. (eds.), Between Mind and Computer, Advances in Fuzzy Systems, World Scientific, London, pp. 139-182. Dubois, D. and Prade, H. (1980), Fuzzy Sets and Systems, Academic Press, New York. Kocherlakota, R., Rosenbloom, E.S., and Shiu, E.S.W. (1988), "Algorithms for cash-flow matching," Transactions of the Society of Actuaries, 40:477-84. Kocherlakota, R., Rosenbloom, E.S., and Shiu, E.S.W. (1990), "Cash-flow matching and linear programming duality," Transactions of the Society of Actuaries, 42:281-93. Okuda, T., Tanaka, H., and Asai, K. (1974), "Decision making and information in fuzzy events," Bull. Univ. Osaka prefect., Ser. A, vol. 23, no. 2, pp. 193-202. Okuda, T., Tanaka, H., and Asai, K. (1978), "A formulation of fuzzy decision problems with fuzzy information, using probability measures of fuzzy events," Inf. Control, vol. 38, no. 2, pp. 135-147. Redington, F.M. (1952), "Review of the principles of life-office valuations," Journal of the Institute of Actuaries, vol. 78, pp. 268-315. Reitano, R.R. (1991), "Multivariate immunization theory," Transactions of the Society of Actuaries, 43:393-428. Reitano, R.R. (1993), "Multivariate stochastic immunization theory," Transactions of the Society of Actuaries, 45:425-61. Shiu, E.S.W. (1988), "Immunization of multiple liabilities," Insurance: Mathematics and Economics, 7:219-24. Tanaka, H., Okuda, T., and Asai, K. (1976), "A formulation of fuzzy decision problems and its application to an investment


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problem, " Kybernetes, vol. 5, pp. 25-30. Tilley, J. A. (1980), "The matching of assets and liabilities," Transaction of the Society of Actuaries, vol. 32, pp. 263-300. Zadeh, L.A. (1965), "Fuzzy sets," Information and Control, vol.8, pp. 338-353. Zimmermann, H.J. and Zysno, P. (1980), "Latent connectives in human decision making," Fuzzy Sets and Systems, vol. 4, pp. 3751.


Industry Issues


Chapter 9 Using Neural Networks to Predict Failure in the Marketplace Patrick L. Brockett, Linda L. Golden, Jaeho Jang, and Chuanhou Yang

This chapter examines the indicators of marketplace failure for business enterprises, with a specific application to predicting insurance company insolvency. Both the US Property and Casualty insurance industry and the US Life Insurance industry are examined. Various approaches are contrasted including discriminant analysis, logistic regression analysis, k-nearest neighbor, expert rating agencies (A. M. Best ratings, and the National Association of Insurance Commissioners' Insurance Regulatory Information System and Financial Analysis Tracking System), and variants of neural networks. It is shown for both industries that the neural network models exhibit high success relative to competing models for predicting market failure of firms.

1

Introduction

The definition and measurement of business risk has been a central theme of financial and actuarial literature for years. Although the works of Borch (1970), Bierman (1960), Tinsley (1970), and Quirk (1961) dealt with the issue of corporate failure, their models did not lend themselves to empirical testing. Other work such as that of Altman (1968), Williams and Goodman (1971), Sinkey (1975), and Altman, Haldeman, and Narayanan (1977) attempted to predict 337

338

P. L. Brockett et al.

bankruptcy or the propensity towards market failure for a particular firm by using discriminant analysis. Failure of firms within certain regulated industries, such as insurance and utility providers, has become a major issue of public debate and concern. Particularly since the terrorist attacks on the world trade center on September 11, 2001, the identification of potentially troubled firms has become a major regulatory research objective. Previous research on the topic of insurer insolvency prediction includes Ambrose and Seward (1988), BarNiv and Hershbarger (1990), BarNiv and MacDonald (1992), and Harrington and Nelson (1986), Brockett et al. (1994), Huang et al. (1994), and Brockett et al. (2002). This chapter gives a review of effectiveness of using neural network models for predicting market failure (insolvency or financial distress) at the firm level. The industry used for application is the insurance industry because of the important social implications of market failure in this industry, as well as the public availability of data at the firm level capable of being analyzed by the public without having to have proprietary data access. The results of neural network models are compared with those of discriminant analysis, logistic regression analysis, A. M. Best ratings, and the National Association of Insurance Commissioners' Insurance Regulatory Information System (NAIC) ratings, and the neural network results show high predictability and generalizability, suggesting the usefulness of neural network approaches for predicting future insurer insolvency.

2

Overview and Background

2.1

Models for Firm Failure

In the context of investigating the financial stability of firms within the insurance industry, the consumer (and regulator) has several

Using Neural Networks to Predict Failure in the Marketplace

339

sources of information. For example, the reporting and rating services from the A. M. Best Company as well as Moody's and Standard and Poors provide the consumer with knowledge useful for determining whether or not the firm considered for potential business is likely to be around when a claim is filed. In addition, the NAIC has developed Insurance Regulatory Information System (IRIS) and Financial Analysis Tracking System (FAST) to provide an early warning system for use by regulators. The NAIC also adopted risk based capital (RBC) formula for insurance prediction. The IRIS system was designed to provide an early warning system for insurer insolvency based upon financial ratios derived from the regulatory annual statement. The IRIS system identifies insurers for further regulatory evaluation if four of the eleven (or 12, in the case of life insurers) computed financial ratios for a particular company lie outside a given "acceptable" range of values. IRIS uses univariate tests, and the acceptable range of values is determined such that, for any given univariate ratio measure, only approximately 15 percent of all firms have results outside of the particular specified "acceptable" range. The adequacy of IRIS for predicting troubled insurers has been investigated empirically and found not to be strongly predictive. For example, one could use the IRIS ratio variables from the NAIC data with more sophisticated statistical methods to obtain substantial improvements over the IRIS insolvency prediction rates (cf., Barrese 1990, Brockett et al. 1994). One of the criticisms of IRIS is that it is too dependent on capital and surplus figures, and another is that IRIS fails to take into account the relationships between the ratios. In response to these criticisms, NAIC developed the FAST system, which is supposed to eliminate some of the problems associated with the IRIS system. Unlike the original IRIS ratios, the FAST system assigns different point values for different ranges of ratio results. A cumulative score is derived for each company, which is used to prioritize it for further analysis. In addition, the risk-based capital systems adopted by the NAIC may enhance regulators' ability to identify problem in-

340


surers prior to insolvency. Evaluations of the accuracy of the riskbased capital system for property-liability insurers are provided in Grace, Harrington, and Klein (1998) and Cummins, Harrington, and Klein (1994). Since the multiple discriminant analysis (MDA) method was introduced by Altman in 1968, it has become a commonly used parametric method for predicting financial distress in various industries (cf., Edmister 1972, Sinkey 1975). However, MDA has received much criticism since the data used to determine the propensity toward marketplace failure of a firm often violate the statistical assumptions of this model. Similar problems exist for the logistic regression model, another important technique used in the previous literature for insurer insolvency prediction. To overcome the problems associated with parametric techniques, nonparametric methods have become popular in recent studies. One such model is the neural network model.

2.2

Neural Network and Artificial Intelligence Background

The neural network model can be represented as a massive parallel interconnection of many simple processing units connected structurally in much the same manner as individual neurons in the brain. Just as the individual neurons in the brain provide "intelligent learning" through their constantly evolving network of interconnections and reconnections, artificial mathematical neural networks function by constantly adjusting the values of the interconnections between individual neural units. The process by which the mathematical network "learns" to improve its performance, recognize patterns and develop generalizations is called the training rule for the network. The learning law proposed by Hebb in 1949 served as the starting point for developing the mathematical training algorithms of neural networks. The subsequent development of the back-propaga-


341

tion "training rule" resolved computational problems outstanding for two decades and significantly enhanced the performance of neural networks in practice. This rule is based on a "feed forward" network that essentially designates that the flow of the network intelligence is from input toward output. The "back propagation" algorithm updates its interconnection weights by starting at the output, determining the error produced with the particular mathematical or logical structure and then mathematically propagating this error backward through the network to determine, in the aggregate, how to efficiently update (adjust) the mathematical structure of interconnections between individual neurons in order to improve the predictive accuracy of the network. The method of updating weights to reduce total aggregate error is one of steepest gradient decent familiar to statistical estimation procedures such as maximum likelihood estimation. 2.2.1

The General Neural Network Model

All neural networks possess certain fundamental features. For example, the basic building block of a neural network is the single "neural processing unit" or neuron, which takes the multitude of individual inputs x = (xi, X2, ..., x„) to the neural unit, determines (through the learning algorithm) optimal connection weights w = (wi, W2, ..., wn), which are appropriate to apply to these inputs, and then aggregates these weighted values in order to concatenate the multiple inputs into a single value A(w,x) = £"=1 Wj-x1 . An activation function, F, is then applied, which takes the aggregated weighted value A(w,x) for the individual neural unit and produces an individual output F(A(w,x)) for the neural unit. The logistic activation function F(z) = l/(l+exp(-r|-z)) is the commonly used activation function, although other sigmoid functions such as the hyperbolic tangent function have been used depending upon the situation. Figure 1(a) graphically displays the configuration of the single neural processing unit or neuron as described above.

342


"Output" of neural unit •£xiwi FjCSxiWi) wx

Aggregation

"Input" to neural unit Figure 1(a). A single neural processing unit (neuron) j . ,0)

"Output" of neural net

O)

•

niwf^cixi.w^)) i=l

Input layer 1

Hidden layer 2

Output layer

Figure 1(b). Multiple neural processing units. Each circle in the hidden layer represents a single neural processing unit.

Until now, we have described a single neural processing unit. The same process, however, can be applied to an array of neural processing units. In practice, the neural processing units are grouped together to form composite topological structures called neural layers, and subsequently the units in sequential layers are


343

connected via interconnection weights to obtain an ultimate topology for the network (see Figure 1(b) above). 2.2.2 Network Neural Processing Units and Layers Artificial neural networks are formed by modeling the interconnections between the individual neural processing units (just as the human brain functions through interconnections between the biological neurons through development of the synapses). The feed forward networks without a hidden layer are networks that have an input layer containing the input data information obtained from an external source and an output layer providing output information. It follows from the neural network topology given in Figure 1 that in the neural network without a hidden layer with a logistic activation function and a single 0-1-output node, the mathematical structure developed is isomorphic to the standard logistic regression model. Multilayer networks, which possess a "hidden" or intermediate processing layers between the input layer and output layer, can overcome many of limitations of the simple network model without hidden layers in representing complex nonlinear rules. The topology of the neural network determines its ability to mathematically simulate the observed process of going between the input and output values. One aspect of this topology is the number of hidden layers to assume. It follows from a theorem of Kolmogorov (cf., Lorentz 1976, Kurkova 1992) that any continuous function of N variables can be computed using only linear summations and a nonlinear but continuously increasing function of only one variable. A consequence of this theorem is that a single hidden-layer network can approximate any continuous function of N variables. It has been proven that the multilayer architecture of the model allows for universal approximations of virtually any mapping relationship (Funahashi 1989, Hornik, Stinchcombe and White 1990). The hidden layers are used to represent non-linearities and interactions between variables (Lippmann 1987). Essentially, these results show that the class of single hidden layer neural network models is

344


"dense" in the space of all continuous functions of N variables, so that whatever the "true" (but unknown) functional relationship between the input variables and the output variable, it can be well approximated by a neural network model. This finding substantially encourages the use of a single hidden layer neural network for most complex behavioral decision problems. Indeed, such three-layer neural networks have been able to "predict" the failure of savings and loan companies (cf, Salchenberger, Cinar and Lash 1992). 2.2.3

The Back-Propagation Algorithm

Much like the techniques used for maximum likelihood estimation, the back-propagation algorithm can be viewed as a gradient search technique wherein the objective function is to minimize a mean squared error between the computed outputs of the network corresponding to the given set of inputs in a multilayer feed-forward network and the actual observed outputs for these same given inputs. A difference is that the back-propagation algorithm sequentially considers data records one at a time, readjusting the parameters (in a gradient search manner) after each observation. On the other hand, the statistical procedures (maximum likelihood, least squares) use an aggregated error of the estimation almost as if in "batch" mode. In back propagation, the parameters are changed after each data point and the process is continued with the same individual observation presented to the algorithm over and over. In statistical methods the data is only presented to the algorithm once in a batch. Specifically, the neural network is trained by presenting an input pattern vector X to the network and computing forward through the network until an output vector O is obtained. The output error is computed by comparing the computed output O with the actual output for the input X. The network attempts to learn by adjusting the weights of each individual neural processing unit in such a fashion as to reduce this observed prediction error. Mathematically, the


345

effects of prediction errors are swept backward though the network, layer by layer, in order to associate a "square error derivative" (delta) with each processing unit, to compute a gradient from each delta, and finally, to update the weights of each processing unit based upon the corresponding gradient. This process is then repeated, beginning with another input/output pattern. After all the patterns in the training set is exhausted, the algorithm examines the training set again and readjusts the weights throughout the entire network structure until either the objective function (sum of squared prediction errors on the training sample) is sufficiently close to zero or the default number of iterations is reached. The precise computer algorithm implementing the back-propagation technique used in this study was obtained from Eberhart and Dobbins (1990), however many commercially available programs are able to do this analysis. To date, neural network mathematical techniques have been applied in many areas, such as pattern recognition, knowledge databases for stochastic information, robotic control, and financial decision-making. Salchenberger, Cinar, and Lash (1992), Coats and Fant (1993), Luther (1993), Huang, Dorsey, and Boose (1994), and Brockett, Cooper, Golden, and Pitakong (1994), and Brockett et al. (2002) have shown that neural network models can perform better than traditional methods in the classification of financial distress.

3

Neural Network Methods for Life Insurer Insolvency Prediction

Huang, Dorsey, and Boose (1994) uses a feed-forward neural network method optimized with a genetic algorithm to forecast financial distress in life insurers. The data is limited to Insurance Regulatory Information System Ratios for the total population of life insurers. The purpose of this study is to use the neural network on the IRIS data for life insurers to see if the use of this new tool is able to


346

measurably improve prediction. It does not attempt to identify an optimal set of variables for forecasting. Neural network forecasts are compared to discriminant analysis, k-nearest neighbor, and logit. The neural network is shown to produce superior estimates to the alternative methods. The IRIS variables for the life insurers are listed in Table 1. Table 1. Description of IRIS variables. Variable Name

Description

Rl R2 R3 R4 R5 R6 R7 R8 R9 RIO Rll R12

Net Change in Capital and Surplus Net Gain to Total Income Commissions and Expenses to Premiums and Deposits Investment Yield Non-Admitted Assets to Admitted Assets Real Estate to Capital and Surplus Investment in Affiliate to Capital and Surplus Surplus Relief Change in Premiums Change in Product Mix Change in Asset Mix Change in Reserving Ratio

The data for their study came from tapes provided by the National Association of Insurance Commissioners (NAIC). The first year of this study uses 1988 data because filing requirements were changed that year and data necessary to calculate the current set of ratios is not available before 1988. The identification of Financially Impaired Companies (FICs) in this study was taken from Best's Insolvency Study (1992), Life/Health Insurers 1976-1991. For the three years of this study, Best classified 42 companies as FIC in 1989, 41 in 1990, and 58 in 1991. Of those, the NAIC data had sufficient information to use 28 in 1989, 28 in 1990, and 44 in 1991. The total number of companies in the data set was 2020 in 1989, 1852 in 1990, and 1801 in 1991.


347

For each year, this study combines the financial classification (FIC or non FIC) of the current year as the dependent variable with 12 IRIS ratios of the prior year as the set of independent variables to form a year-base data set. Each year-base data set is the basis for the empirical study. For the 1989 FICs, 1988 data are used with each statistical tool to estimate the parameters of the model, then the model is used to predict FICs in-sample, 1989, and in each of the next two years where, in each case, the prior year data is used for the independent variables. In the Huang, Dorsey, and Boose (1994) study, the neural network (NN), k-nearest neighbor (KN), discriminant analysis (DA), and Logit methods are applied to the IRIS ratio model for predicting financially impaired insurers. Three criteria, misclassification cost, resubstitution risk, and the level of risk compared to a naive prediction, are used to evaluate the prediction efficiency of NN, KN, DA, and Logit. In general, for the in-sample prediction the misclassification cost is an increasing function of the cost weight. The exception is KN, which is insensitive to the cost weight change, and which is capable in-sample of correctly identifying all FICs as long as the cost weights are not equal. NN is the second best tool, clearly dominating DA and Logit throughout the range of weights, with the difference becoming more important as the relative weight of type I errors increases. Although KN is the best model for predicting in-sample, it deteriorates to the worst for the first year out-of-sample prediction. NN is now clearly the best prediction tool, dominating the other methods at all cost weights. As measured by misclassification cost, the neural network has stable, efficient prediction capacity in both insample and out-of-sample prediction. KN has the smallest insample cost, but also has the highest out-of-sample cost. Logit and DA are considerably less efficient tools in both in-sample and outof-sample prediction. The resubstitution risk of the in-sample prediction is again an

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increasing function of cost weight except for the KN, which again is best in-sample and is unaffected by change in cost weight. The NN is again the second best prediction tool in-sample, and again its advantage over DA and Logit grows with the relative weights. As when using misclassification cost criterion, for the first year out-ofsample prediction, the KN deteriorates from best predictor, and NN dominates DA and Logit. In general, when measured by resubstitution risk, NN is the only tool with reasonable risk on both insample and out-of-sample prediction. All traditional tools have higher risk than NN in out-of-sample prediction. All the methods are efficient in terms of risk saving relative to the naive prediction for in-sample prediction. The KN has the most risk saving except for a cost weight equal to 1. The NN is the second best, and the Logit and DA are less efficient. In out-of-sample prediction, KN, DA, and Logit are less efficient than the naive prediction, and the NN is the only tool having any prediction efficiency compared to the naive prediction. In short, NN is the only prediction method having any gains in prediction efficiency in insample and out-of-sample prediction compared to the naive prediction. Brockett, Golden, Jang, and Yang (2002) provides another comparative analysis of statistical methods and neural networks for predicting life insurers' insolvency. The primary purpose of this study is to examine and compare the performance of two statistical methods (multiple discriminant analysis and logistic regression analysis) and two artificial neural network methods (back-propagation and learning vector quantization) for predicting life insurer insolvency. The second purpose is to investigate and evaluate the usefulness and effectiveness of several variable sets for the two neural network models. The 22 variables, IRIS, FAST, and Texas EWIS variables are compared and evaluated. This study shows that back-propagation (BP) and learning vector quantization (LVQ) outperform the traditional statistical approaches for all four data sets with a consistent superiority across the two different evaluation criteria: total

Using Neural Networks to Predict Failure in the

Marketplace

349

misclassification cost and resubstitution risk criteria. And the results show that the 22-variable model and the Texas EWIS model are more efficient than the IRIS and the FAST model in most comparisons. This study builds a current year model and a one-year prior model. The data are obtained from the Texas Department of Insurance (TDI), from annual statements filed for the years 1991 through 1994, as well as a list of the insurers that became "troubled" companies from 1991 to 1995. All solvent and insolvent life insurance companies whose business domiciles are in Texas and whose data are available for the entire study period (1991-1994) and entire variable sets are included in the samples. Any observation that does not exist throughout the entire period and has a blank data field is deleted from the sample data sets. Four sets of explanatory variables, 22-variable model, IRIS model, FAST model, and Texas EWIS model, are examined in this study. The 22-variable set is used as the benchmark to compare and validate the effectiveness of other variable sets. An examination of the previous studies is conducted to identify the variables that had been found to be more indicative of financial conditions in the context of insolvency. Some of the variables are eliminated because too many of the companies have no data for these variables. Stepwise regression is used to reduce the variables and the final set includes 22 variables. Table 2 presents these 22 variables. The Texas Department of Insurance (TDI) implemented an early warning information system (EWIS) in early 1992. For each company, a binary (0 or 1) variable is valued for each of a set of approximately 393 indicators based on whether the calculated ratio or numerical value of the indicator is above some preselected threshold value. Weights are assigned to each binary indicator by the EW staff according to a subjective assessment of the importance or severity of the indicator. Each binary indicator is then multiplied by its assigned weight and the resulting values are summed across all indicators to obtain an "EWIS company score" for each company.

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P L. Brockett et al.

The rank of the insurers for prioritization is determined based on this score. Table 2. Variable description of 22-variable data set. Variable Name VI V2 V3 V4 V5 V6 V7 V8 V9 V10 Vll V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22

Description Gains/Premiums Liabilities/Surplus Net Gain from Operations after Tax & Dividends Net Investment Income Accident & Health Benefits/Total Benefits (Bonds+Stocks+Mortgages)/Cash & Investment Assets Cash Flow/Liabilities Capital & Surplus/Liabilities Change in Capital & Surplus Delinquent Mortgages/Capital & Surplus Change in Premium Insurance Leverage (Reserves/Surplus) Financial Leverage (Premiums/Surplus) Log of Growth in Assets Log of Growth in Premiums Log of Growth in Surplus Log of Cash Flow from Operations Non-Admitted Assets/Admitted Assets Reinsurance Ceded/Premium Separate Account Assets/Assets Total Benefits Paid/Capital & Surplus Real Estate/Assets

Some of the 393 indicators are automatically input into the EWIS system, while some of them are input manually. Correspondingly, two separate Texas EWIS variable sets are used in this analysis: an automated indicator set (EWIS-automated) and a nonautomated indicator set (EWIS-significant). The variables used in TDI - EWIS are not publicly disclosed. The IRIS variables are listed in Table 1, and the FAST variables for the life insurers are listed below in Table 3.


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Table 3. Description of FAST variables. Var. Name Fl F2 F3 F4 F5 F6 F7 F8 F9 F10 Fl 1 F12 F13 F14 F15 F16 F17

Description Change in Capital & Surplus Surplus Relief Change in Net Prem./Ann. Cons./Dep. Type Funds A&H Bus. to Net Premiums & Annuity Cons. & Deposit Type Funds Change in Dir. &Ass. Annuities & Deposit type Funds Stockholder's Dividends to PY Capital & Surplus Change in Net Income Trending of Net Income Surrenders to Premiums & Deposit Type Funds Grp. SUIT, to Grp. Prem./Grp. Dep. Type Funds Change in Liquid Assets Aff. Investments/Receivables to Capital/Surplus Non Inv. Gr. Bonds & St. Inv. to Capital & Surplus & AVR Collateralized Mortgage Obligations to Total Bonds Problem Real Estate and Mortgages to Capital & Surplus & AVR Sch. BA Assets to Capital & Surplus & AVR Total Real Estate and Mortgages to Capital & Surplus & AVR

This study uses cross-validation tests that employ the weights or parameter estimates developed in the previous year(s) to predict the outcomes in the subsequent year. The learning or training samples consist of the insurance companies in 1992 and 1993. The parameter estimates and weights from the learning samples are used to test the sample that consists of the companies in 1994. The crossvalidation test is practically useful in the sense that we can use the ex post information of the past to predict the ex ante outcomes for the current year. The number of companies in the training and test samples is listed in Tables 4 and 5. Evaluated in terms of their misclassification cost, different prior probabilities and misclassification cost ratios are used. The type I misclassification cost takes on the following values (1, 10, 15, 20, 25, 30), while the type II misclassification cost is fixed at 1. The

352


prior probability of failure is set to equal prior probability and the proportional prior probability in the sample. An evaluation of MDA, logit, LVQ, and BP methods is performed for each of the five data sets using the current year and one-year prior model. Table 4. Data sets used in the training samples. Year

Data Set 22-Var.

1992

IRIS FAST 22-Var.

1993

IRIS FAST

.. , , ivioaei Current Year One-Year Prior Current Year One-Year Prior Current Year One-Year Prior Current Year One-Year Prior Current Year One-Year Prior Current Year One-Year Prior

Number of Insurers Insolvent 64 70 51 51 50 66 55 70 51 51 54 66

Solvent 463 463 463 463 463 463 463 463 463 463 463 463

Table 5. Data sets used in the test samples. Year

Data Set 22-Var.

1994

IRIS FAST

iviouei Current Year One-Year Prior Current Year One-Year Prior Current Year One-Year Prior

Number of Insurers Insolvent Solvent 49 463 70 463 49 463 51 463 48 463 66 463

A consistent pattern of minimum total misclassification costs emerges. For lower cost ratios (under 3), the LVQ method tends to minimize total misclassification costs. This is consistent with its minimization of type II errors. For higher cost ratios (5-30), the back-propagation method is superior in minimizing total misclassi-


353

fication costs because of the smallest numbers of type I errors. For these data sets, the MDA and logit methods (equal and proportional prior probability) fail consistently to minimize total misclassification costs. There is little difference in performance between the current year model and the one-year prior model. This lack of a significant difference between the performance of the current year and one-year prior models is the results of the use of a broader definition of financial distress. The definition of a financially troubled company used in this study includes insurers subject to less severe regulatory supervision that invariably precedes insolvency. This results in a smoothing of the time effect in predicting bankruptcy. Given that cost ratios used by researchers in previous studies have been no lower than 20 (usually, much higher), this study focus on misclassification costs for cost ratios of 20, 25, and 30. For these cost ratios, the MDA with proportional prior probability and logit with equal prior probability consistently yield an unacceptably high percentage of type I errors for all data sets. This leads to extremely high misclassification costs for generally accepted cost ratio ranges. The back-propagation method consistently dominates all other methods for both the current year and one-year prior models. LVQ is a close second. Logit with proportional prior probability is a distant third, but it provides marginally better results than MDA with equal prior probability, which is consistent with the findings of earlier studies. Minimization of resubstitution risk is also used as the criterion for evaluating the MDA, logit, LVQ, and BP methods. A consistent pattern of minimum resubstitution risk emerges for the 22-variable, IRIS, FAST, and the Texas EWIS data sets. Assuming equal costs for type I and type II errors (cost ratio=l), LVQ is the best for both the current year and one-year prior models. For higher cost ratios, the back-propagation works best. For all data sets, the MDA and logit (equal and proportional priors) are almost always inferior to LVQ and BP methods. These results are consistent with those ob-

354


tained using the misclassification cost criterion. MDA with equal prior probability and logit with equal prior probability give unacceptably high levels of resubstitution risks for all data sets. The increased resubstitution risk is due to the additional weights placed on type I errors by using equal prior probability. Given the superiority of neural network methods, this study further evaluates the usefulness of each data set using the neural network methods BP and LVQ. With the back-propagation method and using total misclassification cost as the evaluation criterion, 22-variable and Texas EWISsignificant variables yield the best results for the current year model. FAST model is a close third. For one of the three years (1993), IRIS yields slightly better results than FAST, but only for the lowest two cost ratios (1 and 3). For 1992 data, the FAST and 22-variable data sets give identical results and dominate over the higher cost ratio range. The TDIsignificant data sets results are only slightly worse than these two. Using the IRIS data set yields higher misclassification costs than FAST due to the higher number of type I errors. For 1993 data, the TDI-significant clearly gives the best results. The 22-variable and TDI-automated data sets are distant seconds. For these data, results for IRIS and FAST data sets are almost indistinguishable and are the worst of the five data sets considered. Using 1994 data, the TDI-significant data set again dominates clearly. The 22-variable data set is a distant second. Using these data, IRIS results are somewhat better than FAST results. Looking at all three years, it is difficult to see any consistent improvement in performance using FAST rather than IRIS. Both data sets generally perform worse than the TDI or the 22-variable sets. For the one-year prior model, both TDI and the 22-variable data sets yield the best results. The two TDI sets provide similar results, as is expected given the overlap of certain variables in these two sets. For 1992 data, the TDI-significant performs best, with the 22-


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variable set a reasonably close second. FAST and IRIS are the worst, with FAST slightly better. Using 1993 data, the 22-variable data set is the best and TDI-auto performs equally well. TDI-significant is a fairly close third. Once again, FAST and IRIS give the worst results, with FAST dominating slightly. For 1994 data, the 22-variable data set dominates. TDI-automated and TDI-significant give similar results and are second and third, respectively. In this year, IRIS performs better than FAST, due to a lower number of type I errors. Overall, for the current year and one-year prior models, the TDI and 22-variable data sets perform the best. The FAST and IRIS data sets are clearly inferior. Also, it is important to note that no consistent improvement in performance is obtained using FAST rather than IRIS. Using the resubstitution risk criterion results in a similar evaluation of the data sets to the misclassification cost criterion. With the LVQ method and using the total misclassification cost criterion, TDI-significant and TDI-automated data sets consistently tend to minimize costs for the current year model. For each of the three years of data examined, the 22-variable data set is a close third. Once again, IRIS and FAST data sets produce the worst results, with IRIS dominating in two of the three years. For the one-year prior model, the TDI-significant data set again performs the best. TDI-automated comes in second and the 22variable data set is a somewhat distant third. It is important to note that the results for both the current year and one-year prior models are invariant to the level of cost ratios. Like the current year model, there is a consistently better performance from the IRIS data set than the FAST data set in each of the three years. Generally, for both the current year and one-year prior models, the TDI and 22-variable data sets give the best performance using misclassification cost criterion. It is important to note the consistent superiority of the IRIS data set over FAST, which was designed as an improvement to IRIS. Using the resubstitution risk criterion also lead to a similar evaluation of the data sets.

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4


Neural Network Methods for Property-Liability Insurer Insolvency Prediction

Brockett et al. (1994) introduce a neural network artificial intelligence model as an early warning system for predicting insurer insolvency. To investigate a firm's propensity toward insolvency, the neural network approaches are applied to financial data for a sample of U.S. property-liability insurers. The results of the neural network methods are compared with those of discriminant analysis, A.M. Best ratings, and the National Association of Insurance Commissioners' Insurance Regulatory Information System ratings. The neural network results show high predictability and generalizability. This study focuses on financial variables available from the NAIC annual statement tapes. The Texas State Board of Insurance was involved in variable selection due to their interest in early warning to help firms prevent insolvency. Their rather large list of several hundred insurer financial health indicator variables was used as the first consideration set in the selection of variables. In cooperation with the Texas State Board of Insurance, this list was culled down using published research identifying variables that failed to identify potential insolvencies in previous research. This resulted in 24 variables. These 24 variables were reduced further through a series of statistical analyses using a sample of solvent and insolvent Texas domestic property-liability insurers (using Texas domestic insurer data provided by the Texas State Board of Insurance for insolvencies during the period 1987 through 1990). The first step in the preliminary analysis was to examine each variable separately to see if a significant difference existed between the solvent and the insolvent insurers that could be detected by using that variable alone. Discriminant analysis, canonical analysis, colinearity tests, and step-


357

wise logistic regression were also run to check further which sets of variables might be eliminated due to multivariate considerations. The final subset of eight variables selected using the above techniques is shown in Table 6. Table 6. Final set of variables of the property-liability insurers. Variable Name Description VI Policyholders' Surplus V2 Capitalization Ratio V3 Change in Invested Assets V4 Investment Yields Based on Average Invested Assets Ratio of Significant Receivables from Parent, Subsidiaries, and Affiliates to Capital and Surplus V6 Significant Increase in Current Year Net Underwriting Loss V7 Surplus Aid to Surplus V8 Liabilities to Liquid Assets

In this study, the back-propagation neural network approach is utilized to predict the insolvency of the property-liability insurers, and a model that could detect insolvency propensity using annual statement data from two years prior to insolvency is considered. Two hundred forty-three insurers were used in training sessions consisting of 60 insurers that ultimately became insolvent, and 183 insurers that remained solvent. The list of insolvent insurers was obtained by the Texas State Board of Insurance Research Division through written requests to each state insurance department for a list of all domestic insolvent companies during 1991 or 1992. Follow-up telephone calls ensured complete data records for each insolvency. Accordingly, insolvency data were obtained on firms not included in the A. M. Best list of insolvent firms. NAIC annual statement data for each company two years prior to insolvency were used in the analysis. The data were separated into three subsets. A large data set (n = 145), called set Tl, represents 60 percent of the sample, and is used

358


for training the network. Two smaller sets, T2 and T3, each comprised 20 percent of the sample (n = 49 each). These sets were used, respectively, for determining when to cease training the neural network, and for testing the resultant trained neural network. The set T2 determined when to stop training the network according to the following rule: stop training when the predicting ability of the neural network trained on Tl and as tested on T2 begins to drop. The subset was then used to assess the trained network's outof-sample predictability characteristics. The results of applying the neural networks methodology to predict financial distress based upon our selected variables show very good abilities of the network to learn the patterns corresponding to financial distress of the insurer. In all cases, the percent correctly classified in the training sample by the network technique is above 85 percent, the average percent correctly classified in the training samples is 89.7 percent, and the bootstrap estimate of the percent correctly classified on the holdout samples is an average of 86.3 percent (see Table 7). The bootstrap result shows that the calculated ability of the neural network model to predict financial distress is not merely due to upwardly biased assessments (which might occur when training and testing on the same sample as in the discriminant analysis results). The ability of the uncovered network structure to generalize to new data sets is also a very important component of the learning process for the network and is crucial to the importance of the results for managerial implementation. The bootstrap estimate of the percent correctly classified on the various testing samples shows that generalizability is obtained. Overall, when applied to the entire sample of 243 firms, the neural network method correctly identified the solvency status of 89.3 percent of the firms in the study. Of the firms that become insolvent in the next two years, the neural network correctly predicted 73 percent, and it correctly predicted the solvency of 94.5 percent of the firms that remained solvent for the two-year duration.


359

Table 7. Neural network training and predictive accuracy for the property liability insurers. Sample Training Sample (Tl) Results Sample Sizes Percentage Correctly Classified Stopping Rule Sample (T2) Results Sample Sizes Percentage Correctly Classified Bootstrapped Test Sample (T3) Sample Sizes Mean Percentage Correctly Classified Entire Sample Results Sample Size Overall Percentage Correctly Classified Percentage of Insolvent Firms Correctly Classified Percentage of Solvent Firms Correctly Classified

Predictive Accuracy 145 89.7 49 87.8 49 86.3 243 89.3 73.3 94.5

To compare the results of the neural network model to other methods, several other classification techniques were examined. This study performed a linear discriminant analysis using the same eight variables obtained from the stepwise logistic regression procedure described above. Such an approach assumes that the explanatory variables in each group form a multivariate normal distribution with the same covariance matrix for the two groups. The results of the discriminant analysis are presented in Table 8. As shown, the statistical method, although not doing as well as the neural networks methodology, showed a respectable 85 percent correct classification for insolvent firms and an 89.6 percent correct classification rate for solvent firms. An additional analysis was performed using the NAIC's IRIS ratios (see Table 9). The IRIS system correctly identifies only 26 insolvencies out of the 60 firms that eventually became insolvent within two years, or 43.3 percent of the insolvent companies On the other hand, 96.7 percent of the solvent firms were correctly identified.


360

Table 8. Discriminant analysis classification. True Status

Insolvent 51/60=85% 19/183=10.4% 70

Insolvent Solvent Total

Predicted Status Solvent 9/60=15% 164/183=89.6% 173

Total 60 183 243

Table 9. NAIC IRIS classification table. Predicted Status Insolvent Solvent 34/60=56.7% 26/60=43.3% 6/183=3.3% 177/183=96.7% 70 173

True Status Insolvent Solvent Total

Total 60 183 243

Table 10. A.M. Best ratings. Insolvent Companies Solvent Companies

A+

A

1

3

1

56

22

17

B B - C+

C

c-

2

0

1

4

1

0

47

60

4

3

1

1

0

0

79

183

A- B+

Not Rated Total

Another source of frequently used insurer rating information is the A.M. Best Company. Best's ratings for are presented in Table 10. These ratings were not very useful for predicting insolvencies, partly because of the very large percentage of nonrated companies (78.3 percent) in the insolvent group. A large subset of solvent firms (43.2 percent) also was not rated by A.M. Best.

5

Conclusion and Further Directions

The neural network artificial intelligence methods show significant promise for providing useful early warning signals of market failure (insolvency) within the insurance industry, and the results of using neural networks dominates the IRIS, FAST, and some other statis-


361

tical methods. In addition, other characteristics positively differentiate neural network models from alternatives techniques. For one, the neural network model can be updated (learning supplemented) without completely retraining the network. The current version of the interconnection weights can be used as a starting point for the iterations once new data become available. Accordingly, the system of learning can adapt as different economic influences predominate. In essence, the neural network model can evolve and change as the data, system, or problem changes. Other "nonlearning" or static models do not have these built-in properties. Despite the positive classification results obtained, we still believe that further study is warranted. There are several avenues of research that might potentially enhance the predictability, the robustness, or the interpretability of the neural networks approach. These avenues include the development and inclusion of qualitative data and trend data, which could add significantly to the robustness and accuracy of the model developed. In addition, because of states' different economic climate and regulatory requirements, a comparison of the appropriate models for insolvency prediction in different states (and nationwide) should be investigated to ascertain the impact of certain state-controlled regulatory requirements. These results would then suggest public policy directives concerning these issues for the purpose of decreasing insolvency propensity.

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References Altaian, E.I. (1968), "Financial ratios, discriminant analysis and the prediction of corporate bankruptcy," Journal of Finance, 23, pp. 589-609. Altman, E.I., Halkeman, R.G., and Narayanan, P. (1977), "ZETA analysis: a new model to identify bankruptcy risk of corporations," Journal of Banking and Finance, 55, pp. 229-244. A.M. Best Company (1992), "Best's insolvency study, life-health insurers," A.M. Best Company, Oldwick, NJ. Ambrose, J.M. and Seward, J.A. (1988), "Best's ratings, financial ratios and prior probabilities in insolvency prediction," Journal of Risk and Insurance, 55, pp. 229-244. BarNiv, R. and Hershbarger, R.A. (1990), "Classifying financial distress in the life insurance industry," Journal of Risk and Insurance, 57, pp. 110-136. BarNiv, R. and McDonald, J.B. (1992), "Identifying financial distress in the insurance industry: a synthesis of methodological and empirical issues," Journal of Risk and Insurance, 59, pp. 543573. Barrese, J. (1990), "Assessing the financial condition of insurers," CPCU Journal, 43(1), pp. 37-46. Bierman, H. Jr. (1960), "Measuring financial liquidity," Accounting Review, 35, pp. 628-632. Borch, K. (1970), "The rescue of insurance company after ruin," ASTIN Bulletin, 6, pp. 66-69. Brockett, P.L., Cooper, W.W., Golden, L.L., and Pitaktong, U. (1994), "A neural network method for obtaining and early warning of insurer insolvency," Journal of Risk and Insurance, 61, pp. 402-424. Brockett, P.L., Golden, L.L., Jang, J., and Yang, C. (2002), "Comparative analysis of statistical methods and neural networks for predicting life insurers' insolvency," Working Paper, the University of Texas at Austin.


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Coats, P.K. and Fant, L.F. (1993), "Recognizing financial distress patterns using a neural network tool," Financial Management, 22, pp. 142-155. Cummins, J.D., Harrington, S.E., and Klein, R. (1995), "Insolvency experience, risk-based capital, and prompt corrective action in property-liability insurance", Journal of Banking and Finance, 19, pp. 511-527. Eberhart, R.C. and Dobbins, R.W. (1990), Neural Network PC Tools: A Practical Guide, Academic Press, New York. Edmister, R.O. (1972), "An empirical test of financial ratio analysis for small business failure predictions," Journal of Financial and Quantitative Analysis, 7, pp. 1477-1493. Funahashi, K. (1989), "On the approximate realization of continuous mappings by neural networks," Neural Networks, 2, pp. 183192. Grace, M., Harrington, S., and Klein, R (1998), "Risk-based capital and solvency screening in property-liability insurance: hypotheses and empirical tests," Journal of Risk and Insurance, 65 (2), pp. 213-243. Harrington, S., Nelson, J.M. (1986), "A regression-based methodology for solvency surveillance in the property-liability insurance industry," Journal of Risk and Insurance, 53, pp. 583-605. Hebb, D.O. (1949), The Organization of Behavior, Wiley Press, New York. Hornik, K.M. and Stinchcombe, H.W. (1990), "Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks," Neural Networks, 3, pp. 551-560. Huang, C.S., Dorsey, R.E., and Boose, M.A. (1994), "Life insurer financial distress prediction: a neural network model," Journal of Insurance Regulation, 13, pp. 133-167. Kurkova V. (1992), "Kolmogorov theorem and multilayer neural networks," Neural Networks, 5, pp. 501-506. Lippmann, R.P. (1987), "An introduction to computing with neural nets," IEEEASSP Magazine, 1, pp. 4-22.

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Lorentz, G.G. (1976), "The 13m problem of Hilbert," in Browder, F.E. (ed.), Mathematical Developments Arising From Hilbert's Problems^ American Mathematical Society, Providence. Luther, R.K. (1993), "Predicting the outcome of Chapter 11 bankruptcy: an artificial neural network approach," Ph.D. Dissertation, The University of Mississippi. Odom, M.D. and Sharda, R. (1990), "A neural network model for bankruptcy prediction," Proceedings of International Joint Conference on Neural Networks, volume 2, pp. 163-168. Quirk, J.P. (1961), "The capital structure of firms and the risk of failure," International Economic Review, 2(2), pp. 210-228. Salchenberger, L.M., Cinar, E.M., and Lash, N.A. (1992), "Neural networks: a new tool for predicting thrift failures," Decision Science, 23(4), pp. 899-915. Sinkey, J.F. Jr. (1975), "Multivariate statistical analysis of the characteristics of problem banks," Journal of Finance, 30, pp. 21-36. Texas Department of Insurance (1993), "Report on early warning," Texas Department of Insurance, Austin, TX. Tinsley, P.A. (1970), "Capital structure, precautionary balances and the valuation of the firm: the problem of financial risk," Journal of Financial and Quantitative Analysis, 5(1) pp. 33-62. Williams, W.H. and Goodman, M.L. (1971), "A statistical grouping of corporations by their financial characteristics," Journal of Financial and Quantitative Analysis, 6(4), pp. 1095-1104.

Chapter 10 Illustrating the Explicative Capabilities of Bayesian Learning Neural Networks for Auto Claim Fraud Detection S. Viaene, R.A. Derrig, and G. Dedene This chapter explores the explicative capabilities of neural network classifiers with automatic relevance determination weight regularization, and reports the findings from applying these networks for personal injury protection automobile insurance claim fraud detection. The automatic relevance determination objective function scheme provides us with a way to determine which inputs are most informative to the trained neural network model. An implementation based on Mackay's (1992) evidence framework approach to Bayesian learning is proposed as a practical way of training such networks. The empirical evaluation is based on a data set of closed claims from accidents that occurred in Massachusetts, USA during 1993.

1

Introduction

In recent years, the detection of fraudulent claims has blossomed into a high-priority and technology-laden problem for insurers (Viaene 2002). Several sources speak of the increasing prevalence of insurance fraud and the sizeable proportions it has taken on (see, for example, Canadian Coalition Against Insurance Fraud 2002, Coalition Against Insurance Fraud 2002, Comite Europeen des Assurances 1996, Comite Europeen des Assurances 1997). September 2002, a special issue of the Journal of Risk and Insurance (Derrig 2002) was 365

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S. Viaene et al.

devoted to insurance fraud topics. It scopes a significant part of previous and current technical research directions regarding insurance (claim) fraud prevention, detection and diagnosis. More systematic electronic collection and organization of and company-wide access to coherent insurance data have stimulated data-driven initiatives aimed at analyzing and modeling the formal relations between fraud indicator combinations and claim suspiciousness to upgrade fraud detection with (semi-)automatic, intelligible, accountable tools. Machine learning and artificial intelligence solutions are increasingly explored for the purpose of fraud prediction and diagnosis in the insurance domain. Still, all in all, little work has been published on the latter. Most of the state-of-the-art practice and methodology on fraud detection remains well-protected behind the thick walls of insurance companies. The reasons are legion. Viaene et al. (2002) reported on the results of a predictive performance benchmarking study. The study involved the task of learning to predict expert suspicion of personal injury protection (PIP) (no-fault) automobile insurance claim fraud. The data that was used consisted of closed real-life PIP claims from accidents that occurred in Massachusetts, USA during 1993, and that were previously investigated for suspicion of fraud by domain experts. The study contrasted several instantiations of a spectrum of state-of-the-art supervised classification techniques, that is, techniques aimed at algorithmically learning to allocate data objects, that is, input or feature vectors, to a priori defined object classes, based on a training set of data objects with known class or target labels. Among the considered techniques were neural network classifiers trained according to MacKay's (1992a) evidence framework approach to Bayesian learning. These neural networks were shown to consistently score among the best for all evaluated scenarios. Statistical modeling techniques such as logistic regression, linear and quadratic discriminant analysis are widely used for modeling and prediction purposes. However, their predetermined functional form and restrictive (often unfounded) model assumptions limit their use-

Bayesian Learning Neural Nets for Auto Claim Fraud and Detection

367

fulness. The role of neural networks is to provide general and efficiently scalable parameterized nonlinear mappings between a set of input variables and a set of output variables (Bishop 1995). Neural networks have shown to be very promising alternatives for modeling complex nonlinear relationships (see, for example, Desai et al. 1996, Lacher et al. 1995, Lee et al. 1996, Mobley et al. 2000, Piramuthu 1999, Salchenberger et al. 1997, Sharda and Wilson 1996). This is especially true in situations where one is confronted with a lack of domain knowledge which prevents any valid argumentation to be made concerning an appropriate model selection bias on the basis of prior domain knowledge. Even though the modeling flexibility of neural networks makes them a very attractive and interesting alternative for pattern learning purposes, unfortunately, many practical problems still remain when implementing neural networks, such as What is the impact of the initial weight choice? How to set the weight decay parameter? How to avoid the neural network from fitting the noise in the training data? These and other issues are often dealt with in ad hoc ways. Nevertheless, they are crucial to the success of any neural network implementation. Another major objection to the use of neural networks for practical purposes remains their widely proclaimed lack of explanatory power. Neural networks are black boxes, it says. In this chapter Bayesian learning (Bishop 1995, Neal 1996) is suggested as a way to deal with these issues during neural network training in a principled, rather than an ad hoc fashion. We set out to explore and demonstrate the explicative capabilities of neural network classifiers trained using an implementation of MacKay's (1992a) evidence framework approach to Bayesian learning for optimizing an automatic relevance determination (ARD) regularized objective function (MacKay 1994, Neal 1998). The ARD objective function scheme allows us to determine the relative importance of inputs to the trained model. The empirical evaluation in this chapter is based on the modeling work performed in the context of the baseline benchmarking study of Viaene et al. (2002).

368

S. Viaene et al.

The importance of input relevance assessment needs no underlining. It is not uncommon for domain experts to ask which inputs are relatively more important. Specifically, Which inputs contribute most to the detection of insurance claim fraud? This is a very reasonable question. As such, methods for input selection are not only capable of improving the human understanding of the problem domain, specifically, the diagnosis of insurance claim fraud, but also allow for more efficient and lower-cost solutions. In addition, penalization or elimination of (partially) redundant or irrelevant inputs may also effectively counter the curse of dimensionality. In practice, adding inputs (even relevant ones) beyond a certain point can actually lead to a reduction in the performance of a predictive model. This is because, faced with limited data availability, as we are in practice, increasing the dimensionality of the input space will eventually lead to a situation where this space is so sparsely populated that it very poorly represents the true model in the data. This phenomenon has been termed the curse of dimensionality (Bellman 1961). The ultimate objective of input selection is therefore to select a minimum number of inputs required to capture the structure in the data. This chapter is organized as follows. Section 2 revisits some basic theory on multilayer neural networks for classification. Section 3 elaborates on input relevance determination. The evidence framework approach to Bayesian learning for neural network classifiers is discussed in Section 4. The theoretical exposition in the first three sections is followed by an empirical evaluation. Section 5 describes the characteristics of the 1993 Massachusetts, USA PIP closed claims data that were used. Section 6 describes the setup of the empirical evaluation and reports its results. Section 7 concludes this chapter.

2

Neural Networks for Classification

Figure 1 shows a simple three-layer neural network. It is made up of an input layer, a hidden layer and an output layer, each consisting of a


369

number of processing units. The layers are interconnected by modifiable weights, represented by the links between the layers. A bias unit is connected to each unit other than the input units. The function of a processing unit is to accept signals along its incoming connections and (nonlinearly) transform a weighted sum of these signals, termed its activation, into a single output signal. In analogy with neurobiology, the units are sometimes called neurons. The discussion will be restricted to the use of neural networks for binary classification, where the input units represent individual components of an input vector, and a single output unit is responsible for emitting the values of the discriminant function used for classification. One then commonly opts for a multilayer neural network with one hidden layer. In principle, such a three-layer neural network can implement any continuous function from input to output, given a sufficient number of hidden units, proper nonlinearities and weights (Bishop 1995). We start with a description of the feedforward operation of such a neural network, given a training set D = {(x l ,£i)} i=1 with input vectors x i = (x\,..., < ) T 6 I T and class labels U G {0,1}. Each input vector component is presented to an input unit. The output of each input unit equals the corresponding component in the input vector. The output of hidden unit j G {1,..., h}, that is, Zj (x), and the output of the output layer, that is, y (x), are then computed as follows: n

Hidden Layer : Zj (x) = fi I bij + \]ui>k

Xk

)

Output Layer : y (x) = f2 lb2 + J ] Vj Zj (*) I ,

^ (2)

where bij G IR is the bias corresponding to hidden unit j , u^k G 1R denotes the weight connecting input unit k to hidden unit j , b2 G IR is the output bias, and Vj G IR denotes the weight connecting hidden unit j to the output unit. The biases and weights together make up

370

S. Viaene et al.

Figure 1. Example three-layer neural network.

weight vector w. / i (•) and / 2 (•) are termed transfer or activation functions and essentially allow a multilayer neural network to perform complex nonlinear function mappings. Input units too have activation functions, but, since these are of the form f(a) = a, these are not explicitly represented. There are many possible choices for the (nonlinear) transfer functions for the hidden and output units. For example, neural networks of threshold transfer functions where among the first to be studied, under the name perceptrons (Bishop 1995). The antisymmetric version of the threshold transfer function takes the form


371

of the sign function: sign(a) = {

'

" " J "

(3)

Multilayer neural networks are generally called multilayer perceptions (MLPs), even when the activation functions are not threshold functions. Transfer functions are often conveniently chosen to be continuous and differentiable. We use a logistic sigmoid transfer function in the output layer—that is, sigm (a) = 1+ex p ( _ a) • The term sigmoid means S-shaped, and the logistic form of the sigmoid maps the interval [—00,00] onto [0,1]. In the hidden layer we use hyperbolic tangent transfer functions—that is, tanh (a) = exP(a)~exp(-Q). 0

'

\ y

exp(a)+exp(—a)

The latter are S-shaped too, and differ from logistic sigmoid functions only through a linear transformation. Specifically, the activation function f(a) = tanh (a) is equivalent to the activation function f(a) — sigm(a) if we apply a linear transformation a = | to the input and a linear transformation / = 2 / — 1 to the output. The transfer functions in the hidden and output layer are standard choices. The logistic sigmoid transfer function of the output unit allows the MLP classifier's (continuous) output y (x) to be interpreted as an estimated posterior probability of the form p(t = l|x), that is, the probability of class t = 1, given a particular input vector x (Bishop 1995). In that way, the MLP produces a probabilistic score per input vector. These scores can then be used for scoring and ranking purposes (as, for example, in applications of customer scoring and credit scoring) and for decision making. The Bayesian posterior probability estimates produced by the MLP are used to classify input vectors into the appropriate predefined classes. This is done by choosing a classification threshold in the scoring interval, in casu [0,1]. Optimal Bayes decision making dictates that an input vector should be assigned to the class associated with the minimum expected risk or cost (Duda et al. 2000).

372

S. Viaene et al.

Optimal Bayes assigns classes according to the following criterion: 1

argmin Y % (fc|x) Ljik (x),

(4)

where p(k\x) is the conditional probability of class k, given a particular input vector x, and Lj:k (x) is the cost of classifying a data instance with input vector x and actual class k as class j . Note that Ljtk (x) > 0 represents a cost, and that Ljtk (x) < 0 represents a benefit. This translates into the classification rule that assigns class 1 if ,. ,i x ^ £i,o(x) -L 0 ) o(x) p(*=lx) >

'

s . r f

\

T

( V

(5)

L0,i(x) - Li,i(x) + Ii,o(x) - L0,o(x) and class 0 otherwise, assuming that the cost of labeling a data instance incorrectly is always greater than the cost of labeling it correctly, and that class 0 is the default in case of equal expected costs. In case LJjfc is independent of x, that is, there is a fixed cost associated with assigning a data instance to class j when it in fact belongs to class k, (5) defines a fixed classification threshold in the scoring interval [0,1]. Weight vector w needs to be estimated using the training data D = {(x l , ti)}i=v Learning works by randomly initializing and then iteratively adjusting w so as to optimize an objective function ED, typically a sum of squared errors. That is: 1 N 2 ED = ^(ti-yi) ,

(6)

i=l

where yi stands for y (x l ). The backpropagation algorithm, based on gradient descent on ED, is one of the most popular methods for supervised learning of MLPs. While basic backpropagation is simple, flexible and general, a number of heuristic modifications to gradient descent have been proposed to improve its performance. For an overview of alternative training schemes, see (Bishop 1995).


373

The sum of squared errors criterion is derived from the maximum likelihood principle by assuming target data generated from a smooth deterministic function with added Gaussian noise. This is clearly not a sensible starting point for binary classification problems, where the targets are categorical and the Gaussian noise model does not provide a good description of their probability density. The crossentropy objective function is more suitable (Bishop 1995). At base, cross-entropy optimization maximizes the likelihood of the training data by minimizing its negative logarithm. Given training data D = {(xz, U)}i=l and assuming the training data instances are drawn independently from a Bernouilli distribution, the likelihood of observing D is given by: N

nâ-i/i) 1- * 4 ,

(7)

where we have used the fact that we would like the value of the MLP's output y(x) to represent the posterior probability p(t = l|x). Maximizing (7) is equivalent to minimizing its negative logarithm, which leads to the cross-entropy error function of the form: N

ED = - J 2 (** 00

~

to

I

CO

O CO to I

rH

00

to cT to cT

I

rH

o I

00 rH T—
-^

o < H

O

60 B P CO

O

^ 60

H OH

H

^H

i—I

CT> OS ' _ oo - oo _ 00 O - O O C 5 t>- O i t— G>

-

rH

o~ 1-1 CM b-

o (M O

i—I

v to

U

03

O

~ * ~ ^ '

* > °« (5)

where Kv(u) = \ l°°xv-le-%(x+*)dx 2 Jo is the modified type 3 Bessel function with index v. For v = —1/2, QXQ{\/2, fi', j3') is called the inverse Gaussian distribution, lQ(fi, (5). Its probability density function is given, therefore, by

h-irt*\K0)=2li-.1/2K_i/2{fi/py

(6)

The means of the QTQ{v, /J,', (3') and the TQ(ii, 0) are, respectively,

and E LY1 - a Kl'2

(/x//3)

- a

hence, it is simple to prove that Kv{u) = K-V(u),

\/v € R.

439

Robustness in Bayesian Models for Bonus-Malus Systems

2.3

Conjugate Posterior Structure Functions

Under the Bayesian point of view adopted in this chapter, the contributions of both the observational data (in the form of the likelihood function) and prior opinion or information (in the form of the structure function) are essential. The Bayes' theorem then acts as the formal instrument that transforms the observed data y, the initial information about a parameter of interest, 0, into the final opinion by means of the rule:

hf(y\e)*(e)deIn the BMS setting, and considering models for the frequency component, the portfolio is usually considered heterogeneous, with all policyholders having constant but unequal underlying risks of having an accident. Then, if ki, fc2,..., kt is a i.i.d. random sample from a distribution with a probability density function f(k \ 6) the posterior probability density function of 6 given fci, fc2,.-., fet, corresponding to the prior probability density function ir(0), is

SeI\Lif(h\eM9)d0 (?)

ocn/(fci|0M0) which results from applying the Bayes' theorem. 2.3.1 Negative Binomial-Generalized Pareto Model

The number of claims k, given the parameter A, is considered to be distributed according to BM (r,

P(k\r,\)=

u

—T

—T

, fc = 0,l,...

440

E. Gomez-Deniz and F. J. Vazquez-Polo

The parameter A > 0, denotes the different underlying risk of each policyholder making a claim. Assume for the structure function that A is distributed according to a Generalized Pareto distribution, QV(r*, (*, s*), where £* = s(, r* = r and s* = sr + 1. Therefore, 7T0(A) =

r « ) r ( s r + 1) (r + A) s C + s r + 1 '

With this probability density function E[A]= 1 J

r
/(3/)

•

Premium Calculation in a Bonus-Malus System

This section considers bonus-malus system (BMS) premiums in automobile insurance. In general, BMS are based on the distribution of the number of accidents (or claims) K, irrespective of their size. The methodology of a BMS consists of ensuring that the premium increases with the number of claims and decreases with the period t in which the policyholder does not make a claim. This can be achieved by dividing a posterior expectation by a prior expectation according to an estimate derived by means of an appropriate loss function. The use of standard Bayesian analysis in credibility theory (viewed as the experience-rating process using a statistical decisionmaking model based on loss functions) has been considered in several actuarial applications (Biihlmann 1970, Heilmann 1989, Makov

442


1995, among others). In this chapter we consider the case in which the distribution of K is specified up to an unknown parameter, and where ratemaking incorporates individual claim experience. Now, therefore, we consider the case where a risk K within a given collective of risks is characterized by an unknown parameter A which is thought to be a realization of a parameter space A. Using the notation described in Section 2, the Bayes premium is defined (Heilmann 1989) to be the real number V(ki,k2,...,kt) minimizing the posterior expected loss E7r0(A|fe1,fc2,...,fct)[^(^(A),'P(A;i,A;2,...)A;t)], i.e., the posterior expected loss sustained by a practitioner who takes the action V(ki,k2,..., kt) instead of the 'P(A), the risk premium1, of which is unknown. Let the quadratic loss function be L [P(X),V(h,

k2,...,h)]

= (V(X) - V(k1}

k2,...,kt))2,

then V{k1,k2,...,kt)=

/

V(\)-K0(\\ki,k2,...,kt)d\,

is the Bayes net premium. Using the Bayes net premium, Lemaire (1979) introduces the following formula to be applied in a BMS

this is the ratio between the posterior expectation and the prior expectation multiplied by 100, where V££ (k) denotes the bonus-malus premium to be loaded when we use the prior 7r0, with observed sample result k. Observe that the premium the policyholder has to pay if the initial premium (t = 0, k = 0) is 100. 'Usually V{\) is obtained by minimizing %(fc|A) [£(&, P(A))] using the same loss function that we use to obtain the Bayes premium.

443


Now, using (11) in the models described in Subsections 2.3.1 and 2.3.2, we obtain the Bayes bonus-malus premium by V2 (k) =

1 0

0 ^ < A ' ^ " ^ = 100 * ± * Eno(X) (s + t)C

(12)

and V* (k) = 100*"° (A ' ku **' "' kt) = 1 0 0 ^

+ 1

^/f3>)

(13)

respectively. Our approach in this chapter is based on the assumption that the practitioner is unwilling or unable to choose a functional form of the structure function, ir, but that he may be able to restrict the possible prior to a class that is suitable for quantifying the actuary's uncertainty. Therefore it is of interest to study how the premium for priors in such a class behaves. We use the classical e—contamination class of priors, T£ =

{TT(0)

= (1 - e) 7TO(0) + eq{9) \ q e Q},

where e reflects the amount of probabilistic uncertainty in a base prior 7T0 and Q is a class of allowable contaminations (Berger 1985, Gomez et al. 2000, 2002a and 2002b, Rios and Ruggeri 2000, Sivaganesan and Berger 1989, Betro et al. 1994, Goutis 1994, among others). These classes have been used in several situations to measure the sensitivity of quantities which can be expressed in terms of the posterior expectation of parameter functions. Nevertheless, when a relativity as in (11) is used, few papers have treated the problem (Gomez et al. 2002b). We present basic results to study the range of posterior ratio quantities in the form S* = (/ A ^( A ) 7 r ( A I data)dA) / ^g(X)Tr(X)d\^

,

444


as 7r varies over an Ts. For Q* = {All probability distributions}, and Q** = {All unimodal distributions} we determine the range of 5* as 7r varies over i \ . The variational problems involved are reduced to finding the extremes of the functions of one variable to be solved numerically and the explicit solutions of {inf Sn; re € Te} and {sup 5T; TT € T£} to be obtained. These technical results are shown in the next section.

4

Technical Results

Let K be a random variable with probability function f(k | A); given a sample k = (ki,k2,...,kt) of K, the experimental evidence about A is expressed by the likelihood function f(k \ A) = 4(A). Let Q be the space of the probability measures in the parameter space (A, J7). Let us define

Q = \q : f Hi(\)q(\)d\

< au

i=

l,2,...,n\

(14)

where Hi are given g-integrable functions and en,, i = 1 , . . . , n are fixed real numbers. Let §1 = f g(X)q(\ | k)d\ = JA

IAg(X)lk(X)g(X)d\ fAlk(X)q(X)dX

be an a posteriori functional, where g is a given function on A such that 5g is well defined. The conditions in (14) are called generalized moment conditions and our aim is to find the range of Sn when n runs over T. Theorem 1 (Winkler 1988). Let (A, J7) be a measurable space and suppose that Q is a simplex of probability measures whose extreme points are Dirac measures S\, i.e., the zero-one function taking value 1 in A, and 0, elsewhere. Fix measurable functions Hi, H2, •••, Ht and real numbers ai,a2, ...,at. Consider the set Q as defined in (14). Then


445

(a) Q is convex and the set ex Q of its extreme points satisfies (

m

ex Q C Q' = \q e Q : q = J> 0,

^ *i = 1, Aj 6 A, 1 < m < n + 1 i=i

Furthermore, the vectors (Hi(Xi),..., are linearly independent.

Ht{\))',

1 < i < m

(b) If the moment conditions in Q are given by equalities, then equality of sets holds in (a). Lemma 1 (Sivaganesan and Berger 1989). Let ^ be the convex hull of a set $ / of probability measures on the real line given by $ / = {//T, r € / } , where / C Rfc is an index set. Assume that / and g are both real-valued functions on R, F-integrable for any F 6 i Assume that B + g(x) > 0 for a constant. Then A + Jf(x)dF(x) _ Fey B + / g(x)dF{x)

u TeI

A + Jf{x)fir(dx) B + / g(x)/iT(dx)'

The same result holds with sup replaced by inf. Lemma 2 (Sivaganesan and Berger 1989). For unimodal q and g such that / A g(\)lk(X) \ \)q(\)d\ < oo, then / g{\)lk(\)

| X)q(X)d\ = f"'

JA

JO

HS(z)dF(z),

where F is some distribution function and ' 9

H (z) =

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