Active Fixed Income and Credit Management

Active Fixed Income and Credit Management Frank Hagenstein and Tim Bangemann ACTIVE FIXED INCOME AND CREDIT MANAGEMENT...

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Active Fixed Income and Credit Management Frank Hagenstein and Tim Bangemann

ACTIVE FIXED INCOME AND CREDIT MANAGEMENT

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Active Fixed Income and Credit Management Frank Hagenstein and

Tim Bangemann, PhD

© Frank Hagenstein and Tim Bangemann 2002 Language Translations copyright © Palgrave Publishers Ltd 2002 Operational Risk copyright © Schäffer-Poeschel Verlag GmbH & Co. KG 2001 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The authors have asserted their right to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2002 by PALGRAVE Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world PALGRAVE is the new global academic imprint of St. Martin’s Press LLC Scholarly and Reference Division and Palgrave Publishers Ltd (formerly Macmillan Press Ltd). Published by arrangement with Schäffer-Poeschel Verlag GmbH. ISBN 0–333–99368–3 hardcover This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. A catalogue record for this book is available from the British Library. A catalogue record for this book is available from the Library of Congress. 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06 05 04 03 02 Editing and origination by Aardvark Editorial, Mendham, Suffolk Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire

Contents

List of figures

vii

List of tables

xi

Foreword

xiii

Preface

xiv

List of acronyms

xvi

1

Active diversification of fixed-income portfolios The investment process and benchmark selection Tactical allocation and its building blocks Top-down approach for a Euroland portfolio

1.1 1.2 1.3

2

3

4

1 1 4 9

2.1 2.2

Duration management Factors influencing duration management Decision-making methods for duration

13 13 14

3.1 3.2 3.3 3.4 3.5

Yield curve management Market directionality of yield curves Barbell analysis Strips Rolldown analysis Box trade analysis

23 23 33 44 56 60

Basis management Cheapest-to-deliver analysis Delivery option

75 75 89

4.1 4.2

v

CONTENTS

4.3 4.4

5

97 101

5.1 5.2 5.3 5.4 5.5

Volatility management Volatility and yields Yield volatility Option risk parameters Efficient Gamma trading Options markets and economic data releases

106 106 110 112 114 119

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Credit management Introduction Characteristics of corporate bonds The top-down approach Merton’s approach to evaluating a corporate bond Market drivers (methodology) of credit spreads Efficient frontiers and the Sharpe ratio Industry selection The bottom-up approach Indentures of corporate bonds Corporate bonds and defaults

124 124 126 136 147 156 179 189 201 221 224

Further reading

227

Index

229

6

vi

Calendar spreads Portfolio replication

List of figures

1.1 1.2 1.3 1.4 1.5 1.6 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Diversification of risk factors in fixed-income portfolio management Correlation of risk classes DEM/Euro, 1990–2000 Outperformance of Euro corporate bonds vs. Euro government bonds January 2001–June 2001 Monthly excess returns vs. a government benchmark (JPM World) by the formulation of ‘bets’ – x-axis: volatility; y-axis: excess return Positioning of the duration and the curve in years vs. benchmark Weighting of the countries in Euroland in years vs. benchmark Basis point volatility of four government bonds (Bunds): year 2000 Adjusted basis point values by volatility: year 2000 Relationship between German 3-month Libor minus 2-year yields and German repo rate: January 1990–October 2001 Yields of 2-year swap rates/2-year swap rates 1–12 months forward (dotted lines): January 1991–January 2001 10-year Bunds vs. DAX Index: July 1998–October 2001 Bollinger bands on the Bund future December 2001 IFO (4 months forward) and slope 10y–2y: October 2001 Yield histories of the 2-year bond, the 10-year bond, and the 10y–30y yield spread in Germany: January 1994–June 2000 Regression of the 10y–30y spread vs. 2y yields in Germany (r2=0.868): January 1997–June 2000 10–30 years Treasury spread USA vs. 2-year T-Notes: January 1999–October 2001 An increase in curvature 2y–5y–10y DEM/EUR swaps barbell: positive (negative) spread indicates 5y are cheap (expensive) 2y–5y–10y barbell vs. 10y yields: a certain market directionality is apparent (the increase in 10y yields leads to increased curvature): June 1998–July 2000

5 6 7 8 10 11 15 16 17 19 20 22 24 26 27 28 29 30

31 vii

LIST OF FIGURES

3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 4.1 4.2 4.3 4.4 4.5

4.6 4.7 4.8 4.9 4.10 4.11 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 viii

Relationship between 10-year yield volatility Bunds and Butterfly 2–5–10 Bunds Macauley duration for strips (continuous black line) and bonds Comparison of strips (continuous longer line) and conventional bonds using the equivalent-duration method A negatively sloped, or inverted, curve (strips: black line) A positively sloping curve (strips: black line) A U-shaped curve (strips: continuous black line) Curve-steepening (strips: continuous black line) Curve flattening (strips: continuous black line) Relative value: principal strip–cash bond vs. 10y–30y regression: January 1999–October 2001 Rolldown components Rolldown of German government bonds Jan-07B–Jan-03T spread vs. Oct-02B–OBL122 spread: 6 months Jan-07B–Jan-03T spread vs. Oct-02B–OBL122 spread: 3 months Oct-02B–OBL122 spread: 3 months Jan-07B–Jan-03T spread: 3 months 5y–10y spread (Jul-07B–Sep-01B) vs. Oct-02B–OBL122 spread: 6 months 5y–10y spread (Jul-07B–Sep-01B) vs. Jan-07B – Jan–03T spread: 6 months Gross basis histories of CTD/non-CTD bonds Change in net basis for parallel shifts in the yield curve: Sep–00 Bund future The basis as an option on bond futures The net basis as a function of the overall market level Change in the net basis for different parallel shifts and a simultaneous steepening of 5 basis points of the deliverables basket: Sep-00 Bund futures contract The net basis as a function of the slope of the curve Probability distribution of yields for parallel shifts: 5 slices Probability distribution of yields for parallel shifts: 10 slices Parallel shifts and curve twists with probabilities Mar/Jun-00 Bund calendar spread and Mar-00 Euribor future Calendar spread and yield spread (CTD Sep-00–CTD Jun-00) The left axis shows the annualized, daily yield changes in per cent (logarithmic); the shadowed areas represent bear markets Implied volatility of Bund futures and yield levels of 10-year Bunds Bund implied volatility and OATi discounted CPI (Consumer Price Index) Regression of basis point volatility/10-year Bund yield: 1999–June 2000 Directionality of basis point volatility High daily volatility of the Bund future Trend line of the Bund future Volatility and hedging frequency Regression between 10-year Treasuries and Bunds

32 45 48 49 49 50 51 51 53 57 59 63 64 65 66 70 71 81 84 85 86

86 87 93 94 95 99 100 107 107 108 109 110 116 116 118 121

LIST OF FIGURES

5.10 6.1 6.2

6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22

6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33

Cumulative profit and loss (January 1999–June 2000) by shorting strangles Investment grade and high-yield average path to default, August 2001 Total return correlation between triple-B rated (Merrill Lynch C0A4 Index; US Corporates) and double-B rated (Merrill Lynch H0A1 Index; US High Yield) corporate bonds: December 1996–January 2001 Relationship between spread and rating class (global): August 2001 Credit curve of Fiat Spa: October 2001 Top-down and bottom-up approaches for lower investment grade and non-investment grade credits Baa-spread vs. Fed funds: monthly, February 1977–June 2001 Baa-Corporate bond spread vs. Treasuries Relationship between US GDP growth and credit ratings of the ‘Big 3’ US car manufacturers: April 1982–June 2001 Range of industrial spreads in the US: 1960–99 US high-yield (BB–CCC) spread over Treasuries vs. NAPM PMI: January 1987–September 2001 Distribution of non-investment grade default rate: US 1970–99 Corporate bonds as a ‘short put option’ Viatel’s 12.5% 08 $ spread vs. TD/(TD + MCap) Xerox 7.15% 04 $ spread vs. TD/(TD + MCap) KPN 4.75% 08 EUR spread vs. TD/(TD + MCap) KPN equity vs. KPN debt: April 2000–October 2001 KPN spread and call implied volatility: July 2001–November 2001 Credit spread term structure in Merton model Pfandbrief-Swap spread and Bund-Swap spread Swap spread vs. AAA spread in US Swap spreads and the spread-level of double-B rated corporate bonds Relationship of spread products – left axis: ratio spread products (Corporate bonds, Agencies, ABS and MBS)/Government bonds; right axis: 10-year swap spread Implied swap default probabilities ‘Fair-value model’ DEM/EUR swap spreads Correlation between swap spreads (10 years) and the steepness of the curve (2/10 years asset swap differential): January–October 2000 Distribution of the deviation from the fair-value model Correlations between Euroland bonds and Bunds in the 10-year bracket High correlation between the Bund swap spread and Bund/OAT spread Spain and Belgium in comparison with the Bund swap spread Yield curve and the spread level in the non-investment-grade market Relationship between the steepening or flattening of the curve and swap spreads in Euroland, January 1999 – July 2000 Enron Corp, (Baa1/BBB+; defaulted in December 2001) credit default swap spreads: July 1999–October 2000 Yield Betas of various US industries (average benchmark rating: A3)

122 129

133 134 135 136 139 140 141 141 143 146 148 149 150 152 153 154 155 158 159 160

161 164 165 166 166 168 169 170 171 172 173 174 ix

LIST OF FIGURES

6.34 6.35 6.36 6.37 6.38 6.39 6.40 6.41 6.42 6.43 6.44 6.45

6.46 6.47 6.48

6.49 6.50 6.51 6.52 6.53 6.54 6.55

6.56 6.57 6.58 6.59 6.60 6.61

x

Yield Betas for the rating classes double-A through triple-B in Euroland (average benchmark rating: Aa3) 174 Relationship between NASDAQ volatility and the Euro triple-B spread levels 176 Implied equity volatility as a function of risk aversion 177 Steepening of the US credit curve in 1998 (the Vega effect) 178 Credit spreads and equity market volatility: January–September 2001 179 Correlation between NASDAQ and US high-yield market: 02/02/2000–30/09/01 180 Efficient frontier of a US-fixed-income portfolio: 1989–2000 182 Strategy 1: min. 25% Governments, max. 10% European High Yield, January 1998–July 2001 186 Strategy 2: max. 30% Pfandbriefe max. 10% European High Yield, January 1998–July 2001 186 Strategy 3: no investment restrictions: January 1998–July 2001 187 Comparison chart of efficient portfolios: January 1998–July 2001 187 Six months’ trailing standard deviation vs. current spread levels – y-axis: spread vs. 10-year Treasuries in basis points; x-axis: 6-months’ trailing standard deviation in basis points: 16 August 2001 191 Swap spreads of various sectors in the USA: 8 October 2001 192 One month’s volatilities of different sectors in the USA on a rolling basis: 8 October 2001 192 Three months’ Z-scores of different sectors in US – formula: (actual ASW spread – average ASW spread)/standard deviation: 8 October 2001 193 Telecom spread history, 5 years: June 2000–August 2001 194 Corporate single-A spreads vs. telecoms spreads, 5 years: June 2000–August 2001 194 Multiple comparisons – KPN vs. Telcos vs. Corporates, 5-year spreads: June 2000–August 2001 195 Spread curve for selected European telecoms: October 2001 196 Evaluation of the coupon step-up 198 Bottom-up approach for lower rated credits (A–B) 202 Comparison of the internal rates of return (IRRs) of investments to the cost of capital; NPV (net present value) = PVCFAT (present value of cash flows after taxes) – I (original cost of investment) 208 The main categories of financial ratios 212 Relationship between coverage and spread level of European and US double-B and single-B corporates 214 SWOT analysis 217 The competitive position of a firm 218 Business strategy 221 Structural subordination of corporate bonds 222

List of tables

1.1 1.2

Risk factors of a fixed income portfolio Excess return Euro corporates vs. government bonds: September 2001

3.1

Relationship between butterflies and market level: January 1999– September 2001 Type (a) 2y–5y–10y barbell in Germany Type (b) 2y–5y–10y barbell in Germany Type (c) 2y–5y–10y barbell in Germany Type (d) 2y–5y–10y barbell in Germany Prices and modified durations for bonds in 2y–5y–10y barbell Profit/loss for type (a) barbell for different curve scenarios Profit/loss for type (c) barbell for different curve scenarios Profit/loss for type (b) barbell for different curve scenarios Durations and weightings for barbell Yield pick-up example for type (a) barbell Yield pick-up example for type (b) barbell Strips convexity Duration-neutral switch examples: 1997 A barbell strategy (duration- and cash-neutral): 1997 A cross-currency strip strategy: 1997 Rolldown of German government bonds: June 2000 Rich/cheap analysis values Asset-swap spread analysis Rolldown analysis Repo rates Spot and forward spreads Outstanding issue sizes of bonds Summary of box trade analysis results Box trade weightings Profit and loss simulation (in DEM 1000)

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26

2 7 33 35 35 36 36 37 38 38 39 40 41 42 46 55 55 56 58 62 67 67 67 68 69 72 73 73

xi

L I S T O F TA B L E S

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

The CTD bond for different curve scenarios: Sep-00 Bund futures contract Probabilities of different curve scenarios Sample net basis values: Bund 4.50% Jul-09 Sample probability-weighted net basis values: Bund 4.50% Jul-09 Duration- and cash-neutral barbell Information on CTD bonds Futures barbell Bonds investment Futures portfolio to replicate a bond portfolio

88 95 96 96 102 102 102 103 104

5.1 5.2

Volatility and hedging frequency Economic data in USA

119 120

6.1 6.2

Corporate average rating transition matrix, in per cent: 1980–99 The likelihood for two consistent rating changes for US and European companies: 1983–November 2000 Downgrades and upgrades of corporate issuers (global): 1997–2000 Cumulative average default probability, over 2 years, 5 years and 8 years, by rating class, in per cent: 1983–99 (global) Selected rating changes: October 2001 Changes in bond prices over a 12-month horizon Total return correlation between various asset classes: 1985–2000 Total return correlation between all rating classes: 1989–2000 Selected bonds in financial distress with an inverted credit curve: 12 October 2001 Percentage increase or decrease in absolute and percentage levels between highs and lows of various market indicators; values in parentheses represent the starting points Swap spreads: July 1998–August 2000 Characteristics of US government and corporate bonds: 1990–2000 Returns, volatility and the Sharpe ratio: January 1988–May 2000 Correlation of returns between various asset classes: January 1998–July 2001 Weighting of asset classes – MVP and TP of the three strategies: January 1998–July 2001 Ratios of the three strategies: January 1998–July 2001 Breakeven spread = spread vs. Government x holding period in years/forward duration at the end of the holding period Selected industries and their main event risks Income statement US–GAAP Important balance sheet positions US–GAAP Cash flows US–GAAP Computation of free cash flow using the indirect method US–GAAP

126

6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22

xii

127 127 128 130 130 132 133 156

162 163 183 183 185 188 189 197 200 203 205 209 210

Foreword

This book aims to apply the decision-making process using quantitative methods – already established in the USA, and being used also in the area of investment banking – to the European fund-management sector. In a period of continually low interest rates in Europe and overseas, especially, alternatives are sought for the tactical management of bond portfolios. Using a top-down approach, we examine the process of decision making – a process that can also be used, with slight modifications, in the treasury departments of banks and in asset/liability management. Special thanks are directed to two colleagues: at Union-Investment GmbH, Alexander Mertz (Research and Development), and at Barclays Capital, Paul Hanau (Head of Product Control Germany) who, with their constructive comments, have provided us with significant support. Thanks are due also to Jan Seifert (Union Investment Fund Management Credits) for his excellent editorial input for Chapter 6. Last but not least, we are immensely grateful to Antti Ilmanen (Schroder Salomon Smith Barney) for the Preface and for his careful pre-reading of this book.

FRANK HAGENSTEIN TIM BANGEMANN

xiii

Preface

Not so long ago, bond portfolio management focused on just one question – maturity/duration – and decision making was based on qualitative analysis. The investment world has changed a lot. Now portfolio managers spend most of their time on various spread trades, and often hedge away the market-directional risk. Moreover, the decision-making process has become increasingly quantitative. Active Fixed Income and Credit Management will be a valuable guide to bond investors and market analysts in this new world. The book discusses several types of key trade, and the tools for analysing them. The range is quite comprehensive for bond investors in one country – currency risk and cross-country spread trades are deliberately left out. The book also recognizes the growing importance to bond investors of derivative contracts (futures, options and swaps) and of credit diversification. Separate chapters focus on: ■ duration management (market-directional trades) ■ yield curve shape management (duration-neutral trades, such as

barbells versus bullets with three bonds and box trades with four bonds) ■ basis management (trading bond futures versus cash) ■ volatility management (option trading); and

xiv

P R E FA C E

■ credit management (spread trades between ‘riskless’ government

bonds and ‘risky’ credits, as well as trades across credit rating classes or across industry sectors). For each trade type, the main influences and the analytical tools are presented, with extensive illustrations from recent market experience. Analysing individual trades is just a part of smart portfolio management. Appropriately, Active Fixed Income and Credit Management begins by discussing the broader insights behind successful portfolio management. First, investors should determine the strategic long-run investment goals, and perhaps encapsulate these in a neutral benchmark portfolio. Second, investors should develop a process that gives them the best chances of consistently outperforming the benchmark through active deviations from it (tactical trades). The process naturally requires good forecasting ability but, whether the decision-making process is quantitative or qualitative, it is difficult to enhance forecasting accuracy for a single trade much above 50%; for most of the time, 60% accuracy may be the best realistic goal that can be targeted. However, diversifying across several (somewhat uncorrelated) tactical trades can significantly raise the consistency with which the active portfolio outperforms the benchmark. The magic of diversification smooths portfolio performance and may magnify the odds of outperforming the benchmark to above 60%, perhaps even above 70% of months. Finally, well-structured organization and a ‘top-down’ approach to various trade types are both helpful. ANTTI ILMANEN

xv

List of acronyms

ASW bpv b/e rate bn CAPEX CAPM CBOE

asset-swap spread basis point value breakeven rate billion capital expenditure capital asset pricing model Chicago Board Options Exchange CDS credit default swaps CPI Consumer Price Index CTD cheapest-to-deliver DelOptVal delivery option value EBITDA earnings before interest, taxes, depreciation and amortization ECB European Central Bank ECI Employment Cost Index EPS earnings per share FED Federal Reserve Bank FOMC Federal open market committee FX foreign exchange GAAP General Accepted Accounting Principles GDP Gross Domestic Product GB gross basis IO interest only IPO Initial Public Offering IRR implied repo rate/internal rate of return (in Chapter 6) xvi

M&A m Mod. MVP NAPM n/c OAT OATi OTC p.a. PO PP&E PPI PWYP PVCFAT ROE STRIP SWOT TMT UMTS WACC

Mergers & acquisitions million Modified minimum-variance portfolio National Association Purchasing Management Index net of carry French government bond inflation-linked French government bond over the counter per annum principle only property, plant and equipment Producer Price Index proceeds-weighted yield pick-up present value of cash flows after taxes return on equity Separately Traded Receipts of Income and Principal strengths, weaknesses, opportunities and threats telecom-media-technology universal mobile telecommunications system weighted average cost of capital

CHAPTER 1

Active diversification of fixed-income portfolios

1.1 THE INVESTMENT PROCESS AND BENCHMARK SELECTION Portfolio management is a structured investment process which comprises the following sections: ■ strategic asset allocation (fundamental, long-term perspective) ■ tactical asset allocation (short-term assessment, and hence tactical) ■ performance and risk analysis (performance measurement).

In general, asset allocation denotes the systematic investment of monetary assets into the various asset classes (in this case, fixed-income bonds and their respective derivatives). Strategic asset allocation defines the objectives and orientation of portfolios – that is, it determines the benchmark. In contrast, tactical asset allocation addresses how objectives will be achieved, by formulating tactical short-term (0–3 months) views, based on current market conditions. The goal of tactical asset allocation is to achieve an excess return over the benchmark. Chapters 2–6 describe how this can be achieved, assuming a single-currency portfolio. Performance 1


Table 1.1 Risk factors of a fixed income portfolio Risk factor

Assessment of risk factor

Change in market conditions

Directional market exposure (duration risk) Chapter 2

Duration

Parallel moves in yield levels

Yield curve risk Chapter 3

Steepening/flattening barbell (convexity)

Change in steepness or convexity of the curve

Bond basis risk Chapter 4

Bond basis (gross and net)

Change of the basis (through a parallel yield shift, steepening/flattening and changes in moneymarket/repo rates)

Market volatility risk Chapter 5

Volatility and convexity (Vega and Gamma)

Market volatility (increase/decrease) – historical volatility – implied volatility

Credit risk Chapter 6

Average credit rating and duration

Change in credit spreads (for example, government spreads vs. double-A or triple-B credits)

Allocation risk: countries/sectors – macro level (all countries and sectors, for example governments, agencies, corporates) – micro level (sector and company selection) Chapter 6

Allocation of various Change in spreads on the sectors (in percentages; macro and micro levels macro and micro levels) and aggregate sector spreads

and risk analysis delivers the assessment, measurement and valuation of investment results (for example, by implementing the concepts of ‘tracking error’ as a risk measure, or Sharpe ratio as a risk-adjusted performance measure; see section 6.6). In order to evaluate the performance of investment management, a specific benchmark has to be selected. This could be a market index (for example, Lehman Aggregate, JPM Government Indices – see the Appendix in section 1.3.1) or a self-defined benchmark. In looking at fixed-income portfolios and their respective benchmarks, it is useful to focus on the characteristics that determine the performance of these port2

folios. These characteristics can be defined as risk factors, and are introduced in Table 1.1. Performance measurement compares achieved portfolio returns with the benchmark. Returns are especially similar when the risk factors of the managed portfolios and the risk factors of the benchmark are the same (equivalent). This form of portfolio management is regarded as a passive investment strategy. The full spectrum of possible strategies is as follows.

Replication of all risk factors

Deviation of at least one risk factor from the benchmark

totally passive Indexing (Modelling all risk factors)

Deviation of all risk factors

totally active For example, replication/modelling of the duration risk factor, deviation of other risk factors

Deviation of all risk factors from the benchmark

Having selected a benchmark, the portfolio manager needs to decide whether he or she wants to replicate all risk factors (passive strategy), or whether the deviation of one or more risk factors from the benchmark is allowed (active strategy). Passive strategies do not require a process that would forecast changes in market conditions. An active strategy, on the other hand, is based on the forecasting of changes in market variables. Quantitative techniques are being implemented more and more in the investment process, and can be expressed using an index, as follows: ■ formulation of an investment policy that puts limits on the allocation

of asset classes, diversification needs and duration goals ■ choice of a standardized benchmark ■ quantification of risk exposure, using historical correlations and

volatilities ■ performance and risk analysis of the portfolio versus the benchmark,

by simulations of various scenarios (shift of the yield curve, changes in spread levels, and so on) ■ development of a methodology for the replication of portfolios during

periods of a neutral market view ■ evaluating (comparing) the risk–return relationship. 3

chapter one

A C T I V E D I V E R S I F I C AT I O N OF FIXED-INCOME PORTFOLIOS


In order to manage a portfolio relative to a benchmark, it is important to analyse the composition of the benchmark. The average rating, duration, and allocation – split-up – of the benchmark (see the Appendix, section 1.3.2) into various risk dimensions are of particular relevance; a benchmark consisting merely of corporate bonds can be divided into sectors, rating classes and duration.

1.2 TACTICAL ALLOCATION AND ITS BUILDING BLOCKS The goal of an active diversification strategy is to beat the benchmark: that is, to realize excess returns relative to the market index. Having defined the risk factors in section 1.1, this and the following chapters demonstrate how an outperformance can be achieved via the diversification of various ‘bets’ on the risk factors. Volatility analysis (Chapter 5), together with the relative-value approach (Chapters 3–4), leads to a diversification of independent investment ideas – where possible, uncorrelated bets – and, hence, to improved risk–return relationships. The variety of investment ideas generated in different areas (see Table 1.1) is a result of macroanalysis, valuation models, flows, technical analysis, and strategic over- or underweighting of different asset classes. The following chapters do not focus in detail on macroeconomic analysis but on its importance for the investment process, and any strategic decision over a 3–12-month period should be acknowledged. The macroeconomic analysis leads to long-term interest-rate and currency projections; analyses of political events are also included in this category. With respect to duration management, discussed in Chapter 2, it is important to note that diverse, fundamental factors support the decisionmaking process. The other building blocks (see the risk factors in Table 1.1) – yield curve, basis, volatility and credits – will be discussed in detail in Chapters 3–6. The country and sector selection is done first by countries, and second by different sectors (government bonds, agencies, corporates, swaps, and so on), and leads to corresponding over- and underweightings. Credit analysis incorporates the 'top-down' and ‘bottom-up’ parts and concentrates on individual company selection. Quantitative research develops pricing, yield-curve and fair-value models, and serves both as a bottom-up and a top-down element because of its ability to evaluate the complete market from a top-down perspective apart from the fundamental single-issue analysis, which is bottom-up. 4

The organization of a fixed-income department can be divided into generalists and specialists covering different areas (see Figure 1.1). Diversification of different instruments such as bonds, futures, swaps and options is required to achieve an optimal mix. Figure 1.1 illustrates a topdown approach (see section 1.3), in which specialists bring in their knowhow and this is bundled by one or more generalists. The various risk parameters – such as duration, yield curve, volatility and credits – are correlated in Figure 1.2, which illustrates whether they have a perfect relationship or allow for potential diversification. Uncorrelated risks represent an advantage for the risk profile of a portfolio (section 6.6). Duration is represented by the 5-year DEM/EUR-swap rate, and the yield curve is expressed by the spread between 2-year and 10-year swap rates. The credit spread is the interest rate differential between the REX10 index (an index of German government bonds) and the 10-year swap rate. The volatility is defined as the absolute change in the 10-year swap rate (annualized). It is interesting to observe that only two risk classes, duration and yield curve, have a moderate correlation over a 10-year period. Only a low correlation exists among other risk classes. However, it is worth noting that although the long-term correlation between duration and volatility is not strong, in certain economic phases it may be significant (see Chapter 5). The deviation in duration versus the benchmark (for example, 3 months+) leads to minimal results even if the yield level decreases by 50

Macroanalysis Duration

Yield curve

Market selection Sector selection

Basis/futures

Volatility

Credits

Technical analysis Generalist/sample portfolio

Quantitative models

Credit analysis

Figure 1.1 Diversification of risk factors in fixed-income portfolio management 5

chapter one



-60 -30 0 30 60 90 120

-60 -30 0 30 60 90 120

Figure 1.2 Correlation of risk classes DEM/Euro, 1990–2000 Source: Morgan Stanley, Bloomberg 6

2.40%

1.98% 1.78%

2.00% 1.60%

1.16% 0.97%

1.20% 0.80% 0.40% 0.00%

es rat rpo o AC AA

es rat rpo o C AA

es rat rpo o BC BB

es rat rpo o AC

Figure 1.3 Outperformance of Euro corporate bonds vs. Euro government bonds January 2001–June 2001 Source: Union Investment

basis points (0.50% • 0.25 = 0.125% p.a.). In contrast to this, excess returns (additional yield versus governments) using triple-B Euro corporates during the first half of 2001 were slightly below 2% – see Figure 1.3. For example, assuming a government portfolio contains 30% single-A rated corporate bonds which achieve an excess return of 2.5% per year, this implies an excess portfolio return of 75 basis points (2.5% • 30% = 0.75% p.a.) versus the benchmark (government benchmark). This example demonstrates the high risk/chance characteristics of credit products as additional mix for portfolios with a government benchmark. Clearly, the excess return can also be negative: for example, after the tragic events in New York and Washington on 11 September 2001, an extreme risk aversion and flight to quality lead to an underperformance of corporate bonds versus government bonds (see Table 1.2). So a neutral positioning relative to the government benchmark should lead to a 0%

Table 1.2 Excess return Euro corporates vs. government bonds: September 2001 Excess return Euro corporates vs. governments September 2001

AAA

AA

A

BBB

+0.172%

−0.015%

−0.892%

−3.461%

Source: Union Investment and Bloomberg

7

chapter one


.

. . .

.

. .

.

. . .

.

.

.

. .

.

.

.

.

.

Figure 1.4 Monthly excess returns vs. a government benchmark (JPM World) by the formulation of ‘bets’ – x-axis: volatility; y-axis: excess return (MVP= minimum-variance portfolio; FX = foreign exchange) Source: Union Investment

.

weighting of credits; only a small percentage of credits could be added in the form of a diversified corporate bond fund. Figure 1.4 shows the excess returns versus a government benchmark achieved by deviations from the benchmark. The spread returns comprise only high rating classes (AAA–AA), and do not reflect the high risk/chance in the case of a strong overweighting of lower-rated (A–BB) spread products. A mixture of different ‘bets’ leads to a minimization of the portfolio volatility and an increased return, which is discussed in more detail in Chapter 6.

1.3 TOP-DOWN APPROACH FOR A EUROLAND PORTFOLIO The top-down approach defines the strategic and tactical allocation of a portfolio. At this point credit risks are neglected, because the following example deals with a Euroland portfolio (the benchmark is, for instance, JPM EMU Governments) that consists only of government bonds; the process of credit decision will be described in detail in Chapter 6. Recommendations are based on a fundamental analysis, which consists of an overall economic view and expectations of central bank policy. Furthermore, quantitative indicators and relative-value considerations (Chapter 3) are a vital part of the process, while technical factors such as issuance volumes by government bodies, investor flows and market sentiment are also considered. Duration management (Chapter 2) is applied by taking a position against the benchmark (here, Euro governments), and plays a part in the whole investment process. The shape of the yield curve is the next important factor: it can be described as flattening, steepening, or by formation of curvature (as, for example, barbells – see Chapter 3). The country allocation leads to an over- or underweighting of a country in Euroland. Curve anomalies between two countries are also considered. The individual company selection completes the decision-making process.

1.3.1 Example The following deviations versus the benchmark have been undertaken. ■ Market direction: +0.3 years. This means benchmark 5.2 years + 0.3

years = 5.5 years (see the Appendix, section 1.3.2 below).

9

chapter one



0.35 0.35

0.30 0.3 0.25 0.25 0.20 0.2 0.15 0.15 0.10 0.1 0.05 0.05

–

+ 10

7–

10

ye ar s

ye ar s 7 5–

3–

5

ye ar s

ye ar s 3

0.10 -0.1

–-0.15 0.15

1–

–

ye ar s

00 –-0.05 0.05

0.20 -0.2

Figure 1.5 Positioning of the duration and the curve in years vs. benchmark (sum of the bars = 0.3 years; left axis)

The portfolio duration is positioned over the benchmark duration (here 0.3 years). The positioning will be in the middle maturity bracket. The bet is on decreasing yields (increasing prices) in the market. ■ Curve: butterfly positioning: short, 1–3 years; long, 5–10 years; short,

30 years (10+) (see Chapter 3). Underweighting the short end and the very long end, and overweighting the middle part of the duration spectrum. The view is for a decrease in curvature around the middle maturity sector. ■ Country selection: overweighting: Germany, Italy and Austria.

Overweighting government bonds from the named countries, and underweighting France and Belgium. The goal is an outperformance of the selected countries versus other countries. The individual bond selection concludes the investment process. The aim is to pick or identify ‘cheap’ bonds (see Chapter 3.5) in the preferred countries, and to choose the desired terms to maturity.

10

0.5 0.4 0.3 0.2 0.1

str ia Au

m iu Be lg

N

et h

er lan

ds

Sp ain

ly Ita

ce Fr an

Ge rm

– -0.2

an

y

0 – -0.1

– -0.3

Figure 1.6 Weighting of the countries in Euroland in years vs. benchmark (sum of the bars in years = 0.3 years; left axis)

1.3.2 Appendix: Examples for Euroland benchmarks (October 2001) JP Morgan EMU Government: ■ consists of only government bonds in Euroland (100%) ■ rating classes: AAA, AA, A ■ modified duration: 5.49 years ■ maturity classes: 1–30 years.

Lehman Euro Aggregate: ■ consists of government bonds: Euroland (67%), Collateralized (15%),

Corporate (15%), and Supranationals (3%) ■ rating classes: AAA–BBB ■ modified duration: 4.74 years ■ maturity classes: 1–30 years. 11

chapter one



Merrill Lynch EMU Corporate (February 2002): ■ consists of only Corporate Investment grade bonds (100%) ■ rating classes: AAA–BBB, average Rating A+ ■ modified duration: 4.32 years ■ maturity classes: 1–15 years ■ sectors are divided into Financials (54%), Industrials (16%), Telecoms

(13%), Automotives (9%), Utilities (6%), and ABS-Structures (2%).

12

CHAPTER 2

Duration management

2.1 FACTORS INFLUENCING DURATION MANAGEMENT The process of duration management is a combination of different components. Economic trends play an important role, as do expectations regarding central bank policy. Technical analysis, which can be employed to improve the timing of trading decisions and to help set disciplined stop-loss levels, is also useful. The issuance calendar of government bonds (for example, Bund), investor flows and positioning, and the market sentiment are all relevant. It is worth noting that in past times of longlasting interest-rate trends, mainly the decreasing rates during the 1990s, investors used the low-inflation environment to overweight duration (versus benchmark), but this turned out to be counterproductive during the bear markets in 1994 and 1999. The steepness of the yield curve (a positive-carry situation) plays an important role for the investor entering into and holding long-duration positions. Here the rolldown analysis (section 3.4) supports the decisionmaking process. Real interest rates are an important factor, too. High real yields can lead to moderate or even high confidence levels among investors, which result in bets on duration. Inflation-linked bonds issued by France are considered to be a good proxy for real interest rates and future inflation rates in Euroland; inflation is inversely correlated to the nominal bond returns. A stronger currency and declining commodity 13


prices (CRB Commodity Price Index) are disinflationary forces, which can also be relevant drivers of bond markets. Market momentum – trend movements – must be mentioned as an important component. If interest rates have already been trending lower in the past, the probability for declining future yields increases: the reasons for this could be either technical or fundamental. Momentum can be interpreted, for example, as a gradual reaction of the market to a change in monetary policy (fundamental background). Determining confidence levels in decision making is of particular importance for the duration-management decision process. Here, one differentiates between three different confidence levels: low, moderate, and high. A low confidence for an outright long strategy would lead to a hedge of the long position (versus benchmark duration), through put options or a reduction in the risk position via a sale. A moderate confidence should be reflected in a positioning corresponding to the forecasts. A high level of confidence results in extreme ‘bets’ made by applying instruments with large leverage effects (options). An example would be the additional sale of put options on a 30-year government bond (high confidence for declining interest rates at the long end, despite the fact that a long position already exists), or the purchase of a 30-year zero bond or strips (see section 3.3). In general, weak economic growth, low inflation, decreasing volatility (see Chapter 5), government-bond buybacks and an expansive monetary policy support a positive environment for bond markets.

2.2 DECISION-MAKING METHODS FOR DURATION 2.2.1 Duration Duration can be interpreted as the average length of time that a bond investment is outstanding (the weighted average term to maturity of a stream of cash flows, with the weights being the fraction of total value represented by each cash flow): it is thus a measure of interest-rate risk. If duration were defined as the only measure of portfolio risk, three assumptions would need to be made: ■ that the price/yield relationship of bonds is linear ■ that the changes in bond yields are perfectly correlated ■ that the yield volatilities of bonds are identical.

14

None of these three assumptions holds true in reality. If they were valid at all times, portfolio risk would not be affected by a duration-neutral switch out of 2-year Bunds (government bonds) into 30-year Bunds. It is important to understand that all three assumptions are incorrect. In this case, the switch would lead to an increased portfolio convexity; the correlation of the 2-year and 30-year government bond yields does not equal 1 (r2=0.56 for January 1999–June 2000); and the implied annualized basis point volatility of the two bonds is not identical (see section 5.2 on yield volatility). The implied annualized basis point volatility for 2-year bonds is 83 basis points, whereas it is only 74 basis points for the 30-year bonds (May 2000). This empirical proof of a non-parallel yield change has led to the use of so-called yield Betas. For example, a regression analysis implies that a change of 10 basis points in the yield of the 10-year Bund results in only an 8 basis point move in the yield of the 30-year Bund (r2=0.91 between the yields of these bonds from January 1999 to June 2000). This means that if the yield Beta for the 10-year Bund equals 1.0, the 30-year Bund has a yield Beta of 0.8. Implied volatility can be derived from the term structure of OTC options between 2 and 30 years (see Figure 2.1). In the recent past, the short end of the curve has been more volatile than the long end, resulting in the use of volatility-adjusted duration (see Figure 2.2), which needs to be readjusted in the curve over the course of time, given changing

86 84 82 80 78 76 74 72 70 2 years

5 years

10 years

30 years

Basis point volatility

Figure 2.1 Basis point volatility of four government bonds (Bunds): year 2000 15

chapter two

D U R AT I O N MANAGEMENT


14 12 10 8 6 4 2 0 0

1 2 years

2 5 years

10 3years bpv

4 30 years

5

adj.bpv

Figure 2.2 Adjusted basis point values by volatility: year 2000

volatilities. For example, the duration (here the basis point value) of the maturity range with the highest volatility (2 years) will be kept unchanged and all other maturities will be adjusted according to their volatility (example 30 years: bpv 74/bpv 83 = 0.89; 0.89 × 12 bpv = 10.70 adjusted basis point value). By over- and underweighting various parts of the yield curve, volatility is synthetically bought and sold. If there is a high degree of confidence for decreasing interest rates, portfolio duration will be significantly increased over the benchmark by overweighting 2-year bonds and by avoiding long positions in callable bonds. To achieve the targeted duration one can mix long-maturity strips (see Chapter 3) into the portfolio. Because duration and yield-curve management cannot be separated over a longer time period, examples are included in Chapter 3 to demonstrate the non-parallel behaviour of yield-curve shifts. Another example for duration positioning is presented in the following. Expectations regarding the central bank policy have to be determined. The forward 3-month Euribor rates point to strong interestrate increases over the coming months, which in the year 2000 were already reflected in the yields of 2-year bonds. Since the fund management also has positive expectations regarding interest-rate increases, an analysis needs to be performed to find out whether the interest-rate increases already priced in by the market will actually be achieved in the future (see Figure 2.3). 16

10

1.5 9

1.0 8

0.5 7

0 6

0.5 5

– – 1.0

4

––

1.5 3

– –2.0

2

Source: CSFB

3-month Libor minus 2y yields

Figure 2.3 Relationship between German 3-month Libor minus 2-year yields and German repo rate: January 1990–October 2001 03.09.01

03.05.01

02.01.01

01.09.00

03.05.00

03.01.00

02.09.99

04.05.99

01.01.99

02.09.98

04.05.98

01.01.98

02.09.97

02.05.97

01.01.97

02.09.96

02.05.96

02.01.96

01.09.95

03.05.95

02.01.95

01.09.94

03.05.94

31.12.93

01.09.93

03.05.93

31.12.92

01.09.92

01.05.92

01.01.92

02.09.91

02.05.91

01.01.91

31.08.90

02.05.90

–

01.01.90

2.0

German repo rate (RHS)


2.2.2 Central bank policy There is a strong correlation between the 3-month Euribor rates and the 2-year bond yields (or swap rates), and expectations by the market are interpreted as too bearish (in year 2000). This leads to an overweighting of corresponding bonds and an increased duration in the short end (for example for fixed-income funds with short duration). The yield differential between the 3-month and 2-year yields tends to decline (flatten) in times of an active, restrictive central bank policy (increasing interest rates), which favours 2-year yields – and vice versa (in year 2000; Figure 2.3). The duration will be increased, despite expectations of an increase in interest rates, because forward rates price in a more negative expectation than the fund management expects. Figure 2.4 explains this strategy. Here, yields of 2-year swaps were combined as a time series with computed forward rates. For example, in the beginning of 1997, 2-year swaps were computed on a 1–12 month forward basis. It is observed that in most of the cases, actual realized 2-year rates were below forward rates, implying that future rates for 2-year swaps very often incorporated a forecast of too large an increase in interest rates. Empirical tests show that they are bad proxies for future rates. Since they are traded and liquid, one can lock in an overstated interest rate rise. The chart shows that over the last 10 years, the forwards were exceeded by realized rates only in a few instances (1992, 1994 and 1999), resulting in the market building up a time-risk premium, which proves in many cases to be too high. In a steep yield-curve environment, the forward rates always increase across all maturities. Equity markets in the US came under severe pressure in November 2000 as the economic outlook was revised downwards and companies began announcing lower than anticipated profits. US interest rates decreased, due to high expectations of further interest-rate cuts. In this environment, moves in US equity markets and US interest rates were highly correlated; this was also true in Germany (see Figure 2.5). The strong correlation between equity and fixed income markets was also evident after the terrorist attacks against the US on 11 September 2001: these caused US stock markets to close for four days, after which they opened markedly lower, while fixed-income markets received ‘safe haven’ flows.

18

Figure 2.4 Yields of 2-year swap rates/2-year swap rates 1–12 months forward (dotted lines): January 1991–January 2001 Source: Salomon Brothers

13 /0 7/ 04 98 /0 9 29 /98 /1 0 23 /98 /1 2/ 98 16 /0 2/ 99 12 /0 4/ 99 04 /0 6/ 99 29 /0 7/ 99 22 /0 9/ 99 16 /1 1/ 99 10 /0 1/ 00 03 /0 3/ 00 27 /0 4/ 00 21 /0 6/ 00 15 /0 8/ 00 09 /1 0/ 00 01 /1 2/ 00 25 /0 1/ 01 21 /0 3/ 01 15 /0 5/ 01 09 /0 7/ 01 31 /0 8/ 01

DAX Index 8000

6000

5000 4.5

4000 4

3000 3.5

2000 3

German Stock Index DAX

Source: Union Investment and Bloomberg 10-year Bunds Yield

Figure 2.5 10-year Bunds vs. DAX Index: July 1998–October 2001

Yield in %

9000 6

5.5

7000 5

2.2.3 Bollinger bands Bollinger bands allow another way of making decisions on duration. They are a combination of a charting technique and a statistical method, and are introduced here by a further example. Bollinger bands (Figure 2.6) are defined as a specified number of standard deviations above and below a moving price average. Bollinger bands are bands of variable width: they tend to tighten in less volatile periods, and vice versa. They are useful for determining ‘probable’ support and resistance levels in the market. The most common set-up is 2 standard deviations (assets move with a 95.5% probability between the upper and lower band range), and the moving price average period is 20 days. Should the upper (lower) band be touched, market resistance (support) could be expected. (Short-term overshooting is possible.)

21

chapter two

D U R AT I O N MANAGEMENT

Figure 2.6 Bollinger bands on the Bund future December 2001 Source: Bloomberg

CHAPTER 3

Yield curve management

3.1 MARKET DIRECTIONALITY OF YIELD CURVES 3.1.1 Drivers of yield curve steepness The German IFO Index is a general business-climate index compiled monthly by consulting approximately 10,000 companies throughout Germany. In the October 2001 IFO survey, German business confidence showed the largest monthly decline since November 1973. A fall of this magnitude was partly attributable to the terrorist attacks of 11 September 2001. The weaker the economy, the lower short-term rates have to fall to give rise to a recovery in the economy – that is, the yield curve steepens. The IFO measures not only current conditions but also expectations, so it tends to anticipate movements in the slope (4 months, in this case). As the IFO – the economy – gets weaker, the market starts pricing in rate cuts and the curve steepens. The interesting part of this chart is the significant fit up to 1998, clearly indicating that the Bundesbank’s monetary policy reacted to German weakness or strength. From 1999, the ECB has reacted to Euro-11 data, and for this reason the fit cannot be expected to be as good as pre-EMU. It follows, however, that ECB monetary policy at time of writing (October 2001) is too tight for Germany. The flattening or steepening and the changes in the convexity or concavity of yield curves are examined here using different examples. 23

10y–2y slope

IFO (4 months forward)

Figure 3.1 IFO (4 months forward) and slope 10y–2y: October 2001 Source: Merrill Lynch

From these, it becomes apparent that the direction of the market – or the yield levels of certain maturity sectors – is decisive for the shape of the curve. In the first half of 2000, yield curves flattened strongly on a global basis. In the US, the phenomenon of an inverted yield curve in the long end – that is, 30-year yields lying below 10-year yields – was seen again after a long time. Also, in Germany the curve flattened across all maturity sectors and especially in the 10y–30y area. Several factors were influential. In the US in particular, structural changes via bond buy-back programs caused an accelerated flattening in the longer end of the yield curve. Additionally, short-end interest rate increases by central banks in the US and Europe helped long bonds outperform, which caused the curve to flatten. This correlation between short-end interest rates and the 10y–30y yield spread is shown in Figure 3.2, which uses an example with German bonds. The regression analysis between the 10-year and 30-year bond yield spread and the absolute level of 2-year yields illustrates this significant relationship (see Figure 3.3). The higher the level of 2-year yields, the narrower the yield difference between the 10y and 30y bonds becomes, and vice versa. The 2-year bond is often used by central banks as an indicator of expected future monetary policy. The coefficient of determination (r2) for the previous 5 years is greater than 80%, and was calculated over the time period January 1997 to June 2000 as r2=0.87 (see Figure 3.3). An increase in 2y yields of 100 basis points would therefore imply a flattening in the 10y–30y yield spread by 37 basis points. A correlation was also observed between the 10y–30y spread and the 10-year yields (r2=0.40; January 1997–June 2000). It was, however, significantly lower than the correlation with 2-year yields. Long 30-year Bund issues typically followed the 10-year yields with a time lag; such a lag leads to a narrowing of the spread as yields increase. This implies that the interest rate sensitivity for 30-year bonds was lower than that of 10-year bonds. The yield Beta of 30y bonds thus lies below 1, if the 10y yield Beta is set at 1. A yield change of 10 basis points in a 10y bond would therefore entail a yield change of only 8 basis points for the 30y bond (see the example in section 2.1, with a slope of 0.8 of the regression line using 30y bonds versus 10y bonds). If the view for short-end (2-year) yields is seen as positive, an underweighting of the 30-year sector (against the 10-year sector) is recommended, and vice versa, provided that one expects the correlation to hold (see Figure 3.4). Note that the strongest curve-flattening position is obtained via the duration-weighted position of long 30-year bonds and a long cash position. 25

chapter three

YIELD CURVE MANAGEMENT

. . . . .

.

.

.

.

.

.

.

Figure 3.2 Yield histories of the 2-year bond, the 10-year bond, and the 10y–30y yield spread in Germany: January 1994–June 2000 Source: JP Morgan

DEM benchmark 10s30s since 1997

EM benchmark 10s30s since 97

1,20 .

1,00 .

0,80 .

0,60 .

0,40 .

0,20 . y = -0,3682x + 2,0788 . . 2

R = 0,8685 . -– . 2,50

3,00 .

3,50 .

. 4,00

4,50 .

5,00 .

5,50 .

2-year yield

Figure 3.3 Regression of the 10y–30y spread vs. 2y yields in Germany (r2=0.868): January 1997–June 2000 Source: JP Morgan

When unforeseen extraordinary events take place, the markets often overreact and yield curves can move out of line. Two examples follow. ■ The dramatic terrorist attack in the US on 11 September 2001.

Euroland 2y yields declined by 3.5 bps, while the 10y–30y yield spread widened by 7 bps, moving the 10y–30y yield spread significantly further away from the mean regression line (similar moves were also experienced in the US market). During the course of the days after the event (up to 18 September 2001), Euroland 2y yields declined by 27 bps and the 10y–30y yield spread widened by 18 bps. Subsequently a flattening in 10y–30y yield spread occurred, and the regression moved back into line. ■ The announcement by the US Treasury on 31 October 2001 that it

would not issue any new 30-year government bonds. The 10y–30y yield spread flattened during the following two days by 21 bps, while 2y yields increased by 4 bps (two circled points in Figure 3.4 – it was not possible to follow the spread movement further in this book because of the timing of its publication). It is likely that, going forward, the 30y bonds will remain at expensive levels on the curve, as 27

chapter three


Spread 10y-30y in basis points –

–

–

2-year yield in per cent

Figure 3.4 10–30 years Treasury spread USA vs. 2-year T-Notes: January 1999–October 2001 Source: Salomon Smith Barney

demand for long-maturity bonds will outstrip supply. Therefore, although Figure 3.4 shows the 10y–30y spread to be currently out of line relative to the level of the 2y yield, it is not expected to move back into line as the Treasury announcement has permanently changed the market supply/demand balance. The important point to take from the above examples is to understand when yield curve anomalies can be taken advantage of and when there are permanent changes in regimes, which make the use of historical data of little use.

3.1.2 Market directionality of barbells A barbell position is defined as being overweight in the short and long sectors (wings), and underweight in the intermediate sector (centre or body), and vice versa (butterfly); for example, being long 2y and 10y but short 5y. The weighting in Figures 3.5–3.8 uses equal risk weightings (50%:50% wings:body). Barbells allow for the establishment of a wide variety of curve positions, which are analysed in this chapter. In bear markets – markets with high volatility – the curvature around the 5-year sector of the yield curve tends to increase, that is, there is an underperformance of the body against the wings (see Figures 3.5 and

4.75 4,75 4,5 4.50 4.25 4,25 4.00 4 3.75 3,75 3.50 3,5 3.25 3,25 3.00 3 2.75 2,75 2.50 2,5 2,25 2.25 2 2.00

Flattening 5 -10

Steepening 2 - 5 yield 1 yield 2

22years years 2

55years years 5

10 10 years years 10

Figure 3.5 An increase in curvature 29

chapter three


.

Germany: 5s - 0.5*(2s + 10 s) swaps .

. .

. . .

. –

.

–

.

–

.

–

.

.

.

Figure 3.6 2y–5y–10y DEM/EUR swaps barbell: positive (negative) spread indicates 5y are cheap (expensive) Source: CSFB

Germany: 5s - average 2s and 10s vs. 10s swaps, June 1998–July 2000 0.20 0.15

5s - Aug 2s, 10s

0.10 0.05 0 –

0.05

–

0.10

–

0.15

–

0.20 3.75

4.25

4.75

5.25

5.75

6.25

10 year swaps

Figure 3.7 2y–5y–10y barbell vs. 10y yields: a certain market directionality is apparent (the increase in 10y yields leads to increased curvature): June 1998–July 2000 Source: CSFB

10y yld volat

Butterfly 2–5–10 (SBS)

25

15 20

10

%

5

5y underperf.

20

15

10

5 Sep–99

Dec–99

Mar–00

Jun–00

Sep–00

Dec–00

Mar–01

–

5

–

10

–

15

5y outperf.

0

Jun–01

Figure 3.8 Relationship between 10-year yield volatility Bunds and Butterfly 2–5–10 Bunds Source: Merrill Lynch

Table 3.1 Relationship between butterflies and market level: January 1999–September 2001 Duration and cash-neutral butterfly

Selling weights

Wing weights in bpv

Correlation with steepness (2y–10y spread)

Correlation with 2y yield

2–5–10

58/42

25/75

−0.70

0.89

5–10–30

63/35

37/63

−0.79

0.82

2–10–30

52/48

13/87

−0.73

0.90

2–5–30

80/20

35/65

−0.80

0.92

Source: Salomon Smith Barney

3.6). If in addition the bear market is led by speculations of interest rate increases, the impact is even greater. Positively sloping yield curves are typically concave, while inverse (negatively sloping) curves tend to be convex. The barbell trade 2y–5y–10y shows a strong correlation against the spread of the current 3-month Euribor versus 3-month Euribor one year forward (future Euribor rate) (r2=0.75 for the time period 1 January 1999 to 1 June 2000). Similarly, in bull markets where yield curves steepen – as was seen in most parts of the year 2001, as central banks decreased their key interest rates – real-money investors with a bullish, steepening view could position themselves in butterfly structures: that is, they could buy the body bond against selling the wing bonds, for example buying 5y versus selling 2y and 10y, or buying 10y versus selling 2y and 30y. These butterflies have a strong correlation with the overall market level and benefit from overall curve steepening: see Table 3.1 (for explanations on weights, see the following section).

3.2 BARBELL ANALYSIS It is possible to put on various types of barbell positions, in order to express different views on the interest rate curve. In this section, the methods for weighting barbell trades and their performance are examined in detail. Consider these four barbell positions:

33

chapter three



(a) Duration-neutral and cash-neutral – that is, neutral against parallel shifts in the yield curve: the duration of wing positions (for example 2y and 10y) is equal but opposite to that of the body (for example 5y). These barbells have a curve-flattening bias, as the 5y–10y leg requires a greater weighting than the 2y–5y leg of the barbell: that is, more long 10y than 2y (the BBS function in Bloomberg). (b) Duration-neutral and twist-neutral – that is, neutral against parallel shifts and neutral against curve twists. These barbells represent a hedge against flattening or steepening of the curve: in a curvesteepening scenario, the gain on the 2y–5y leg compensates for the loss on the 5y–10y leg, and is thus a pure-curvature trade (immunizing for duration differences and basis point values of the legs 2y–5y and 5y–10y). (c) Duration-neutral with equal wings – that is, neutral against parallel shifts, with 2y and 10y weighted equally. These barbells represent curve trades where the 2y–5y spread is expected to move in line with the 5y–10y spread (the BBA function in Bloomberg). (d) Duration-neutral with unequal wings – that is, neutral against parallel shifts and any chosen weighting of 10y versus 2y. These barbells represent curve trades, whereby the investor can express his or her subjective view on the market, depending on how the 2y–5y spread is expected to move versus the 5y–10y spread. It is important to observe that only the type (a) barbell is cash-neutral: all other barbell types require either the borrowing or lending of cash. If the investor has no strong view on the curve and he or she wishes to take advantage of increasing or decreasing convexity in the curve, he or she should prefer the type (b) barbell, since its property of being twist-neutral provides a natural hedge against curve moves (flattening or steepening). In practice, however, the type (a) barbell is frequently preferred, because its property of cash neutrality simplifies the funding of the position. Also, entering into a 2y–5y–10y cash-neutral barbell position is not complicated, since the nominal weights for the 2y and 10y are almost equal. The strategy profits mainly form a general curve flattening in the 5y–10y sector. Barbells of type (c) and (d) are mainly used for curve trades, where the investor has a strong view on the curve, or where he or she prefers one wing position to the other. For example, if it is expected that the 5y–10y sector will outperform the 2y–5y sector, a barbell weighting of 70% in the 5y–10y leg and 30% in the 2y–5y leg could be used. Clearly, the type (c) barbell is a special case of the type (d) barbell, in which both wings are 34

Table 3.2 Type (a) 2y–5y–10y barbell in Germany Barbell type (a) Duration-neutral and cash-neutral

(long 2y and 10y vs. 5y)

Nominal (in 1,000)

Duration

Cash

Wing long

BKO 4.5% Mar-02

57,143

0.86

−57,623,203

Body short

OBL 5% May-05

−100,000

−4.21

101,892,466

Wing long

Bund 5.25% Jul-10

43,534

3.35

−44,269,263

Total

0.00

0

weighted by 50%. A barbell of type (d) with weightings of 0% and 100% constitutes a straight yield spread position. Using as an example the German 2y–5y–10y barbell (long 2y and 10y versus short 5y), Tables 3.2–3.5 show weights used for the different barbell types (market data as of 4 August 2000, with some values rounded). As is apparent from the above examples, all barbells are durationneutral, and the type (a) barbell is the only one that is cash-neutral and has similar nominal weights for the wing bonds. It is worth noting also that the type (c) barbell has equal duration weights for both wing bonds. The twist-neutral barbell, type (b), initially seems underweight in the 10y bond (24m nominal as compared with 158m nominal in 2y). The reasons for this are twofold: first, the duration of the 10y issue is significantly greater than that of the 2y bond; and second, the ratio of the durations between the wing bonds and the body bond is greater for the 10y bond. So it is only possible to achieve the twist-neutrality, whereby for example in a steepening scenario of the 2y–10y sector, the losses made on

Table 3.3 Type (b) 2y–5y–10y barbell in Germany Barbell type (b) Duration-neutral and twist-neutral


Nominal (in 1,000)

Duration

Cash

Wing long

BKO 4.5% Mar-02

157,876

2.38

−159,202,939

Body short

OBL 5% May-05

−100,000

−4.21

101,892,466

Wing long

Bund 5.25% Jul-10

23,782

1.83

−24,183,957

Total

0.00

−81,494,430 35

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Table 3.4 Type (c) 2y–5y–10y barbell in Germany Barbell type (c) Duration-neutral with equal wings


Nominal (in 1,000)

Duration

Cash

Wing long

BKO 4.5% Mar-02

139,582

2.10

−140,755,626

Body short

OBL 5% May-05

−100,000

−4.21

101,892,466

Wing long

Bund 5.25% Jul-10

27,369

2.10

−27,831,534

Total

0.00

−66,694,694

the 5y–10y leg of the barbell are exactly immunized by gains on the 2y–5y leg. Mathematical derivations for the weightings for the different barbell types are given in Appendix 1 at the end of section 3.2.2. For example, the weightings NS, for the wing bond with shorter maturity, and NL, for the wing bond with longer maturity, for a duration- and cash-neutral barbell (type (a)), are derived as follows (using a given weighting NC for the body bond): NS = NC • (PC / PS) • ((DL – DC) / (DL – DS)) NL = (NC • PC – NS • PS) / PL

Prices (including accrued interest), PC, PL and PS, and modified durations, DC, DL and DS, for the individual bonds are shown in Table 3.6. Thus, for NC = 100m (partially rounded values):

Table 3.5 Type (d) 2y–5y–10y barbell in Germany Barbell type (d) Duration-neutral with weightings 5–10 (70%) and 2–5 (30%)

36


Nominal (in 1,000)

Duration

Cash

Wing long

BKO 4.5% Mar-02

83,749

1.26

−84,453,376

Body short

OBL 5% May-05

−100,000

−4.21

101,892,466

Wing long

Bund 5.25% Jul-10

38,317

2.95

−38,964,147

Total

0.00

−21,525,057

Table 3.6 Prices and modified durations for bonds in 2y–5y–10y barbell Prices and duration of bonds

Price

Duration

Wing long

BKO 4.5% Mar-02

100.84

1.495

Body short

OBL 5% May-05

101.89

4.129

Wing long

Bund 5.25% Jul-10

101.69

7.558

NS = NC • (PC / PS) • ((DL – DC) / (DL – DS)) = 100m • (101.89/100.84) • ((7.56 – 4.13)/(7.56 – 1.49)) = 100m • (1.01043) • (3.43 / 6.06) = 100m • 0.57143 = 57.143m

With NC = 100m and NS = 57.143m one then gets: NL = (NC • PC – NS • PS) / PL = (100m • 101.89 – 57.143m • 100.84) / 101.69 = (10,189m – 5,762.3m) / 101.69 = 4,426.9m / 101.69 = 43.534m

These calculated barbell weights – NC = 100m, NS = 57.143m and NL = 43.534m – are the same as those given in Table 3.6 for the duration- and cash-neutral barbell. Naturally, it is also possible to set up a barbell trade using bond futures (for example, long Schatz and Bund futures with short Bobl futures). How and when to substitute cash bonds with bond futures is analysed in Chapter 4, which discusses basis management.

3.2.1 Performance of barbells in different curve scenarios The performance of the different barbell types can be calculated relatively easily using parallel shifts in the curve, or by simulating curve 37

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Table 3.7 Profit/loss for type (a) barbell for different curve scenarios Barbell type (a) Duration- and cash-neutral

(long 2y and 10y vs. 5y) Flattening by 10 bps

No change

Steepening by 10 bps

Yield increase by 50 bps

23,479

183

−22,790

No increase

24,748

0

−24,409

Yield decrease by 50 bps

29,048

2,762

−23,166

steepening or flattening scenarios, or both. Table 3.7 shows the profit/loss for the duration- and cash-neutral barbell, using the same weightings as in the example above. The special case of the type (d) barbell is not examined further. The ‘steepening by 10 bps’ (‘flattening by 10 bps’) refers to a scenario where both yield spreads, 2y versus 5y and 5y versus 10y, widen (flatten) by 10 basis points. The table confirms that the cash-neutral barbell profits from a curve flattening – approximately EUR 25,000 for the position, with nominal values EUR 57m in 2y, EUR 100m in 5y, and EUR 43m in 10y. The example also shows that the barbell is durationneutral, as parallel shifts of 50 bps do not provide any gains or losses. The profit/loss for parallel shifts of 50 bps is not exactly equal to zero because duration-neutrality provides a first-order hedge; and for large market moves, higher order factors, in particular convexity, also play (small) roles.

Table 3.8 Profit/loss for type (c) barbell for different curve scenarios Barbell type (c) Duration-neutral with equal wings Yield increase by 50 bps No increase Yield decrease by 50 bps 38


No change


−2,141

−1,379

−376

−40

0

292

2,868

2,136

1,668

Table 3.9 Profit/loss for type (b) barbell for different curve scenarios Barbell type (b) Duration- and twist-neutral


No change


Yield increase by 50 bps

−7,826

−1,725

4,598

No increase

−5,541

0

5,774

Yield decrease by 50 bps

−2,942

1,997

7,179

The type (c) barbell is now discussed, before the type (b) barbell. Since both the 2y–5y and 5y–10y spreads are twisted by 10 bps (steepening or flattening), it is expected that the barbell with equal wings will show no profit/loss. Table 3.8 confirms this. Table 3.9 shows the profit/loss for a duration- and twist-neutral barbell. As expected, the profit/loss for the twist-neutral barbell is smaller (compared with the duration- and cash-neutral barbell), since the barbell was weighted so that it was hedged against both parallel shifts and steepening or flattening of the curve. The fact that a small profit/loss is still shown here is due to the fact that the differences in the durations between the wing bonds and the body bond are not identical. Note from Table 3.10 that the ratio of the duration differences is 1.30 (= 3.43/2.63). The inverse of the ratio of the wing bond weights is also 1.30 (= 1/(1.83/2.38)). This equality is expected, since the condition for twist-neutrality is that the product of (a) and (b) must be equal for both wing bonds, which is confirmed by Table 3.10 (for a mathematical derivation, see Appendix 1 at the end of section 3.2.2). In practice, it is not necessarily expected that, for example, a flattening by 10 bps will take place in both the 2y–5y and the 5y–10y sectors. Rather, it is assumed that a flattening of 10 bps in the 2y–5y spread will entail a flattening of approximately 13 bps in the 5y–10y spread. The maturity difference between the 5y and 10y bonds is 5 years, while the difference between the 2y and 5y bonds is only 3 years. It is therefore to be expected that the 5y–10y sector will flatten more relative to the 2y–5y part of the curve. In particular, it can be expected that the flattening will take place on a duration-weighted basis (with the movement in 5y–10y spread being 1.3 times that of the 2y–5y spread: see Table 3.10). The twist-neutral barbell is constructed so that exactly this curve move of 10 bps in the 2y–5y spread 39

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Table 3.10 Durations and weightings for barbell Durations, differences and weightings (Duration and twist-neutral barbell)

Duration

Wing long

BKO 4.5% Mar-02

1.49

Body short

OBL 5% May-05

4.13

Wing long

Bund 5.25% Jul-10

7.56

Absolute Weighting Product difference of barbell difference of durations (b) times to body weighting bond (a) (a) × (b) 2.63

2.38

6.27

4.21 3.43

1.83

6.27

and 13 bps in the 5y–10y spread entails a profit/loss of zero on the position. The same twist-neutral barbell can also be set up using the type (d) barbell, by choosing 57% (short wing) and 43% (long wing) as weights.

3.2.2 Yield pick-up Often the decisive factor in whether or not to enter into a barbell trade is not just a view on the curve, but also the aim of achieving a higher portfolio yield. If the yield of the portfolio consisting of the two wing bonds is higher than the yield of the body bond, a higher yield can be obtained by setting up the barbell position – that is, by selling the body bond (underweighting this maturity sector) and by buying the wing bonds (overweighting these maturity sectors). The yield of the body bond is known from market prices, but the yield of the portfolio consisting of the two wing bonds is not easily determined. Theoretically the yield of a portfolio equals the internal rate of return of its individual cash flows. This is a complex iterative calculation, so in practice two methods are used as approximations: ■ proceeds-weighted yield pick-up method ■ duration-weighted yield pick-up method.

The proceeds-weighted yield pick-up method calculates the yield of the portfolio with the two wing bonds by weighting their cash components. 40

Table 3.11 Yield pick-up example for type (a) barbell Barbell type (a) Duration- and cash-neutral


Proceeds-weighted yield pick-up (bps)

4.3

Duration-weighted yield pick-up (bps)

8.0

Internal rate of return yield pick-up (bps)

7.8

The duration-weighted yield pick-up method weights the portfolio by taking into account the two bonds’ durations. In general, the proceedsweighted yield pick-up method is easily represented, while the durationweighted yield pick-up method provides a theoretically better approximation. The mathematical formulae for determining the yield pick-ups for the two methods are given in Appendix 2 at the end of this section. For the duration and cash-neutral barbell the results in Table 3.11 are obtained. For example, the proceeds-weighted yield pick-up, PWYP, is calculated using the following equation: PWYP = (NS • PS • YS + NL • PL • YL) / (NS • PS + NL • PL) – YC

Thus one obtains: PWYP = (57.143m • 100.84 • 5.090% + 43.534m • 101.69 • 5.193%) / (57.143m • 100.84 + 43.534m • 101.69) – 5.092% = (293.302m + 229.890m) / (5,762m + 4,427m) – 5.092% = (523.192m) / (10,189m) – 5.092% = 5.135% – 5.092% = 0.043% = 4.3 basis points

In the same way, for the twist-neutral barbell one gets the results shown in Table 3.12. The following observations can be made: ■ In both cases the pick-up is greater for the duration-weighted method

than for the proceeds-weighted method. This can be expected, since 41

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Table 3.12 Yield pick-up example for type (b) barbell Barbell type (b) Duration- and twist-neutral


Proceeds-weighted yield pick-up (bps)

1.2

Duration-weighted yield pick-up (bps)

4.3

Internal rate of return yield pick-up (bps)

3.5

the high duration of the 10y issue is not taken into account when calculating the proceeds-weighted pick-up. ■ The pick-up for the cash-neutral barbell is greater. This feature should

not lead to a preference in favour of the cash-neutral barbell rather than the twist-neutral barbell. The reason for the better pick-up stems from the fact that the cash-neutral barbell has a greater weighting in the 10-year bond. For a positively sloping curve it is true to say that a higher weighting in the 10-year issue than in the 2-year issue leads to a greater yield pick-up for the 2y–5y–10y barbell. Simultaneously this also means that the increasing weighting in the 5y–10y leg of the barbell incorporates a stronger curve-flattening position, which would lose value in a curve-steepening scenario. (The extreme example is a straight 5y–10y flattening position in which the wing weight for the 10-year bond is 100%.)

Appendix 1: Mathematical derivation of barbell weightings Definitions: C: body bond S: wing bond (shorter maturity) L: wing bond (longer maturity) Ni: nominal amount of barbell bond i (i ∈ {C,S,L}) Di: modified duration of barbell bond i (i ∈ {C,S,L}) Pi: dirty price of barbell bond i (i ∈ {C,S,L}). The mathematical conditions for the different types of barbell are then as follows:

42

(a) Duration- and cash-neutral barbell: PS• NS • DS + PL • NL • DL = PC • NC • DC (duration-neutral) NS• PS + NL • PL = NC • PC (cash-neutral)

(b) Duration- and twist-neutral barbell: PS• NS • DS + PL • NL • DL = PC • NC • DC (duration-neutral) (DC − DS) • NS • DS • PS = (DL − DC) • NL • DL • PL (twist-neutral)

(c) Duration-neutral barbell with equal wing weightings: PS• NS • DS + PL • NL • DL = PC • NC • DC (duration-neutral) PS• NS • DS = PL • NL • DL (equal wing weightings)

(d) Duration-neutral barbell with any chosen wing weightings: PS• NS • DS + PL • NL • DL = PC • NC • DC (duration-neutral) (1−RS) • PS • NS • DS = (1−RL) • PL • NL • DL (any chosen wing weightings − RS: the weighting for S; RL: the weighting for L; RS + RL =1)

In all four cases, one obtains a set of linear equations with two equations and three unknowns, the respective weightings for the individual bonds. One can now fix one weighting and thus solve the set of equations for the other weightings. For example, suppose the weighting of the body bond is fixed at 100m nominal. The two equations for the type (a) barbell can then be rearranged to read: NL = (PC • NC • DC − PS • NS • DS) / (PL • DL) (duration-neutral) NL = (NC • PC − NS • PS) / PL (cash-neutral)

Substitution of NL gives: NS = NC • (PC / PS) • ((DL − DC) / (DL − DS))

and NS can immediately be calculated for a given weighting NC. By insertion into one of the above equations, one can then determine the third weighting, NL. The weightings for all different barbell types can be determined in the same way.

43

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Appendix 2: Calculation of the barbell yield pick-ups Definitions – as above for barbells, and additionally: YPF: yield of the portfolio consisting of the two wing bonds Yi: yield of the barbell bond i (i ∈ {C,S,L}). The yield for the portfolio consisting of the two wing bonds is then calculated for the two methods, as follows. ■ Proceeds-weighted yield pick-up method: YPF = (NS • PS • YS + NL • PL • YL) / (NS • PS + NL • PL) ■ Duration-weighted yield pick-up method: YPF = (NS • PS • DS • YS + NL • PL • DL • YL) / (NS • PS • DS + NL • PL • DL)

The resulting barbell yield pick-up, BYPU, is then determined in both cases via: BYPU = YPF – YC

If BYPU is positive (greater than zero), the barbell trade achieves an increase in yield compared with the original position (long body bond).

3.3 STRIPS 3.3.1 Differences between strips and conventional bonds The strip market in Europe is still relatively young. In Germany it has existed since 4 July 1997, while in France bond strips were introduced in 1991. STRIP, an acronym, stands for Separately Traded Receipts of Income and Principal. Stripping of bonds means the separation of the coupon and nominal payments of the bond into its individual components. The coupon strip is often called Interest Only (IO), and the final payment of the nominal is the principal strip or Principal Only (PO). Each strip is thus a zero-coupon bond. Strips are almost predestined for the implementation of duration and curve views, as will become apparent later in this section.

44

Because strips are zero-coupon bonds, there is an obligation to pay out a certain cash amount at a certain date in the future. In particular, strips have the following characteristics: ■ strips have a Macauley duration equal to their life time in years ■ strips do not have any reinvestment risk, since no intermediate cash

flows take place ■ prices and yields for strips are easily calculated and do not require any

iterations. Stripping bonds and reconstituting bonds using strips is recommended, provided that the sum of the market values of the individual components (coupon strips and principal strip) is not equal to the market value of the bond. In this case an arbitrage is possible. The existence of this arbitrage relationship ensures that strips are priced effectively and fairly in the market. Because of supply and demand issues it happens that strips of certain maturities are more expensive in relation to a theoretical curve, and this in turn implies that strips of other maturities trade more cheaply. In order to purchase a certain nominal amount, a smaller investment is required for strips than for conventional bonds – this is because zero-

35.0

Macauley duration (years)

30.0

Zero 4% Coupon coupon 6% Coupon coupon coupon 8% Coupon

25.0 20.0 15.0 10.0 5.0 0.0 0

5

10

15

20

25

30

Time until maturity (years)

Figure 3.9 Macauley duration for strips (continuous black line) and bonds 45

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Table 3.13 Strips convexity Yield %

Mod duration

Convexity

% Price change −100 bps

% Price change +100 bps

Bund 6.25% Jan-24

5.44

12.29

2.26

13.97

−11.63

PO strip Jan-24

5.43

22.18

5.13

24.98

−19.82

IO strip Jul-13

5.41

12.22

1.61

13.08

−11.47

coupon bonds, by definition, do not pay intermediate coupons, so the price of a strip represents its discounted nominal amount. Strips have a significantly higher duration for a given maturity, which gives investors an increased amount of flexibility for matching cash flows. The relationship between price and yield of a zero-coupon bond is convex. This means that a bond increases in value more as yields decrease, compared with the loss in value as yields increase. Convexity is therefore a desirable characteristic. Strips allow for the construction of strategies for given maturities with higher convexity than is possible using only conventional bonds. Strips can therefore profit from high volatility. This statement is true even though the convexity of strips is lower than the convexity of conventional bonds with the same duration, since a duration-neutral switch requires a larger amount of strips. For example, both the Bund 6.25% Jan–24 and the coupon strip 0% Jul–13 have the same duration of approximately 12.2 years, but the convexity of the Bund Jan–24 (2.26) is higher than that of the strip (1.61), as is shown in Table 3.13 (using market data in August 2000). For a durationneutral switch, 22.7m nominal of the Jul–13 strips must be bought against a sale of 10.0m Bund Jan–24. An increase in portfolio convexity is achieved, even though the convexity of the Bund is roughly 40% (2.26/1.61) higher, since more than twice the nominal amount of strips (2.27) are bought against the sale of the bond. Strips can also be used to decrease the foreign exchange exposure. Because of the higher duration and the lower price of strips compared with conventional bonds, an investor outside Europe could purchase a lower nominal amount of German zero-coupon bonds while keeping the exposure to overall yield levels unchanged.

46

3.3.2 Valuing and pricing strips In order to identify relative-value opportunities, it is important to understand the behaviour of strips in different curve scenarios. The yield of a strip is predominantly a function of the shape of the yield curve, although other factors – such as supply and demand, market view, and future interest rate expectations – also play a role. In general, there are three methods for valuing strips: ■ using the conventional bond yield curve ■ using the equivalent-duration method ■ using the theoretical zero curve (bootstrapping).

The yield spread between a strip and a coupon-bearing bond with a similar maturity is often used as an indication of whether a strip is rich or cheap. This spread can only be used as a rough valuation tool. The obvious problem with this method is that two instruments with different risk profiles are being compared; this is the case especially for strips with long maturities. The equivalent-duration method compares strips and bonds with similar durations, instead of those with similar maturities. This method is significantly better, as is confirmed by Figure 3.10. The method has the drawback, however, that no indication is given as to the change in the value of the strip as the yield curve changes shape. In order to be able to identify this change, the third method, which employs a theoretical zerocoupon curve, is used. The theoretical zero curve is derived using a theoretical par bond curve. The latter can be constructed by charting conventional bond yields against their maturities and by fitting a (smooth) line through these points.1 The theoretical zero curve is then obtained via bootstrapping: that is, by calculating the yields for the individual zero-coupon bonds for different maturities, as implied by the par yields of the theoretical par bond curve. The slope of the par curve is an important factor when determining a strip yield. In order to understand this better, four different curve examples are discussed below: ■ a flat curve ■ an inverted curve

47

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7.50% 7.00%

Yield

6.50% 6.00% 5.50% 5.00% Bonds Strips STRIPs

4.50% 4.00% 3.50% 3.00% 0

5

10

15

20

25

30

Modified duration

Figure 3.10 Comparison of strips (continuous longer line) and conventional bonds using the equivalent-duration method

■ a positively sloping curve ■ a U-shaped curve.

A flat curve

Each bond can be represented as the sum of zero-coupon bonds. Therefore, in the case of a flat curve the theoretical zero curve is identical to that of the theoretical par curve.

An inverted curve

The theoretical zero curve has to lie below that of the par curve, as the yield of the bond is impacted by the facts that the investor receives coupon payments before maturity and the discount factors for these coupon payments are higher, when the nominal amount of the bond is paid out, than at maturity. Additionally, the spread between zero-coupon bond yields and coupon-bearing bond yields should increase negatively, so zero-coupon bonds will always have a lower yield compared with conventional bonds. 48

0.09 0.08 Theozero Zerocurve curve Theo. Theo. Theopar Parcurve curve

Yield

0.07 0.06 0.05 0.04 0.03 0.02 0

5

10

15

20

25

30

35


Figure 3.11 A negatively sloped, or inverted, curve (strips: black line) A positively sloped curve

Yield

The theoretical zero curve lies above the theoretical par bond curve, for reasons opposite to those governing the inverted curve. The slope of the curve is of interest: the steeper the bond curve, the steeper the zero curve.

0.12 0.11 0.1 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02

Theozero Zerocurve curve Theo. Theo. Theopar Parcurve curve

0

5

10

15

20

25

30

35


Figure 3.12 A positively sloping curve (strips: black line) 49

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0.08

Yield

0.07 0.06 0.05 Theo Zero curve Theo. zero curve Theo. par curve Theo Par

0.04 0.03 0

5

10

15

20

25

30

35


Figure 3.13 A U-shaped curve (strips: continuous black line)

It is worth noting that each annual point on the theoretical par curve is used for deriving the theoretical zero curve – that is, the yields for 1y, 10y and 30y bonds could all be exactly the same, and at the same time the yield of the 30y strip could be significantly higher or lower than that of the 30y bond. This depends upon what path the bond yields follow across the maturity spectrum. This point also illustrates the drawback of the bootstrapping method, as it can lead to highly sensitive theoretical zero rates.

A U-shaped curve

A U-shaped curve can be seen as a combination of a negatively and a positively sloping curve. It is interesting to note in Figure 3.13 that because of lower yields in the 1y–3y sector, the 30y zeros underperform strongly compared with 30y conventional bonds. Were short-end yields to fall significantly, an underperformance of 30y strips would be likely to follow.

3.3.3 Curve dynamics As will have become apparent above, changes in strip prices are driven in practice by changes in prices in conventional coupon-bearing bonds. Figures 3.14 and 3.15 show changes in strip and bond prices in curve50

5

Change (bps)

4

Theo. zero curve curve Theo Zero change change Theo. par curve curve Theo Par change

3 2 1 0 -1

0

5

10

15

20

25

30

35

-2 Time until maturity (years)

Figure 3.14 Curve-steepening (strips: continuous black line)

steepening and curve-flattening scenarios. In each case it can be seen that the changes in the par curve cause the changes in the zero curve. Figure 3.14 shows a stronger steepening of the strip curve than the par bond curve – for example, the yield of a 25-year bond increased by 0.9 basis points, while the strip yield for a similar maturity increased by 2.5 basis points. This behaviour is expected, since the bond yield represents

10 Theo zero Theo. zero curve curve change change Theo. par curve curve Theo par change change

Change (bps)

8 6 4 2 0 0

5

10

15

20

25

30

35

-2 Time until maturity (years)

Figure 3.15 Curve flattening (strips: continuous black line) 51

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an ‘average yield’ of all its individual cash flows. If yields of early cash flows do not move considerably, changes in later cash flows must be more significant if they are to create a balance. Therefore strips with longer maturities are more sensitive to interest rates and move faster in a flattening rally (a flattening of the curve and a decrease in overall yield levels) or in a steepening sell-off (a steepening of the curve and an increase in overall yield levels). Figure 3.15 represents a flattening sell-off scenario. It is interesting that while all par bond yields have increased, yields for strips with long maturities have actually decreased. This situation could also take place in the opposite way: if the curve were to steepen and yields to decrease, strip prices for long maturities could fall (depending on the levels of overall moves).

3.3.4 Strips strategies in the bond market The reconstitution of bonds using strips is an interesting strategy because of the existence of the arbitrage relationship. However, this opportunity is typically available only to market traders and not to investors.

Strips’ relative value

Relative value in individual strips can be observed via spreads against theoretical strip yields. Supply and demand factors could therefore cause a cheapening (low demand) of a strip against the curve. This fact could then induce an investor to buy the strip. On the other hand, a strip can become very expensive if there is high demand, and this may then lead to selling interest from investors. Relative value in long-maturity principal strips is often analysed using regressions of the yield spread between the principal strip and its underlying cash bond versus the 10y–30y yield spread. Figure 3.16 shows a regression of the yield spread Jul-27 principal strip – Bund 6.5% Jul-27 versus the 10y–30y yield spread in Germany (Bund Jul-27 used here as the 30y bond). It is apparent from the chart that the last point (a thick square) is a significant distance away from the mean regression line. This indicates that the PO Jul-27 strip is cheap on the curve, as the PO cash spread is currently 35 bps, whereas the regression implies that the spread should be 29 bps – that is, the PO Jul-27 strip is 6 bps too cheap compared with the curve. The correlations of the long PO cash spreads against the 10y–30y spread are 52

PO Jul-27 / Bund 6.5% Jul-27

PO Jul-27 / Bund 6.5% Jul-27 vs. Bund 6.5% Jul-27 / 10y

–

10y-30 (Bund 6.5% Jul-27 / 10y)

Figure 3.16 Relative value: principal strip–cash bond vs. 10y–30y regression: January 1999–October 2001 Source: Bloomberg


typically very strong (here r2=0.87), hence these types of regression analyses are very useful in identifying relative value in long-maturity strips.

Rolldown of strips and belly-zone strategies

In a positively sloping curve, as was the case for example in the Eurozone in the summer of 2000, strips offer a superior return: this is due to their greater rolldown impact compared with conventional bonds, because the zero curve lies above the par bond curve, and hence is also steeper. If a change in monetary policy were to be expected, a duration-lengthening strategy via the purchase of strips with long maturities could lead to a good performance because of strong leveraging. Principal strips with longer maturities tend to trade more expensively because of their greater demand; this is because they have a high duration and are more liquid than coupon strips. Coupon strips with intermediate maturities (between 10 and 30 years) therefore typically trade more cheaply, because of the arbitrage condition. This causes the intermediate maturity spectrum of the strip curve, also known as the belly zone, to be partially very steep, allowing for good rolldown returns.

Duration-neutral switch

When switching into strips from conventional bonds, it is important not to change the risk profile against parallel shifts in the curve. Maintaining the risk profile can be achieved via a duration-neutral switch, as shown in Table 3.14 which gives three examples. The optimal strategy is one whereby duration-neutrality is maintained and a pick-up in both yield and convexity is achieved.

Barbell strategies

As described in section 3.1, a barbell strategy consists of the sale of a bond with a medium maturity and the purchase of a short- and a longmaturity instrument. The opposite strategy is called a butterfly. Barbell strategies have the following advantages: ■ barbells can increase the return during the investment period ■ barbells can increase the convexity of the portfolio. 54

Table 3.14 Duration-neutral switch examples: 1997 Nominal

Modified duration

Price

Cash

Deposit Yield

100

7.0

100.00

100

0

5.63

131

9.4

56.48

74

26

5.88

100

7.0

100.00

100

0

5.63

154

25.3

17.94

28

72

6.57

100

7.0

100.00

100

0

5.63

270

2.9

89.60

242

−142

3.73

(1) Switch bond in strip with same maturity Sale Bund 6% Jan-07 Purchase PO Jan-07 (2) Switch bond in strip with longer maturity Sale Bund 6% Jan-07 Purchase PO Jan-24 (3) Switch bond in strip with shorter maturity Sale Bund 6% Jan-07 Purchase IO Jan-00

In order to achieve a yield pick-up, one could for example sell the Bund 6.25% Jan-24 issue and purchase against that the Bund 6% Sep-03 and a long 0% Jul-23 strip, so that the exposure to overall market levels was kept unchanged.

Table 3.15 A barbell strategy (duration- and cash-neutral): 1997 Sale 26.5y bond; purchase 6y bond and 25y strip

Cash

Modified duration

Yield

Sell 6.25% Jan-24

1.000

12.30

6.47

Buy 6% Sep-03

0.618

4.85

6.20

Buy IO Jul-23 strip

0.382

24.36

7.07

Duration-weighted yield pick-up

6 bps 55

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Table 3.16 A cross-currency strip strategy: 1997

Bond 6% Jul-07

Cash

Nominal

Modified duration

Yield

100

100

7.1

5.68

$ Cash (1y)

53

55

0.0

6.00

15y Bund strip

47

149

15.1

6.33

Yield pick-up

47 bps

The relative performance of a barbell depends on general curve moves. If the interest rate curve flattens, a barbell benefits; if the curve steepens, a butterfly would outperform.

Portfolio optimization

Because strips are zero-coupon bonds, they can be used to match cash flows exactly. Alternatively, strips can be employed in order to increase the portfolio yield via duration matching.

Cross-currency strategies

As mentioned above, strips can be used to decrease the exposure to moves in foreign exchange rates, as less cash is required for an investment in strips with the same interest rate sensitivity than is the case for conventional bonds. Table 3.16 illustrates how a US-based investor who owns the Bund 6% Jul–07 issue can decrease his or her foreign exchange exposure by selling the bond and using some of the proceeds to purchase a duration-weighted amount of strips while holding the remaining money in domestic cash.

3.4 ROLLDOWN ANALYSIS 3.4.1 Roll and carry breakdown analysis The rolldown analysis is helpful for identifying which part of the curve offers the best return over a given time period. Usually, this is done by assuming that the yield curve will remain unchanged over the chosen 56

time period. Two types of rolldown analysis are used in practice, which are discussed in more detail below: ■ roll and carry breakdown analysis ■ horizon return analysis.

In a positively sloping curve environment, an investor profits from two factors: ■ The yield of the bond ‘rolls’ down the curve. For example, assume that

the yield of the 7-year issue is 5.14% and the yield of the 6-year issue is 5.05%. In 12 months, the 7-year issue rolls 9 basis points (5.14%–5.05%) down the curve, presuming that the curve does not change shape. ■ The investor has a positive carry situation, since the financing costs of

the bond are lower compared with the achieved yield. This carry benefit is calculated as a breakeven rate (B/E rate) which states by how many basis points the curve can shift upwards in a parallel fashion until a breakeven is obtained. (Breakeven is the determined future interest rate; a loss will be made only if this level is exceeded.) The roll and carry breakdown analysis breaks the rolldown into these components, as shown in Figure 3.17. Table 3.17 shows example rolldown results for German government bonds with different maturities.

5.45 5.40 5.35

B/E rate = 13 bps

5.25 5.20

Total = 22 bps

5.15 5.10

Roll = 9 bps

5.05 5.00

8

7

6

4.95

5

Yield (%)

5.30


Figure 3.17 Rolldown components 57

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Table 3.17 Rolldown of German government bonds: June 2000 spot

2y 4.905

3y 4.998

4y 5.040

5y 4.992

6y 5.134

7y 5.181

8y 5.180

9y 5.161

10y 5.113

15y #N/A

20y 5.185

30y 5.286

1 month

n/c return (tics)

5

6

6

6

7

8

8

7

7

0

8

8

til 07/01/00

b/e rate (bps)

3.4

2.5

1.8

1.4

1.4

1.3

1.2

1.1

0.9

0.0

0.7

0.5

days 30

roll (bps)

1.6

0.8

0.2

–0.5

1.2

0.3

0.0

–0.2

–0.4

0.0

0.1

0.1

total (bps)

5.0

3.3

2.1

0.9

2.6

1.6

1.2

0.9

0.5

0.0

0.8

0.6

total (tics)

8

8

7

4

14

10

8

6

4

0

9

9

10

10

10

12

13

13

12

12

0

13

14

2 months

n/c return (tics)

8

til 08/01/00

b/e rate (bps)

5.5

4.1

3.0

2.3

2.3

2.1

2.0

1.8

1.5

0.0

1.1

0.9

days 61

roll (bps)

3.2

1.5

0.5

–1.1

2.5

0.7

0.0

–0.3

–0.8

0.0

0.2

0.1

total (bps) total (tics)

0.7

0.0

13

8.7

13

5.6

12

3.5

5

1.2

25

4.9

17

2.8

13

2.0

10

1.5

5

0

15

1.3

16

1.0

12

13

11

15

16

16

15

14

0

16

18

3 months

n/c return (tics)

9

til 09/01/00

b/e rate (bps)

6.5

5.0

3.7

2.8

2.9

2.7

2.6

2.3

1.9

0.0

1.4

1.1

days 92

roll (bps)

4.9

2.3

0.7

–1.7

3.8

1.0

0.0

–0.5

–1.2

0.0

0.3

0.2

4.4

1.1

1.8

total (bps)

11.3

0.7

0.0

total (tics)

16

17

15

5

34

22

16

12

5

0

19

22

21

23

20

27

30

30

29

26

0

30

35

7.3

6.8

3.7

2.5

1.7

1.3

6 months

n/c return (tics)

16

til 12/01/00

b/e rate (bps)

13.3

9.9

7.0

5.2

5.6

5.2

4.9

4.4

3.5

0.0

2.7

2.1

days 183

roll (bps)

9.7

4.6

1.5

–3.3

7.6

2.0

–0.1

–1.0

–2.4

0.0

0.6

0.4

total (bps)

23.0

14.5

8.5

1.9

13.3

3.4

total (tics)

28

30

27

7

64

1.1

0.0

41

7.2

29

4.9

22

8

0

37

3.3

41

2.5

55

54

49

0

56

66

12 months

n/c return (tics)

28

37

41

37

51

56

til 06/01/01

b/e rate (bps)

37.0

22.3

14.6

10.4

11.4

10.5

9.7

8.6

6.8

0.0

5.1

4.1

days 365

roll (bps)

19.4

9.2

2.9

–6.6

15.2

4.0

–0.1

–1.9

–4.8

0.0

1.2

0.7

total (bps)

56.3

31.5

17.5

total (tics)

43

53

50

3.8 14

26.6 119

14.5 76

9.6 55

6.7 42

2.0 15

0.0 0

6.4 69

4.8 78

The individual components in Table 3.17 are: ■ N/C return (ticks): the (net of carry) return (in ticks), assuming the

yield of the bond remains unchanged ■ B/E rate (bps): the number of basis points by which the curve can be

shifted upwards until the breakeven is achieved ■ roll (bps): the number of basis points the bond rolls down the curve in

the given time period ■ total (bps): the total rolldown – that is, the sum of the B/E rate and roll ■ total (ticks): the total return (in ticks) – that is, the total (bps) × the basis

point value of the bond (in ticks). From the table it is apparent that the 6-year bond offers the best rolldown. This is the part of the curve where the slope is the steepest. In general, the sector of the curve that is the steepest is the sector with the best rolldown.

3.4.2 Horizon return analysis The aim of the horizon analysis is to identify sectors of the curve that offer a good rolldown return. Figure 3.18 shows sample rolldown returns over 12 months for German government bonds. The 6-year bond offers the best rolldown return, as was also the case using the roll and carry analysis. The chart illustrates that this is the steepest part of the curve, which is otherwise very flat. The reason for the steepness

Yield (%)

6.20 5.70 5.20

Curve Kurve

4.70

12M Horizon Return

4.20 1 2 3

4 5 6

7 8 9 10

15

30


Figure 3.18 Rolldown of German government bonds 59

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in this part of the curve is that the 5-year bond is a benchmark issue, and so trades expensively on the curve; this also becomes obvious by observing the return on the 5-year bond itself, which lies on a low level. The same argument applies for the 10-year benchmark bond: it offers altogether the worst rolldown return, as its yield lies below that of the 9-year bond. Special repo situations are not considered here. The horizon return analysis can be performed in Bloomberg, using the IYC9 function.

Choice of bonds for the rolldown analysis

The results of the rolldown analysis are, at least to some extent, dependent on the bonds that are included in the analysis. In the example above, one issue was chosen per year of maturity. From the 10-year maturity sector onwards, a 16-year issue and a 30-year issue were taken into account. A more detailed rolldown analysis would include more bonds and the calculation of a theoretical par curve, which would then be used for the rolldown. This method eliminates the problem of some bonds being expensive or cheap, either because they have benchmark status or for other idiosyncratic reasons.

3.5 BOX TRADE ANALYSIS 3.5.1 Components of a single-issue analysis The following detailed example shows an intensive analysis of single issues, and their behaviour against each other on the yield curve. It aims to demonstrate the process for choosing different maturity sectors on the curve and for individual bond picking in these sectors. Looking at two yield spreads (bond versus bond) against each other is called a box trade. The purchase of a spread (here a steepening) is analysed and put on against the sale of another spread (here a flattening). The trade idea is defined as the purchase of the spread Bund Oct-02 versus OBL122 Feb-02 (purchase of OBL122, sale of Bund Oct-02), and the sale of the spread Bund Jan-07 versus Treuhand Jan-03 (purchase of Bund Jan-07, sale of Treuhand Jan-03). The data for the example is from 1998 (1.95583 DEM = 1 EUR). This type of analysis is extraordinarily detailed and could for example be performed by arbitrage-orientated hedge funds. The following factors are taken into account in the analysis: 60

■ histories of spreads versus spreads ■ histories of spreads ■ rich/cheap analysis ■ asset-swap values ■ rolldown ■ financing ■ correlation with the curve.

The rich/cheap analysis states whether a bond is expensive or cheap compared with a theoretical par curve. The measure used in a rich/cheap analysis is the z-score, which is defined as follows (assuming a standard normal distribution): current yield difference average yield difference − to fair value to fair value z-score = standard deviation of historical differences to fair value

The z-score always refers to a given time period. A low absolute z-score indicates that the bond is trading within its usual trading range. It does not permit a statement to be made regarding the bonds’ relative value. A high absolute z-score (+2 or higher) indicates that the bond is too cheap, whereas a high negative z-score (−2 or less) indicates that the bond is too expensive in relation to its history.2 There are, however, bonds that continuously trade rich or cheap against their theoretical value, so that after a certain time period the z-score would move towards zero when the bond is no longer becoming more expensive or cheaper. A z-score of +/−2 indicates that the bond lies 2 standard deviations away from its historical mean. Asset-swap spreads state how many basis points it would cost an investor to swap a coupon-bearing bond into a floating-rate note. As for bond yields, z-scores can also be determined for asset-swap spreads, using the same formula. This way an indication can be gained as to whether the bond trades at expensive or cheap levels against the swap curve. This information is useful as the swap curve is smoother than the bond yield curve, and hence the z-score results for asset swaps may be interpreted to give a more reliable measure of bond richness or cheapness. 61

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3.5.2 Histories of spreads versus spreads Figure 3.19 shows a regression of the Bund Jan-07–Treu Jan-03 spread against the spread Bund Oct-02–OBL122. The chart shows a strong correlation over the 6-month period, with R2=0.85. The latest observation (circled point) is almost 3 standard deviations away from the regression line, and shows that the spread Oct02–OBL122 is nearly 6 basis points too flat (at 10.5 bps), compared with the spread level of 44 bps for the spread Bund Jan-07–Treuhand Jan-03. It should be noted that the correlation does not hold over the time period of the last 3 months (r2=0.04; see Figure 3.20), which is why the last point in Figure 3.19 is so far away from its expected value. Figure 3.21 shows a 3-month history of the spread Bund Oct02–OBL122. It also shows that the spread is at its lowest level, with a zscore of –2.2, indicating that the Bund Oct-02 is expensive versus the OBL122. Figure 3.22 shows the 3-month history of the spread Bund Jan07–Treu Jan-03.

3.5.3 Rich/cheap analysis Table 3.18 shows rich/cheap values for the four bonds for the last 10, 30 and 60 days. The table indicates that the OBL122 looks cheap (in particular over the last 30 days), while the other bonds have moved by and large neutrally.

3.5.4 Asset-swap spreads Table 3.19 shows asset-swap spreads and their z-scores over the last 3 months. The table indicates that the Bund Oct-02 is expensive compared with its asset-swap history. The same applies to the Treuhand Jan-03.

Table 3.18 Rich/cheap analysis values Issue

62

10-day z-score

30-day z-score

60-day z-score

Oct-02B

1.1

−0.4

−0.9

OBL122

1.2

2.1

1.6

Jan-07B

−1.4

−0.5

−0.7

Jan-03T

0.4

−0.1

−0.2

DBR 7.25 10/21/02, OBL 4.5 02/22/02 spread

DBR 6 01/04/07, THA 7.125 01/29/03 spread vs. DBR 7.25 10/21/02, OBL 4.5 02/22/02 spread 30.720 29.696 28.672 27.648 26.624 25.600 24.576 23.552 22.528 21.504 20.480 19.456 18.432 17.408 16.384 15.360 14.336 13.312 12.288 11.264 10.240

35

40

45 50 55 60 65 DBR 6 01/04/07, THA 7.125 01/29/03 spread

70

75

Figure 3.19 Jan-07B–Jan-03T spread vs. Oct-02B–OBL122 spread: 6 months

DBR 7.25 10/21/02, OBL 4.5 02/22/02 spread

DBR 6 01/04/07,THA 7.125 01/29/03 spread vs. DBR 7.25 10/21/02, OBL 4.5 02/22/02 spread 21.504 20.992 20.480 19.968 19.456 18.944 18.432 17.920 17.408 16.896 16.384 15.872 15.360 14.848 14.336 13.824 13.312 12.800 12.288 11.776 11.264 10.752 10.240 32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

DBR 6 01/04/07, THA 7.125 01/29/03 spread

Figure 3.20 Jan-07B–Jan-03T spread vs. Oct-02B–OBL122 spread: 3 months

49

Prices/values

DBR 7.25 10/21/02, OBL 4.5 02/22/02 spread 21.5 21.0 20.5 20.0 19.5 19.0 18.5 18.0 17.5 17.0 16.5 16.0 15.5 15.0 14.5 14.0 13.5 13.0 12.5 12.0 11.5 11.0 10.5 10/10/97 10/17/97 10/24/97 10/31/97 11/7/97 11/14/97 11/21/97 11/28/97 12/5/97 12/12/97 12/19/97 12/26/97 1/2/98

Dates

Figure 3.21 Oct-02B–OBL122 spread: 3 months

1/9/98

DBR 6 01/04/07, THA 7.125 01/29/03 spread 48 47 46 45 44

Prices/values

43 42 41 40 39 38 37 36 35 34 33 10/13/97 10/20/97 10/27/97 11/3/97 11/10/97 11/17/97 11/12/97 12/1/97 12/8/97 12/15/97 12/22/97 12/29/97 1/5/98

Dates

Figure 3.22 Jan-07B–Jan-03T spread: 3 months

1/12/98

Table 3.19 Asset-swap spread analysis Issue

Asset-swap spread

z-score of asset-swap spread

Oct-02B

−18.4

−2.0

OBL122

−18.2

0.6

Jan-07B

−21.6

−0.9

Jan-03T

−16.7

−1.5

3.5.5 Rolldown Table 3.20 shows the rolldown, in basis points per month. The steeper 5year sector offers a better rolldown, compared to the Bund Jan–07 (with a remaining lifetime of 9 years).

3.5.6 Financing (repo rates) Table 3.21 shows 3-month repo rates for the four bonds. The above repo rates are interest rates at which the bond could be borrowed or lent for 3

Table 3.20 Rolldown analysis Issue

Roll down the curve (bp/month)

Oct-02B

1.4

OBL122

1.4

Jan-07B

0.7

Jan-03T

1.4

Table 3.21 Repo rates Issue

Repo rate to March

Oct–02B

3.45–3.35

OBL122

3.45–3.35

Jan–07B

3.40–3.30

Jan–03T

3.45–3.35 67

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months, and are used for calculating the financing costs of the bond position. Financing costs and repo rates are discussed further in Chapter 4. The forward price of a bond is determined using the following equation: Forward price = P – k • t/365– c + (P + z) • r • t/360

where P = price of the bond z = accrued coupon interest from the previous coupon payment date until the settlement date t = number of days from the settlement date until the forward date (here 91 days for 3 months)3 k = bond coupon r = repo interest rate for the time period between the settlement date and the forward date c = possible further coupon income, if the bond’s next coupon payment date is between the settlement date and the forward date. The difference between the yield of the bond and the yield of the bond on the forward date (the forward yield), which is the yield change, can be approximated using the following relationship: yield change ≈ (forward price – current price) / bpv where bpv represents the basis point value of the bond. Forward spreads can thus be determined as differences in forward yields. There are no significant differences in the repo rates, which implies that forward spreads should be stable; Table 3.22 confirms this. Forward spreads are virtually identical to current (spot) spreads. The spread Bund Jan-07–Treu Jan-03 should flatten by approximately 0.8 bps in 3 months, in order to achieve its breakeven, whereas the breakeven for the spread Bund Oct-02–OBL122 is equal to the current spread level.

Table 3.22 Spot and forward spreads Spread issues

68

Spot spread (bp)

Forward spread to March (bp)

Oct-02B–OBL122

10.9

10.9

Jan-07B–Jan–03T

43.9

43.1

Table 3.23 Outstanding issue sizes of bonds Issue

Issue size (DEM)

Oct-02B

10bn

OBL122

10bn

Jan-03T

30bn

Jan-07B

14bn

3.5.7 Correlation with the curve Figures 3.23 and 3.24 show the correlation over the last 6 months against the yield curve (5y–10y). Both spreads, Bund Oct-02–OBL122 and Bund Jan-07–Treu Jan-03, are correlated with the 5y–10y spread (Bund Jul-07–Bund Sep-01) over the last 6-month time period. The strong correlation of the spread Bund Jan07–Treu Jan-03 (R2=0.98) was expected, as this represents a 6y–9y spread and is this similar to a 5y–10y spread. It is interesting to note that the spread Bund Oct-02–OBL122 also shows a strong correlation (R2=0.81) with the 5y–10y spread. Also, it is emphasized that the current level (the circled point in Figure 3.23) is nearly 3 standard deviations away from the regression line. The above regressions indicate that for a steepening (flattening) of the 5y–10y spread by 10 bps, the spread Bund Jan-07–Treu Jan-03 should widen (narrow) by 7.2 bps and the spread Bund Oct-02–OBL122 should widen (narrow) by 2.7 bps.

3.5.8 Liquidity Table 3.23 shows the liquidity of the four bonds by listing their outstanding issue sizes. It shows that all four bonds are liquid and are thus suitable for putting on the trade.

3.5.9 Summary of results Table 3.24 summarizes the results of the above analysis. The analysis results clearly indicate that the box trade idea – that is, the purchase of the spread Bund Oct-02 versus OBL122 (purchase of OBL122, sale of 69

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DBR 7.25 10/21/02, OBL 4.5 02/22/02 spread

DBR 6 07/04/07, DBR 8.25 09/20/01 spread vs. DBR 7.25 10/21/02, OBL 4.5 02/22/02 spread 30.72 29.696 28.672 27.648 26.624 25.60 24.576 23.552 22.528 21.504 20.480 19.456 18.432 17.408 16.384 15.360 14.336 13.312 12.288 11.264 10.240

55

60

65

70

75

80

85

90

95

100

105

110

115

120

125

DBR 6 07/04/07, DBR 8.25 09/20/01 spread

Figure 3.23 5y–10y spread (Jul-07B–Sep-01B) vs. Oct-02B–OBL122 spread: 6 months

DBR 7.25 10/21/02, OBL 4.5 02/22/02 spread

DBR 6 07/04/07, DBR 8.25 09/20/01 spread vs. DBR 6 01/04/07, THA 7.125 01/29/03 spread 75 70 65 60 55 50 45 40 35 55

60

65

70

75

80

85

90

95

100

105

110

115

120

125

DBR 6 07/04/07, DBR 8.25 09/20/01 spread

Figure 3.24 5y–10y spread (Jul-07B–Sep-01B) vs. Jan-07B – Jan–03T spread: 6 months


Table 3.24 Summary of box trade analysis results Analysed factor

Result

Spread vs. spread histories

Oct-02B–OBL122 spread too flat; Jan-07B–Jan-03T spread too steep

Spread histories

Oct-02B–OBL122 too flat

Rich/cheap analysis

OBL122 cheap

Asset-swap analysis

Oct-02B expensive Jan-03T expensive

Rolldown analysis

5 years steeper than 10 years

Financing costs

Neutral

Correlation with the curve

Jan-07B–Jan-03T correlated with 5y–10y spread Oct-02B–OBL122 correlated with 5y–10y spread

Liquidity

Good liquidity available

Bund Oct-02) and the sale of the spread Bund Jan-07 versus Treuhand Jan–03 (purchase of Bund Jan-07, sale of Treuhand Jan-03) – is justified.

3.5.10 Setting up the box trade The above analysis showed that over the last 6 months, the spread Bund Oct-02–OBL122, at 10.5 bps, is approximately 6 bps too flat compared with the spread Bund Jan-07–Treuhand Jan-03, at 44 bps. Moreover recently the spread Bund Oct-02–OBL122 has not followed the market trend and has moved away from the regression line (nearly 3 standard deviations) versus the 5y–10y spread. The spread Bund Jan-07–Treu Jan-03, however, has moved with the market. The expectation of the trade is that a spread widening in the Bund Oct-02–OBL122 will take place, back towards the regression line. The spread Bund Jan-07–Treu Jan-03 is a curve-flattening trade and acts as a hedge against a general curve flattening. Other analysed factors also support the trade. The asset-swap analysis supports the idea of selling the Bund Oct-02 and the Treu Jan-03. The rich/cheap analysis identified the OBL122 as cheap. Other analyses, of the rolldown, the financing and the liquidity, did not produce information that argued against the trade idea. 72

Table 3.25 Box trade weightings 1. sell buy

10m

Bund Oct-02

12.25m

OBL122

(duration-neutral steepener)

2. buy

2.5m

Bund Jan-07

(0.39 times duration of Bund Oct-02)

sell

3.9m

Treu Jan-03

(duration-neutral flattener)

Table 3.26 Profit and loss simulation (in DEM 1000) −15

Oct-02B – OBL122 spread chg (bp)

Jan-07B – Jan-03T spread chg (bp) −7.5 0 7.5

15

6

54.0

40.5

27.0

13.5

0.0

3

40.5

27.0

13.5

0.0

−13.5

0

27.0

13.5

0.0

−13.5

−27.0

−3

13.5

0.0

−13.5

−27.0

−40.5

−6

0.0

−13.5

−27.0

−40.5

−54.0

The weightings for the trade are calculated using the regression results, where the slope of the regression line is at 0.39 (see Figure 3.19). The resulting box trade is then determined to be as shown in Table 3.25. Table 3.26 shows the profit/loss of the box trade for different curve scenarios. Note that the anticipated steepening of 6 basis points (spread Bund Oct-02–OBL122) implies a profit of DEM 27,000 using the abovementioned nominal trade amounts. The risks of the trade are that the Bund Oct-02 remains expensive and/or that the spread Bund Jan-07–Treu Jan-03 undergoes a strong steepening. This type of analysis leads to the identification of anomalies that may occur in the yield curve. The extent of this analysis is enormous, however, and thus can only be exercised regularly – for all bonds on the curve – if an automated process is put in place.

Notes 1.

Several different models exist, which can be used for building par bond curves. Since not all bonds lie exactly on the curve, different methods have been devel73

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2.

3.

74

oped for building the theoretical par bond curve through the points. Two known models used in practice are the cubic spline model and the model by Nelson and Siegel. It is possible to calculate z-scores using price differences versus the par curve, rather than yield differences. In this case, a high positive z-score would indicate richness, while a high negative z-score would indicate cheapness. The coupon amount is calculated using the ‘actual/actual’ convention, that is, ‘actual number of days between the coupon payment date and settlement date/actual number of days in year’. Therefore the denominator should read ‘366’ instead of ‘365’ in a leap year.

CHAPTER 4

Basis management

4.1 CHEAPEST-TO-DELIVER ANALYSIS All important government bond markets are driven by bond futures, and cash bonds are priced off these futures. The price of a cash bond is a function of the futures price and the bond basis, which quantifies the relationship between the future and a cash bond. It is therefore important to understand the functionality of the bond basis. As bond futures are being used more and more in the fund-management industry, this section covers in detail the valuation of bond futures and calendar spreads, in order to achieve optimal timing in positioning in bond futures. A bond future is basically a synthetic bond, which is standardized on a fictitious cash bond with a given coupon and a remaining time to maturity. For example, the Bund futures contract is based on German government bonds with a remaining lifetime of 8.5–10.5 years and a nominal coupon of 6%. Cash bonds with a remaining time until maturity in this range and a sufficient liquidity (the outstanding issue amount) are deliverable into the Bund futures contract. The set of deliverable bonds is called the deliverable basket. At contract expiry, the seller of the contract has the obligation to deliver a bond from the deliverable basket, and the buyer of the contract has the obligation to take delivery of this bond. Since not all deliverable bonds are equally expensive, the seller will choose the bond that is cheapest, the so-called cheapest-to-deliver (CTD) bond. 75


The CTD analysis determines the expected bond basis at expiry of the bond futures contract, which is calculated using forward yields. At expiry, at least one bond, the CTD bond, will have a basis equal to zero. The CTD bond can vary between the current date and expiry.

4.1.1 Bond futures contract and price factors A bond futures contract is a legally binding contract, to buy or to sell a specified nominal amount of a bond from the deliverables basket, on a predefined day (or days) in the future. The costs involved in setting up a bond futures position are often only a fraction of the costs of dealing in the underlying bond. Bond futures therefore allow for a more flexible management of cash flows, and thereby broaden the range of available strategies for fund managers. The set of deliverable bonds consists of bonds with similar maturities. At expiry, the seller of a bond futures contract has the option to choose which of these bonds he or she wishes to deliver. In order to maximize his or her profit (or to minimize the loss), the seller chooses the cheapestto-deliver bond. Clearly there are price differences between the deliverable bonds. In order to make these different instruments comparable, so-called price factors, or conversion factors, were introduced. Each deliverable bond has a price factor for a specific contract expiry date, such that the price factor multiplied by one hundred gives the price at which the bond would trade at contract expiry, if its yield was equal to that of the nominal coupon (for example, 6% for the Bund future). At contract expiry, the seller of the bond futures contract can deliver any bond from the deliverable basket for: price factor (PF) • futures price (F)

Evidently, the seller of the contract aims to minimize the expression P – F • PF,

where P is the price of the cash bond. For the seller, therefore, the cash flow at the bond futures’ expiry equals P – F • PF.

76

4.1.2 Identifying the CTD bond The price of the bond future trades in correspondence with that of the CTD bond, as delivery of the CTD bond is expected at expiry. Therefore, the CTD bond trades like the bond future, and is typically liquid. The CTD bond is thus the connecting link between cash bonds and the bond future. Mathematically, the CTD bond is determined using the implied repo rate or the net basis; these expressions are described in more detail below.

4.1.3 Implied repo rate The implied repo rate (IRR) is the annualized yield that can be obtained if simultaneously the cash bond is bought and bond futures corresponding to a similar (price-factor weighted) nominal amount are sold. The bond is held until expiry and then delivered against the short futures position. This strategy is called a cash and carry trade. Even though the IRR provides a riskless return on the trade, a riskless profit is achieved only if the actual market repo rate of the CTD bond is below that of the IRR. In this case, the bought cash bond can be lent in the repo market at a lower rate – that is, cash money can be borrowed at a cheaper interest rate. To illustrate this better, the following formula shows the relationship between the theoretical futures price and the cash bond price. (The coupon amount is determined using the ‘actual/actual’ convention: that is, applicable days/days in the year. In a leap year the denominator should therefore read 366 instead of 365.) F = (P CTD – k CTD • t/365 – c CTD + (P CTD + z CTD) • r • t/360) / PF CTD

where F = futures price P CTD = price of the CTD bond PF CTD = price factor of the CTD bond k CTD = coupon of the CTD bond z CTD = accrued coupon interest since the previous coupon payment date until the settlement date t = number of days from settlement date to futures delivery date r = funding rate c CTD = possible further coupon income, should the next coupon payment date fall in between the settlement date and the futures expiry. 77

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Solving this equation for r gives the formula for the implied repo rate: IRR = (F • PF CTD – P CTD + k CTD • t/365 + c CTD) / (P CTD + z CTD) • 360/t

From this relationship it becomes apparent that if the IRR is below the market repo rate, the resulting theoretical futures price will be below that of the market futures price, and will thus provide an arbitrage opportunity. The IRR can be determined for all deliverable bonds, not for just the CTD bond; and the resulting information can be used to identify whether a bond is rich (expensive) or cheap versus the bond future. As noted above, a bond with a high IRR is seen as cheap, whereas a bond with a low IRR is seen as rich versus the bond future (assuming that all deliverable bonds have the same repo rate). Provided that all deliverable bonds have the same repo rate, therefore, the bond with the highest implied repo rate is the bond for which delivery into the futures contract is cheapest – that is, it is the cheapest-to-deliver bond. In general, however, the repo rates of deliverable bonds are not the same; and in this case the bond with the highest IRR is not necessarily the CTD issue. Another method of identifying the CTD bond is therefore required. The forward price FP of a bond can be determined using: FP = P – k • t/365 – c + (P + z) • r • t/360

where r is the repo rate until contract delivery. The expected cash flow at contract delivery is FP – F • PF, which can be written as: FP – F • PF = [P – k • t/365 – c + (P + z) • r • t/360 ] – [(P – k • t/365 – c + (P + z) • IRR • t/360) / PF] • PF

= (r – IRR) • (P + z) • t/360 As mentioned above, the seller of the bond futures contract aims to minimize this expression. It is apparent that this is the case where the difference between the repo rate and the implied repo rate is the greatest.1 Consequently, the CTD bond can be identified as the bond for which the spread between the repo rate and the IRR is maximized. Here is an example. On 1 June 2000, the September 2000 Bund futures price was 105.97; the bond Bundesanleihe 4.5% Jul–09 had a cash price of 95.37 and a price factor 0.899414; and other bond information was as follows:

78

F = 105.97 P = 95.37 PF = 0.899414 k = 4.5 z = 4.11 t = 102 c=0 Using the formula for the implied repo rate, these figures give: IRR = (F • PF CTD – P CTD + k CTD • t/365 + c CTD) / (P CTD + z CTD) • 360/t = (105.97 • 0.899414 – 95.37 + 4.5 • 102/365 + 0) / (95.37 + 4.11) • 360/102

= 4.25% Another bond, the Bund 5.25% Jul–10, traded on the same day at a price of 101.13 and had a price factor 0.944942. The IRR in this case is calculated as follows: IRR = (F • PF CTD – P CTD + k CTD • t/365 + c CTD) / (P CTD + z CTD) • 360/t = (105.97 • 0.944942 – 101.13 + 5.25 • 102/365 + 0) / (101.13 + 4.79) • 360/102

= 1.57% If only these two bonds had been deliverable into the Sep–00 Bund futures contract, and if both bonds had had the same repo rates, the Bund 4.5% Jul–09 would have been the CTD issue, as its IRR was higher than that of the Bund 5.25% Jul–10. Assuming, however, that the repo rate for the Bund 4.5% Jul–09 was 4.00% and that the repo rate for the Bund 5.25% Jul–10 was 1.25%, for the Bund 4.5% Jul–09 one would get: FP – F • PF = (r – IRR) • (P + z) • t/360 = (4.00% – 4.25%) • (95.37 + 4.11) 102/360

= –0.070

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Similarly for the Bund 5.25% Jul–10 one gets: FP – F • PF = (r – IRR) • (P + z) • t/360 = (1.25% – 1.57%) • (101.13 + 4.79) 102/360

= –0.096 In this example the expression FP – F • PF is minimized for the Bund 5.25% Jul–10, and therefore this bond obtains cheapest-to-deliver status. In general, a valid rule of thumb is that the bond with the highest IRR is the CTD issue. Only in cases where one deliverable bond has a significantly lower repo rate than others may this statement be false. In these exceptional cases it is recommended that the spreads between the IRR and repo rates be watched closely.

4.1.4 Gross basis A position consisting of being long in (bought) cash bonds and short in (sold) bond futures (cash and carry) is described as a long basis position, or being long in the bond basis. The gross basis (GB) is defined as GB = P – (PF • F)

and is positive (negative) if the coupon income until contract delivery date is higher (lower) versus financing costs. As an approximation it can be said that the gross basis is positive (negative) if the coupon of the bond is higher (lower) than the bond’s repo rate. It was noted above that, in general, the CTD bond is the bond implying the highest profit or the lowest loss in a cash and carry trade. The following statements are therefore valid: ■ If the cash bond is the CTD bond at contract delivery date, the return

on the cash and carry trade equals the IRR (the futures price converges with the cash price) ■ If the cash bond is not the CTD bond at contract delivery date, the

return on the cash and carry trade is higher than the IRR (the futures price lies below the cash price). Figure 4.1 shows gross basis histories for two issues, the Bund 4.5% Jul-09 and the Bund 5.25% Jul-10. The Bund 4.5% Jul-09 is the CTD bond into the September 2000 Bund futures contract, and the convergence of 80

1.20 1.00 0.80 0.60 0.40

01.09.2000

17.08.2000

02.08.2000

18.07.2000

03.07.2000

Date

18.06.2000

03.06.2000

19.05.2000

04.05.2000

19.04.2000

04.04.2000

0.00

20.03.2000

0.20 05.03.2000

Gross basis (ticks)

DBR 4.5% 4.5% Jul-09 DBR Jul-09Gross gross basis Basis DBR Jul-10Gross gross DBR 5.25% 5.25% Jul-10 basis Basis

Gross Basis example CTD (Jul-09) and Non-CTD (Jul-10)

Date

Figure 4.1 Gross basis histories of CTD/non-CTD bonds the gross basis towards zero is evident. The second bond, the Bund 5.25% Jul-10, however, which was issued in early May 2000, is far away from being CTD and it is apparent that no trend towards the zero line exists, that is, the probability that this bond will become CTD, is in this case estimated to be small. It is recognized that the formula for the gross basis deals only with today’s profits and losses of the position, and ignores the impact of financing until the futures contract’s expiry, which for example was taken into account when determining the IRR using repo rates. In order to analyse the financing implications of the bond basis, the so-called net basis is used.

4.1.5 Net basis A position in which a cash bond is bought and bond futures are sold (price-factor weighted) against it, and the bond is lent out in the repo market until contract delivery and then delivered against the short bond futures contract, is a long position in the net basis. The difference between the net basis and gross basis is the inclusion of the carry (the financing cost) of the cash bond from the settlement date until the contract delivery. The net basis is therefore defined as: net basis = gross basis – carry 81

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The position can be locked in until the contract delivery date.2 An arbitrage opportunity exists if the resulting net basis is negative, as the theoretical futures price lies below the market futures price (analogous to the situation with the IRR). This arbitrage condition ensures that, in the normal case, the net basis is above zero (positive value). Because of these observations, the net basis of the CTD is often also defined as follows: Net basis = (theoretical futures price – actual market futures price) • PFCTD

where the theoretical futures price is calculated as in section 4.1.3. The net basis is driven by the market’s expectations concerning which bond will be the CTD at expiry. The net basis for bonds that are expected to be delivered against futures converges on zero or remains there. For bonds that are not expected to be delivered, there is no reason why the net basis should decrease. The net basis for bonds that are far from becoming the CTD is primarily a function of the yield spread to the CTD issue and effective market moves. In general, the net basis of the CTD issue decreases as expiry approaches, and the uncertainty over the identity of the CTD issue disappears. The following statements are therefore valid for a long position in the net basis: ■ If the cash bond is the CTD bond at expiry, the investor loses the net

basis (both the futures price and the forward price of the bond converge on the cash price) ■ If the cash bond is not the CTD bond at expiry, the investor cannot lose

more than the net basis, and can achieve an unlimited profit (the forward price converges on the cash price, but the price-factor adjusted futures price lies below the cash price). It follows from this also that the net basis can be used in identifying the CTD issue – the bond with the lowest net basis is the CTD bond (taking into account repo rates for individual deliverable bonds). If market participants are concerned that the outstanding issue size in the CTD issue may not be sufficient, the net basis of the second (or even third) cheapest-to-deliver bond can also decrease to zero (at contract expiry).

82

4.1.6 Delivery option The seller of the futures contract has the obligation to deliver a bond from the deliverable basket at contract delivery. It is, however, his or her choice which one he or she delivers. This ability to choose the bond to be delivered is the seller’s right, though not an obligation: it is therefore called the delivery option. Since it is possible that the CTD bond will change between settlement date and contract expiry, this option has a value, the delivery option value. The long futures position demands compensation against the possible change in the CTD issue. Futures therefore trade cheaper than if no delivery options existed. The value of the delivery option should theoretically be equal to the net basis. If the delivery option value is greater (lower) versus the net basis, the future trades expensive (cheap). The calculation of the delivery option value can be complex, and includes statistical assumptions regarding future interest rates and yield curve moves; section 4.2 describes and derives a method for calculating the delivery option value that has been found to be accurate and reliable in practice. In general, it can be said that the value of the delivery option depends on two factors: ■ The distance of the deliverable bond from being the CTD – that is,

what magnitude of market moves would cause a change in the CTD. This distance can be quantified using the difference between the net bases of the CTD bond versus that of a non-CTD bond: the smaller the difference, the closer the non-CTD bond will be to becoming the CTD ■ The difference in modified durations between the CTD bond and a

non-CTD bond – the greater the difference of the durations between the two bonds, the greater the relative change will be in their bases. These statements are in relation to the overall market level and moves in the curve, and are discussed in detail in the following.

4.1.7 Curve dynamics Changes in the yield curve can lead to a change in the CTD issue. In order properly to risk-manage a bond basis position, it is therefore important to fully understand the implications of curve moves to the profit/loss of the position. 83

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A parallel shift or a steepening or flattening in the curve can each cause a change in the CTD issue. If bond futures are used for hedging, or for managing the duration of a portfolio, the basis point value of the futures contract can change as the CTD bond changes, requiring an adjustment in the number of futures contracts used.

Parallel shifts

The following statements, which were obtained using price-factor weighting, are generally valid (duration-weighted hedging would lead to different results; for example, the exposure to market-directional moves would be immunized): ■ Higher yields result in the delivery of high (long) duration bonds ■ Lower yields result in the delivery of low (short) duration bonds.

Figure 4.2 shows, using the example of the Sep–00 Bund future, how the net basis changes for the four deliverable bonds for different parallel shifts of the curve (the remaining time to expiry of the futures contract is 1 month). Figure 4.2 shows that the Bund 4.5% Jul-09 is the CTD. A parallel shift of approximately 20 bps (an increase in rates) would cause the

Yield Shift Effects on Sep-00 BUND Future Difference from CTD

250 200

Jul-09 4.500 Jul-09 4.000 Jan-10 5.375 Jul-10 5.250

150 100 50 0 –-200 –-160 –-120 –-80 –-40

0

40

80

120 160 200

Yield Shift

Figure 4.2 Change in net basis for parallel shifts in the yield curve: Sep–00 Bund future 84

103.00

Price/conversion factor

102.50

High-duration bond High Duration Bond Medium Duration Bond Medium-duration bond Low-duration bond Low Duration Bond Futures price Futures Price

102.00 101.50 101.00 100.50 100.00 99.50 4.0

5.0

6.0 Yield (%)

7.0

8.0

Figure 4.3 The basis as an option on bond futures Bund 4% Jul-09 to become the CTD. Even though both bonds have exactly the same maturity date, their net bases change in different ways as the yield curve shifts up or down, because the two bonds have different coupons (the 4% coupon bond has a higher duration than the 4.5% coupon bond). Similarly, the Jul-10 issue is the bond with the highest duration in the deliverables basket, and its net basis falls faster than that of any of the other bonds as interest rates are increased. If the curve shifted up by more than 200 basis points, the Jul-10 bond would become the CTD. This important point is further illustrated and clarified in Figure 4.3. This shows how bond durations affect the CTD status at different yield levels, and in particular the fact that higher duration bonds attain CTD status at higher yield levels and lower duration bonds attain CTD status at lower yield levels. For example, if yields were to fall rapidly, the prices of high-duration bonds would increase more than those of lower duration bonds. The chart therefore illustrates also the optionality of the bond futures: that is, that which bond is the CTD bond depends on overall yield levels (parallel shifts starting from 6% yield levels). Figure 4.4 shows how the optionality of the bond futures depends on overall market yield levels. It is noticeable that the profile of the net basis position for a short (long) duration bond resembles that of a long position in a put (call) on the bond future; and a net basis position in a bond with an intermediate duration is similar to a position in a strangle. These behavioural characteristics are stronger the closer the yield levels are to 85

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Short Short duration Duration Medium duration Medium Duration

Net basis

Long duration Long Duration

Futures price

Figure 4.4 The net basis as a function of the overall market level

Difference from CTD

the nominal coupon of the futures contract (for example, 6% for the Bund futures contract). The change in the CTD issue (assuming equal yields and parallel shifts of bonds) illustrates the negative convexity of futures for large market moves. In cases of equal durations, the bond with the highest yield is the CTD.

180 160 140 120 100 80 60 40 20 0 –-200 –-160 –-120 –-80 –-40

Jul-09 4.500 Jul-09 4.000 Jan-10 5.375 Jul-10 5.250

0

40

80

120 160 200

Yield shift

Figure 4.5 Change in the net basis for different parallel shifts and a simultaneous steepening of 5 basis points of the deliverables basket: Sep-00 Bund futures contract 86

Steepening and flattening

In general, the following statements are valid: ■ A curve steepening results in the delivery of high-duration bonds ■ A curve flattening results in the delivery of low-duration bonds.

It is apparent that a curve steepening leads to the cheapening of bonds with longer maturities relative to bonds with shorter maturities. Using the example of the Sep-00 Bund futures contract, it can therefore be expected that a curve steepening will result in a change in the CTD issue. Figure 4.5 confirms this assumption. Figure 4.5 shows how the net bases of the four deliverable bonds change for parallel shifts in the curve, with an additional steepening of the deliverables basket by 5 basis points applied to the basket – in other words, the forward yield of the Bund Jul-10 is increased by 5 bps versus the forward yield of the Bund Jul-09. Compared with the previous example, in which only parallel shifts were applied, it is noticeable that the Bund Jul–10, the bond with the highest duration, now attains the CTD status after a parallel move in the curve of approximately 80 bps (compared with more than 200 bps using only parallel shifts). Figure 4.6 shows the optionality of futures depending on the slope of the curve. It is apparent that the profile of the net basis position for a bond with a long (short) maturity resembles a long put (call) position on

Net basis

Long maturity Long Maturity Medium maturity Medium Maturity Short maturity Maturity Short

flatter

Slope of the curve

steeper

Figure 4.6 The net basis as a function of the slope of the curve 87

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Table 4.1 The CTD bond for different curve scenarios: Sep-00 Bund futures contract −15

rally

sell-off

Flattening −10

−5

0

5

Steepening 10

15

−100

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

−75

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

−50

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jan-10

−25

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-10

0

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-09 (4.5%)

Jul-10

25

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-10

50

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-10

Jul-10

75

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-10

Jul-10

100

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-09 (4%)

Jul-10

Jul-10

the curve, whereas the net basis position for a bond with an intermediate maturity is similar to a strangle position. Table 4.1 provides an overview of which bond would be the CTD given different scenarios of parallel shifts and curve steepening or flattening.

4.2 DELIVERY OPTION 4.2.1 Factors influencing the delivery option value The CTD example described in section 4.1 showed that a parallel shift of 20 bps in the yield curve would entail a change in the CTD issue. An investor can develop his or her own opinion as to whether yields are likely to go up or down, and hence can consider subjectively whether the 20 bps parallel shift (during the lifetime of the futures contract) is probable or improbable, and then position himself or herself in the bond basis accordingly. The sale of the net basis of the CTD (the sale of the CTD issue and the purchase of the future) can here be seen to be similar to the writing of an out-of-the-money option, as the parallel shift of 20 bps until contract expiry is considered to be more or less improbable – to say nothing of other curve moves within the basket! The value of the delivery option has therefore to be determined as the probability that a certain bond in the deliverable basket will become the CTD. When available, this information can be useful in two ways: ■ in verifying that the net basis in the market trades at its ‘fair’ level ■ in allowing improved decision making for bond basis positions, as the

value and the probability of booking a profit or loss are known. The following statements regarding the factors influencing the delivery option value are generally true: ■ The more volatile the market, the higher the value of the delivery

option ■ The more volatile the curve moves (steepening or flattening), the

higher the value of the delivery option ■ The further away the contract delivery date, the higher the value of the

delivery option

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■ The wider the deliverables basket, the higher the value of the delivery

option (that is, the greater the maturity difference between the first and the last bond in the deliverable basket) ■ The more bonds close to being the CTD, the higher the value of the

delivery option (the higher the probability of a change in the CTD status). In the next section, a two-parameter model for calculating the delivery option value is derived. The two parameters are a parallel shift and a steepening or flattening of the curve, similar to the examples given in the previous section. In particular, the value of the delivery option is determined as the sum of probability-weighted expected bases of a parallel shift and a steepening or flattening of the curve.

4.2.2 Terminology The following expressions are used in the derivation below: fyldi = forward yield of bond i Pi = forward price of bond i PFi = price factor of bond i βi = yield Beta of bond i mati = maturity of bond i n = number of bonds in deliverable basket Any other expressions used will be explained in the text.

4.2.3 Parallel shifts of the curve The relationship between the forward price of a bond, the forward yield and its price factor is given by: Fi = Pi (fyldi) / PFi

A given parallel shift of p basis points then gives: Fip = Pi (fyldi + βip/10000) / PFi

Let the minimum of these Fip be defined as: 90

Fp (min) = min (Fip ), i = 1,..,n

for a given parallel shift of p basis points. The bond net basis Bip can now simply be determined as: Bip = Fip – Fp (min)

Since not only a parallel shift of p basis points is possible but a whole range of shifts, the total net basis of a bond i, B(P)i, can be attained by summing over all possible parallel shifts in the curve and by weighting these according to their probabilities of taking place: B(P)i = ∑p Bip W(p)

where W(p) is the probability of a parallel shift of p basis points.

4.2.4 Steepening or flattening of the curve For a given steepening or flattening of s basis points, the relationship between the forward price of a bond and its forward yield and price factor is given as: Fip = Pi (fyldi + γis/10000) / PFi

where γi represents the number of years between the ‘mid-maturity’ bond – the synthetic maturity of a bond with the same time difference as the shortest- and longest-maturity bonds in the deliverable basket – and the maturity (mat) of bond i: γi = [ (matlongest + matshortest) / 2 – mati ] / 365

As with the case of parallel shifts, let Fs (min) = min (Fis), i = 1,..,n

be the minimum of the Fis. From this it follows that the bond net basis Bis for each bond i, assuming a steepening or flattening of s basis points, can be represented as: Bis = Fis – Fs (min) 91

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Since not only a curve steepening or flattening of s basis points is possible but a whole range of shifts, the total net basis of a bond i, B(S)i, can be obtained by summing over all possible curve steepening or flattening scenarios in the curve and by weighting these according to their probabilities of taking place, that is, B(S)i = ∑s Bis W(s)

where W(s) is the probability of a curve steepening or flattening of s basis points.

4.2.5 Delivery option value In order to obtain the full value of the delivery option, one has to allow both for parallel shifts and for curve-steepening or curve-flattening scenarios. The delivery option value, DelOptVal, is therefore shown as: DelOptVal = ∑p ∑s Bip,s W(p) W(s)

where the probabilities W(p) and W(s) are independent.

4.2.6 Yield Beta The Yield Beta states how the yield of a bond moves in relation to yield moves of another bond; for example in Chapter 3 the example was mentioned in which a yield move of 10 bps in a 10-year bond would lead to a move of 8 bps in a 30-year bond – that is, the 30-year bond has a yield Beta of 0.80 versus the 10-year bond (assuming a high correlation). When deriving the delivery option value, yield Betas were included for parallel shifts for bonds. In general, one bond in the deliverable basket is chosen to be the so-called ‘norm bond’ with a yield Beta equal to one. The chosen bond might be the benchmark bond or the CTD bond. For all other bonds, yield Betas can be used to reflect expected yield moves relative to the norm bond. In case the investor wishes to use pure parallel shifts in the analysis, all yield Betas should be set equal to one. Yield Betas allow for further flexibility when determining the delivery option value, as in addition to the two parameters, parallel shifts and curve steepening or flattening, particular moves in the individual bonds can also be implemented. 92

Determining values for yield Betas can be done in the following ways: ■ historical yield Betas: historical yield moves can be used to determine

yield Betas ■ subjective yield Betas: subjective opinions as to future yield moves can

be used to determine yield Betas – for example, the following information could be used: – that a change in the supply and demand in a bond is expected, for instance because of the issuance of a new bond – that a change in the bond repo rate is expected, which would make the bond expensive (‘special’) in the repo market.

4.2.7 Number of ‘slices’ in calculating the delivery option value Theoretically, the value of the delivery option can be determined by solving an integral equation. In practice, however, it is easier, as already mentioned above, to use an approximation with the help of sums. The question arises as to what the best number of ‘slices’ should be – that is, into how many slices should the probability distribution be divided?3 The more slices that are used, the more accurate the end result. On the other hand, increasing the number of slices can significantly increase the computational effort. In practice, therefore, different numbers of slices are tested for accuracy and stability; and it can be said that using

50% Probability

40% 30% 20% 10% 0%

−-54

−-27

0

27

54

Shift (bps)

Figure 4.7 Probability distribution of yields for parallel shifts: 5 slices 93

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25%

Probability

20% 15% 10% 5% 0%

–-61.02 –-47.46 –-33.9 –-20.34 –-6.78

6.78

20.34

33.9

47.46 61.02

Shift (bps)

Figure 4.8 Probability distribution of yields for parallel shifts: 10 slices 50 slices generally provides good results without using too much computational time. Figures 4.7 and 4.8 show possible probability distributions using 5 and 10 slices, respectively.

4.2.8 Determining volatilities The expression for determining the delivery option value can be programmed relatively easily. The question here is: what volatility values should be used for the computation? Since the data sought represent information about market volatility, there are two obvious possible ways: ■ historical volatilities: the volatilities of parallel shifts and curve steep-

ening or flattening can be determined using historical data ■ implied volatilities: market information for traded bond options could

be used for determining the probabilities of a bond reaching a certain yield level (see section 5.2).

4.2.9 Example: delivery option value calculation For simplicity, the number of slices for this example is set equal to 5. Also, it is assumed that the possible parallel shifts and curve twists (steepening or flattening) follow a standard normal probability distribution. Using 94

−88

4%

−27

24%

0

45%

27

24%

88

4%

−12

−4

4%

24%

0

4

12

45%

24%

4%

Flattening (negative values) and steepening (positive values)

Probability of each curve twist Probability of each parallel shift

Parallel shifts in bps

Figure 4.9 Parallel shifts and curve twists with probabilities

volatilites of 23 bps (annualized yield volatility of 15%: see section 5.2) for parallel shifts and a volatility of 3 bps for curve twists (until contract expiry) gives the results shown in Figure 4.9. The probability of the scenario (example from Figure 4.9) of a parallel shift of 27 bps and a steepening of 12 bps is thus approximately 1% (= 24% x 4%). The probabilities for all other possible curve scenarios can be determined in the same way, and shown in a matrix as shown in Table 4.2. The next step is to determine the net bases for these identical curve scenarios. In the example that was analysed in section 4.1, for the bond Bund 4.50% Jul–09, the following results are obtained (in ticks).

Table 4.2 Probabilities of different curve scenarios −12 (%)

−4 (%)

0 (%)

4 (%)

2

1

12 (%)

−88

0

−27

1

6

11

6

1

0

2

11

20

11

2

27

1

6

11

6

1

88

0

1

2

1

0

1

0

95

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Table 4.3 Sample net basis values: Bund 4.50% Jul-09 −12

−4

0

4

12

−88

0.0000

0.0000

0.0000

0.0000

0.0000

−27

0.0000

0.0000

0.0000

0.0000

0.0000

0

0.0000

0.0000

0.0000

0.0000

0.0000

27

0.0263

0.0220

0.0202

0.0183

0.1202

88

0.0868

0.0829

0.0812

0.0794

0.3865

The net basis values are now multiplied with the corresponding probabilities, giving a table with the probability-weighted bases (Table 4.4). The delivery option value is, by definition, the sum of these probabilityweighted bases. In this example the result is: delivery option value = 0.009 (0.9 ticks)

The net basis of the Bund 4.50% Jul-09 trades in the market at 0 ticks. By employing this model, therefore, a result has been obtained that is very close to the actual market price. In using the delivery option model successfully, three points are of crucial importance: 1. the number of slices (5 in the example above) must be large enough to allow for an accurate result 2. the volatilities used for the calculation (23 bps for parallel shifts and 3 bps for curve twists in the example above) must be realistic

Table 4.4 Sample probability-weighted net basis values: Bund 4.50% Jul-09

96

−12

−4

0

4

12

−88

0.0000

0.0000

0.0000

0.0000

0.0000

−27

0.0000

0.0000

0.0000

0.0000

0.0000

0

0.0000

0.0000

0.0000

0.0000

0.0000

27

0.0002

0.0013

0.0022

0.0010

0.0010

88

0.0001

0.0007

0.0013

0.0007

0.0005

3. the probability distribution used for the calculation (a standard normal distribution in the example above) must be realistic.

4.3 CALENDAR SPREADS 4.3.1 Fair value of calendar spreads This section discusses calendar spreads – that is, the price difference between the front (first) and back (second) bond futures contracts. The theoretical value of calendar spreads is derived, and factors influencing its value are examined. This information can be used to improve the timing of futures rolls: that is, when a long (short) position in the front month futures contract should be sold (bought back), and a new long (short) position in the back month futures contract be established. The theoretical fair value of a bond futures contract is given by: Fair Futures Value = (Forward PriceCTD – DelOptValCTD) / PFCTD

This implies that the fair value of a calendar spread is given by: Fair Calendar Spread Value = Fair Futures Value (Contract 1) – Fair Futures Value (Contract 2) = (Forward PriceCTD1 – DelOptValCTD1) / PFCTD1 –(Forward PriceCTD2 – DelOptValCTD2) / PFCTD2

There are two possible reasons why the traded market price of the calendar spread would not be equal to its fair value: ■ The delivery option values are not equal to zero. In this case the market

prices the delivery options differently than the theoretical model: that is, the calculated delivery option values are not equal to the bond net bases. ■ The delivery option values and the net bases of the two CTD bonds are equal

to zero. In this case there is an arbitrage situation, as it is possible to finance the cash bond positions against the futures positions with a profit. Note, however, that the arbitrage can only be achieved if the positions can be entered into with the guaranteed profit – that is, bid and offer prices should be taken into account to verify whether the arbitrage is possible. Frequently, the calendar spread trades only a 97

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couple of ticks away from its theoretical value. In this case no arbitrage is feasible as the costs for entering into the position would exceed the profit taken. More often, movements in the bond repo markets allow for short-term arbitrage possibilities, when one or both CTD issues are in strong demand and the corresponding repo rates change accordingly.4 These situations are taken advantage of by arbitrageurs, eliminating further arbitrage opportunities quickly.

4.3.2 Factors influencing calendar spreads When the first (front month) and second (back month) bond futures contracts have the same CTD issue, the calendar spread is driven by the future 3-month interest rate (Euribor futures contract) for the time period between the expiries of the first and second bond futures contracts. For contracts with insignificant delivery option values, the relative value of a calendar spread can be determined by comparing the implied repo rate (IRR) of the calendar spread with the future 3-month interest rate. The IRR of the calendar spread represents the breakeven financing rate of a position, which consists of a long position in the calendar spread (that is, a long position in the front month contract and a short position in the back month contract), taking delivery of the CTD bond at expiry of the front month contract and delivering it into the back month contract. If the IRR of the calendar spread is especially low compared with the future 3-month interest rate, the calendar spread is considered expensive (for example, front month expensive, back month cheap). In general, increasing interest rates cause the bond futures to become richer relative to cash bonds, because of the tightening of the financing basis until contract expiry. Therefore, the calendar spread reflects changes taking place at the short end of the interest-rate curve. The impact of interest-rate changes is more significant to the back month futures contract, as its financing period is 3 months longer. Two different cases are distinguished: ■ The CTD bond into the front month contract is also the CTD into the

back month contract: in this case the calendar spread is strongly correlated with the future 3-month interest rate ■ The front and back month futures contracts have different CTD bonds:

in this case the calendar spread is strongly correlated with the yield spread between the two different CTD bonds. 98

The two cases are discussed in more detail below, using examples to help illustrate these statements.

CTD 1=CTD 2: calendar spread versus future 3-month interest rates The example of the March–June 2000 Bund future calendar spread confirms the strong correlation between the calendar spread and the March 2000 Euribor 3-month interest rate future. The Bund 3.75% Jan-09 was the CTD into both the Mar-00 and Jun-00 Bund futures contracts. The calendar spread therefore follows the 3-month Euribor rate. If the calendar spread became too cheap compared with the 3-month interest rate, the investor could buy the calendar spread, take delivery of the bond in March against the long futures position, and then deliver it into the June contract, after having lent it for the time period in between the two contract delivery dates. It needs to be kept in mind, however, that the true financing rate for the calendar spread depends on the repo rate of the bond: between the two futures contract delivery dates, the bond can be lent at the repo rate, not at the 3-month Euribor rate. Therefore, if the repo rate is below the

96.8

1.10 Mar/Jun-00 Bund Bund Mar/Jun-00 calendar spread Calendar Spread

96.6

1.00 0.90 0.80

96.4 0.70 96.2

Calendar spread (ticks)

Euribor (points)

MAR-00 EURIBOR Euribor

0.60

27 /0 9/ 1 07 999 /1 0/ 1 19 999 /1 0/ 1 29 999 /1 0/ 1 10 999 /1 1/ 1 22 999 /1 1/ 1 02 999 /1 2/ 1 14 999 /1 2/ 1 27 999 /1 2/ 1 07 999 /0 1/ 2 19 000 /0 1/ 2 31 000 /0 1/ 2 10 000 /0 2/ 2 22 000 /0 2/ 2 03 000 /0 3/ 20 00

99 19 9/ 0 / 15

Date

Figure 4.10 Mar/Jun-00 Bund calendar spread and Mar-00 Euribor future 99

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Euribor rate, the investor can borrow money ‘more cheaply’ by lending the bond than if he or she borrowed money in the money market at the Euribor rate. It is possible that the repo rate of the bond might become ‘special’, if there is strong demand for this particular bond in the repo market: in this case there can be a significant difference between the Euribor rate and the bond repo rate. Thus Figure 4.10 also shows periods where the calendar spread does not exactly follow the Euribor rate, because at these points in time the repo rate for the CTD changed.

CTD 1CTD 2: calendar spread versus yield spread of CTD1–CTD2

3

60

2.5

55

2

50 45

1.5

40

1

35 0.5

30

0 −-0.5 −-1 −-1.5

20 15 10

/0 21 3/2 /0 00 27 3/2 0 /0 00 31 3/2 0 /0 00 06 3/2 0 /0 00 12 4/2 0 /0 00 18 4/2 0 /0 00 24 4/2 0 /0 00 28 4/2 0 /0 00 04 4/2 0 /0 00 10 5/2 0 /0 00 16 5/2 0 /0 00 22 5/2 0 /0 00 26 5/2 0 /0 00 01 5/2 0 /0 00 6/ 0 20 00

00

3/

20

15

0

0 9/

25

Yield YldSpr spread Calendar CalSpr spread

Date

Figure 4.11 Calendar spread and yield spread (CTD Sep-00–CTD Jun-00) 10 0

Calendar spread (ticks)

Yield spread (bps)

The example of the Jun–Sep 2000 Bund future calendar spread confirms the strong correlation between the calendar spread and the yield spread of the two different CTD bonds (Bund 3.75% Jan-09 CTD into the Jun-00 contract, and Bund 4.5% Jul-09 CTD into the Sep-00 Bund futures contract). A long position in the calendar spread (a long position in Jun-00 Bund future, and a short position in Sep-00 Bund future) means implicitly that the Bund 3.75% Jan-09 was purchased and the Bund 4.5% Jul-09 sold

against it. This position benefits from a steepening in the yield curve, since it consists of a long position in a bond with a shorter maturity and a short position in a bond with a longer maturity. Because of this relationship, the calendar spread follows the yield spread of the two CTD bonds (Figure 4.11). Since the two contracts have different CTD bonds, they also do not have identical exposures to yield moves (the basis point values are different), and the rollover of a futures position cannot be performed using a one-to-one relationship. The calendar spread is thus marketdirectional – that is, its value also varies if parallel shifts in the yield curve take place.

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4.4 PORTFOLIO REPLICATION 4.4.1 Futures barbell The previous sections showed when a future is considered expensive or cheap compared with cash bonds. It is possible to construct a barbell using futures (2y–5y–10y barbell strategy) much as a barbell can be constructed using cash bonds. In the German bond market in particular this can be done without difficulty, as three liquid bond futures corresponding to these maturities are readily available. Understanding the relationships between bond futures and cash bonds helps the investor make a decision about whether he or she prefers to use bond futures instead of cash bonds. The 2y–5y–10y barbell, examined in detail in section 3.2, can hence be constructed using futures in the German bond market. The equation for a futures hedge of a cash bond is given by: Number of futures for hedge = (nominalcash bond) / (nominalfutures contract) • (bpvcash) / (bpvfuture)

where the basis point value of the future, bpvfuture, is given by: bpvfuture = bpvCTD / PFCTD

The duration- and cash-neutral barbell from section 3.2 was constructed as shown in Table 4.5. Each individual bond in this barbell can be substituted (hedged) using a bond future. The CTD issues for the three bond futures contracts 101


Table 4.5 Duration- and cash-neutral barbell Barbell type (a) Duration-neutral and equal wings


Nominal (in 1,000)

Duration

Cash

Wing long

BKO 4.5% Mar-02

57,143

0.86

−57,623,203

Body short

OBL 5% May-05

−100,000

−4.21

101,892,466

Wing long

Bund 5.25% Jul-10

43,534

3.35

−44,269,263

Total

0.00

0

Table 4.6 Information on CTD bonds Lifetime

Future

CTD

bpv

PF

2 years

Schatz

BKO 4.5 03/15/02

1.66

0.975442

5 years

Bobl

THA 6.25 03/04

3.44

1.007771

10 years

Bund

DBR 3.75 01/04/09

6.41

0.852420

Table 4.7 Futures barbell Barbell type (a) Duration- and cash-neutral


Number of contracts

Wing

Long

Schatz

Body

Short

Bobl

Wing

Long

Bund

505

−1216 441

(expiring in June 2000), and their basis point values and price factors, are given in Table 4.6. For example, using the information for the 10-year bond and the equation above gives the following hedge: Number of futures for hedge = (nominalcash bond) / (nominalfutures contract) • (bpvcash) / (bpvfuture) = (43.534m / 100,000) • (7.62 / 7.52)

= 441 10 2

where the nominal value of the futures contract is given by:

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BASIS MANAGEMENT

Nominalfutures contract = 100,000

and bpvfuture = bpvCTD / PFCTD

= 6.41 / 0.852420 = 7.52 In the same way, futures hedges can be determined for the 2-year and the 5-year bonds. If bond futures were preferred to cash bonds, the barbell would be constructed as in Table 4.7. When entering into a barbell position, the investor thus has the added flexibility of using bond futures instead of cash bonds, at least to some extent, if he or she prefers these due to liquidity, maturity or other considerations.

4.4.2 Portfolio replication using futures It is also possible to replicate a whole portfolio of cash bonds using bond futures. Both low transaction costs and easy positioning have made bond futures a popular instrument for portfolio management. For example, assume the investor has invested EUR 100 mm in German government bonds. This portfolio could be broken down in maturity sectors as shown in Table 4.8. The situation here is similar to that of the futures barbell. The bonds are arranged in such a way that each of the three sub-portfolios could be replicated with the corresponding bond future – that is, bonds in the 0–3-year maturity range using the Schatz future, bonds in the 3–7-year maturity range using the Bobl future, and bonds in the 7–10-year maturity range

Table 4.8 Bonds investment Time until maturity

0–3 years

3–7 years

7–10 years

Investment (m)

20

50

30

Basis point value

1.50

5.00

7.00 103


Table 4.9 Futures portfolio to replicate a bond portfolio Bond future

Lifetime

Number of futures

Schatz

0–3 years

176

Bobl

3–7 years

732

Bund

7–10 years

279

using the Bund future. Table 4.9 shows the futures portfolio that replicates the cash bond portfolio. This portfolio, consisting solely of futures, has the same duration as the original portfolio consisting of cash bonds: the profit (loss) of decreasing (increasing) yields of the two portfolios are identical. Were a change to take place in the CTD bonds, the hedge ratios of the futures portfolio should be recalculated. This method is suitable for a portfolio consisting mainly of government bonds. If additionally strips (see section 3.3) or Credits (that is, corporate bonds – see Chapter 6) were included, it is recommended that portfolio durations be used for arranging bonds in buckets rather than bond maturities. Strips have a significantly higher duration than conventional coupon-bearing bonds, since by definition they are zero-coupon bonds and so do not pay coupon payments. On the other hand, Credits tend to have higher coupons because the credit ratings of issuers tend to be worse than those of governments. Credits typically have lower durations compared with government bonds with similar maturities. Empirical research (data from Lehman Brothers) has shown that, by substituting the CTD bonds with bond futures for the time period Feb-99 to Mar-00, an outperformance of approximately 3 bps could be achieved. This can be explained by recalling that the implied repo rate of bond futures lies below the repo rate of the CTD issue, which led to the future being identified as cheap relative to the cash bond. In the past few years, the Bund future tended to trade too cheap, since it was being used more and more as a hedging instrument. An IRR below the CTD repo rate means a financing benefit, determined as the difference between the two. The tracking error – the deviation from the benchmark – here was 15 basis points per month (bond futures versus an index of Euro aggregate government bonds).

10 4

Notes 1. 2.

3.

4.

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Exceptions are possible, if the prices including accrued coupon interest (expression P+z ) of two bonds are significantly different. Sometimes it can happen that the net basis is negative before and at contract expiry. In particular this can be the case where the number of open short futures positions is large compared with the outstanding issue size in the CTD bond, thus increasing the pressure for short positions to buy back their sold contracts (‘short squeeze’). In a long net basis position, the repo is typically done until a few days before the contract delivery date, whereas the repo in a short net basis position is usually done until approximately 10 days after expiry. Therefore, in general, there are differences in repo rates, which close before expiry compared with those, which close after futures expiry (repo rates are less special if closed before expiry). The probability distribution used for the calculation of the delivery option value is the standard normal distribution, where the probability of a yield increase is equal to the probability of a yield decrease. Similarly, the probability of a curve steepening is equal to that of a curve flattening. If other assumptions are deemed to be more appropriate, other probability distributions can also be used. The repo rate of a bond is the interest rate, which can be achieved in a repurchase agreement. In a repurchase agreement, an investor A sells a bond to an investor B with the obligation to buy it back on a predefined future date. At this future date, investor A pays investor B an interest rate, the repo rate, for the money he or she borrowed at the beginning of the deal.

105

CCHAPTER H A P T E R 51

Volatility management

5.1 VOLATILITY AND YIELDS Volatility expresses the dispersion of the price or yield of a financial instrument from its mean, which can be measured for example on a daily, weekly or annual basis. One differentiates between implied (projected or expected) and historical volatility. Implied volatility is the volatility as implied by options prices traded in the market: in the following the focus will be mainly on implied volatility and its possible applications. Section 5.2 describes in detail how the dispersion from a mean, the standard deviation, is computed. Implied volatility on the Bund futures (Treasury futures), measured by the implied volatility of Bund options, is a good proxy for overall market uncertainty – uncertainty regarding future inflation expectations, as well as central bank policy and economic growth. This uncertainty increases the daily standard deviation of Bund futures and 10-year Bunds, respectively, which then leads to an increase in implied volatility. Increasing volatility often goes together with lengthy bear markets (increasing interest rates; see Figures 5.1 and 5.2). Increasing volatility leads to a widening of bid-offer spreads and an increased risk aversion among investors, and consequently to overall market instability. The Russian crisis in 1998 represents an exception, because the extreme increase in volatility was accompanied by decreasing yield levels. The ‘flight to quality’ (government bonds as a ‘safe haven’) and the widening 10 6

4 3

GD: German German Bund Bund(Eurex) (Eurex)FRONT FRONT Implied Volatility implied volatility

4/6/01 06.04.01

01.07.99

01.01.99

01.07.98

01.01.98

Gilts

10/6/00 06.10.00

4/6/00 06.04.00

10/6/99 06.10.99

4/6/99 06.04.99

10/6/98 06.10.98

01.07.97

01.01.97

01.07.96

01.01.96

01.07.95

Treasuries

4/6/98 06.04.98

10/6/97 06.10.97

4/6/97 06.04.97

10/6/96 06.10.96

4/6/96 06.04.96

10/6/95 06.10.95

01.01.95

01.07.94

01.01.94

01.07.93

01.01.93

01.07.92

01.01.92

Bear mkts

4/6/95 06.04.95

10/6/94 06.10.94

4/6/94 06.04.94

10/6/93 06.10.93

4/6/93 06.04.93

10/6/92 06.10.92

4/6/92 06.04.92

10/6/91 06.10.91

4/6/91 06.04.91

10/6/90 06.10.90

06.04.90 4/6/90

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V O L AT I L I T Y MANAGEMENT

Bunds

Figure 5.1 The left axis shows the annualized, daily yield changes in per cent (logarithmic); the shadowed areas represent bear markets Sources: Merrill Lynch and own data

12 11

10 9 8

7

6 5

Bund benchmark Benchmark10 10yield Yield

Figure 5.2 Implied volatility of Bund futures and yield levels of 10-year Bunds

Source: JP Morgan

107


of credit spreads led in 1998 to a significant increase in government bond volatility. At the same time, the volatility term-structure curve (depicting the volatility of options for 1 month to 12 months) became inverted, indicating that high fluctuations were expected in the near future, but that over time these fluctuations would decrease again – the volatility of short-term options is higher than the volatility of long-term options. Market participants’ expectations typically determine the shape of the term structure (positively sloped or inverted). The dependence of volatility on overall market levels can also be explained by trends in inflation. Higher inflation rates are accompanied by higher and more volatile interest rates. Investors demand a higher risk premium for holding nominal bonds as inflation increases or when it is already relatively high, and this then leads to an increase in the overall levels of interest rates and volatility. Current prices already reflect several events expected in the future, because the market anticipates that these actions will take place. Inflation-linked government bonds in France (OATi) serve as a good example (see Figure 5.3): in 1999, there existed a strong relationship between the breakeven inflation (the expected infla-

Bund implied volatility

OATi discounted CPI

Figure 5.3 Bund implied volatility (left axis) and OATi discounted CPI (Consumer Price Index) (right axis) Source: Merrill Lynch 10 8

.

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. .

.

.

.

.

.

. .

.

.

.

.

.

.

Basis point value

Figure 5.4 Regression of basis point volatility/10-year Bund yield (circled data points year 2000): 1999–June 2000 Source: JP Morgan

tion + the risk premium), as implied by inflation-indexed OATis, and the volatility of 10-year Bunds. Also, the relationship to swap spreads (see Chapter 6) can illustrate market instability and the level of investors’ risk aversion, and hence can be used as an alternative proxy for volatility. Basis point volatility, defined as yield volatility multiplied by the yield level (see section 5.2), is clearly higher in bear markets (see Figures 5.4 and 5.5) compared with flat or rising markets (prices here). The direction of the trend is a decisive factor here. In an environment of falling yields (at the same interest-rate level), basis point volatility is markedly lower than in a state of rising interest rates (1998 interest rates 5%; 1999 interest rates 5% as well): basis point volatility was 30 basis points higher in 1999 (rising yields) than in 1998. This behaviour leads to observed differences in the volatility of out-of-the-money options. Out-of-the-money put options (on Bund futures) are quoted with a higher volatility than corresponding out-of-the-money call options. The desire of investors to hedge their positions, and their experience that large price movements in the market are more likely to occur to the downside (an increase in interest rates, a fall in bond prices), results in skewed implied volatilities of put options (out-of-the-money puts are more expensive than out-of-the109


BPV

DEM 10year benchmark yield

Yield . . . . . . . . .

Bp implied vol 3 month into 10year option

. . .

Date

Figure 5.5 Directionality of basis point volatility Source: JP Morgan

money call options). In general, it has been empirically observed that strong market movements occur more frequently than expected by the Black option model using a normal distribution (outliers).

5.2 YIELD VOLATILITY Price volatility denotes the annualized dispersion of the price of an asset – for example, a future or a bond – from its mean. If the asset has a price of 100 and a volatility of 6%, the price in one year’s time is expected to be between 94 and 106 with a probability of 68.3%, assuming a standard normal distribution. The reason why yield volatility represents a significantly better measure for deviations in the market than price volatility is that yield volatility removes the effects of different durations of individual bonds – for example, longer duration bonds have a higher price volatility than shorter duration bonds; the decisive factor is the basis point volatility. The price volatility of a high duration bond lies significantly over the price volatility of a short duration bond, so no statement can be made regarding which bond is effectively more volatile. If yield volatility is 11 0

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used, however, it can be determined which bond has had higher fluctuations or should expect higher fluctuations in future. A yield volatility of 15%, given an interest-rate level of 5% for 10-year bonds, should be interpreted in the following way. With a probability of 68.3%, the yield of the bond in one year’s time will be between 4.25% and 5.75% (5% • 15% = 75 basis points; 1 standard deviation). The 75 basis points are defined as the annual basis point volatility. Basis point volatility considers also the overall interest-rate levels, which can be biased for yield volatility if, for example, yield levels decrease substantially. A yield volatility of 20%, given interest rates of 3%, appears very high compared with the case of yield volatility being at 10% with interest rates at 6%. However, the annual basis point volatility in both cases is exactly the same: 60 basis points. The following calculation shows how to obtain the daily price and the yield volatility of an OTC-option on the Bund Jan-2010 (OTC options are computed via the forward price on the maturity of the option) with 3 months to expiry. Price volatility: 5.20% Forward price: 100 Basis point value (bpv): 7.25 Forward yield: 5.30% Yield volatility can then be determined using these parameters, as: price volatility • forward price Yield volatility = forward bpv • forward yield • 100 When the values above are inserted into the equation, a yield volatility of 13.533% is obtained. The annual price or yield volatility can be converted into a daily standard deviation by dividing the implied price or yield volatility by the square root of the number of business days in a year (here 252 days). The resulting figure is multiplied by the forward price or forward yield, and the forward basis point value of the bond. Price volatility:

5.20%/ √252 = 0.328 0.328 • 100 = +/−32.8 Ticks/day

Yield volatility:

13.533%/ √252 = 0.853 0.853 • 5.30% = 4.52 Basis points/day 4.52 • 7.25 = +/−32.8 Ticks/day

111


5.3 OPTION RISK PARAMETERS The risk parameters – or ‘Greeks’ – of an option are introduced briefly below, and are used in further examples in the text, where the main focus is on practical applications. Option price sensitivities can be categorized into five different terms: Delta, Gamma, Vega (Kappa), Theta, and Rho.

5.3.1 Delta An option’s Delta is a measure of the absolute change in the option’s price due to a 1 unit change in the underlying – for example, a change of 1 point or 100 ticks in the Bund futures, increasing from 104 to 105. ■ Delta is positive for calls and negative for puts (long options) ■ Delta varies between zero and +/–1 ■ The value of Delta is highest for options, which are deep-in-the-money;

and is lowest for options, which are far out-of-the-money. Options, which are at-the-money, have a Delta of approximately +/–0.5. The following example will clarify this. A call option on the Bund futures contract has a price of EUR 0.50 and a Delta of 0.50, with the exercise price being 104. The corresponding price of the Bund future rises from 104 to 104.10. This implies that the price of the option increases from EUR 0.50 to EUR 0.55 (0.50 • 0.10 = 0.05). (The change of only 10 ticks in the future is chosen for ease of illustration.)

5.3.2 Gamma An option’s Gamma is a measure of the absolute change of the option Delta due to a 1 unit change in the underlying. Delta underestimates price increases of options, because the curve of the Delta values in relation to the futures price is not linear. Gamma is a measure for the nonlinearity of this relationship (‘Delta’s Delta’). ■ Gamma is positive for long call options and long put options ■ The value of Gamma is highest for at-the-money options. The value

converges towards zero for options either deep in-the-money or far out-of-the-money. 11 2

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Again, an example will clarify this. The Bund futures call option, with a strike price 104, has a Gamma value of approximately 0.30. If the future increases by 10 ticks, from 104 to 104.10, then the option’s Delta would increase from 0.50 by a Gamma (0.30 • 0.10 = 0.03) to 0.53. This gives a total of 0.58, as the Delta itself increases by 0.05, as seen in the example above. (The change of only 10 ticks in the future is chosen for ease of illustration.)

5.3.3 Vega (Kappa) An option’s Vega is a measure of the absolute change of the option’s price due to a 1% change in volatility. ■ Vega is positive for long call options and long put options ■ The value of Vega is highest for at-the-money options. The value

converges towards zero for options either deep in-the-money or far out-of-the-money. The following assumptions are made: let the option have a Vega of 0.10 and volatility increases by 1%. For the 104 call option from the example above, the change in Vega is 0.10 • 1.0 = 0.10: this increases the value of the option to EUR 0.60 (from EUR 0.50), while keeping the futures price constant at 104.

5.3.4 Theta An option’s Theta is a measure of the absolute change of the option’s price due to a 1-day change in expiry (here a measure for the time decay of the long option). ■ Theta is negative for long call options and put options (daily time

value loss) ■ The value of Theta is highest for at-the-money options. The value

converges towards zero for options either deep in-the-money or far out-of-the-money. (At-the-money options with short maturities have the highest Theta.)

113


In the example above, the value of the 104 call option decreases by approximately EUR 0.01 per day, and this decrease will increase sharply towards the option’s expiry date.

5.3.5 Rho An option’s Rho is a measure of the absolute change of the option’s price due to a change in the risk-free interest rate. Note that for Bund futures options a negligible Rho value is obtained, because of the insignificance of the discount factor in the valuation formula (futures positions only require payments of margin, not the full asset price). Rho is of major importance for OTC options on Bunds (change of the riskfree rate by 0.10%).

5.4 EFFICIENT GAMMA TRADING 5.4.1 Gamma trading and strategies In very volatile markets or in particularly quiet interest-rate environments, it makes sense to pursue an active Gamma strategy. Bought option positions are Gamma long positions, which may require permanent hedging to compensate for the daily time decay. On the other hand, a short option position requires control of the Delta of the position in order to avoid a possible loss of the underlying (here sale of a call – loss of the underlying if the market price exceeds the strike price at maturity). It is also evident that a defensive hedging strategy can generate additional value in less volatile interest-rate periods, using futures options as in the examples above. In order to be able to implement a successful hedging strategy, it is crucial to determine the mid-point of the daily trading range. The goal of a Gamma short position is to stay Delta neutral, exactly at this predetermined level. With a Gamma long position, on the other hand, the goal is to keep the options portfolio (the option) in the mid-point, either Delta long or Delta short. This means that futures can be either bought or sold. The Gamma of a position of various options with the same or different strike prices is defined here in futures contracts. For example, a position consisting of 533 sold calls (Gamma short) has a Gamma equivalent to 88 futures contracts. Should the Bund future rise by 1 point (100 ticks), 88 futures would have to be bought in order to achieve a Delta 11 4

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neutral position. If the Bund future falls by 1 point, 88 futures would have to be sold in order to preserve Delta neutrality. The straddle is a simple strategy in the options market – buy or sell at-the-money call and put – and is characterized by high liquidity and tight bid-offer spreads. The straddle is often used to realize efficient Gamma trading. To build a Gamma short position, in this example, a straddle on the Bund future with 1 month to expiry is sold. The resulting position of 254 sold straddles (sale of 254 calls and puts) with 1 month to expiry has a Vega of approximately EUR 51,050, a Gamma equivalent to 153 futures per 100 ticks change in the future, and a daily Theta of EUR 5,105. If volatility decreases by 1%, the position would lock in a profit of EUR 51,050. If the futures increase by 100 ticks, 153 futures would need to be bought in order to achieve Delta neutrality (approximately 15 futures every 10 ticks), which would be accompanied by costs. The daily profit due to the decrease in time value is initially EUR 5,105 (and increases over time). The goal of the strategy is to undertake Delta hedges, which are less costly than the profit obtained from the daily time decay. This straddle will be substituted by a strangle position one or two weeks prior to maturity – sale of call, put out-of-the-money – in order to limit the increasing Gamma risk.

5.4.2 Gamma short strategies Gamma short strategies can be described using three different scenarios. SCENARIO: High daily volatility expected See Figure 5.6. Here, the hedge limits are set to 15–20 ticks movements in the Bund future from the starting point (for example, 104). If the barriers are hit, Delta should be hedged narrowly, which implies that a Delta adjustment occurs every 5–10 ticks. If no hedges are necessary for a couple of days, the hedging range limits can be increased to 25–30 ticks. SCENARIO: No strong opinion or view regarding the sizes of daily moves in the market If undecided, Bund futures are hedged every 10 ticks until a pattern develops in the future’s behaviour, after which hedging is performed accordingly. 115


104,5 . 104,4 .

. 104,3 . 104,2 104,1 . 104 103,9 . Futurepreis Futures Price Futures price

103,8 . 103,7 . 8.00

9.00

10.00

11.00

12.00

13.00

14.00

15.00

16.00

17.00

18.00

19.00

Trading time Time of of Bundfutures Bundfutures

Figure 5.6 High daily volatility of the Bund future

SCENARIO: A clear trend can be monitored See Figure 5.7. If there is a clear trend, hedging is performed every 10 ticks (tight Delta hedge), ignoring possible outliers.

104,7 . 104,5 . 104,3 . 104,1 .

Futurepreis Futures Price Futures price

103,9 . 103,7 .

. 103,5 8.00

9.00

10.00

11.00

12.00

13.00

14.00

15.00

16.00

17.00

Trading time Timeof ofBundfutures Bundfutures Trading

Figure 5.7 Trend line of the Bund future 11 6

18.00

19.00

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5.4.3 Gamma long strategies Gamma long strategies are preferably applied in a volatile market environment. These strategies can be expressed using the following three scenarios. SCENARIO: High daily volatility expected See Figure 5.6. A hedge position should be built every 10–15 ticks until the loss in time value is compensated for (a sample calculation will follow). Thereafter strong price eruptions should be anticipated (and a hedge tactic used every 20–30 ticks). SCENARIO: No opinion or view regarding the sizes of daily moves in the market As mentioned above, here the Delta will also be neutralized every 10 ticks in the Bund futures. In the case that volatile, unexpected economic data are released, a widening of the Gamma hedging range should follow (hedging only after a movement of 30 ticks in one direction). After the market has calmed down, an adjustment to approximately 20 ticks should be undertaken. SCENARIO: A clear trend can be monitored See Figure 5.7. The Gamma hedging range is widened such that a hedge is only performed after 30–40 ticks. After the hedge, the range can be reduced to 10–20 ticks.

5.4.4 Frequency of Gamma hedging The computation of the frequency of Gamma hedges in relation to the size of implied volatility is now discussed. Given an implied price volatility of 6% and the futures price at 104, the daily standard deviation is approximately 39 ticks (for the computation, see section 5.2). The 39 ticks are squared, which results in 1,521. In order to determine the number and timing of Delta hedges – which need to be performed daily, such that Theta is neutralized – the following equation is used: √(1,521/number of trades) = × 117


For example: how many ticks (x) are required as the hedge distance from the neutral starting point of 104? √(1,521/10) = 12.33

The result of 10 hedges per 12 ticks means simply that the Bund future (starting at 104) has to be hedged for the first time at 104.12, and then again at 104.24, and so forth. (A decline from 104.24 to 104.12 is possible as well.) If this procedure is repeated 10 times, the impact of the daily Theta will be offset. Should the daily Theta be achieved through a single hedge, the computation shows that the market should move 39 ticks before the position is neutralised (for example, by increasing the futures to 104.39). It is irrelevant whether the options were bought or sold, so that either Gamma long or Gamma short positions could be in place. A further example will illustrate the hedge. Assume that the options portfolio is Delta neutral at a futures price of 104. The Gamma is 100 futures per 100 ticks movement – that is, only 10 futures per 10 ticks increase or decrease (from 104 to 104.10 or 103.90). Assume also that implied volatility trades currently at 4.55%, which is equivalent to a daily standard deviation of 30 ticks; the square is 900. If it is necessary to determine how many hedges are required, given the volatility level, the above equation is applied; and shows that either a single hedge of

40 35

h Ticks/Hedge (6%)

Ticks Ticks

30 25 20

Ticks/Hedge (5%) h

15 10 5 1

2

5 Number of Hedges

10

Number of hedges

Figure 5.8 Volatility and hedging frequency 11 8

25

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Table 5.1 Volatility and hedging frequency Number of hedges

1

2

5

10

25

Ticks/hedges (5% volatility)

33

23

15

10

7

Ticks/hedges (6% volatility)

39

28

17

12

8

30 ticks, or 9 hedges per 10 ticks (302 • 1 = 900; 102 • 9 = 900), would be necessary. The intention of short Gamma positions lies in the disciplined, close adjustment of the Delta, whereas Gamma long positioning typically aims to profit from significant market moves. Since compensation is obtained via the gained or lost time value, the goal of sold option positions (short Gamma) is to minimize hedge losses; whereas long Gamma positions seek to maximize profits from hedge results, which can be achieved in volatile markets through just one or two hedge operations using wider hedge distances. Figure 5.8 and Table 5.1 show the relationship between the range width, which lies between the pairs of individual hedges, and their corresponding frequency.

5.5 OPTIONS MARKETS AND ECONOMIC DATA RELEASES Volatility is typically higher on days when important economic data are released than on ‘normal’ trading days. An analysis of recent years shows the development described below. The study describes the impact on US T-Bond futures, with the numbers representing the average annualized volatility of the market in per cent (Table 5.2). Table 5.2 shows that unemployment data figures have been of great importance for T-Bonds. This indicator has also been at the top of the list in past studies, considering time periods before August 1995. Over the same time period, the relevance of CPI and PPI figures for the standard deviation of the T-Bonds decreased. This could be explained by the ‘New Economy’ phenomenon, which has led to a low-inflation environment. The average annual volatility can be converted into daily standard deviations using the following relationship: V / √252 • FP • 32/100 = σ

119


Table 5.2 Economic data in USA Published data1

08/1995–08/1998 (%)

Unemployment

10.5

FOMC meetings

6.4

CPI

6.3

PPI

4.6

Retail sales

6.4

ECI

7.8

Real GDP

4.6

Other days (w/o data)

4.3

where: V = annualized volatility FP = futures price σ = standard deviation in ticks with tick size = 1/32.

For example: consider a 10% annualized volatility (employment volatility), with a futures price of 97: 10 / 15.87 • 97 • 32/100 = +/–20/32 ticks per day

Given a volatility of 4% (on other trading days), the daily high or low spread would only be +/–8/32 ticks per day. The question is whether this type of analysis, as applied to the US market above, is also applicable in the German market. The regression analysis (see Figure 5.9) between the yield of 10-year T-Notes and corresponding 10-year Bunds shows a relatively high correlation (r=0.85, r2=0.73) for the time period covering the last 2.5 years (weekly observations). This relationship declined to 0.70 when weekly yield differences were tested for correlation. A high correlation is also existent using daily observations, including days with and without economic data releases. This implies that the release of US economic indicators has a considerable impact on price movements of Bund futures. At the same time, no significant relationship could be established between the implied volatilities of the two futures contracts. 12 0

Figure 5.9 Regression between 10-year Treasuries and Bunds Source: Bloomberg


The following conclusions can be drawn: ■ On days when important economic data releases take place, Gamma

short positions can potentially cause significant losses, due to costly hedging operations required when large market moves take place ■ An options portfolio with a Euro Vega position, which is mainly deter-

mined by long dated expiries, is largely unaffected by economic data releases.

5.6 Yield enhancement via premium income The analysis in this section aims to provide evidence for yield enhancement in portfolio management via the sale of calls and puts on the Bund future (strangle position). Standardized options with 3 weeks to expiry and a Delta of +/–0.25 (out-of-the-money options) are chosen in the following example. The profit and loss was computed at expiry and summed (Figure 5.10). The time horizon of the analysis was 16 months.

Cumulative trading profit Call profit

Per cent

Total profit

Put profit

– – – – –

Date

Figure 5.10 Cumulative profit and loss (January 1999–June 2000) by shorting strangles Source: Lehman 12 2

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Despite significant losses, realized by the short put options (bear market 1999), an overall gain could be achieved. Legal issues, which can restrict the use of short option positions in the fund management industry, are not discussed here. The same analysis was performed using straddles (short the call and put at-the-money; Delta approx. 0.50). In this case, the cumulative profit was 322 ticks (3.22 Euro = 3.22%) over the same time period. The information ratio (the annualized average of the overperformance versus the benchmark divided by the annualized standard deviation of the overperformance) was higher than 0.5 for both strategies (straddles 0.70; strangles 0.65), which is considered an enrichment.

Note 1.

Abbreviations in Table 5.2 are: FOMC = Federal Open Market Committee; CPI = Consumer Price Index; PPI = Producer Price Index; ECI = Employment Cost Index; GDP = Gross Domestic Product; w/o = without.

123

CCHAPTER H A P T E R 61

Credit management

6.1 INTRODUCTION In general, credits (all non-government bonds) offer investors a yield pick-up and additional diversification potential, and this has resulted in a sharp increase in demand for higher-yielding corporate bonds relative to government bonds. Companies, on the other hand, consider the issuance of bonds as an alternative financing source, which gives them more financial flexibility. Banks try to reduce their exposure to corporate loans because they are not willing to carry the credit risk any more; also, the margins in syndicated loans have been diminishing over the last couple of years. This chapter describes the investment process for corporate bonds, which supports the decision making of portfolio managers. A variety of analysis tools will be introduced, and the classical definition of investment grade (AAA–BBB) and high yield (BB–B) will be abandoned throughout the text where possible, because, for example, low investment grade bonds (BBB–) show similarities with better rated non-investment grade bonds (BB+). In this context, it should be mentioned that ratings are in many cases a slightly lagging indicator for a company’s credit quality (spreads are more meaningful). Swap spreads are applied to the analysis of strong investment grade credits (AAA–AA). One comprehensive investment approach for the rating classes single-A through to single-B will be discussed throughout the text. 12 4

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CREDIT MANAGEMENT

In a top-down approach we start with the assessment of the global situation for credits. Macroeconomic variables like GDP growth, central bank policies as well as the level, trend and migration velocity of default rates and market drivers like swap spreads, the correlation with equity markets and equity volatilities, and the shape of the yield curve will together determine a strategy of overweighting or underweighting corporate bonds in general. In the next step the same criteria will be used to make an allocation between the different sectors and rating classes. The industry selection incorporates both top-down and bottom-up approaches, because macroeconomic as well as quantitative and qualitative factors can have a large impact on the selection process. Finally, the company selection is based to a large extent on the bottom-up approach. The fundamental analysis incorporates information from the cash flow and income statement, the balance sheet and financial ratio analysis (coverage, leverage, liquidity, profitability), the business strategy, the competitive environment, and the management quality. In order to achieve a better investment performance, the fundamental analysis on the company level has to be accompanied by relative-value analysis, by comparing a company’s credit spreads with those of its peer group. Any corporate bond team has to divide its human resources between top-down and bottom-up. The competence of the various specialists and the fact that performance can be measured against a corporate benchmark, a government benchmark or a mixed benchmark (for example, 60% Governments, 20% Corporates and 20% Jumbopfandbriefe) will determine the appropriate mix. If the performance is measured against a government benchmark, the top-down approach will be given more importance because in the first step a global picture for credits (economic environment and so on) must be assessed. It can be stated that even corporate bonds with weak financial ratios and balance sheets (relative to their peer group) could dramatically add to the performance of a portfolio if, for example, the economy is in a growth period (positive credit environment). A fund with a corporate benchmark should allocate a large part of the fixed-income department to the industry and company selection (bottom-up). On the other hand, one has to consider that the mix of a few corporate bonds in a fund with a government benchmark can lead to a high over- or underperformance versus the competitors, whereas the chances for an underperformance remain significant because of nonexisting diversification. Instead, corporate bond funds should be considered as a suitable mix for funds with a government benchmark. The accompanying risk, of the fund performance versus the benchmark, has to be evaluated by the fund management. 125


A combination of the quantitative (top-down) and qualitative (bottomup) approach will optimize the investment decision and increase the performance of any fixed-income portfolio at the same time.

6.2 CHARACTERISTICS OF CORPORATE BONDS The following paragraphs will describe some general trends and relationships in the corporate bond markets.

6.2.1 Migration of corporate ratings Corporate bond ratings are not static but rather tend to move over time. The lower the rating class, the more likely it is that a rating will change over a 1-year period (see Table 6.1). Corporate ratings will usually be assigned an outlook (such as credit watch positive, positive outlook, stable outlook, developing, negative outlook or credit watch negative). It is important to notice that the rating agencies Moody’s and S&P will change the rating of a corporate issuer with a stable outlook immediately when an event occurs that justifies the rating action. The rating migration between lower investment grade (BBB–) and non-investment grade (BB+) is very important for the quality of the whole credit market. In times of deteriorating quality in the credit market, the number of ‘Fallen Angels’ (companies that lost their investment grade status) will outweigh

Table 6.1 Corporate average rating transition matrix, in per cent: 1980–99 In %

Rating Aaa to:

Aa

A

Baa

Ba

B Caa–C Default WR (rating withdrawn)

Rating from:

Aaa Aa

85.9 0.9

9.8 84.9

0.5 9.6

0 0.4

0 0.2

0 0

0 0

0 0

3.8 4.0

A

0.1

2.2

86.2

6.1

0.8

0.2

0

0

4.4

Baa

0.1

0.4

6.0

79.2

6.5

1.3

0.1

0.2

6.3

Ba

0

0

0.5

4.0

76.8

7.9

0.5

1.4

8.9

B

0

0

0.2

0.5

5.9

76.1

2.7

6.6

8.0

Caa–C 0

0

0

1.0

2.8

5.4

56.7 25.4

8.7

Source: Moody’s

12 6

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CREDIT MANAGEMENT

Table 6.2 The likelihood for two consistent rating changes for US and European companies: 1983–November 2000 First rating action

Probability for the second rating action Upgrade % Downgrade %

Upgrade

64.10

35.90

Downgrade

29.10

70.90

Source: Moody’s and Goldman & Sachs

the number of ‘Rising Stars’ (companies that were upgraded to investment grade status). It is worthwhile mentioning that the probability for two consecutive downgrades exceeds the probability for two consecutive upgrades (Table 6.2). The ratio downgrades/upgrades for all rated companies by Moody’s and S&P is a good estimate for the general trend in the credit market (see Table 6.3), but an even better indicator for the direction of near-term credit trends is the proportion of ratings for a ‘potential’ downgrade or upgrade. At times of an increasing downgrades/upgrades ratio, the lower rating classes especially will experience a widening of the credit spreads. The condition of the overall credit market is also determined by the aggregate leverage on the company level. The aggregate leverage of companies reached its most recent bottom at the end of 1997. Since then companies have been using leverage as a means of increasing their earnings and maximizing shareholder value, while at the same time neglecting bondholders’ needs. Equity share buy-back programmes counted as a main reason for a downgrade by Moody’s in 2000. Mergers & Acquisitions (M&A)-related

Table 6.3 Downgrades and upgrades of corporate issuers (global) 1997–2000 Moody’s

Standard & Poor’s

1997

1998

1999

2000

1997

1998

1999

2000

Downgrades

116

227

317

325

84

168

416

442

Upgrades

168

153

137

99

118

109

119

134

0.69×

1.48×

2.31×

3.28×

0.71×

1.54×

3.50×

3.30×

Ratio

Source: Moody’s and S&P

127


Table 6.4 Cumulative average default probability, over 2 years, 5 years and 8 years, by rating class, in per cent: 1983–99 (global) Rating/ defaults

Cum. average 2 years %



Aa1

0.00

0.23

0.39

Aa2

0.00

0.45

0.79

Aa3

0.10

0.41

0.55

A1

0.03

0.66

0.97

A2

0.03

0.60

1.24

A3

0.13

0.40

0.88

Baa1

0.26

1.28

2.27

Baa2

0.33

1.80

2.93

Baa3

0.81

2.84

5.66

Ba1

2.13

8.49

13.67

Ba2

2.58

9.16

12.76

Ba3

6.96

20.98

32.61

B1

9.29

24.48

38.35

B2

13.95

28.45

34.39

B3

20.71

37.54

47.84

Caa1–C

28.37

43.37

51.33

Source: Moody’s

debt-financing and post-acquisition integration problems were the next major causes for corporate downgrades. B3 and worse-rated credits made up 34% of downgrades by Moody’s between 1998 and 2000. Table 6.4 shows the cumulated default probabilities for US corporate issuers. The default probability is increasing progressively with a decreasing rating class. Figure 6.1 shows the average path to default for investment grade and high-yield credits. A critical barrier is the rating category between low investment grade (Baa3/BBB–) and high non-investment grade (Ba1/BB+) categories. The deterioration of credit quality usually starts long (12–24 months) before the actual default occurs. However there will always be companies that will run into financial distress during a much shorter timeframe because of unpredictable event risks. For some examples, see Table 6.5. 12 8

Figure 6.1 Investment grade and high-yield average path to default: August 2001 Source: Salomon Smith Barney


Table 6.5 Selected rating changes: October 2001 Industry

Company

Rating (date)

Transportation

Railtrack Group Plc

A (08.10.01)

CC (09.10.01)

Airlines

Swissair Group

A3 (28.03.01)

Ca (02.10.01)

Technology Services

Comdisco Inc.

BBB+ (01.10.00)

D (16.07.01)

Source: Bloomberg

High-yield issuers have less room for rating downgrades (average rating single-B). They can experience a severe deterioration in credit quality in short periods.

6.2.2 Migration of corporate bond prices Another way of showing the predominant risks in lower rated (BBB–B) corporate bonds is to observe the price migration over a 1-year horizon after the bond price fell below 90% of par for the first time. Table 6.6 shows a study done by CSFB for 1247 corporate bonds (global) between 1993–2000.

Table 6.6 Changes in bond prices over a 12-month horizon % of par after the price dropped the first time below 90% in %

90–80%

80–70%

70–60%

60–50%

50%–

12.83

3.47

2.78

0.00

8.30

28.38

13.19

11.11

6.67

0.00

80–90%

22.22

24.31

27.78

20.00

8.30

70–80%

13.13

18.06

2.78

6.67

0.00

60–70%

7.58

11.81

22.22

26.67

8.30

50–60%

5.45

9.72

8.33

13.33

0.00

10.40

19.44

25.00

26.67

75.00

Defaults in %

8.48

20.83

27.78

26.67

58.33

Number of issues

1247

% of par 100+% 1 year later 90–100%

50–%

Source: CSFB

13 0

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Table 6.6 shows the downside risk faced by every bond investor. In fact, there is a high probability for a bond price to decline further once it has fallen below a certain threshold (for example, 80% of par). This probability increases further in a very volatile environment when the risk appetite for credit products is declining.

6.2.3 Correlation with other asset classes The total returns of non-investment grade bonds show the highest correlation with small cap stocks, but it cannot be said that the relationship is very strong. A similar but slightly weaker correlation exists between noninvestment grade bonds and investment grade bonds as well as large cap stocks. In contrast to investment grade bonds, the non-investment grade bonds show a low correlation with Treasuries (see Table 6.7). Table 6.8 shows the total return correlation between various rating classes. Single-B rated bonds show a high correlation with double-B rated bonds, which have an even higher correlation with triple-B, and single-A rated bonds. Triple-Bs have a high correlation with the whole investmentgrade rating spectrum. The strong relationship between double-B (noninvestment grade) and triple-B rated (investment grade) corporate bonds is presented in Figure 6.2, which is based on Merrill Lynch index data for triple-B and double-B rated US companies. This brings us to the main topic of this chapter where the two investment approaches (top-down and bottom-up) will be applied simultaneously in order to achieve the best investment decision for lower rated investment grade bonds (A–BBB) and non-investment grade (BB–B) bonds. Especially during times of deteriorating credit quality in the bond markets, a lot of the triple-B bonds behave more like double-B bonds and a clear distinction between the investment grade status and noninvestment grade status cannot be drawn. It is not very uncommon for single-A rated companies that face a negative trend in their respective industries to reach credit spread levels associated with triple-B or double-B rating categories. The European investment grade telecommunications sector is a good example for deteriorating average industry ratings. In May 2000, the telecommunications industry was a low double-A industry: a couple of months later the average rating was low single-A/high triple-B, and the credit spreads were already trading on triple-B levels of comparable companies. Other sectors, like steel or textiles, can experience such severe deterioration of their industry 131

Table 6.7 Total return correlation between various asset classes: 1985–2000 Non-investment grade1

Mortgagebacked2

10-year Treasuries

3-month Treasuries

Large cap stocks3

Small cap stocks4

Non-investment grade

1.000

Mortgage-backed

0.419

1.000

10-year Treasuries

0.340

0.872

1.000

3-month Treasuries

0.009

0.364

0.324

1.000

Large cap stocks

0.508

0.268

0.286

0.017

1.000

Small cap stocks

0.571

0.112

0.100

–0.084

0.769

1.000

High-grade corporates

0.529

0.900

0.934

0.299

0.373

0.217

1. 2. 3. 4. 5.

Merrill Lynch High Yield Master Index (BB–CCC) Merrill Lynch Mortgage-Backed Master Index S&P Index of 500 Common Stocks Russell 2000 Index Merrill Lynch High Grade Corporate Index

Source: Merrill Lynch

High-grade corporates5

1.000

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Table 6.8 Total return correlation between all rating classes: 1989–2000 10-year AAA Treasuries 10-year Treasuries

AA

A

BBB

BB

B

CCC/CC/C

1

AAA

0.967

1.000

AA

0.957

0.990 1.000

A

0.940

0.983 0.992 1.000

BBB

0.890

0.949 0.962 0.979 1.000

BB

0.581

0.675 0.704 0.747 0.817 1.000

B

0.119

0.244 0.266 0.325 0.420 0.737 1.000

−0.073

0.029 0.060 0.119 0.210 0.549 0.838

CCC/CC/C

1.000

Source: Merrill Lynch

fundamentals that they fall from investment grade to non-investment grade. On the other hand, there is always a migration from non-investment grade to investment grade, and the strong non-investment grade (BB+)

5.00%

Monthly Returns Corp BBB

4.00%

y = 0.9131x + 0.0007 R 2 = 0.6832

3.00% 2.00% 1.00%

-3.00%

-2.00%

-1.00%

0.00% 0.00%

1.00%

2.00%

3.00%

4.00%

-1.00% -2.00% -3.00% Monthly Returns Corp BB

Figure 6.2 Total return correlation between triple-B rated (Merrill Lynch C0A4 Index; US Corporates) and double-B rated (Merrill Lynch H0A1 Index; US High Yield) corporate bonds: December 1996–January 2001 Source: Union Investment 133


issuers already trade at triple-B levels long before they actually get the upgrade to investment grade status. The long-term advantages of an investment grade status (access to money markets, lower financing costs and higher financial flexibility) are an incentive for a lot of non-investment grade companies to improve their credit ratings. One trend, which can be monitored over the last years, is that an increasing number of companies consider a rating between single-A and triple-B as the appropriate target. Industrial companies especially have little incentive to maintain a triple-A or double-A rating, and their managements accept a lower rating (A–BBB) in exchange for more balance sheet flexibility.

6.2.4 Rating classes and spreads The yield differential between triple-A rated bonds (for example, government bonds and supranationals to a lesser extent) merely indicates a liquidity premium, because both bonds should have ‘no’ default risk. The spread between a government bond (AAA) and lower rated corporate bonds is defined as a credit spread, and this expresses the uncertainty regarding default or rating migration. The increasing default risk, evaluated by a decreasing rating class, leads to wider spread levels.

Spread vs Governments in basis points

Various Rating Classes and average Spreads August 2001 900 787

800 700 600 500

413

400 300 185

200 100

105 47

56

AAA

AA

0 A

BBB

BB

Figure 6.3 Relationship between spread and rating class (global): August 2001 Source: Bloomberg and Union Investment 13 4

B

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The relationship between average spreads and ratings is roughly linear within the investment grade category, and average spread levels increase exponentially for the non-investment grade rating classes (Figure 6.3). There is a relationship between spreads and time to maturity (duration) for every rating class and issuer (higher uncertainty, higher cumulative default probability). Spread levels increase as duration increases. A financial distress scenario is an exception to this relationship, because the issuer’s credit curve becomes inverted. Figure 6.4 shows an isolated credit curve of an issuer at a point in time. There is also a relationship between the shape of the yield curve (see section 6.5.3) and the shape of the credit spread curve. Given a steeper yield curve, a lot of new issuance will be concentrated in the shorter maturities: because of the supply and demand dynamics the credit spread curve will flatten, particularly at the short end of the credit spectrum. The credit spread curve for corporates with 5-year maturities and more depends mainly on company-specific risk. Credit default swaps (CDS) with 5-year maturities are more liquid than the corresponding bonds and react faster to credit events. During periods of higher market risk CDS will drive the market up. The 5-year bonds will follow the CDS, moving the credit spread curve in 5 years upwards. This leads to a flattening of the credit spread curve between 5 years and 10 years.

FIAT (Euro) Spa (Euro) Creditcurve Curve 07.10.2001 FIAT Spa credit 07.10. 2001

SwapSwap spreads in basis points Spreads in basis points

225 200

FIAT 11 FIAT 10

175 150

FIAT 05 FIAT 06

125 FIAT 04 FIAT 03

100 75 50

FIAT 02

25 0 0

1

2

3

4

5

6

7

8

Modified Durationin in years years Modified duration

Figure 6.4 Credit curve of Fiat Spa: October 2001 Source: Bloomberg and Union Investment 135


6.3 THE TOP-DOWN APPROACH The investment process for corporate bonds begins with a top-down approach (see Figure 6.5). Strategic asset allocation defines the objective and orientation of portfolios (benchmark determination). In contrast, tactical asset allocation addresses how objectives will be achieved (outperformance of the benchmark). Accordingly, first a global approach for credits is developed: credits are over- or under-weighted depending upon their respective benchmark determinations. In the next step, the allocation between investment grade bonds (AAA–BBB) and noninvestment grade bonds (BB–B) is determined under the consideration of macroeconomic variables like GDP growth and the central bank (FED and ECB) policy cycles. The direction of swap spreads, and market

TOP-DOWN approach Strategic asset allocation Benchmark selection

Tactical asset allocation Global picture for credits (overweight/underweight) Selection of the sector/rating class (AAA/AA/A/BBB/BB/B)

Single company selection BOTTOM-UP approach

Figure 6.5 Top-down and bottom-up approaches for lower investment grade and non-investment grade credits Source: Union Investment 13 6

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drivers such as implied equity market volatility and the shape of the yield curve, are also important for the spread movement. The industry selection is based on the top-down approach, but decision variables associated with the bottom-up approach have an effect on the industry selection as well. The selection of companies with lower rating classes (A–B) is mainly based on the bottom-up approach, because firm-specific risks (financial and business risk) are very good explanatory variables for the movement of their credit spreads.

6.3.1 Global situation for credits Factors to consider include: ■ economic outlook and the monetary policy (central bank positions) of

FED and ECB ■ technical factors, such as client flows and issue activity (supply) ■ default rates of credits, according to rating classes (Moody‘s) ■ spread differentials, between AA–BBB, BBB–BB, BB–B and all rating

classes versus the market index ■ rating change expectations, based on the history of upgrades and

downgrades ■ repurchase programmes or reductions in government offerings with

respect to government bond issues ■ event risks like M&A transactions, share buy-back programmes,

extreme profit warnings or litigation risks ■ leveraging/deleveraging of balance sheets (rather deleveraging in

recessions). The quantitative drivers for credit spread performance are: 1. the shape of the yield curve 2. the equity market volatility 3. the equity market performance 4. swap spreads (until July 2001) 5. the ratio of Governments to Corporates. 137


6.3.2 Macroeconomic factors and spread movements The following macroeconomic factors have a significant impact on the direction of credit spreads. Central bank policy

The central bank (mainly FED and ECB) policy appears to be a dominant factor for corporate credit spreads. The level and direction of short-term interest rates usually indicate the central bank’s relative tightness or looseness of monetary policy, which will definitely have an impact on economic performance, financing costs and credit spreads. Triple-B spreads usually reach their peak when the FED has done approximately two-thirds of the interest-rate cuts. Figure 6.6 shows that this relationship holds for the period 1977–2001. Economic growth

The economic cycle is an important factor in the performance of the bond market. Empirical studies have shown that corporate credit spreads in particular demonstrate a negative correlation with GDP growth in their particular countries. A slower economic growth was consistent with wider credit spreads, and vice versa (Figure 6.7). The described relationship was true for a long period of time, but appears to have become somewhat weaker recently than in the past. It can be concluded that after the events in 1998 (Russia’s default and Long-term Capital Management), a structural change occurred in the credit markets. Despite the strong economic growth in the US, higher default rates can be observed and the credit spreads tend to widen because of leveraging and less financial flexibility. Recession is not always bad for corporate bonds because it imposes balance sheet discipline on over-levered companies once debt and equity markets restrict the supply of new capital (for example European Telecoms in 2001). Figure 6.8 shows the strong long-term relationship between US GDP growth and credit quality of the US automobile industry. Credit ratings of the ‘Big 3’ US car manufacturers (Ford, GM and Chrysler) are heavily dependent on the US economy, so GDP growth is a good leading indicator for a rating trend in the automobile sector (high cyclicality). During periods of economic growth, consumer confidence rises and car manufacturers will improve their profitability by increasing car sales numbers: this in turn leads to better credit profiles and tighter spread levels (often already priced in forward). 13 8

Baa Spread (bps, LHS)

420

Fed Funds (%, RHS)

25

380 20 340 300

15

260 10

220 180

5 140 100 1977

0 1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

Figure 6.6 Baa-spread vs. Fed funds: monthly, February 1977–June 2001 Source: Deutsche Bank

2001

Figure 6.7 Baa-Corporate bond spread vs. Treasuries Source: Moody’s

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Figure 6.8 Relationship between US GDP growth and credit ratings of the ‘Big 3’ US car manufacturers: April 1982–June 2001 Source: Union Investment

Figure 6.9 Range of industrial spreads in the US: 1960–99 Source: JP Morgan 141


To help make this clear, the behaviour of industrial spreads during different economic cycles is shown in Figure 6.9, which shows the last six economic cycles, from 1960 – the exception was an Aaa low at 10 basis points, which is considered an outlier. The spread levels in November 1998 were wider than in typical recession periods and show the effects of drastic deleveraging of hedge funds in the summer of 1998. The credit curve was also flatter than in previous weak economic cycles, which points more to a liquidity fear than to a fear of recession.

NAPM Index

The National Association Purchasing Management index (NAPM) compares the changes in various market areas on a month-to-month basis. (The breakdown is: 30% new orders; 25% production; 15% delivery; 10% inventories; 20% employment.) The trend in the NAPM index can serve as another means of explaining the relationship between economic growth and credit spreads. Generally, a decrease in the NAPM index is consistent with widening credit spreads (Figure 6.10). A strong relationship could be observed during the economic crisis in the 1970s and 1980s. This relationship broke during the period 1995–97. Credit spreads were flat, although the NAPM index clearly decreased to a new low around December 1995, and reached its preliminary peak around July 1997. This divergence from the close relationship during the 1970s and 1980s can be explained by an overall better credit quality (relatively more double-B credits) in the US non-investment grade market. One good way to measure aggregate credit quality is by looking at the amount of outstanding corporate issues trading over 1000 basis points over the comparable Treasury. Additional financing is almost impossible at those levels, which leads to sharply increasing default rates. The overall quality in the credit markets deteriorates with an increasing number of corporate bonds trading at the described levels. Generally speaking, the spreads in the corporate bond market reached clearly higher levels after August 1998, and the levels in November 2000 were only seen during the crisis in 1990–91, despite a better macroeconomic environment than during the Gulf War. Two reasons can be given for the higher risk premium: ■ A higher risk premium is required because default rates show an

increasing trend, implying even higher expected future default rates. At the same time, recovery rates are decreasing 14 2

Figure 6.10 US high-yield (BB–CCC) spread over Treasuries vs. NAPM PMI: January 1987–September 2001 Source: Merrill Lynch


■ The liquidity in the bond markets decreased dramatically after August

1998 because brokers reduced their positions in lower rated corporate bonds because of a much higher volatility. This policy resulted in a wider bid-offer spread (higher premium for increasing illiquidity).

Default rates

The following paragraphs refer mainly to the high-yield segment. Nevertheless we have to be aware of the fact that companies with an investment grade rating (for example, A/BBB) might face a default scenario due to fast-deteriorating credit fundamentals (see section 6.2.1). The level of default rates is a good indicator for the state of the credit market. The default risk premium should rise or fall in accordance with the prevailing probability of default. More importantly, the expected default rates are important for the future development of the credit market. The expectations about future default rates depend on several factors: ■ GDP growth, because slower economic growth is accompanied by

lower corporate earnings expectations, so fewer companies will generate necessary cash flows to serve their debt obligations ■ The distribution of all outstanding issues on the rating spectrum. The

default probability increases with proportionally more lower rated (B– and worse) issues than better rated issues (BBB–BB) in the corporate bond market ■ The age effect of bonds – the likelihood of payment default by a bond

issuer can change with decreasing time to maturity. At the time of issuance, the new issuer has abundant cash, and payment default is easily avoided. In addition it is quite common to put some restricted cash from the issuance into an escrow account to cover the first three or four coupon payments. Over time the ‘hazard rate’ grows, as cash reserves gained through the bond issue are used. The critical period is reached when the success of the firm is least certain and the hazard rate is at a maximum. However, after corporate plans have been successfully implemented and sufficient profit has been generated to offset debt, the critical phase has passed and the likelihood of payment default declines rapidly. Several empirical studies show that the critical time after first issues is approximately 3 years after issuance for single-B rated bonds, and climbs to approximately 4 years for double-B rated bonds

14 4

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■ Financing alternatives during an economic downturn can have severe

implications on the credit market. Deteriorating credit statistics in the credit market will force the banks to tighten their lending policies (more security and higher risk premiums) resulting in less available liquidity for the corporate sector. The lower rated companies (for example, BBB–B) are the first to be affected. Lending policies of commercial banks can be an indicator for future default rates in the corporate sector. Over the last 30 years the default rates in the US crossed the 9% mark only three times (see Figure 6.11). The comparison of annual default rates and annual total returns is of special interest, because years with high default rates also experience the highest total returns. During the years 1991 and 1995, the default rates peaked at 10.3% and 3.8% respectively; and during the same periods the total returns reached their maximum highs, at 39.2% and 20.5% respectively. A strong negative correlation between default rates and economic growth could always be expected, but during the economic crisis in the US in the 1970s and 1980s a sharp increase in the default rates was not observed. It was the economic slowdown in 1990 that was accompanied by extremely increasing default rates. The same acceleration of default rates occurred in 2001, due to a prolonged downturn of the economy, high political uncertainty after 11 September 2001, and aggregate weak credit fundamentals (for example, in October 2001 approximately 60% of the European high-yield market traded on distressed levels). Structural changes in the credit markets can explain the weaker than expected relationship between the default rates and the economic cycle. This leads us to the conclusion that the relationship between default rates and the spread level in the corporate bond market is a very complex one, and that a good performance in the corporate bond market (tightening of credit spreads) can be consistent with increasing default rates at times when the market anticipates future decreasing default rates. The change in default rates lags behind the change in credit spreads. History shows that most rallies in the corporate bond market happened in times of increasing default rates. This means that credit spreads begin to tighten some time before the default rates reach their peak and begin to fall. The crisis period in 1989–90 serves as a good example. Moody’s default rates reached their peak of 11.57% in June 1991, while credit spreads in the non-investment grade market had already tightened almost 600 basis points at the same time. The highest correlation between

145

1970–1999 US HY Annual Default Rate

Times of occurrence

8 Mean: 3.6% Median: 3.4% Standard deviation: 2.6%

7 6

1991 Default Rate: 10.3%

5 4

1970: 9.4% 1990: 9.8%

3 2 1 0 1% and 2% and3% and4% and5% and6% and7% and8% and9% and10%

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credit spreads and Moody’s default rate exists with a lag of 5–6 months. Two arguments can explain this time lag: ■ The decrease of short-term interest rates has a lagging effect on

economic growth, which determines the trend in default rates ■ There is a positive effect of low short-term interest rates on non-

investment grade bond mutual fund flows, because investors can react fairly fast to an interest rate change (a cut). The spreads of lower rated corporate bonds will tighten if there is a reallocation of capital towards their asset class. The recovery rate is another important figure when evaluating corporate bond spreads. This is the amount the bondholders will get after a company defaults and its debt is restructured. The recovery in the noninvestment grade market tends to decrease over the last years (from 40% to 30%). Depending on the ranking in the capital structure (senior secured, senior unsecured, junior) and the industry (telecom companies, for example, point to very low recovery values), the recovery rate can vary between 25% and 50%.

6.4 MERTON’S APPROACH TO EVALUATING A CORPORATE BOND Credits can easily be described by using terms of the options analysis: corporate and credit spread issues can be defined as a long position in a risk-free asset (a bond) and the sale (a short position) of a put option on the assets of the issuer. The strike price equals the nominal value of the issuer’s liabilities. In the case where the value of the corporate assets falls below the value of its debt, the corporate becomes bankrupt. If the leverage of a company increases (for example due to share buy-back programmes, or new issuance of bonds), the strike price of the option increases automatically, meaning that the sold option moves closer ‘atthe-money’. For high-yielding bonds (high-yield debt), the strike price – the value of indebtedness – is normally closer to at-the-money than in the case of investment grade bonds (Figure 6.12). The rating, or rather the spread, should simultaneously determine how far a company is from the strike price. Assuming average spread levels, a single-A or better rating implies that the put option is far out-of-the-money, whereas a double-B or singleB rating indicates that it is slightly out-of-the-money. 147


Debt in-the-money

B

BB

BBB

A

AA

$100m out-of-the-money

$100m

$500m

Assets

Strike price

Figure 6.12 Corporate bonds as a ‘short put option’ Source: Union Investment and Robert C. Merton

Lower rated bonds and bonds with higher spreads especially behave more like equities in falling equity markets (a crisis scenario), and their correlation with government bonds increases during moderately increasing equity markets. There is a positive relationship between credit spreads and implied equity volatilities, because falling equity markets are compatible with increasing volatilities. A strong increase in equity volatility could result in a steepening of the credit curve (the yield differential between highand low-rated corporate bonds). Generally, a credit investor, in contrast with the equity investor, sells volatility. The equity price is a good proxy for the value of the assets of a company. The following definition establishes a relationship between the equity price of a company and its leverage: Leverage ratio = total debt / (total debt + market capitalization) = TD/(TD + MCap) Market capitalization = (number of common shares • equity price)

If the leverage ratio is around 40%, the credit spreads show a moderate sensitivity to the movement in equity prices. If levels of 40–50% are reached, this sensitivity tends to increase. A general rule is that once the 14 8

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leverage ratio TD/(TD + MCap) has exceeded approximately 60–70%, the default probability of an issuer increases progressively. It should be noted that this rule does not hold for banks and financial institutions. Banks usually maintain a leverage ratio of over 80%, while their credit spreads correspond with spreads of industrial companies that have leverage ratios of around 30%.

6.4.1 Some company examples Viatel, an alternative pan-European telecommunications company, lost over 90% of its equity value during the downturn in equity markets in 2000, and provides an example (Figure 6.13). Viatel’s bond price remained unaffected by the negative equities trend, up to a price of about EUR18–20. After the equity price continued its fall, the credit spreads began a massive widening, due to the increased credit risk (the deterioration of the leverage ratio, TD/(TD + MCap). In Viatel’s case the trend in the equity price and spread level was a very good indicator of the company’s severe business situation. Finally, Viatel defaulted on its May 2001 coupon payment, and filed for Chapter 11 (bankruptcy protection).

Figure 6.13 Viatel 12.5% 08 $ spread vs. TD/(TD + MCap) Source: Union Investment 149


Spread vs. T 7,25% 05/15/04

At a higher ratio of debt to market capitalization, issuers run a greater risk of credit meltdown and the credit spreads become more sensitive to further declines in equity prices. Moreover, a rising implied equity volatility reflects the market’s expectations about probable extreme price movements, and this leads to further spread widening risk. If a crisis scenario for a company occurs, the credit spreads will behave independently of the company’s rating class. A good example is Xerox Corp., which saw its rating going down from an A-rating in July 2000 to Ba1/BB in December 2000/October 2001 (Figure 6.14). The credit spreads reached distressed levels during this period, due to a lack of visibility about Xerox’s earnings outlook. The Dutch incumbent telephone operator KPN (Baa3/BBB–) is another example, even though its financial situation should not be compared with Xerox for which at some point Chapter 11 (bankruptcy protection) was a possible outcome. The example of KPN should demonstrate that event risks like 3G licence costs could have a large impact on the credit quality (spread movement) of previously strong investment grade (AA) companies. KPN used to have an Aa1/AA rating back in May 2000, and was continuously downgraded to Baa3/BBB– with a negative outlook in October 2001. In the course of becoming a dominant pan-European

–300 –800 –1300 –1800 –2300 –2800 –3300 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

Figure 6.14 Xerox 7.15% 04 $ spread vs. TD/(TD + MCap) Source: Union Investment 15 0

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telecom operator, the company has stretched its balance sheet. The debt levels increased from EUR7.5bn at the end of 1999 to EUR24bn at the end of 2000, because of higher wireless (3G) licence costs and the acquisition of the mobile unit E-Plus in Germany. KPN’s equity price dropped from EUR72 in March 2000 to EUR11 in December 2000. The leverage ratio (TD/(TD + MCap)) did not reach critical levels, as it did at Xerox, but Figure 6.15 shows that the credit spread began a steep widening (from 70 bps to more than 320 bps over Governments) much earlier. The sharp spread widening began when the equity price reached a level of around EUR25 in September 2000 (see Figure 6.16). Despite a steady fall in the equity price – in August 2001 to below EUR5 – the spreads of KPN’s bonds were quoted at a stable level (the circled data points fall in June–August 2001), because during this period KPN was supposed to merge with the Belgium incumbent Belgacom (Aa3/AA). A successful merger was considered as credit positive for KPN. On 31 August both parties abandoned their merger talks because they could not agree on the valuation. During the first week of September KPN’s market capitalization dropped below EUR 3bn and all analysts started pointing towards a possible funding gap that KPN was facing. The company had EUR 3.5bn maturing in June 2002 and would need to cut CAPEX (capital expenditure: this increases the property value of a company, and hence the earnings power of a business) by about EUR1.4bn in order to repay the maturing notes. Moody’s downgraded KPN to Baa3, with further credit watch negative. The fear of a downgrade into high yield led to a sell-off of all KPN bonds, which began trading on a price basis. Clearly the European high-yield market is not large enough to absorb an issuer like KPN, and forced sellers could initiate a free-fall of the bonds. Fortunately, however, in the afternoon hours of 10 September a new CEO was introduced to the investor community, and at the same time it was announced that a new bank credit facility of EUR2.5bn had been secured. Furthermore KPN had by this time realized some $600m from asset sales. This series of positive news led immediately to a spread tightening (around 200 basis points) during the next few days (11–14 September). During the following weeks, KPN’s stock price increased by 50% to EUR4.5 and the bonds stabilized at a level of around 700 basis points over Treasuries. This example shows clearly that a fundamental analysis (bottom-up) of an investment grade company will bring additional value to the investment process. In Figure 6.17 KPN is used to show the relationship between a company’s credit spreads and its implied equity volatility. KPN’s implied equity volatility is a leading factor for the direction of KPN’s credit spreads. While the implied volatility increased sharply (with peer group 151

Figure 6.15 KPN 4.75% 08 EUR spread vs. TD/(TD + MCap) Source: Union Investment

Figure 6.16 KPN equity vs. KPN debt: April 2000–October 2001 Source: Union Investment

Figure 6.17 KPN spread and call implied volatility: July 2001–November 2001 Source: Union Investment

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volatility around 60%) beginning of September 2001, the spreads widened to extreme levels and began to tighten again as the implied volatility came from the highs at the end of October 2001.

6.4.2 Credit spread term structure The implications from the Merton model can be applied to the credit spread term structure (Figure 6.18). The credit spread curve should be upward sloping with increasing maturity, but Figure 6.18 shows that a situation with an inverted credit spread curve can occur with increasing leverage or decreasing rating class. ■ Low-leveraged firm: Most credits exhibit an upward-sloping credit curve,

because the quality of the credit is expected to remain constant over the short term. However, the future credit quality becomes less certain, and leads to increased credit spreads in order to compensate the investor for the increased probability of a deteriorating credit quality ■ Medium-leveraged firm: This is a commonly seen form of the credit

curve for companies that are likely to experience a worsening in credit

Figure 6.18 Credit spread term structure in Merton model Source: Lehman Brothers 155


Table 6.9 Selected bonds in financial distress with an inverted credit curve: 12 October 2001 Industry

Company

Telecom equipment

Marconi Plc (Ba1/BB)

Telecom Technology

Bonds

Price

Spread in bps

Euro 5.625% 03/05 Euro 6.375% 03/10

36 32

4220 2420

Sonera Group (Baa2/BBB)

Euro 5.625% 05 Euro 4.625% 09

77 71

1100 600

Xerox Corp (Ba1/BB)

$7.15% 08/04 $7.2% 04/16

85 71

1120 670

Source: Union Investment

quality over the medium term, as the chance of defaulting in the very short term is extremely low. As the maturity increases, the credit spread then falls, to reflect the view that, should the credit survive in the medium term, it will be more likely to survive the long term ■ Highly leveraged firm: The inverted curve is usually associated with

credits that have experienced or will experience a significant deterioration, to the extent that a default is probable. The bonds begin to trade on a price basis. If defaulting is a very likely scenario, bonds of the same seniority trade with the same price, irrespective of their maturity and coupon. This has the effect of elevating short-maturity spreads and inverting the spread curve. Table 6.9 shows some examples. If an investor is bullish on a company, and strongly believes that it will manage a turnaround and that the credit spread curve will reverse to its normal shape, he or she should invest in the lower-priced bonds regardless of yield or spread levels. (The very high spread levels in the shorter maturities are not a good measure of risk, because the bonds trade on a price basis.) One cannot say that the shorter-maturity bonds are riskier than the long-maturity bonds, because if the short-maturity bonds default, all other bonds will default at the same time.

6.5 MARKET DRIVERS (METHODOLOGY) OF CREDIT SPREADS The following paragraphs will identify the most important market drivers for credit spreads. Particular weight will be assigned to swap spreads, and to their effect mainly on credit spreads of better rated 15 6

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(AAA–AA) bonds. Market drivers, like implied equity volatility and the shape of the yield curve, certainly have a large impact on corporate bond spreads and will be explained in detail.

6.5.1 Swap spreads Swap spreads represent a bank credit risk, which is generally in the range from a low double-A to a high single-A area (in Europe). As a determinant of swap spreads, however, credit quality is somewhat misleading. The definition of credit is the event (the risk) of default and swaps cannot default. Counterparties of swaps can, but they will be replaced. A swap spread can be defined, for example, as the spread in yield between a 10year Bund and a 10-year swap (a Bund swap spread). This spread is not static and depends on a variety of factors. Its influence on, and its correlation with, credit spreads (AAA–AA) are significant, and will be analysed further. On calculating the correlations between Pfandbriefspreads, semi-government spreads (such as Deutsche Hypothekenbank or Kreditanstalt für Wiederaufbau) and swap spreads, it becomes apparent that there exists a strong relationship: for 1997–99 the correlation ranges between 0.8 and 0.9. These groups of issuers – mortgage banks, supranational and semi-governmental financial services – have specific spread targets, and having achieved these (normally negative spread to swaps) they introduce new issues to the market. This new issuance posts a limit to further overperformance of the Pfandbrief sector, which results in relatively stable spread movements versus swaps (see Figure 6.19). The movement of swaps versus Bunds is more volatile (see Figure 6.19). A good example is the situation that occurred in August 1998 when there was a Flight-to-Quality (Bunds). During market crisis scenarios especially (until August 2001), the swap spreads tended to move in line with credit spreads. They are a good proxy, particularly for higher rated credits (AAA–AA), whose arbitrage relationship versus swaps is an important factor (see Figure 6.20). The correlation of swap and credit spreads increased after 1998 from around 40% to approximately 70%, because the increased volatility of spreads to Treasuries encouraged traders as well as investors to start hedging their credit exposures with interest-rate swaps rather than Treasuries. In September 2001 the correlation fell to levels prevailing before the crisis in 1998. It may be that the reduced fear of a future shortage of Treasuries, combined with the market’s earlier adjustment to an envisaged world ‘without’ Treasuries, led to a decoupling of the swaps and 157


05.09.01 9/5/01

7/5/01 05.07.01

05.05.01 5/5/01

05.03.01 3/5/01

05.01.01 1/5/01

05.11.00 11/5/00

9/5/00 05.09.00

05.07.00 7/5/00

05.05.00 5/5/00

05.03.00 3/5/00

05.01.00 1/5/00

05.11.99 11/5/99

05.09.99 9/5/99

05.07.99 7/5/99

05.05.99 5/5/99

3/5/99 05.03.99

05.01.99 1/5/99

05.11.98 11/5/98

05.09.98 9/5/98

05.07.98 7/5/98

05.05.98 5/5/98

05.03.98 3/5/98

05.01.98 1/5/98

05.11.97 11/5/97

05.09.97 9/5/97

Bund Swapspreads and Jumbo Pfandbriefe Swapspreads 05.09.97 - 14.09.01

20 10 0 -10 -20 -30 -40 -50 -60 -70 -80

Bund Swapsspreads in bp 7-10 year Jumbo Pfandbriefe Swapspreads in bp All Jumbo Pfandbriefe Swapspreads in bp

Figure 6.19 Pfandbrief-Swap spread and Bund-Swap spread Source: Bloomberg

credit markets, because traders and investors have returned to the Treasury market as a hedging and relative-value tool. Another explanation for the decoupling of swaps and credit markets can also be considered. After the tragic events of 11 September 2001, swap spreads did not widen as they had done in August 1998; instead we have seen rather a tightening of swap spreads. At the same time, credit spreads for the rating classes A–B widened significantly, and implied equity volatility reached higher levels (on 20 September 2001, the CBOE OEX Volatility Index was 49.04%) than during the 1990–91 recession. During this time most of the new issues came from supranationals, agencies and Pfandbriefe. Since a majority of these issuers can be categorized as floating rate borrowers, we can conclude that they will swap their new-issue proceeds, and this will have a tightening effect on the swap spread. Another reason for the good performance of swap spreads was the strong support by central banks, as opposed to August 1998 when central banks added liquidity only in stages. This time financial intermediaries could maintain balance sheet leverage because central banks immediately provided sufficient liquidity to the banking system. The strong participation by central banks in the swaps market was a clear sign that they will ring-fence the banking system’s current liabilities. Libor became basically a benchmark for the pricing of corporate credit. 15 8

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Credit Spreads vs Swap Spreads 23.08.96 - 17.08.01

140 120

AAA Spread

100 80 60

y = 0.9797x - 6.7129 R2 = 0.8878

40 20 0 0

20

40

60

80

100

120

140

160

10 year swap spread

Figure 6.20 Swap spread vs. AAA spread in US Source: CSFB

Swap spreads were dependent on the market tolerance or market aversion for risk and were therefore a good measure of liquidity. During times of high swap spreads, the market tolerance for non-investmentgrade bonds is lower than during times with decreasing swap spreads (see Figure 6.21). This is true because the widening of corporate credit spread is accelerated through a downturn in the equity market and lack of risk appetite on the buy side. A decoupling of this relationship was observed during the weeks after 11 September 2001 (see before). Which factors influence swap spreads? Above all, the reduced issuing activity of government bonds is an influential factor for swap spreads. This can easily be observed in the US, as one example. There is a strong tendency towards a decreasing supply of Treasuries and an increasing supply of spread products, thereby enforcing the impact on swap spreads (Figure 6.22). In this example the ratio of spread products to Treasuries was determined by taking into account the last four years. This shows that swap spreads widened considerably with a decline in the supply of government bonds. The decreased issuance is a result of a global reduction of budget deficits (especially in Euroland), and the budget surplus in the US due to a strong economic growth in the US in the past. Of course, there are also other relevant factors, such as implied volatilities. As regards swap spreads, the implied volatility of the fixed-income 159

Figure 6.21 Swap spreads and the spread-level of double-B rated corporate bonds Source: Union Investment and Bloomberg

04.01.99 20.01.99 04.02.99 22.02.99 09.03.99 25.03.99 12.04.99 26.04.99 14.05.99 01.06.99 17.05.99 05.07.99 21.07.99 05.08.99 23.08.99 08.09.99 24.09.99 12.10.99 28.10.99 12.11.99 30.11.99 16.12.99 03.01.00 19.01.00 04.02.00 22.02.00 09.03.00 27.03.00 12.04.00 30.04.00 16.05.00 01.06.00 19.06.00 05.07.00 21.07.00 06.08.00 24.08.00 11.09.00 27.09.00 12.10.00 30.10.00 15.11.00 01.12.00 19.12.00 05.01.01 23.01.01 08.02.01 26.02.01 14.03.01 30.03.01 17.04.01 03.05.01 21.05.01 06.06.01 22.06.01 09.07.01 25.07.01 10.08.01 28.08.01 18.09.01

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. . . . . . . . .

Figure 6.22 Relationship of spread products – left axis: ratio spread products (Corporate bonds, Agencies, ABS and MBS)/Government bonds; right axis: 10-year swap spread Source: JP Morgan

markets has also proven to be a significant factor in the past (up to 1999), which cannot be neglected, especially during volatile market periods. Increasing volatility in the bond markets – which is also a proxy for market uncertainty regarding future monetary policy and the economic situation – causes a widening in the bid-offer spreads of bond prices and leads to a general preference for liquid issues (government bonds, benchmark bonds ‘on-the-run’ issues). Volatility is a key driver of the term ‘liquidity premium’, the premium that is demanded from investors for holding riskier assets. Also, there should be less interest from risk-averse investors attempting to earn a positive carry by receiving swaps instead of buying bonds. The consequence was a widening of swap spreads. This relation was disrupted during certain time periods. While recently, in October 2000–September 2001, some financial markets turned more volatile (see Table 6.10), the bond markets did not experience any significant increase in movement intensity. The assumption that unstable equity and currency markets lead to a ‘Flight-to-Quality’ situation in the fixed-income markets and therefore to price movement intensity increases did not prove right. The comparison with 1998 leaves us with the 161


following insight: the capital inflows into the fixed-income markets were increased by the deleveraging effect of the hedge funds, for example covering of short positions of governments versus swaps. In an environment of falling equity markets (between August 2000 and October 2001) swap spreads – bank credit risk – could hold up better than corporate spreads, which were negatively affected by profit warnings from New and Old Economy companies and a negative ratings trend in some sectors (for example telecoms, technology, automobiles). Table 6.10 shows the differences between August 1998, October 2000 and September 2001 (differences in the highs and lows of equity indices and implied volatilities). The preferred position to receive fix swap rates in a scenario of steep yield curves in order to achieve a positive carry situation leads to riskaverse behaviour and to covering the position given a prevailing swap spread volatility or generally high fixed-income market volatility. The consequence is the widening of swap spreads and credit spreads. The equity volatilities, mentioned above, are important as well, because a correlation did exist between equity volatility and swap

Table 6.10 Percentage increase or decrease in absolute and percentage levels between highs and lows of various market indicators; values in parentheses represent the starting points Indices and volatilities

August 1998 (03.08.98)

October 2000 (02.10.00)

September 2001 (11.09.01)

S&P 500 Index (percentage change)

–12% (1100)

–8% (1440)

–14.7% (1130)

S&P 500 Index implied volatility (absolute change)

+18% (22%)

+7.5% (20%)

+19% (23%)

Nasdaq Composite Index (percentage change)

–19% (1850)

–14% (3570)

–19.6% (1770)

DAX implied volatility (absolute change)

+24% (18%)

+6% (22%)

+28% (26%)

Bund future implied volatility (absolute change)

+3.5% (3%)

+0.40% (4.4%)

+1.03% (4.38%)

Bund swap spread (percentage change)

+100% (35)

+12% (59)

+22% (36)

Source: Bloomberg

16 2

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Table 6.11 Swap spreads: July 1998–August 2000 Mean (basis points)

Half-life (weeks)

Volatility of the spread change (annually in basis points)

German 10-year spread

44.4

5.7

36.0

USA 10-year spread

90.1

25.4

34.2

Source: Lehman Brothers

spreads. During the Russia crisis in 1998 especially, US bank stocks showed a strong correlation with swap spreads. The overall interest-rate levels can also have an impact on the level of swap spreads. Over a long period, the swap spreads increased or decreased (exception 1998 – Treasuries have proved to be ‘safe havens’ during crisis scenarios) with the yield levels of the 10-year Treasuries. Investors are keen on increasing their returns, especially during low interest-rate periods. They can achieve this goal by investing in spread products, which generally leads to a spread tightening. Low yield levels were a sign of lower inflation expectations and resulted in lower risk premiums. A clear destabilization of this relationship can be observed since the financial crisis in 1998. In the past, swap spreads were subject to the phenomenon of mean reversion (an approximation to the average long-term rate). The half-life, which is the time needed for the spread to move halfway towards its long-term average, was much lower for DM-Swap spreads than for USSwap spreads. A lower value for the estimated half-life implies a stronger mean reversion (Table 6.11). Implied default probabilities for swap spreads can be computed with a simple model, which makes general assumptions. Thus, using the data for 10-year benchmarks and 10-year swap rates (Figure 6.23), the implied default probability for swaps was computed with the following formula: P = default probability p.a. R = risk-free rate (Governments) X = risky rate (Swaps) Recovery rate of 50% (Moody’s) P = (X – R) / [(1+X) • (1 – 50%)] 163


. . . . . . . .

Figure 6.23 Implied swap default probabilities Source: Deutsche Bank

For example: P = (0.042 – 0.038) / [(1 + 0.042) • (1 – 0.50)] P = 0.77%

The determination of implied default probabilities is an approximation because in this study only the 10-year data were used. In this context it may be mentioned that Union Investment has developed a model for estimating the fair value of swap spreads (Euro). This model supports the investment process for high rated (AAA–AA) credits because these credits prove to have a good and stable correlation with swap spreads. The factors used to determine the fair value are volatilities, stock returns, the ratio of credits to Governments, the shape of the yield curve, and parameters for autocorrelation. The correlation is very high, with r2 = 0.90. The residual spread (the line around zero: see Figure 6.24) explains the difference between the actual and the estimated value. The distribution is not symmetrical, so the probability of a spread widening was clearly higher than the probability for a tightening in spreads. If the model shows a significant deviation from the actual level, which is included in the investment process, it could lead to an over- or under16 4

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CREDIT MANAGEMENT

Figure 6.24 ‘Fair-value model’ DEM/EUR swap spreads Source: Union Investment

weighting of better rated credits. The model cannot recognize or predict special effects, event risks like mergers and acquisitions, or shocks to the system like Russian default and the LTCM crisis. The distribution is skewed to the right for the last two and half years, partly due to the financial crisis in 1998 (see Figure 6.26). All factors and variables applied in the fair-value model are periodically monitored for their validity and substituted by other variables if necessary (volatility, for example). The correlation between the US and German swap spreads increased continuously over the last couple of years; it was r2 = 0.68 during January 1999 through June 2000. The slope of the straight line can be used to determine that a change in US swap spreads by 3 basis points would result in a change in German swap spreads by just 1 basis point. The correlation tends to be higher with a high volatility of swaps spreads. If the correlation decreases again, the forecast quality would worsen. The steepness of the credit curve can be an indicator of the certainty or uncertainty in the market system concerning an improvement or a worsening of the overall credit environment. The steepness of the better rated credit curve, demonstrated on the 2-year and 10-year swap spreads, shows a steepening tendency, given a widening of the 10-year swap spreads (Figure 6.25). 165


10yr swap spread less 2yr swap spread

Slope of the swap spread curve vs. 10-year swap spread

10-year swap spread (bp)

Figure 6.25 Correlation between swap spreads (10 years) and the steepness of the curve (2/10 years asset swap differential): January–October 2000 Source: Union Investment

Kernel density (Epanechnikov, h = 3.3838)

Residuum

Figure 6.26 Distribution of the deviation from the fair-value model Source: Union Investment 16 6

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There is a relationship between implied Bunds volatility and swap spreads during times of extreme, hectic market movements, which implies the need for a hedge. Fixed-income market volatility was a relevant variable in the fair-value model, but has since been replaced.

6.5.2 Hedge strategies for government bonds and Jumbopfandbriefe in Euroland Euroland portfolios are a mix of government bonds of different countries, Jumbopfandbriefe and Corporates. In order to hedge the government bond position, the average credit quality corresponds more with Spanish government bonds (Aa2/AA+) than with Bunds (AAA), because the portfolio also contains Italian (Aa3/AA) and Greek (A2/A) government bonds (October 2001). Government bonds differ mainly in two factors: ■ liquidity ■ credit quality.

Liquidity can be determined by the bid-offer spread or by the depth of the market – its ability to absorb new supply. The German Bunds up to 10-year maturity have a substantial size and can be hedged with liquid futures contracts. A benchmark role can be assigned to them in Euroland (implying generally lower repo rates for benchmark bonds in Germany). Other countries, such as France and Italy, also have very liquid issues in the government sector. Italy, for example, received a lower credit rating from the rating agencies than German Bunds, so Italian government bonds have a slightly higher downgrading risk and interest-rate differential risk during periods of high market uncertainty and political crisis. This higher risk is subjectively compensated by a higher spread versus German government bonds. The correlation between Euroland government bonds and Bunds decreases during periods in which certain events – like UMTS-auctions in Germany, or the Bunds issuance calendar – have a major influence on the Euroland spreads (Figure 6.27). The wireless spectrum auctions (UMTS) resulted in an accelerated decrease in deficit because of higher income in Germany and wider spreads in Euroland. On the other hand, the surprisingly high issuance calendar for the Bunds was one reason for the temporary spread tightening in Autumn 2000.

167

Figure 6.27 Correlations between Euroland bonds and Bunds in the 10-year bracket Source: Merrill Lynch and Bloomberg

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CREDIT MANAGEMENT

Figure 6.28 High correlation between the Bund swap spread and Bund/OAT spread Source: Merrill Lynch

Hedging (or duration management) of non-AAA Euroland government bonds or portfolios with Eurex Bund futures

In order to extend the analysis for highly rated credits (here Jumbopfandbriefe), the correlation between the Bund swap spread – a good proxy for Jumbobundspreads – and the Bund/OAT spread is computed (Figure 6.28). A deteriorating hedging efficiency of the Bund futures can be observed, especially during market periods in which Bunds are considered as liquid and a ‘safe haven’. The bond buy-back programmes of the French treasury can lead to a shortage of French government bonds (OAT). Spain did achieve a higher acceptance (a tighter yield differential versus Bunds) in the capital market over the time because of a higher liquidity of Spanish government bonds and a better rating (Figure 6.29). Nevertheless, both spreads over Bunds roughly followed the trend of the Bund swap spreads.

6.5.3 Yield curve The steepness of the yield curve has a major impact on credit spreads. They tend to widen if the yield curve flattens, and to tighten in times of a 169


80

80

70

70

60

60

50

50

40

40

30

30

20

20

10

10

0

0

Figure 6.29 Spain and Belgium in comparison with the Bund swap spread Source: Bloomberg

yield curve steepening (this is especially true for non-cyclicals; see Figure 6.30). This relationship holds better for higher rated (AAA–AA) credits and weakens when applied to lower rated (BBB–B) credits. This statement has to be put in the perspective of the cyclicality of an industry, because not only the rating class but also the cyclical characteristics of an industry determine the relationship between the shape of the yield curve and the sector spreads, as explained below. The slope of the yield curve can be used as a good proxy for future economic growth and corporate profits. Steep yield curves also imply that future rates are implied to be higher than at present, which represents an optimistic view on the part of government bond investors about future economic growth: that economic conditions tomorrow will be more robust than today. If government bond investors are at their most optimistic about future economic growth, credit investors might be the opposite, particularly in demanding a high risk premium for overweighting cyclical credits. This can be explained by the fact that the credit market is sometimes slower to re-price the economic outlook than are the government bond and equity markets. Due to the asymmetric return distribution, credit investors care more about the current prospects of a 17 0

10.00%

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CREDIT MANAGEMENT

–1.00%

–0.50%

8.50%

0.50% 5.50%

Yield curve

High yield spread

0.00% 7.00%

1.00% 4.00%

High yield spread

01.07.99 01.09.99 01.11.99 01.01.00 01.03.00 01.05.00 01.07.00 01.09.00 01.11.00 01.01.01

01.05.99

01.01.99 01.03.99

01.09.98 01.11.98

01.07.98

01.03.98 01.05.98

01.11.97 01.01.98

01.05.97 01.07.97 01.09.97

01.01.97 01.03.97

01.03.96 01.05.96 01.07.96 01.09.96 01.11.96

01.05.95 01.07.95 01.09.95 01.11.95 01.01.96

2.50%

01.01.95 01.03.95

1.50%

2.00%

Yield curve (10YR–3MO)

Figure 6.30 Yield curve and the spread level in the non-investment-grade market Source: Union Investment

company, and less about future earnings. This means that the cyclical stocks have to show an outperformance over non-cyclical stocks before credit investors are more attracted to the cyclical sectors. On the other hand, steep yield curves increase the attraction of swaps (which are less volatile) compared with government bonds for banks, because of the positive carry situation (the refinancing rate is below the expected coupon). Yield curves react also to structural changes, such as the buy-back programmes of long-maturity government bonds in the US (a shortage situation) and are therefore a relevant factor for the shape of the yield curve and hence for swap spreads (Figure 6.31). During times of a strong equity market downturn (such as October 2000) and resulting overall market instability, the so-called ‘Flight-toQuality’ situation occurs. Government bond yields as well as swap yields will decrease: both yield curves are subject to a steepening, but swap spreads will widen relative to government bonds. An increase in implied volatilities of short-term options (less than 3 months) in the equity markets and to a lesser extent in the bond markets can be observed at the same time. This scenario might disrupt the strong relationship between the shape of the yield curve and the swap spreads. 171


180 160 140

Govt 2-10s

120 100 80 60 40

y = -3.5265x + 250.1 R2 = 0.6473

20 0 -20

25

30

35

40

45

50

55

60

65

70

75

10 yr Swap Spread

Figure 6.31 Relationship between the steepening or flattening of the curve and swap spreads in Euroland, January 1999 – July 2000 Source: CSFB

6.5.4 Credit default swaps (protection) A default swap is a contractual agreement between two counterparties in which one party pays a periodic fee in return for a one-off payment by the counterparty if a credit event (default) takes place. Figure 6.32 shows the credit default spreads and bond spreads of Enron. The credit default spreads are a very good leading indicator for Enron’s bond spreads, as market participants purchase default protection on this company to hedge against non-debt counterparty risks like commodities trading, project finance and structured transactions. The party paying fees is the protection buyer (‘short’ the credit) while the seller of protection is going ‘long’ the credit (taking credit exposure). Bond investors can take advantage of looking at the credit default market. The price of a default swap is typically quite close to the spread over Libor at which a credit trades in the corporate market. There are situations in which the default swaps lag or lead bond spread changes, but, over the long term, the two markets should be mean reverting. The pricing of bond spreads and credit default spreads do drift away at times, because of differences in demand and supply and 17 2

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CREDIT MANAGEMENT

120 105

Spread (bp/year)

100

88 75

80

80

80

76 75

71

75

74 73

81 76 76 77

59 60

50

40 20

6.625% 2005 asset swap spread 5-year default swap premium

0 7/23/99

10/15/99

1/14/00

4/7/00

6/30/00

10/6/00

Figure 6.32 Enron Corp, (Baa1/BBB+; defaulted in December 2001) credit default swap spreads: July 1999–October 2000 Source: Goldman Sachs

regulatory constraints on investors. Bond investors should find indications in the default market if spreads are too narrow or too wide. Arbitrage is not risk-free because the position between a default swap and a bond has a refinancing risk, as it may be extremely difficult to fund the position in the cash instrument to the end of the contract. It should also be kept in mind that counterparty default risk is prevalent with credit default swaps. Credit curve analysis using default swaps: a steep credit curve for lower investment grade credits usually exists because of stronger demand by investors for shorter-dated bonds. A steep credit curve today implies a steep credit curve in the future (forward). A strategy using default swaps to create a long forward spread position is ideal for investors who have concerns over the short period but believe that if the credit overcomes the problems, the firm has inherent value. This can be expressed by buying short-dated default protection and selling longerdated default protection (for example, buying protection for one year and selling protection for five years represents a long position in one year for four years).

173

0.8 0.8

17 4

AAA

AA

A


30.08.01 8/30/01

16.08.01 8/16/01

02.08.01 8/2/01

19.07.01 7/19/01

7/5/01

05.07.01

21.06.01 6/21/01

07.06.01 6/7/01

24.05.01 5/24/01

10.05.01 5/10/01

Telecom

26.04.01 4/26/01

12.04.01 4/12/01

29.03.01 3/29/01

15.03.01 3/15/01

01.03.01 3/1/01

15.02.01 2/15/01

Industrial ex Telecom/Transportation

01.02.00 2/1/01

1/18/01

18.01.00

04.01.00 1/4/01

21.12.00 12/21/00

07.12.00 12/7/00

23.11.00 11/23/00

09.11.00 11/9/00

26.10.00 10/26/00

12.10.00 10/12/00

28.09.00 9/28/00

14.09.00 9/14/00

31.08.00 8/31/00

β

03.04.00 4/3/00

20.08.01 8/20/01

06.08.01 8/6/01

23.07.01 7/23/01

09.07.01 7/9/01

25.06.01 6/25/01

11.06.01 6/11/01

28.05.01 5/28/01

14.05.01 5/14/01

30.04.01 4/30/01

16.04.01 4/16/01

02.04.01 4/2/01

19.03.01 3/19/01

05.03.01 3/5/01

19.02.01 2/19/01

05.02.01 2/5/01

22.01.01 1/22/01

08.01.01 1/8/01

25.12.00 12/25/00

11.12.00 12/11/00

27.11.00 11/27/00

13.11.00 11/13/00

30.10.00 10/30/00

16.10.00 10/16/00

02.10.00 10/2/00

18.09.00 9/18/00

04.09.00 9/4/00

21.08.00 8/21/00

07.08.00 8/7/00

24.07.00 7/24/00

10.07.00 7/10/00

26.06.00 6/26/00

12.06.00 6/12/00

29.05.00 5/29/00

15.05.00 5/15/00

01.05.00 5/1/00

17.04.00 4/17/00

β 1.15 1.15

17.08.00 8/17/00

03.08.00 8/3/00

20.07.00 7/20/00

06.07.00 7/6/00

22.06.00 6/22/00

08.06.00 6/8/00

25.05.00 5/25/00

11.05.00 5/11/00


Yield Betas of US Bonds by Sectors Yield Betas of UScorporate Corporate Bonds by Sectors

1.1 1.1

1.05 1.05

1 1

0.95 0.95

0.9 0.9

0.85 0.85

Banks

Figure 6.33 Yield Betas of various US industries (average benchmark rating: A3)


1.4 1.4

Yield Betas of Euroland-Indices byClasses Rating Classes Yield Betas of Euroland-Indices by Rating

1.3 1.3

1.2 1.2

1.1 1.1

11

0.9 0.9

BBB

Figure 6.34 Yield Betas for the rating classes double-A through triple-B in Euroland (average benchmark rating: Aa3)

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6.5.5 Yield Betas Yield Betas here are defined on the basis of rolling Betas (250 trading days) and computed using daily changes of spreads to Treasuries. The formula is: Beta = cov(dSi,dSm)/var(dSm)

where dSi = daily absolute changes in option-adjusted spreads of the asset class i dSm = daily absolute changes in option-adjusted spreads of the market index.

The concept of yield Betas can also be applied to corporate bonds. We can make a distinction between yield Betas of different industries or rating classes. The benchmark can be defined as ‘yield Beta = 1’, and the various industries will have a yield Beta either above or below 1 depending on their volatility. Empirical studies show that bonds from telecommunications issuers have a much higher yield Beta than the overall market and accordingly fall into the category of ‘high-Beta bonds’ (see Figure 6.33). The increased yield Beta levels of the industrials between April–August 2001 can be explained by the automobile bonds, which experienced a high volatility during this time. It should be said that the average rating of the telecommunications companies (A–BBB+) is between 1 and 2 notches lower than the average rating of industrials (A), and between 2 and 3 notches lower than the average rating of banks (AA–A+). Bonds with a higher credit rating usually have a lower yield Beta than lower rated bonds. Figure 6.34 shows this relationship by comparing the yield Betas of double-A and triple-B rated bonds. The better rated double-A bonds clearly have a lower yield Beta.

6.5.6 Implied equity volatility Falling equity markets are accompanied by increasing volatility, which allows a positive relationship to be drawn between credit spreads and equity market volatility (see Merton 1974). A rise in equity market volatility leads to a steepening of the credit curve – the yield differential between high and low rated credits – and will have a larger effect on 175


lower rated credits (for example BB–B). Companies with deteriorating credit statistics are more likely to experience high equity price volatility than companies with a stable credit trend. Figure 6.35 illustrates that equity volatility is an important driver for triple-B rated credit spreads (in this case the Euro). The strongest correlation between the Euro triple-B index and 10-day NASDAQ volatility exists with a 1 month lag (in year 2000). This means that triple-B spreads showed the strongest reaction 1 month after the move in the historical equity volatility (NASDAQ in this case) occurred. Figure 6.35 shows clearly that the triple-B spreads widen when the NASDAQ volatility increases and the spreads tighten with decreasing NASDAQ volatility. Xerox breaks this relationship a little but the overall message is clear, and it can be concluded that the widening of credit spreads over all rating classes in 2000 can be explained to a large extent by the increased volatility in the equity markets during the same period. Implied equity volatility can also be considered as a function of risk appetite or risk aversion. A purchased option represents a reduction of risk, and so the increase in the option price (of the implied volatility) can be seen as being equivalent to higher risk aversion. Correspondingly, higher risk aversion implies the existence of wider spreads in the

Xerox taken out of BBB index after downgrade

6.00%

180 170

5.00%

160 4.00%

150

3.00%

140 130

2.00%

120 1.00%

10-dayNASVOL (1month fwd)

BBB (govts)

Figure 6.35 Relationship between NASDAQ volatility and the Euro triple-B spread levels Source: Barclays 17 6

2/23/01 23.02.01

2/9/01 09.02.01

1/26/01 26.01.01

1/12/01 12.01.01

12/29/00 29.12.00

15.12.00 12/15/00

01.12.00 12/1/00

17.11.00 11/17/00

03.11.00 11/3/00

10/20/00 20.10.00

10/6/00 06.10.00

9/22/00 22.09.00

9/8/00 08.09.00

8/25/00 25.08.00

8/11/00 11.08.00

7/28/00 28.07.00

7/14/00 14.07.00

6/30/00 30.06.00

6/16/00 16.06.00

6/2/00 02.06.00

5/19/00 19.05.00

5/5/00 05.05.00

4/21/00 21.04.00

4/7/00 07.04.00

3/24/00 24.03.00

3/10/00 10.03.00

2/25/00 25.02.00

11.02.00 2/11/00

28.01.00 1/28/00

1/14/00 14.01.00

110 12/31/99 31.12.99

0.00%

100

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CREDIT MANAGEMENT

credit markets. Besides this, the implied equity volatility provides important information concerning the uncertainty of future corporate revenues. Since triple-B rated bonds, which can be defined as a long position of a risk-free bond and a short put on the assets of the firm, are closer to at-the-money than considerably higher rated triple-A bonds, it can be expected that the increase of equity market volatility will lead to a widening of the return differential between these rating classes (AAA–BBB). This is due to the fact that their sensitivity to changes in volatility differs (steepening of the credit curve; see Figure 6.36). One can argue that the implied equity volatility is a good estimate for the steepness of the credit curve (between ratings) and can be a leading indicator for credit spreads. A significant rise in implied equity volatility can introduce a crisis in the credit market. Options that trade close to at-the-money levels react more strongly given a change in volatility (at the maximum of the Vega curve) compared with options, which trade far out-of-the-money. The relationships described above can be witnessed particularly well during crash scenarios in equity markets. In October 2000 the extreme rise in equity volatility, which was initiated by increasing profit warnings by companies, was a leading indicator for credit spreads. Figure 6.37 points to the Vega effect (volatility effect).

-3.5

40

-3 -2.5

35

-2 -1.5

30

-1 -0.5

25

0 0.5

20

1 1.5

S&P 100 Vol

13.01.01

1/12/01

12/30/00

30.12.00

12/17/00

16.12.00

02.12.00

12/4/00 18.11.00

11/21/00 04.11.00

21.10.00 11/8/00

07.10.00 10/26/00

9/30/00 10/13/00 23.09.00

9/4/00

9/17/00

09.09.00

26.08.00

8/22/00 12.08.00

8/9/00 29.07.00

7/1/00

15.07.00 7/27/00

01.07.00 7/14/00

6/5/00

6/18/00

17.06.00

5/23/00

03.06.00

20.05.00

5/10/00 06.05.00

4/27/00 22.04.00

08.04.00 4/14/00

25.03.00 4/1/00

3/6/00

3/19/00

11.03.00

2/9/00 2/22/00

26.02.00

12.02.00

1/27/00 29.01.00

1/14/00 15.01.00

01.01.00 1/1/00

15

Underperformance of BBBs vs AAAs

Figure 6.36 Implied equity volatility as a function of risk aversion Source: Barclays 177

Aaa-Baa Spread

Equity Volatility


Figure 6.37 Steepening of the US credit curve in 1998 (the Vega effect) Source: CSFB

Figure 6.38 is another example of the relationship between implied equity volatility and credit spreads under the scenario of an external shock. It contains the period after the tragic events of New York and Washington on 11 September 2001. At this time the volatility index (CBOE OEX) reached higher levels than during the Gulf War in 1990 and the LTCM bailout in August 1998. The likelihood of a company failing to pay off its bond debt is thus approximately equal to the likelihood of the value of its corporate assets falling below the value of the corporate’s short-term liabilities. This likelihood can be increased either by a decrease in market capitalization or by an increase in the volatility of market capitalization, or by both at the same time. Credit spreads should therefore be highly sensitive to large changes in equity prices or volatilities. This phenomenon could be observed also in the European non-investment grade market during October–December 2000. This market has a bias towards ‘New Economy’ type companies, mainly out of the TMT sector (over 50% of all issues at the end of 2000) and was therefore negatively affected by the high NASDAQ implied volatility during this time. NASDAQ serves as a good proxy for the TMT sector, because it is heavily weighted towards ‘New Economy’ companies. Figure 6.39 shows the strong relationship between the NASDAQ prices and the US high-yield market. A negative correlation of over 72% 17 8

chapter six

CREDIT MANAGEMENT

Credit spreads and Volatility January 01 - September 01 250

55 50 45 40 35

200 30

VIX in %

Spread in bps

225

25 175

20 15 27.09.01

10.09.01

28.08.01

CBOE OEX Volatility Index

15.08.01

02.08.01

20.07.01

09.07.01

27.06.01

14.06.01

01.06.01

21.05.01

08.05.01

$-BBB Spread

$-BBB Spread

25.04.01

11.04.01

30.03.01

19.03.01

06.03.01

21.02.01

08.02.01

26.01.01

15.01.01

02.01.01

150 10 1/2/01 1/29/01 2/23/01 3/22/01 4/18/01 5/15/01 6/11/01 7/5/01 8/1/01 8/28/01 9/28/01

CBOE OEX Volatility Index

Figure 6.38 Credit spreads and equity market volatility: January–September 2001 Source: Union Investment and Bloomberg

was observed between NASDAQ prices and the spread movements of the high-yield market when NASDAQ was in a downward trend between March 2000 and September 2001. A weaker correlation can be expected when NASDAQ is rising.

6.6 EFFICIENT FRONTIERS AND THE SHARPE RATIO Corporate bonds have proven to be a long-term investment with attractive yields and acceptable risk, as evidenced by their long history in the US market – especially in a fund context. Harry Markowitz, who was awarded the Nobel prize for economics in 1990 for developing modern portfolio theory, identified the volatility of a portfolio, not just that of individual investments, as the relevant risk. The volatility of a portfolio, which is measured by the standard deviation (dispersion) of portfolio returns, does not depend solely on the volatility of individual investments, but also on the correlation between individual investment returns. When the returns of individual investments are not perfectly positively correlated, it is 179

Figure 6.39 Correlation between NASDAQ and US high-yield market: 02/02/2000–30/09/01 Source: Union Investment and Bloomberg

chapter six

CREDIT MANAGEMENT

possible to lower the volatility of a portfolio below that of the individual investments through proper diversification, for example with a small position in high-yield bonds (see Figure 6.40). Risk diversification plays a central role in fund management, so it makes sense to define upper limits for the weighting of sectors and single companies in a portfolio. Usually the weighting limit is tied to the market capitalization of the respective industries and companies. Nevertheless fund management policy should not apply too-tight rules regarding the investment limits because these restrict the allocation decision and achievable risk-return combinations. Figure 6.40 shows that the maximum-return portfolio (3) comprised over 42% non-investment grade bonds. This portfolio maximizes its return under the premise that portfolio volatility cannot exceed the volatility of a government bond portfolio. The maximum-return portfolio also contains financials ex banks (11%), utilities (24%) and governments (23%). The extra yield over Governments averaged 82 basis points, and the Sharpe ratio was 0.89. The high weighting of non-investment grade bonds does not take into consideration the underestimation of this asset class’s volatility. Noninvestment grade bonds are illiquid, and this illiquidity leads to nonupdated prices in the secondary market. On a risk-adjusted basis the performance (Sharpe ratio) of non-investment grade should be lower. If non-investment grade bonds were traded and priced more often (roughly 25% are priced once a month), then the volatility would increase and result in a lower Sharpe ratio than the statistical data actually shows. Due to this fact, investors – even risk- and growth-oriented investors – should not hold more than 20–25% of non-investment grade bonds in their mix-portfolios, especially when they are already invested in equities of the same or similar industry sectors. Mix-bond portfolios have a better risk–reward relationship than all other asset classes on a separate basis. The Sharpe ratios of the various asset classes were between 0.59 (telecommunications) and 0.80 (financials). Diversification strategies, which are quite common in investment funds, achieved Sharpe ratios of up to 0.90. The risk–reward profile of a fixed-income portfolio is positively impacted by overweighting investment grade and non-investment grade corporate bonds. The Sharpe ratio is a measure for the risk/return relationship of individual investments as well as portfolios. The formula to determine the historic Sharpe ratio of a portfolio is as follows: – SR (Rp) = Rp–rf σ (Rp)

– Rp : average portfolio return σ (Rp) : standard deviation of portfolio return rf : risk-free interest rate. 181

Yield (%) Efficient Portfolios

Volatility (%)

Figure 6.40 Efficient frontier of a US-fixed-income portfolio: 1989–2000 Source: Boersen Zeitung 18.02.01

chapter six

CREDIT MANAGEMENT

The Sharpe ratio can be graphically interpreted as the slope of a tangent on the efficient frontier, stemming from the risk-free rate of interest. Consequently, the Sharpe ratio is dependent not only upon the return and the standard deviation, but also upon the risk-free interest-rate level. The tangency portfolio is the portfolio on the efficient frontier with the maximum Sharpe ratio, or with the most favourable risk/ return ratio. The characteristics of US government and corporate bonds during the last 10 years are shown in Table 6.12. Individual non-investment grade bonds have a high volatility, but most of the risk is company- and sector-specific and hence is uncorrelated. Non-investment grade bonds have a lower volatility than equities, due to their coupon payments and senior ranking in the capital structure. Table 6.13 shows that during January 1988–May 2000 the non-

Table 6.12 Characteristics of US government and corporate bonds: 1990–2000 USA

Average rating

Average return %

Volatility %

Sharpe ratio

Government bonds

Aaa

7.75

4.16

0.69

Industrials

A3

8.40

4.94

0.72

Financials

A1

7.97

3.97

0.78

Baa1

8.43

4.48

0.80

Telecommunications

A2

8.44

5.54

0.65

Risk-minimal portfolio

A2

8.15

4.12

0.80

Utilities


Table 6.13 Returns, volatility and the Sharpe ratio: January 1988–May 2000 In %

Treasuries

Corporate bonds

High-yield index

S&P 500

Average total return

7.88

8.68

10.16

19.16

Return over Treasuries

0.00

0.80

2.28

11.28

Average volatility

4.19

4.68

5.78

13.31

Sharpe ratio

0.59

0.68

0.78

0.94

Source: DLJ

183


investment grade market index had a higher Sharpe ratio than other bond classes. The non-investment grade index cannot be built synthetically out of the S&P 500 and US Treasuries. A regression of returns between those asset classes explains only 25% of non-investment grade bond returns. A portfolio with a high degree of Jumbos, Agencies and high rated (AAA–AA) credits will engage in highly correlated ‘bets’ in the longer end (8–10 years), due to the high correlation among those asset classes. Triple-B and double-B rated corporate bonds offer a significant diversification and performance potential during periods of a positive credit environment and should be added as a mix to an investmentgrade portfolio. Table 6.14 shows the high correlation of returns between the various asset classes except triple-Bs. This implies that the lower rated Bonds (BBB) are well suited for diversification. The risk-return profile of three different portfolio strategies is now demonstrated on a mixed portfolio containing Euroland asset classes. In practice, several restrictions regarding the mix of portfolios and riskreturn profiles have to be considered. The Sample 1 portfolio will invest at least 25% of its assets in government bonds and a maximum of 10% in European High Yield Bonds (Strategy 1, Figure 6.41). The Sample 2 portfolio will be a corporate bond portfolio with a maximum of 30% invested in Pfandbriefe and 10% in European High Yield (Strategy 2, Figure 6.42). Triple-B financials are very rare in the Euroland universe and will be limited to only 1% in each strategy. The Sample 3 portfolio has no investment restrictions (Strategy 3, Figure 6.43) and shows the efficiency loss resulting from the investment limitations of portfolios 1 and 2. It should be noted that the unconstrained portfolio constitutes a theoretical upper limit for the risk–return profile of diversified portfolios. It is not realistic because of the perceived extreme weightings, liquidity constraints and the benchmark orientation of the fund manager. Figures 6.41 and 6.42 show that because of investment restrictions, the risk of the variance-minimum, efficient portfolio is higher than the risk of the Pfandbriefe. This is a first sign of costs resulting from the use of restrictions in a portfolio context. It can be clearly observed that the efficient frontiers of portfolio strategies 1 and 2 lie below the efficient frontier of the strategy with no investment restriction (Figure 6.44). Furthermore they cover a lesser risk–return spectrum. Two portfolios with special features on the efficient frontier are now introduced briefly. The portfolio with the least risk is the minimum18 4

chapter six

CREDIT MANAGEMENT

Table 6.14 Correlation of returns between various asset classes: January 1998–July 2001 Govt AAA AA Fin. Fin.

A BBB AAA AA A BBB Pfand. Quasi Fin. Fin. Non- Non- Non- NonFin. Fin. Fin. Fin.

Govt

1.00 0.98 0.97 0.97 0.81 0.96 0.98 0.93 0.82

0.96

0.98

AAA Fin.

0.98 1.00 0.99 0.98 0.84 0.97 0.99 0.96 0.86

0.98

1.00

AA Fin.

0.97 0.99 1.00 0.99 0.86 0.97 0.99 0.97 0.87

0.98

1.00

A Fin.

0.97 0.98 0.99 1.00 0.85 0.96 0.98 0.96 0.88

0.97

0.99

BBB Fin.

0.81 0.84 0.86 0.85 1.00 0.79 0.85 0.82 0.75

0.85

0.84

AAA Non-Fin. 0.96 0.97 0.97 0.96 0.79 1.00 0.96 0.94 0.83

0.95

0.97

AA Non-Fin.

0.98 0.99 0.99 0.98 0.85 0.96 1.00 0.96 0.87

0.97

0.99

A Non-Fin.

0.93 0.96 0.97 0.96 0.82 0.94 0.96 1.00 0.91

0.95

0.96

BBB Non-Fin. 0.82 0.86 0.87 0.88 0.75 0.83 0.87 0.91 1.00

0.86

0.86

Pfand.

0.96 0.98 0.98 0.97 0.85 0.95 0.97 0.95 0.86

1.00

0.99

Quasi

0.98 1.00 1.00 0.99 0.84 0.97 0.99 0.96 0.86

0.99

1.00


variance portfolio (MVP). No portfolio can be constructed with less risk without the addition of other asset classes in the decision process. The practical use of this portfolio is of minor importance, because it represents a strategy solely based on the minimization of portfolio risk. Bond indices help to determine the optimal weights of the various asset classes in a fixed-income portfolio. The 1-month Libor was used as an approximation for the risk-free rate. For the previous three strategies, the following weights can be found in the minimum-variance portfolio (MVP) and the tangency portfolio (TP; see Table 6.15). The performance is very important, in addition to the weightings of the various asset classes which show whether a portfolio adequately translates the underlying investment idea or strategy. Expected return, risk and correlations among the various asset classes clearly determine 185


Figure 6.41 Strategy 1: min. 25% Governments, max. 10% European High Yield, January 1998–July 2001 Source: Union Investment

Figure 6.42 Strategy 2: max. 30% Pfandbriefe max. 10% European High Yield, January 1998–July 2001 Source: Union Investment 18 6

chapter six

CREDIT MANAGEMENT

Figure 6.43 Strategy 3: no investment restrictions: January 1998–July 2001 Source: Union Investment

Figure 6.44 Comparison chart of efficient portfolios: January 1998–July 2001 Source: Union Investment 187


the weightings in the minimum-variance and tangency portfolios (maximum Sharpe ratio). It is worth noting that the different risk appetites of investors are reflected in the portfolio mix. Risk-averse investors, who favour the minimum-variance portfolio, would strongly overweight Pfandbriefe. On the other hand, in the tangential portfolios lower rated investment grade Financials would be overweight because of their better risk–return profile. Table 6.15 also shows the effects of different restrictions on the portfolio mix. Table 6.16 contains the average annual returns, the standard deviations and the Sharpe ratios for the different portfolios. The minimum-variance portfolios always have a lower risk than the tangential portfolio with the corresponding strategy. The tangential portfolio has a higher Sharpe ratio. Both results were expected. One thing to

Table 6.15 Weighting of asset classes – MVP and TP of the three strategies: January 1998–July 2001 >25% Govt,

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