Financial Risk Management for Pension Plans

Financial Risk Management for Pension Plans Lesaw Gajek Krzysztof M. Ostaszewski ELSEVIER Financial Risk Management...

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Financial Risk Management for Pension Plans

Lesaw Gajek Krzysztof M. Ostaszewski

ELSEVIER

Financial Risk Management for Pension Plans

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Financial Risk Management for Pension Plans

Lesław Gajek Dean, College of Technical Physics, Computer Science and Applied Mathematics, Technical University of Ło´dz´, Ło´dz´, Poland

Krzysztof M. Ostaszewski Actuarial Program Director, Illinois State University, Normal, Illinois, USA

2004

Amsterdam † Boston † Heidelberg † London † New York † Oxford Paris † San Diego † San Francisco † Singapore † Sydney † Tokyo

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q 2004 Elsevier B.V. All rights reserved. Exercises from the past actuarial examinations are copyrighted by the Society of Actuaries and/or Casualty Actuarial Society and are used here with permission. Cover illustration: Copyright 2004 q Brad Perks PC Image Network. This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.

First edition 2004 Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data A catalogue record is available from the British Library. ISBN: 0-444-51674-3 1 The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). W

Printed in The Netherlands.

v

Abstract This book is devoted to modern financial risk management of pension plans, mostly defined benefit plans. It presents main actuarial funding and valuation methods for defined benefit plans. Modern investment theory is used to discuss ways to value both marketable and nonmarketable assets as well as liabilities. Finally, financial risk management for pension plans is presented in detail, with emphasis on applicable asset –liability management methodologies.

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vii

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Part 1. Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 1.

Chapter 2.

Retirement plans as a part of economic security system . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Genesis . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Classification of pension plans . . . . . . . . 1.3 The role of the actuary in the management of pension plans . . . . . . . . . . . . . . . . . . .

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

3 3 5

...........

10

Fundamental concepts of theory of interest . . . . . . . . . . . . . . . 2.1 Accumulation function . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Rate of return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Measurement of interest . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cash flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Interest measurement in a fund . . . . . . . . . . . . . . . . . . . 2.5.1 Discrete-time model . . . . . . . . . . . . . . . . . . . . . 2.5.2 Continuous-time model . . . . . . . . . . . . . . . . . . . 2.5.3 Allocation of interest to accounts . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Interest measurement in a group of funds . . . . . . . . . . . . 2.6.1 Nonrandom continuous-time model . . . . . . . . . . 2.6.2 A discrete-time stochastic model . . . . . . . . . . . . 2.6.3 Deriving the formula for the average rate of return . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Martingale-fairness of the average rate of return r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Axiomatic approach to the investment efficiency index . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6 Merger of funds . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13 14 17 19 22 23 23 24 25 26 29 30 32 34 36 38 42 44

viii

Chapter 3.

Contents

Life insurance and annuities . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Loan amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Term structure of interest rates . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Depreciation of fixed assets . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Capitalized cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Random future lifetime . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Mortality tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Life insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Life annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Life insurance premiums . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Life insurance reserves . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Multiple lives models . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Multiple decrements . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 52 56 59 62 64 65 67 68 72 74 75 80 82 85 90 96 101 108 118 121 125 128 139 141 144 148

Part 2. Valuation of Assets and Liabilities . . . . . . . . . . . . . . . . . . . . . . . .

157

Chapter 4.

159 159 161 162 163 165 165 166 167 172

General principles of pension plan valuation . . . . . . . . . . . . . . 4.1 Objectives and principles of pension funding . . . . . . . . . 4.2 General principles of pension valuation . . . . . . . . . . . . . 4.2.1 Accrued liabilities . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The normal cost versus accrued liabilities . . . . . 4.2.3 Unfunded actuarial liabilities . . . . . . . . . . . . . . . 4.2.4 Actuarial gain . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Some additional remarks . . . . . . . . . . . . . . . . . . 4.3 Optimal payoff of a liability . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Contents

Chapter 5.

Chapter 6.

Valuation of pension plan liabilities . . . . . . . . . . . . . . . . . . . . 5.1 Unit credit method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Accrued liabilities . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Normal cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Unfunded actuarial liability . . . . . . . . . . . . . . . . 5.1.4 Actuarial gain . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Entry Age Normal method of funding . . . . . . . . . . . . . . 5.2.1 Normal cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Accrued liabilities . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Unfunded actuarial liabilities . . . . . . . . . . . . . . . 5.2.4 Actuarial gain . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Individual level premium method of funding . . . . . . . . . 5.3.1 Normal cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Accrued liabilities . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Unfunded actuarial liabilities . . . . . . . . . . . . . . . 5.3.4 Actuarial gain . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The method of Frozen Initial Liability . . . . . . . . . . . . . . 5.4.1 Normal cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Accrued liabilities . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Unfunded actuarial liabilities and the actuarial gain . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 A recursive formula for the unit normal cost . . . 5.4.5 Iterative method of normal cost calculation . . . . 5.5 Valuation of plans paying pension benefits . . . . . . . . . . 5.5.1 Accrued liabilities . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Unfunded actuarial liabilities . . . . . . . . . . . . . . . 5.5.3 Actuarial gain . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 New plan participants . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 The method of Frozen Initial Liability . . . . . . . . 5.6.2 Unit credit method . . . . . . . . . . . . . . . . . . . . . . 5.6.3 The Entry Age Normal method . . . . . . . . . . . . . 5.6.4 Individual level premium method . . . . . . . . . . . . 5.7 Aggregate pension funding methods . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random fluctuation of pension liabilities . . . . . . . . . . . . . . . . . 6.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . 6.2 Fluctuation of the present value of a life annuity . . . . . . 6.2.1 The expected value of the random variable a T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The variance of the random variable a T . . . . . . . 6.2.3 The coefficient of variation of the random variable a T . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 180 181 182 183 185 186 186 189 190 191 191 192 195 195 196 196 197 199 200 203 204 204 207 207 208 209 212 214 214 215 217 231 231 232 233 233 235

x

Contents

6.2.4 The Gompertz Law of Mortality . . . . . . . . . . . . Confidence intervals for accrued liabilities of a pension plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Cohorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

238

Valuation of pension plan assets . . . . . . . . . . . . . . . . . . . . . . . 7.1 Theories of capital markets . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Optimal consumption decision versus the level of interest rates . . . . . . . . . . . . . . . . . . . . . 7.1.2 Optimization of an investment portfolio . . . . . . . 7.1.3 Capital assets pricing methodologies . . . . . . . . . 7.1.4 The Principle of No Arbitrage . . . . . . . . . . . . . . 7.1.5 The Fundamental Theorem of Asset Pricing . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 General methodology of arbitrage-free valuation of capital assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Derivative securities . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 A survey of asset valuation methods . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253 253

6.3

Chapter 7.

242 242 246

253 256 260 263 267 271 273 275 278 280 283

Part 3. Financial Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287

Chapter 8.

General rules of risk classification and measurement . . . . . . . . 8.1 Risk classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Total risk measurement and capital requirement . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Asset-side risks and investment benchmarks . . . . . . . . . 8.4 Stability of total annual premium . . . . . . . . . . . . . . . . .

289 289 290 295 298 300

Chapter 9.

Interest rate risk—nonrandom approach . . . . . . . . . . . . . . . . . 9.1 The relationship between the price of capital assets and the interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Duration of a security . . . . . . . . . . . . . . . . . . . . 9.1.2 Convexity of a financial asset . . . . . . . . . . . . . . 9.1.3 Nominal interest rates . . . . . . . . . . . . . . . . . . . . 9.1.4 The Babcock Equation . . . . . . . . . . . . . . . . . . . 9.2 Classical immunization . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 A critical discussion of classical immunization for a pension plan . . . . . . . . . . . . . . . . . . . . . . .

303 303 303 307 311 313 315 317

xi

Contents

9.2.2 9.2.3

9.3

Multivariate immunization . . . . . . . . . . . . . . . . . Simulation approach to asset – liability management . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom of Solvency and surplus immunization . . . . . . . . 9.3.1 Monotone shift factors . . . . . . . . . . . . . . . . . . . . 9.3.2 Arbitrary shift factors . . . . . . . . . . . . . . . . . . . . 9.3.3 Conservative asset –liability management . . . . . .

Chapter 10. Stochastic approach to financial risk management . . . . . . . . . . 10.1 One-period analysis of the plan surplus and funded ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Risk management by shortfall control . . . . . . . . 10.1.2 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Funded ratio return . . . . . . . . . . . . . . . . . . . . . . 10.2 Axiom of Solvency and random interest rates . . . . . . . . 10.3 Immunization when the portfolio is not correlated with term structure of interest rates . . . . . . . . . . . . . . . . 10.3.1 Monotone shift factors . . . . . . . . . . . . . . . . . . . . 10.3.2 Arbitrary shift factors . . . . . . . . . . . . . . . . . . . . 10.4 Immunizing arbitrary portfolios . . . . . . . . . . . . . . . . . . . 10.4.1 Monotone shift factors . . . . . . . . . . . . . . . . . . . . 10.4.2 Arbitrary shift factors . . . . . . . . . . . . . . . . . . . . 10.5 Conservative strategy of risk management for pension plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Conservative choice of TSIR . . . . . . . . . . . . . . . 10.5.2 Portfolio with a minimal immunization risk . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 11. Conclusion

320 321 323 325 325 326 329 329 332 334 336 337 338 338 340 341 341 343 344 344 346 348

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353

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357

Index

361

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xiii

Introduction The key objective of pension plans is the delivery of retirement benefits, typically payable for life or a set period of time, to the specified group of recipients. Such plans are usually organized by employers, although there also exist multi-employer plans, or plans organized by various levels of government. First private retirement plans were created in the United States in the nineteenth century. In both the United States and Canada, private pension plans play a crucial role in the overall financial security system. While other countries initially relied mostly on state pension systems, the last quarter century had brought about new growth in private retirement systems worldwide. Chapter 1 contains more detailed discussion of the role of pension plans in financial security systems. Private pension plans can have a very diverse structure—they may be directed towards a small group, or a group of several thousand employees, or even more in the case of multi-employer plans—in all cases providing for the financial needs of the members of the group. This results in a diversity of forms of pension plans, and creates a need for understanding of a variety of possible funding mechanisms for such funds. These issues are addressed in Chapters 4 and 5. Chapter 4 contains general rules for calculation of the actuarial liabilities of a defined benefit pension plan, its normal cost and possible amortization of initial liability, or unfunded liability. Chapter 5 presents basic methods of actuarial pension plan valuation, discussing both funding and calculation of accrued liabilities. Chapter 6 contains analysis of stability of pension plan liabilities, especially in relation to the plan size. Chapter 7 is devoted to fundamental results of modern investment theory. These results are later used in the analysis of marketable and nonmarketable assets of pension plans, and financial approach to valuing assets and liabilities. Chapter 8 presents general rules of risk classification and measurement for pension plans. In particular, we describe in detail capital requirements to cover the risks. In Chapter 9 we present nonrandom approach to asset –liability management. Chapter 10 is devoted to stochastic approach to financial risk management. Chapter 11 contains a brief presentation of a final conclusion. The whole book is divided into three parts: Part 1 (Chapters 1 –3) contains auxiliary results and information; Part 2 (Chapters 4 –7) is on methods of assets and liabilities valuation; Part 3 (Chapters 8– 11) concerns the methodologies of financial risk management for pension plans. This part is, in our opinion, of particular importance for defined benefit plans, for which we analyze their investment risk, funding ratio risk, and asset – liability management practices.

xiv

Introduction

This book aims to integrate the analysis of assets and liabilities for pension plan. We study the dependence of both assets and liabilities on interest rates, and other random variables, e.g., longevity of plan participants. We present general results concerning asset – liability risk management as well as practical problems and requirements. The intended audience of this work are all decision makers involved in management of assets, liabilities, and integrated portfolios of both sides of the balance sheet, for pension plans. This may include actuaries, investment managers, plan fiduciaries, and government regulators. We assume that our readers are familiar with basic probability theory, as well as mathematical analysis. Chapters 2 and 3 introduce basic terminology and notation of actuarial mathematics for life contingencies, as applied to life insurance and pensions. The authors hope that this work can also become an academic textbook for courses in finance, mathematics, economics, and actuarial science. For that reason we have incorporated plenty of examples and exercises. Early versions of this text have been used in lectures at the Technical University of Ło´dz´, Poland, Summer Actuarial School at the University of Warsaw, Poland, University of Louisville and Illinois State University. These experiences have resulted in numerous improvements of the text. Some of the ideas of this book have appeared in a Polish text Plany emerytalne: Zarza¸dzanie Aktywami i Pasywami, published in 2002 by Wydawnictwo NaukowoTechniczne in Warsaw, Poland, by the same authors. We express our gratitude to the Society of Actuaries and the Casualty Actuarial Society for allowing us the use of past actuarial examination problems as exercises. We dedicate this work to our wives: Elz˙bieta and Patricia, who offered us infinite patience and support and whom we offer love and gratitude. Lesław Gajek Krzysztof M. Ostaszewski

Part 1 Fundamentals

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3

Chapter 1 Retirement plans as a part of economic security system

1.1 Genesis Retirement plans constitute one of the key elements of every society’s financial security system. Of course they are of utmost importance for the elderly, but they have great effect on the functioning of the national economy and everyone’s welfare. Let us note that one could divide human life into three main phases: youth, productive years, and retirement. There are many perspectives on this division, but in this work we are interested in the economic one. Youth is therefore the period when a person, through education, acquires human capital, i.e., the ability to earn income throughout one’s productive life. Acquisition of education, i.e., investment in human capital, is among the most important financial issues in a person’s life. Another such issue is, of course, the accumulation of financial capital for retirement, the subject of this work. Human capital acquired mostly during one’s youth is gradually used up during one’s productive life. One may argue that a person should re-acquire education later in life, as the human capital accumulated early in life does become obsolete eventually, as the technology of production changes. Ironically, the blessing of longer life enjoyed by most of the world’s population in the twentieth century has made such a threat of a need for reacquisition of human capital more likely. Nevertheless, eventually one’s physical and mental ability to work is depleted, and somewhere before or at that moment, one needs to replace human capital with financial assets allowing for comparable standard of living. The problem of financial security in the late stages of life has assumed increasing significance as human lifespan expanded. In the prehistoric times, the antiquity, or the Middle Ages, retirement protection was never a social issue, as few people reached the retirement age (defined as the age when productive ability is no longer available). On the other hand, in the twentieth century, providing for retirement became a very important social issue. As that century drew to a close, it also has become a major challenge globally. For example, the life expectancy for a person aged 65, calculated

4

Chapter 1

on life tables1 for 1900 and 1950, respectively, increased by 25.28% in England and Wales and by 29.86% in France. This seems to be a permanent universal tendency; in the United States the life expectancy for an analogous person increased between 1960 and 1999 by 23.40%. The World Bank publication [6] goes as far as to call it the Old Age Crisis. The gravity of the problem results from the fact that many of the world economies appear to be unprepared for increasing longevity of their populations, the resulting ageing of their societies, and increased retirement needs. There are various possible means of support in the old age. Traditionally, the elderly had depended on support from their family members, mostly their children. Churches and charitable organizations had assumed a larger role over time. But major social transformations of the twentieth century had given those two forms of support a secondary role. The simplest solution to the problem of old age is the one used once by the aristocracy: a person can retire as soon as this person has acquired enough wealth to retire on. In short, the wealthy ones can retire. Interestingly enough, this may not be as difficult as it appears, due to the increased longevity we all enjoy. A person earning 4000 a month, saving 10% of income over the span of 40 years, and earning 7% real on the income saved, will accumulate slightly over 1 million monetary units by the end of the period. With a rough approximation of a life annuity factor of 20, this provides enough income to replace one’s wages for the rest of the lifetime. This calculation ignores the effects of inflation, but in a free-market economy both wages and capital market instruments generally grow at rates not only matching inflation but also exceeding it. For example, in the United States, long-term average rate of return of stocks in the twentieth century was roughly 7% above inflation [59]. At the end of the nineteenth century, new state-run retirement schemes were created, beginning with the German social insurance system created by Chancellor Bismarck. New private retirement also started around the same time. American Express created a company pension plan in 1875, and Baltimore and Ohio Railroad Company started one in 1880 [2]. In the United States, private pension plans grew rapidly during and following World War II, when wage increases were limited due to wartime wage and price controls, but collective bargaining for pension benefits was allowed. In 1974, another watershed event for pension plan history in the US occurred—the federal law named Employee Retirement Income Security Act (ERISA) was passed. This law created a requirement for qualified pension plans (i.e., those receiving tax assistance in the form of tax exemption for contributions and tax deferral for investment income) of having a fund appropriate for paying the benefits and making a regular payment of the plan normal cost, as established by a qualified actuary. 1

Source: Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de (data downloaded on April 17, 2004).

Retirement plans as a part of economic security system

5

Most developed economies in the world have gradually evolved towards a system of what is commonly called a three legged stool in the retirement security system. The three legs in this concept refer to: state-sponsored, employer-sponsored, and individual retirement benefits. The first leg is commonly created with a system of social insurance, such as the United States Social Security System (Old Age, Survivors and Disability Insurance or OASDI), or with a privatized mandatory system of individual accounts, the first of which was created in Chile in 1981. In the United States, the second leg is represented by employer-sponsored pension plans of either the defined benefit or the defined contributions variety. The third leg in the US consists of the variety of personal retirement accounts, such as Individual Retirement Accounts (IRA), Roth IRA accounts, and others. In what follows, we will also refer to the three “legs” of the stool as the tiers of the pension system.

1.2 Classification of pension plans In this section we will present a short review of the most important types of pension plans, which may be, for the purpose of our classification, part of any of the three legs of the retirement security stool. We will use various criteria for the classification. The first criterion is the asset base for the liabilities for benefits promised to plan participants. Here we can point out two types of plans: † pension plans without a fund ( pay-as-you-go plans) and † pension plans with a fund ( funded pension plans). A pay-as-you-go plan does not accumulate specific assets designated to create income needed to pay benefits. Instead, they depend on future income generated by the plan sponsor to pay future benefits. In effect, such plans are backed by the credit of the plan sponsor, as opposed to funded plans, where benefits are backed by the portfolio of capital assets. Employer-sponsored pay-as-you-go cannot be qualified for tax purposes in the United States: this is a provision of ERISA. The reasoning behind pay-as-you-go private plans prohibition in ERISA is quite sound. In a pay-as-you-go plan, participants depend on their employer for the security of their benefits. This dramatically increases the risk to plan participants. Such increased risk is present even for participants who are still working. Without legal protection, in the case of a bankruptcy, plan participants’ claims would become a part of all creditors’ claims and may not be safe. It is even possible, that participants’ claims would not even be recognized as employer’s liability. Even if recognized, if, in the course of bankruptcy proceedings, other liabilities were determined to have priority over pension obligations, employees would not have any recourse and any retirement protection. Funded pension plans avoid the potential pitfalls of pay-as-you-go plans. The fund accumulated for the purpose of providing benefits allows for a substantial

6

Chapter 1

increase in the security of participants’ benefits. In the United States, qualified pension plans are funded. However, pay-as-you-go plans are still a dominant form of pension plans among social insurance plans, such as OASDI in the United States. Social insurance is administered and guaranteed by the state, or, in the case of the United States, by the federal government. This makes the credit problems described above practically nonexistent. Insolvency of social insurance would be basically equivalent to insolvency of the state itself. Such an insolvency is not only extremely unlikely, it is also an uninsurable event: one cannot find a realistic protection against it. The second criterion for classification of pension plans is the method of correction of an imbalance between assets and liabilities. From that perspective, we distinguish between: † defined benefit plans and † defined contribution plans. If a defined benefit plan exhibits an imbalance between its assets and liabilities, its normal cost (regular payment required from the contributor to the plan, i.e., the analog of an insurance premium) is adjusted. If a defined contribution plan shows such an imbalance, its future benefits are automatically adjusted. These definitions of defined benefit and defined contribution plans are nonstandard. They were originally proposed by Dr Wojciech Otto of the University of Warsaw in Poland and are adopted by this work. In other works, it is more common to define defined contribution plans as the plans for which only the contributions are prescribed in advance (and benefits are determined by the performance of the assets of the plan), while defined benefit plans are defined as plans for which benefits are prescribed in advance, and asset performance affects contribution levels needed to fund benefits. A pay-as-you-go plan must be, by definition, a defined benefit plan, as it specifies a benefit which will be delivered to a participant without any fund so it shows a permanent imbalance. Private pension plans are also built upon a promise of future benefits, but the promise is backed by a fund, and contributions (normal costs) paid regularly into the fund. The levels of plan liabilities and contributions needed are established by an actuary. There are four basic forms of a promise of future pension benefits that exist in private retirement plans: † † † †

a set dollar amount of monthly (or annual) benefit, a set dollar amount of monthly benefit for each year of employment service, a set percentage of (final or average) wages paid as a monthly pension benefit, a set percentage of (final or average) wages for each year of employment service paid as a monthly pension benefit.

Retirement plans as a part of economic security system

7

There are also combinations of the four above, or variations of them, which are used in practice. For example, both private and social insurance programs set the maximum for the number of years of service counted towards granting of benefits, if such benefit depends on service length. This design serves to encourage workers to retire when they reach a certain age believed to be the maximum age desired for a given professional group. Some professional groups also enjoy certain retirement privileges, e.g., social insurance systems of many countries grant special early retirement benefits to police, military, or miners. Private defined benefit plans are very common in the United States and Canada, but not in other countries. In these two countries, defined benefit plans must be funded. Benefits are typically calculated as the number of years of service (subject to a certain maximum) times a percentage of final year wages, or average wages of the last 5 or 3 years of service. For example, if a plan grants 2% of the average wages of the last 3 years of employment and if the worker’s salaries during the last 3 years of service were: $4000 per month, $4500 per month, and $5000 per month, respectively, and the worker has 40 years of service, this worker’s retirement benefit will be 40 £ 0.02 £ ((4000 þ 4500 þ 5000)/3) ¼ $3600. Private defined contributions plans in the United States and Canada developed rapidly beginning in the late 1970s. The 1970s were marked by high inflation and increased investment risk. These factors may have caused the employers to want to shift the risk to plan participants. In a defined contribution plan, investment risk is absorbed by plan participants, and the only obligation of the employer is the contribution itself. Over the last quarter of the century, a dramatic shift in the relative role of the two types of plans in the United States has occurred: in 1975, defined benefit plans had a near monopoly in the pension provision market, but by the beginning of the twenty-first century, defined contribution plans achieved dominance. The reason for this change has been hotly debated, especially among actuaries. Let us try to understand the positions of this debate. In a defined contribution plan, the employer makes a regular contribution to an individual account of a given employee. Typically, the contribution equals a set percentage of the employee’s wages (although in the United States there are limits due to nondiscrimination rules). Such contributions are subsequently invested based on a choice made by the employee, although the variety of choices available to the employee is generally defined by the employer. In the United States, an employer who wants to fully shift the responsibility for investment choices to the employee must provide the following four investment choices: money market equivalent (very shortterm bonds, considered to be cash equivalent), diversified portfolio of bonds, balanced portfolio (stocks and bonds), and a diversified portfolio of stocks. Employer stock may be an investment choice, but it must be in addition to the above four. If the employer does not provide these four choices, the plan may still be qualified for tax purposes, but the employer will have to accept fiduciary duty and responsibility for investment choices by the employees. Otherwise, such responsibility rests fully with the employee.

8

Chapter 1

This shifting of the investment risk to the employees is often considered to be the main explanation for the seminal shift from defined benefit to defined contribution plans in the United States. Another hypothesis ventured to explain this phenomenon is the relative ease of transfer from one defined contribution plan to another contribution plan. If an employee terminates employment from an employer offering a defined benefit plan, such employee will typically receive pension benefit upon retirement based on a relatively shortened period of service and on the last wages with the said employer. This means that if there are periods of high inflation between termination and retirement, the value of benefits will deteriorate substantially. Defined contribution plans, on the other hand, offer relatively easy transfer opportunities. In the United States, fund balance in the participant’s account can be transferred either to the new employer’s plan or to an IRA. Both defined benefit plans and defined contribution plans can be financed solely by the employer, solely by the employee, or jointly by both parties. Plans which require employee financing are called contributory. Contributory defined contribution plans are fairly common in the United States and Canada alike. On the other hand, contributory defined benefit plans do not exist in the United States (the reason is simple—US tax law does not make employee contribution tax-exempt), while they are quite common in Canada (where they are generally tax-exempt). Pension plan contributions are invested in various capital assets (and sometimes, although generally rarely, in real assets, e.g., real estate or commodities). Regulations usually limit the investment universe. In the United States it is fairly common for defined benefit pension plan to hold a balanced portfolio of approximately 60% stocks and 40% bonds. Investments in employer’s stocks or bonds are generally either prohibited or limited. It is quite common for plan participants in defined contribution plans to choose their own asset allocation. Too often, it seems, many of them choose to invest predominantly in money market instruments, very low-risk asset class, thus limiting their long-term investment opportunities. On the other hand, a typical defined benefit plan portfolio of 60% stocks and 40% bonds may be a good choice for an individual with an intermediate risk tolerance profile. But even the debate concerning the investment portfolio of defined benefit plans is not free from controversy. Some finance theoreticians claim that defined benefit plans, instead of a balanced portfolio of stocks and bonds, should invest only in bonds as such bonds will provide a better match for the liabilities. The problem of construction of an optimal portfolio for a retirement plan is discussed by Leibowitz et al. [42]. Immediately following World War II, and in the 1950s, defined contribution plans constituted virtually the entire second tier of the pension system in the United States. When ERISA was passed, assets of defined benefit plans amounted to 75% of all pension plan assets. As of this writing, they account for less than half of pension plan assets, the rest having been taken over by the defined contribution plans. Over 75% of new contributions flow into defined contribution plans. Even though defined benefit plans have a significantly longer history of existence, more than half the pension

Retirement plans as a part of economic security system

9

benefits are paid by defined contribution plans. And, most importantly, employers no longer start new defined benefit plans. This massive decline in the role of defined benefit plans has received significant attention from researchers, who proposed several explanations for it. We had already mentioned the shifting of risk onto employees as one of them, and the portability of defined contribution plans as another. But there are other theories, some of them contradictory. Some hypothesize it was the ERISA law itself that caused the demise of defined benefit plans, by putting additional reporting and valuation requirements on them, thus causing increasing costs and bureaucratic burdens. Requirement of actuarial valuation for defined benefit plans is not to be ignored here, as it increases costs substantially. Other explanations proposed are: greater ease of explanation and understanding of defined contribution plans (which are quite similar to bank accounts or mutual fund accounts) and high rates of return available to participants’ accounts in the years 1983– 1999 that have encouraged participants to chase those stellar returns [54]. In the face of the shift away from defined benefit plans, their proponents have worked to create new alternative designs, which may help stem the tide of the escape to defined contribution plans. Two main innovations in design are: † target benefit plans and † cash balance plans. Target benefit plans are designed as a hybrid between defined benefit plans and defined contribution plans. They are officially qualified as defined contribution plans, but their funding is based on projected retirement benefits. Normal cost is calculated based on such a projection and estimated by the plan actuary to be at the level which would provide for the expected benefit. However, unlike in a defined benefit plan, the employer does not assume an obligation to deliver that level of benefit. Thus, as in the defined contribution plan, investment risk in a target benefit plan is assumed by the employees. It should be noted, however, that target benefit plans have not been accepted by the retirement markets in the United States. The second new alternative design is the cash balance plan. The first cash balance plan was created by the Bank of America in 1984. In a cash balance plan, a worker has exactly the same benefits and rights to them, as in a defined benefit plan. The key difference lies in the fact that the actuarial value of all benefits, accrued as a result of his/her service up to date, is presented to the worker as an account balance. Each year, this account additionally increases by interest credited according to a set formula and credits due to additional service. The method of crediting due to service is established by the plan actuary, and reported to the Department of Labor. Interest crediting, on the other hand, is set by the employer, although there is a series of Internal Revenue Service (IRS), the US tax authority, regulations concerning

10

Chapter 1

methods of crediting approved by the IRS (such an approval is essential for the plan to be qualified).

1.3 The role of the actuary in the management of pension plans Pension funding mathematics exhibits many similarities with the mathematics of life insurance. In both cases the actuary must rationally value products, which provide the customer with future income stream, in return for a premium stream paid to the provider. There are, however, some fundamental differences. First, most pension plans have fewer participants than the typical number of customers of an insurance company. Pension plans vary in their member count from as few as one or several participants to, rarely, as many as hundreds of thousands, but a typical customer base of a life insurance company is at least in tens of thousands, if not hundreds of thousands. This smaller number of participants means that random fluctuations of assets and liabilities of pension plans may have a more profound effect on plan funded status than in the case of a life insurance company. This increased uncertainty must be taken into account by the actuary when establishing pension plan liabilities and normal cost. The second key difference lies in the timing of benefit payments. Pension plan participants may withdraw from the plan early, due to termination of employment. They also generally have great latitude in choosing their retirement age, within the bounds set by the early retirement age and the latest age allowed by the plan. The actual amount of the benefit will be directly influenced by the date chosen and, additionally, indirectly, the date will affect the final salary (or final salary average), again influencing the benefit amount. This makes the work of the plan actuary more challenging, especially if one wants to achieve stable normal cost, a common desire among employers. Pension plan management requires substantial involvement of the plan actuary. In the case of defined contribution plans, the actuary must assure that all applicable regulations are followed, and that existing plan assets provide appropriate level of projected benefits for plan participants. In the case of defined benefit plans the role of the plan actuary is especially pronounced: it is the actuary who values plan benefits granted and calculates the normal cost. In general, a pension plan actuary has the following responsibilities: † † † † † † † †

to know generally the accepted pension valuation and funding methods, to know which methods are applicable to the plan under consideration, to establish appropriate assumptions for valuation, to estimate the effect of plan size on the stability of its funding, to value benefits other than retirement benefits, if granted (e.g., disability benefits), to model future cash flows of the plan, to value plan assets appropriately and to model sensitivity of the plan to changing parameters such as interest rates, mortality, or general economic variables.

Retirement plans as a part of economic security system

11

The above list illustrates that the plan actuary must possess vast knowledge and experience in order to meet such a variety of responsibilities. In the following chapters we will lay the foundation for such a knowledge. We hope that this book can serve as a, naturally limited, substitute for experience. We will present methods of valuation of plan liabilities, assets and normal cost calculation. We will next study the methods of financial risk management applicable to pension plans, with emphasis on asset – liability management.

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13

Chapter 2 Fundamental concepts of theory of interest

2.1 Accumulation function If a unit amount is placed in an account, its accumulated value at time t is denoted by aðtÞ and called the accumulation function. It must be equal to 1 (amount invested) plus the interest earned over the period ½0; t: We assume generally that aðtÞ is an increasing continuous function. If the amount invested is different than 1, we typically denote the accumulated value in the account at time t by AðtÞ and call it the amount function. The two functions are related via AðtÞ ¼ Að0Þ · aðtÞ;

ð2:1:1Þ

which effectively states that every monetary unit invested earns interest at the same rate.

2.2 Rate of return Assume at first that rate of return (interest rate) i is constant in the period of time considered. We will use year as a unit of time. If the interest is compounded (i.e., added to the principal invested) once a year (i.e., annually), the rate of return i is called the effective rate of return (over a year). If the compounding frequency is different than once a year, the annual rate is called a nominal rate of return and denoted as iðmÞ ; where m is the compounding frequency per year. The relationship between these two measurements of interest is !m iðmÞ 1þi¼ 1þ : ð2:2:1Þ m If the interest rate is not constant, we define the effective annual interest rate as the interest earned over a given year divided by the amount invested at the year’s

14

Chapter 2

beginning. Thus the effective annual rate it during the tth year, i.e., in the period ½t 2 1; t; is defined by it ¼

aðtÞ 2 aðt 2 1Þ AðtÞ 2 Aðt 2 1Þ ¼ ; aðt 2 1Þ Aðt 2 1Þ

ð2:2:2Þ

assuming that the only principal invested is either 1 for the function aðtÞ or Að0Þ for the function AðtÞ: We typically assume that the period of time over which the interest rate is defined is a year, but this is not in any way required, it is merely a convention. We should, however, note that the unit of time over which the interest rate is defined should be the same as the one used for actual counting of time.

2.3 Measurement of interest If the effective rate of interest i is constant every year and the interest earned is reinvested in the account, we call the resulting accumulation function the compound interest accumulation function. Its form is aðtÞ ¼ ð1 þ iÞt ;

ð2:3:1Þ

i.e., it is exponential. If, however, the interest earned every year does not earn additional interest, the resulting form of the accumulation function aðtÞ ¼ 1 þ ti

ð2:3:2Þ

is termed the simple interest accumulation function. The two functions are equal for t ¼ 0 or t ¼ 1; and the compound interest function increases much more rapidly for t . 1: However, it should be noted that for 0 , t , 1 simple interest actually exceeds compound interest. Under compound interest, the expression 1 þ i is called the accumulation factor, and equals the accumulated value of a monetary unit after 1 year. Its reciprocal, v¼

1 ; 1þi

ð2:3:3Þ

is called the discount factor, and is the present value of a monetary unit paid a year from now. Note also that vt ; the t-year discount factor, is the present value of a monetary unit paid t years from now (t need not be an integer). The effective annual rate of discount in the year t, dt, is defined as the interest earned over a year divided by the accumulated amount at the year-end. In the case of the compound interest accumulation function, the effective rate of discount is constant and written as d. We have then d¼

að1Þ 2 að0Þ aðtÞ 2 aðt 2 1Þ i ¼ ¼ : að1Þ aðtÞ 1þi

ð2:3:4Þ

15

Fundamental concepts of theory of interest

Note that vþd ¼

1 i 1 þ ¼ 1; v ¼ 1 2 d; d ¼ 1 2 v; and vi ¼ i ¼ d: 1þi 1þi 1þi

It is also interesting to observe that if i ¼ 1=n for an integer n, then d ¼ 1=ðn þ 1Þ: The accumulation function produced by a constant rate of discount is aðtÞ ¼ ð1 2 dÞ2t ¼

1 : ð1 2 dÞt

ð2:3:5Þ

Two measurements of interest are said to be equivalent if for any amount of principal invested for any length of time they yield equal accumulated values at any time. It can be shown easily that two measurements of interest are equivalent if, and only if, they produce the same accumulation function. The following identities hold for equivalent measurements of interest: !m 1 1 iðmÞ 1þi¼ ¼ ¼ 1þ : ð2:3:6Þ v 12d m For the nominal annual interest rate iðmÞ compounded m times a year, the number m need not be an integer, as the formula works equally well for an annual rate compounded, e.g., every 8 months, with m ¼ 1:5 in such a case. Nominal rates of discount can also be considered. Constant nominal annual rate of discount compounded m times a year is denoted by d (m), and the accumulation function produced by it is !2mt dðmÞ ð2:3:7Þ að t Þ ¼ 1 2 m so that the following holds for equivalent measurements of interest: !m !2r 1 1 iðmÞ dðrÞ ¼ 1þ ¼ 12 1þi¼ ¼ v 12d m r

ð2:3:8Þ

(we are using a different notation for the compounding frequency of nominal interest and nominal discount to indicate that the two need not be equal). We also have ðmÞ

12

d ¼ v1=m ; m

iðmÞ d m ¼ ; m iðmÞ 1þ m ðmÞ

v1=m ·

iðmÞ d ðmÞ ¼ : m m

ð2:3:9Þ

As the compounding frequency for a deposit earning a fixed annual nominal interest rate i (m) increases, the amount of interest earned in a year increases, but only to a

16

Chapter 2

certain upper bound. The accumulated value at the end of 1 year will be: lim

m!1

iðmÞ 1þ m

!m ¼ lim

m!1

1þ

1 m=iðmÞ

m=iðmÞ !iðmÞ

ðmÞ

¼ ei :

ð2:3:10Þ

We obtain the same kind of limit result when working with compound discount. The limiting nominal annual interest rate is said to be compounded continuously. In actuarial literature, it is commonly called the force of interest and denoted by d. The accumulation function given by it is aðtÞ ¼ edt :

ð2:3:11Þ

We have the following identities for equivalent measurements of interest: !m !2r 1 1 iðmÞ dðrÞ ¼ 1þ ¼ 12 ¼ ed : ð2:3:12Þ 1þi¼ ¼ v 12d m r It should be noted that for equivalent measurements of interest we have i . ið2Þ . ið3Þ . · · · . d . · · · . dð3Þ . dð2Þ . d:

ð2:3:13Þ

In general, the force of interest can vary over time and in such general case it is defined as

dt ¼

a0 ðtÞ d A0 ðtÞ ¼ lnðaðtÞÞ ¼ : aðtÞ dt AðtÞ

ð2:3:14Þ

Note that Aðt þ DtÞ 2 AðtÞ < AðtÞ · dt · Dt:

ð2:3:15Þ

Under varying force of interest aðtÞ ¼ exp

ðt 0

ds ds :

In particular, if

dt ¼ c

f 0 ðtÞ f ðtÞ

for some function f ðtÞ then aðtÞ ¼

f ðtÞ f ð0Þ

c :

ð2:3:16Þ

Fundamental concepts of theory of interest

17

Also, if we move money forward in time (accumulate) from time t1 to time t2, with t2 . t1 ; we multiply it by ðt 2 aðt1 Þ ¼ exp 2 ds ds : aðt2 Þ t1 Sometimes it is useful to know the rate of change, i.e., the derivative, of one measurement of interest with respect to another. This is particularly important when considering the derivatives of financial assets’ prices with respect to the interest rate or the force of interest. We have, e.g., dd d 1 lnð1 þ iÞ ¼ ¼ v ¼ 1 2 d; ¼ di 1þi di

ð2:3:17Þ

dv d 1 1 ¼ ¼2 ¼ 2v2 ; di di 1 þ i ð1 þ iÞ2

ð2:3:18Þ

dvt ¼ 2tvtþ1 ; di

ð2:3:19Þ

dd dð1 2 vÞ ¼ ¼ v2 ¼ ð1 2 dÞ2 : di di

ð2:3:20Þ

and

Exercises Exercise 2.3.1 May 2003 Society of Actuaries (SOA)/Casualty Actuarial Society (CAS) Course 2 Examination, Problem No. 1. Bruce deposits 100 into a bank account. His account is credited interest at a nominal rate of interest i convertible semiannually. At the same time, Peter deposits 100 into a separate account. Peter’s account is credited interest at a force of interest of d. After 7.25 years, the value of each account is 200. Calculate ði 2 dÞ: Solution Bruce’s account after 7.25 years is worth: 100ð1 þ i=2Þ2£7:25 ¼ 200: Therefore, 1 þ i=2 ¼ 21=14:5 so that i ¼ 9:79285%: Peter’s account after 7.25 years is worth: 100e7:25d ¼ 200: Thus, 7:25d ¼ ln 2 and d ¼ 9:56065%: We conclude that ði 2 dÞ ¼ 0:2322%: A Exercise 2.3.2 May 2003 SOA/CAS Course 2 Examination, Problem No. 12. Eric deposits X into a savings account at time 0, which pays interest at a nominal rate of i, compounded semiannually. Mike deposits 2X into a different savings account at time 0, which pays simple interest at an annual rate of i. Eric and Mike earn the same amount of interest during the last 6 months of the 8th year. Calculate i.

18

Chapter 2

Solution Eric’s interest during the last 6 months of the 8th year is: i 15 i X 1þ · : 2 2 Mike’s interest in the last 6 months of the 8th year is 2X · ði=2Þ: Therefore i 15 i i i 15 X 1þ · ¼ 2X · ) 1 þ ¼2 2 2 2 2 so that i ¼ 9.46%.

A

Exercise 2.3.3 May 2003 SOA/CAS Course 2 Examination, Problem No. 50. Jeff deposits 10 into a fund today and 20 fifteen years later. Interest is credited at a nominal discount rate of d compounded quarterly for the first 10 years, and at a nominal interest rate of 6% compounded semiannually thereafter. The accumulated balance in the fund at the end of 30 years is 100. Calculate d. Solution Writing the equation of value we have d 240 10 1 2 ð1:03Þ40 þ 20ð1:03Þ30 ¼ 100: 4 This implies d 240 d 10 1 2 ¼ 15:77; 1 2 ¼ 0:98867052 4 4 and therefore d ¼ 0:0453:

A

Exercise 2.3.4 November 2001 SOA/CAS Course 2 Examination, Problem No. 1. Ernie makes deposits of 100 at time 0, and X at time 3. The fund grows at a force of interest

dt ¼

t2 ; t . 0: 100

The amount of interest earned from time 3 to time 6 is X. Calculate X. Solution The accumulation function is ðt 2 t 3 3 aðtÞ ¼ exp s =100 ds ¼ es =300l0 ¼ et =300 : 0

19

Fundamental concepts of theory of interest

The amount in the account at time 3 is 3

100að3Þ ¼ 100 e3 =300 ¼ 100 e0:09 ¼ 109:41743: Thus between times 3 and 6 the accumulation process can be summarized by the equation 3

3

ð100 e0:09 þ XÞe6 =30023 =300 2 ð100 e0:09 þ XÞ ¼ X: This is a linear equation that solves to 96.025894 ¼ 0.1223894X, and then X ¼ 784.59. A

2.4 Cash flows In practice, deposits are made more frequently than just at the beginning of the period considered. Withdrawals can be made, as well. Consider an account in which an amount of C0 is initially invested, and at the end of the kth year, additional amount of gk ; k ¼ 1; …; n is invested (the amounts could, in general, be positive or negative). Such a combination of payments will be termed a cash flow C0, g1 ; …; gn of this account. Calculation of the combined value of such cash flows is a common practical financial problem. In general, when we value a cash flow or a set of cash flows, its value today is called the present value, while its value at some future time is termed the accumulated value. If the value is calculated at a time somewhere in between now and the future time horizon, it is called the current value. Let Ck be the account balance at the end of the kth year. Then Ck ¼ Ck21 þ iCk21 þ gk ;

k ¼ 1; …; n:

After a simple rearrangement of the above formula we obtain Cn ¼ ð1 þ iÞn C0 þ

n X

ð1 þ iÞn2k gk :

ð2:4:1Þ

k¼1

The formula (2.4.1) basically tells us that the balance in the account at the end of the nth year will be the accumulated value of the initial deposit plus the accumulated value of all additional deposits made up to the time n. Example 2.4.1 Suppose that we invest $1000 a year at the beginning of every year over a span of 3 years. Assume also that the effective annual interest rate is 10% throughout this period of time. Interest earned is compounded annually, i.e., it is added to the account at the end of each year. The amount accumulated after

20

Chapter 2

3 years will be C3 ¼ ð1:10Þ3 · 1000 þ ð1:1Þ2 · 1000 þ ð1:1Þ · 1000 ¼ 3641:00:

A

Using the discount factor v, we can also rewrite (2.4.1) as vn Cn ¼ C0 þ

n X

vk gk :

ð2:4:2Þ

k¼1

Example 2.4.2 The present value at time 0 of the cash flow described in Example 2.4.1 is: 1000 þ 1000ð1:1Þ21 þ 1000ð1:1Þ22 ¼ 2735:54:

A

Now assume that the payments made into an account are made continuously at a rate of gðtÞ; a function of time t. This means that the payment amount in the infinitesimal period of time ðt; t þ dtÞ is equal gðtÞdt: Let CðtÞ be the account value at time t. Assume also that the interest is compounded continuously with the force of interest equal to dt ¼ dðtÞ: This means that if at the time t the account balance is C, then after the infinitesimal period of time ðt; t þ dtÞ the account value will increase by C dðtÞdt (assuming dðtÞ is positive; if dðtÞ is negative, this is a decrease). The total increase in the account value in the time period ðt; t þ dtÞ is therefore equal to dCðtÞ ¼ CðtÞdðtÞdt þ gðtÞdt:

ð2:4:3Þ

This gives us a differential equation C0 ðtÞ ¼ CðtÞdðtÞ þ gðtÞ:

ð2:4:4Þ

Ð By multiplying both sides of (2.4.4) by exp 2 t0 dðsÞ ds we arrive at the following ðt ðt C0 ðtÞexp 2 dðsÞds 2 CðtÞdðtÞexp 2 dðsÞds 0

0

ðt ¼ gðtÞexp 2 dðsÞds :

ð2:4:5Þ

0

We now note left-hand side of (2.4.5) is the derivative of the function Ð that the CðtÞexp 2 t0 dðsÞds and by integrating both sides of (2.4.5) from 0 to t; and moving Cð0Þ to the right-hand side, we conclude that ðt ðt ðt CðtÞexp 2 dðsÞds ¼ Cð0Þ þ exp 2 dðsÞds gðtÞdt: 0

0

0

ð2:4:6Þ

21

Fundamental concepts of theory of interest

Observe that (2.4.6) is a continuous version of (2.4.2). Thus the present value of an amount Ð t CðtÞ at time 0 can be obtained by multiplying it by the discount factor exp 2 0 dðsÞds ; and that present value is equal to the sum of Cð0Þ and the integral of the discounted value of the continuous payment stream gðtÞ: We also note that by accumulating (2.4.6) we obtain ðt ðt ðt dðsÞds þ gðtÞexp dðsÞds dt: ð2:4:7Þ CðtÞ ¼ Cð0Þexp 0

0

t

In particular, the accumulated value at time t of a single payment C0 made at time t, with t . t; is: ðt dðsÞds : ð2:4:8Þ CðtÞ ¼ C0 exp t

If dðsÞ ¼ d is constant and the effective annual interest rate is i, (2.4.8) leads to

d ¼ lnð1 þ iÞ

ð2:4:9Þ

Example 2.4.3 Assume that in a span of 2 years, initial deposit in an account has increased by 21%. The effective annual interest rate i is then calculated from the equation ð1 þ iÞ2 ¼ 1:21; and we conclude that i ¼ 0:10: Using (2.4.9) we can also calculate the force of interest equivalent to i ¼ 0:10 as d ¼ 0:09531: A One practical problem in dealing with cash flows is the question of an interest rate that makes certain cash flows equivalent. For example, suppose that we have 30 monetary units paid every year at the end of the year for 4 years in return for 100 paid today. What is the interest rate that makes these two equal? It is called the internal rate of return (IRR) or the dollar-weighted rate of return. In order to find that interest rate i, we set up the equation 100 ¼

30 30 30 30 þ þ þ : 1þi ð1 þ iÞ2 ð1 þ iÞ3 ð1 þ iÞ4

ð2:4:10Þ

If the payments are level, most financial calculators can find the interest rate. The problem can be also solved using the Solver function in Microsoft Excel software. The solution is i < 7:71%: The IRR is also called the yield rate for the cash flow analyzed. It is important to note that finding the yield rate involves solving a polynomial equation, and such an equation may have multiple solutions, no solution, or solutions, which are complex numbers. There are situations when a unique real solution exists: if all outflows occur before all inflows or if the cash outflows can be treated as a ‘deposit’ and outflows as ‘withdrawals’, with the resulting account balance remaining positive throughout the period under consideration [40].

22

Chapter 2

There is also a problem with the use of the IRR as a tool for evaluating a project in which the outflows are invested and from which the inflows are returns—using IRR implies that cash flows received from the project are reinvested at that IRR, which may or may not (usually is not) be possible as one has to generally reinvest at the prevailing market rates. Note that if two cash flows are equivalent at one point in time, they are equivalent at any other time. Also, a value of a linear combination of cash flows is a linear combination of values of individual cash flows. Finally, if a value VðtÞ is established at time t then the value VðsÞ at any other time may be calculated as: VðsÞ ¼ vt2s VðtÞ: One important consequence is also that you can compare cash flows or calculate their values, at various points in time, and it is often convenient to pick a different reference date than just today or a future horizon. Exercises Exercise 2.4.1 November 2001 SOA/CAS Course 2 Examination, Problem No. 24. David can receive one of the following two payment streams: (i) 100 at time 0, 200 at time n, and 300 at time 2n, (ii) 600 at time 10. At an annual effective interest rate of i, the present values of the two streams are equal. Given that vn ¼ 0:75941; determine i. Solution As we know, the two streams are equal, therefore 100 þ 200vn þ 300v2n ¼ 600v10 : We are given that vn ¼ 0:75941; and using this, we obtain 100 þ 151:882 þ 173:10 ¼ 600v10 : This solves to v10 ¼ 0:708; so that i < 3:50%:

A

Exercise 2.4.2 November 2001 SOA/CAS Course 2 Examination, Problem No. 28. Payments are made to an account at a continuous rate of 8k þ tk; where 0 # t # 10. Interest is credited at a force of interest

dt ¼

1 : 8þt

After 10 years, the account is worth 20 000. Calculate k.

23

Fundamental concepts of theory of interest

Solution Note that ð 10 exp n

1 dt 8þt

10

¼ elnð8þtÞln ¼

18 : 8þn

Therefore, by (2.4.7), 20 000 ¼

ð 10

kð8 þ tÞ

0

18 dt ¼ 18ktlt¼10 t¼0 ¼ 180k: 8þt

This implies that k¼

20 000 : 180

A

2.5 Interest measurement in a fund 2.5.1 Discrete-time model In the practice of insurance companies a problem of interest measurement in a fund over a year arises naturally. First of all, it is used to reflect the assets’ increase by showing the investment efficiency. But it is also an important indicator for the participants of defined contribution pension plans for the increase of their share in the plan assets is usually defined via the rate of return of the participation (accounting) unit. Let us introduce the following notation At ¼ fund balance at time t [ ½0; 1; I ¼ interest earned during the year, between t ¼ 0 and t ¼ 1; Ct ¼ principal contributed (if positive) or withdrawn (if negative) at time t, t [ ½0; 1; P C ¼ t Ct ¼ total net contributions to the fund (if negative, this is an indication of net withdrawals), i ¼ effective interest rate between t ¼ 0 and t ¼ 1: We have A1 ¼ A0 þ C þ I;

and

I ¼ iA0 þ

X

Ct ðð1 þ iÞ12t 2 1Þ:

0#t#1

In order to find i we would have to solve the second equation for it, and the first equation would actually be only the source of the value of I. This is not an easy problem, and in practice one often uses a simple interest approximation ð1 þ iÞ12t 2 1 < 1 þ ð1 2 tÞi 2 1 ¼ ð1 2 tÞi;

24

Chapter 2

which allows us to obtain a solution i<

A0 þ

I X

Ct ð1 2 tÞ

:

ð2:5:1Þ

0#t#1

The denominator of the formula for i so obtained is commonly called the exposure associated with i. It represents the net amount-time exposed to earning interest. The approximation of the IRR obtained with this simple interest assumption is what is also commonly called the dollar-weighted rate of return, and this concept will mean such an approximation in this section. If the net principal contributions occur at time k on the average, i.e., k¼

X 0#t#1

t·

Ct C

then we can use a simpler approximation i<

I : A0 þ Cð1 2 kÞ

It would seem that these approximations provide a good practical measure of the return earned by a fund. But investment professionals do not use dollar-weighted rate of return because it is very sensitive to the timing of cash flows and such timing is often beyond their control, thus should not be used to measure their performance. Instead, investment performance is measured with the time-weighted rate of return, which is the return earned by a participation (accounting) unit of a fund under consideration. If we write At for the account balance just before the contribution Ct is made, then the effective rate of return between two contributions Cs and Ct, with s , t; is given by 1 þ jt ¼

At : As þ Cs

If the contributions are made at times t1 ; t2 ; …; tm then the time-weighted rate of return for the entire year is the number i such that 1 þ i ¼ ð1 þ jt1 Þð1 þ jt2 Þ· · ·ð1 þ jtm Þ:

ð2:5:2Þ

Note that the amounts and timing of contributions do not affect the time-weighted rate of return. 2.5.2 Continuous-time model Let us denote the assets at time t by AðtÞ and assume that the assets are divided into kðtÞ equal participation (accounting) units, with participants’ property rights expressed by ownership of a certain number of those units. Let wðtÞ denote value

25

Fundamental concepts of theory of interest

of the participation unit at t, so that AðtÞ ¼ kðtÞwðtÞ: Assume that both kð · Þ and wð · Þ are differentiable functions. The infinitesimal relative change of the assets of the fund during the time interval ðt; t þ dtÞ is dAðtÞ dkðtÞ dwðtÞ ¼ þ : AðtÞ kðtÞ wðtÞ The first summand corresponds to the allocation of units (e.g., making new contributions, appearing of new participants or disappearing of old ones). The second summand, describing a pure investment effect at the fund, is equal to dðtÞdt; where dðtÞ is the force of interest rate at t. Hence dðtÞdt has two interpretations: it is the infinitesimal rate of return for the accounting unit in the fund and, simultaneously, it is the infinitesimal rate of return for the assets of this fund, due to the pure investment effects (we shall use this duality in Section 2.6 to define the average return rate for a group of pension funds). Let rðs; uÞ denote the rate of return of the fund during a given time period ½s; u: Clearly rðs; uÞ ¼ exp

ð u

dðtÞdt 2 1:

ð2:5:3Þ

s

The rate rðs; uÞ informs a plan participant what would be his return at time u if he/she bought one accounting unit of the fund at time s. If t0 ¼ s , t1 , · · · , tm ¼ u are such that dðtÞ ¼ dk ; for t [ ½tk21 ; tk Þ; k ¼ 1; …; m; then " # m m Y X dk ðtk 2 tk21 Þ ¼ exp½dk ðtk 2 tk21 Þ rðs; uÞ þ 1 ¼ exp k¼1

¼

m Y

ð1 þ jtk Þ;

k¼1

ð2:5:4Þ

k¼1

where 1 þ jtk ¼ exp½dk ðtk 2 tk21 Þ: Observe that (2.5.4) concurs with (2.5.2). So (2.5.2) is a discrete-time approximation of (2.5.3). 2.5.3 Allocation of interest to accounts One more practical problem encountered by insurance and investment companies, as well as trust divisions of banks, is allocation of interest to accounts. If all participants get pro-rate share of the fund investment performance (this is the standard for mutual funds), the following method is commonly used.

26

Chapter 2

† Participation Unit Method. Each account is credited accordingly with a common interest rate per participation unit (calculated by formula (2.5.2) or (2.5.3)) times the initial unit value times the number of units deposited on the account. If, however, assets of various customers are commingled, and they do not all get pro-rate share of the fund’s performance (which happens for insurance companies, stable value funds, or commingled trust accounts), Participation Unit Method is not applicable. There are two standard methods of allocation in this case: † Portfolio Method. Each account is credited with an average rate based on earnings of the entire fund for the period. The drawback of this method is that when interest rates are rising, there is a disincentive for new deposits, as fixed coupon bonds in the portfolio bought with previous deposits are now declining in value, and this negative performance contribution is received by new money, as well. † Investment Year Method. ‘New money’ (i.e., new deposits) receive a different rate than ‘old money’; this rate is based on current market conditions. However, once investments stay in the account for a specified period of time (for insurance companies offering fixed annuities, this is typically 5 – 7 years or the period during which a surrender charge is assessed), they eventually are credited the portfolio rate, just as the ‘old money’. The transition period from the ‘new money rate’ to the portfolio rate can be handled either through a declining index system (the rate is gradually changed from the ‘new money’ rate to the portfolio rate) or the fixed index system (the rate credited remains the ‘new money’ rate until the end of period during which this money is treated differently than the entire portfolio).

Exercises Exercise 2.5.1 A fund has 500 000 participation units each of value 1 monetary unit. After 1 year the assets grow to 550 000 monetary units and additional 100 000 of participation units are sold at the valuation price. At the end of second year the assets of the fund are valued as 706 200 monetary units. What is the 2-year average rate of return rð0; 2Þ of that fund? Solution The rate of return during the first year is rð0; 1Þ ¼ 10%: The rate rð1; 2Þ during the second year is rð1; 2Þ ¼

706 200 2 ð550 000 þ 100 000 · 1:1Þ ¼ 0:07: 550 000 þ 100 000 · 1:1

Fundamental concepts of theory of interest

27

Using (2.5.2), we get rð0; 2Þ þ 1 ¼ ð1 þ 0:1Þð1 þ 0:07Þ; hence rð0; 2Þ ¼ 17:7%:

A

Exercise 2.5.2 May 2003 SOA/CAS Course 2 Examination, Problem No. 17. An association had a fund balance of 75 on January 1 and 60 on December 31. At the end of every month during the year, the association deposited 10 from membership fees. There were withdrawals of 5 on February 28, 25 on June 30, 80 on October 15, and 35 on October 31. Calculate the dollar-weighted rate of return for the year. Solution The standard approach for the dollar-weighted rate of return calculations is to assume simple interest. We have: † total deposits ¼ 120, † total withdrawals ¼ 145, † investment income ¼ 60 þ 145 – 120– 75 ¼ 10. Therefore the dollar-weighted rate of return is 10 10 ¼ 11%: ¼ 1 11 10 6 2:5 2 90:833 þ ··· þ 80 2 35 75 þ ·10 2 5 2 25 2 12 12 12 12 12 12

A

Exercise 2.5.3 May 2003 SOA/CAS Course 2 Examination, Problem No. 30. You are given the following information about interest rates credited to an account of customers of a certain insurance company issuing fixed deferred annuities, in which y is the calendar year of the original investment, and iy1 ; iy2 ; iy3 ; iy4 ; iy5 are investment year interest rates credited in the years following the investment, with the ultimate portfolio rate iyþ5 becoming effective 5 years after the original investment i1992 ¼ 8:25%; i1992 ¼ 8:25%; i1992 ¼ 8:4%; i1992 ¼ 8:5%; 1 2 3 4 i1992 ¼ 8:5%; i1992þ5 ¼ 8:35%; 5 i1993 ¼ 8:5%; i1993 ¼ 8:7%; i1993 ¼ 8:75%; i1993 ¼ 8:9%; 1 2 3 4 i1993 ¼ 9%; i1993þ5 ¼ 8:6%; 5

28

Chapter 2

i1994 ¼ 9%; i1994 ¼ 9%; i1994 ¼ 9:1%; i1994 ¼ 9:1%; 1 2 3 4 i1994 ¼ 9:2%; i1994þ5 ¼ 8:85%; 5 i1995 ¼ 9%; i1995 ¼ 9:1%; i1995 ¼ 9:2%; i1995 ¼ 9:3%; 1 2 3 4 i1995 ¼ 9:4%; i1995þ5 ¼ 9:1%; 5 i1996 ¼ 9:25%; i1996 ¼ 9:35%; i1996 ¼ 9:5%; i1996 ¼ 9:55%; 1 2 3 4 i1996 ¼ 9:6%; i1996þ5 ¼ 9:35%; 5 i1997 ¼ 9:5%; i1997 ¼ 9:5%; i1997 ¼ 9:6%; i1997 ¼ 9:7%; 1 2 3 4 i1997 ¼ 9:7%; 5 i1998 ¼ 10:0%; i1998 ¼ 10:0%; i1998 ¼ 9:9%; i1998 ¼ 9:8%; 1 2 3 4 i1999 ¼ 10:0%; i1999 ¼ 9:8%; i1999 ¼ 9:7%; 1 2 3 i2000 ¼ 9:5%; i2000 ¼ 9:5%; 1 2 i2001 ¼ 9:0%: 1 A person deposits 1000 on January 1, 1997. Let the following be the accumulated value of the 1000 on January 1, 2000: P : under the investment year method, Q : under the portfolio yield method, R : where the balance is withdrawn at the end of every year and reinvested at the new money rate. Determine the ranking of P, Q, and R. Solution We have P ¼ 1000 · 1:095 · 1:095 · 1:096 ¼ 1314:13; Q ¼ 1000· 1:0835· 1:086· 1:0885 ¼ 1280:82; R ¼ 1000 · 1:095 · 1:10 · 1:10 ¼ 1324:95: Therefore R . P . Q:

A

29

Fundamental concepts of theory of interest

2.6 Interest measurement in a group of funds In this section we investigate below an index formula (2.6.3), which accounts for the rate of return achieved by the group of funds. We will show that this index is martingale-fair (as well as it satisfies several other axioms). The martingale fairness axiom is based on the Arbitrage Free Pricing Theory due to Ross [62]. According to his Fundamental Theorem of Asset Pricing the securities market is arbitrage free if and only if there exists a risk-neutral probability measure. But then the discounted value of the portfolio corresponding to a self-financing strategy is a martingale under a risk-neutral probability measure (see Theorem 7.1.1 of Section 7.1 and the discussion following it). This shows a central role of martingale market models as well as the importance of the notion of martingale itself. Therefore we start this section with a definition of martingales. Let ðV; F ; PrÞ be a probability space and F ¼ {F 0 ; F 1 ; …} be a filtration, i.e., a nondecreasing stream of s-subalgebras of F . The filtration describes an increasing stream of possible information, available to investors on the market. A vector-valued stochastic process {SðtÞ; t ¼ 0; 1; 2; …} is called a martingale (with respect to the stream F) if: 1. St is Ft -measurable for every t ¼ 0; 1; …; 2. the following equality holds almost surely (a.s.): EðStþ1 lFt Þ ¼ St

ð2:6:1Þ

for every t ¼ 0; 1; … Analogously, a real-valued stochastic process {SðtÞ; t ¼ 0; 1; 2; …} is called supermartingale (submartingale) if it is Ft -measurable and EðStþ1 lFt Þ # St ð$Þ

a:s:

ð2:6:2Þ

for all t ¼ 0; 1; … In the latter definition we have restricted our attention to realvalued processes in order to avoid complications concerned with comparing vectors. For the sake of this book, however, the above definition is sufficiently general. Observe also that one can equivalently verify if (2.6.1) or (2.6.2) holds with EðSt lFt Þ; for any natural t . t; in place of EðStþ1 lFt Þ; respectively. An important consequence of the Fundamental Theorem of Asset Pricing is that any index which reflects properly the value of the whole market should inherit the martingale property. In fact, if it would not, one could replicate the index using the Arrow –Debreu securities and construct a self-financing trading strategy, which is an arbitrage opportunity. For that reason any index properly reflecting investment efficiency of the group of investment funds should behave like a martingale if the whole market is a martingale. Following [30] we call this property the martingale fairness axiom (shortly, the fairness axiom) and say shortly that the index is fair. To enlighten the name of the axiom a bit, recall that

30

Chapter 2

a random game with two players is called fair if the expectation of its payoff is zero. Martingale fairness would mean that a multiperiod game has, after each period, a payoff with a conditional expectation zero subject to a given result from the previous periods. This shows that the notion of a fair index is both mathematically consistent and has appropriate economic implications. Now let us consider a group of n pension funds which report on their investment results using accounting (participation) units. Let Ai ðtÞ denote the balance of the ith pension fund at time t. The problem we are dealing with in this section is how to define an average rate of return to meet several requirements in a discrete-time stochastic model. To make the analysis simpler, we start with deriving the formula for the average rate of return in a continuous-time nonrandom financial market model. Next we introduce a simple stochastic model with discrete time. This allows us to derive a proper definition of the average rate of return. The definition we arrive at is as follows: 0 rðs; uÞ ¼

uY 21 B B

n X

B1 þ B t¼s @

i¼1

1 ri ðt; t þ 1ÞAi ðtÞ C C C 2 1: n C X A A ðtÞ

ð2:6:3Þ

i

i¼1

Afterwards, we show that rðs; uÞ is martingale-fair as well as is satisfying several other axioms. Finally, we investigate in detail the problem of merger of several funds. The results of this section are mainly based on [29,30]. The axiomatic approach to index numbers was developed, e.g., in [19,22 – 24]. 2.6.1 Nonrandom continuous-time model Consider a group of n pension (or investment) funds, which use accounting (or participation) units to report on their investment results. Denote by ki ðtÞ; i ¼ 1; 2; …; n; the number of all units possessed by the participants of the ith fund at the moment t, and by wi ðtÞ the value of the ith fund unit at the moment t. The value wi ðtÞ is established by dividing the total assets of the ith fund, Ai ðtÞ; by the number of the units ki ðtÞ: The assets Ai ðtÞ can change due to the change of ki ðtÞ or of the unit’s value wi ðtÞ according to the formula Ai ðtÞ ¼ ki ðtÞwi ðtÞ: Let AðtÞ denote the total assets of the group at the moment t [ ½s; u; i.e., AðtÞ ¼

n X i¼1

Ai ðtÞ ¼

n X i¼1

ki ðtÞwi ðtÞ:

31

Fundamental concepts of theory of interest

Assume that both ki ð · Þ and wi ð · Þ are differentiable functions. Then n n X X d d d AðtÞ ¼ ki ðtÞ wi ðtÞ þ ki ðtÞ wi ðtÞ: dt dt dt i¼1 i¼1 After rescaling (2.6.4) by total assets, we get n n X X d d d k ðtÞ w ðtÞ ki ðtÞ wi ðtÞ i i AðtÞ dt dt dt ¼ i¼1 X þ i¼1 : n n X AðtÞ ki ðtÞwi ðtÞ ki ðtÞwi ðtÞ i¼1

ð2:6:4Þ

ð2:6:5Þ

i¼1

The first sum on the right side of (2.6.5) corresponds to the influence of fluctuations of the number of units on the total assets value at each fund; the second sum corresponds to the influence of fluctuations of the units values. The second sum corresponds only to the effects of investing the assets, not to allocating clients between the funds. This is exactly what we are interested in when defining the average return rate. The second sum on the right side of (2.6.5), denoted in the sequel by d(t), may be written as

dðtÞ ¼

n X i¼1

n X ki ðtÞwi ðtÞ Ai ðtÞ d ðtÞ ¼ d ðtÞ; i n X AðtÞ i i¼1 ki ðtÞwi ðtÞ i¼1

where

di ðtÞ ¼

d ½ln wi ðtÞ; dt

for i ¼ 1; 2; …; n;

denotes the rate of return, expressed as the force of interest, achieved by the ith fund. Similarly as in Section 2.5, the infinitesimal rate of return for the group of funds during the time interval ðt; t þ dtÞ is equal to

dðtÞdt ¼

n X Ai ðtÞ d ðtÞdt; AðtÞ i i¼1

and the weighted average rate of return of the group, during a given time period ½s; u; is "ð # n u X Ai ðtÞ rðs; uÞ ¼ exp di ðtÞdt 2 1: ð2:6:6Þ s i¼1 AðtÞ Note the analogy between the formulae (2.6.6) and (2.5.3) in the previous section. In the next sections we shall show that (2.6.3) is a discrete version of (2.6.6), which establishes a relationship analogous to that between (2.5.2) and (2.5.3).

32

Chapter 2

2.6.2 A discrete-time stochastic model Throughout this section we assume that ki ðtÞ and wi ðtÞ are observed in discrete moments t ¼ 0; 1; 2; … Since we investigate investment performance in an arbitrary time period, our analysis is quite general, as we can choose the time to be a year, a month, a day, or even a nanosecond. The problem is to find a proper definition of the average rate of return for the group of funds during a given time period ½s; u: Let us denote it by rðs; uÞ: Clearly, rðs; uÞ should take values in ð21; 1Þ: We shall use the following state-variables: cj ðtÞ ¼ price of asset j of the financial market at time t, j ¼ 1; …; N; uij ðtÞ ¼ number of units of asset j possessed by the ith fund at time t, i ¼ 1; …; n; j ¼ 1; …; N; wi ðtÞ ¼ value of participation unit of the ith fund at time t, ki ðtÞ ¼ number of units of the ith fund at time t, di ðtÞ ¼ the amount of contributions of the ith fund’s liabilities minus the drawdown amount of ith fund’s liabilities at time t, P dðtÞ ¼ ni¼1 di ðtÞ; Ai ðtÞ ¼ value P of ith fund’s assets, AðtÞ ¼ ni¼1 Ai ðtÞ; Api ðtÞ ¼ Ai ðtÞ=AðtÞ—the relative value of assets of the ith fund at time t, Df ðxÞ ¼ f ðx þ 1Þ 2 f ðxþÞ: We assume that: † † † †

All investments are infinitely divisible. There are no transaction costs or taxes and the assets pay no dividends. Plan member does not pay for allocation of his/her wealth. There is no consumption of funds.

Let ðV; F ; PrÞ be a probability space. Let {F ¼ F 0 ; F 1 ; F 2 ; …} be a filtration, i.e., each F t is a s-algebra of subsets of V with F 0 # F 1 # F 2 # · · · # F : Without loss of generality, we assume F 0 ¼ {Y; V}: The filtration F describes how information is revealed to the investors. Given a moment t, we have wi ðtÞki ðtÞ ¼ ui1 ðtÞc1 ðtÞ þ · · · þ uiN ðtÞcN ðtÞ; i ¼ 1; …; n:

ð2:6:7Þ

Here and subsequently, the symbol X ¼ Y means that the random variables X; Y are defined on ðV; F ; PrÞ and PrðX ¼ YÞ ¼ 1: We assume that: † cj ðtÞ is F t -measurable for each j; t: † Each random variable wi ðtÞ is adapted to F; which means that wi ðtÞ is F t -measurable for each i; t:

33

Fundamental concepts of theory of interest

† Both ki ðtÞ and uij ðtÞ are adapted to F; i.e., we are allowing the investor to buy (sell) units after the values ci ðtÞ are observed. † Split of units is allowed at any time. The new price of one unit and the number of units of ith fund are denoted by wi ðtþÞ and ki ðtþÞ; respectively. Clearly, wi ðtþÞki ðtþÞ ¼ wi ðtÞki ðtÞ; i ¼ 1; …; n:

ð2:6:8Þ

At time t þ 1; market prices of financial assets change and we obtain wi ðt þ 1Þki ðtþÞ ¼ ui1 ðtÞc1 ðt þ 1Þ þ · · · þ uiN ðtÞcN ðt þ 1Þ; i ¼ 1; …; n: ð2:6:9Þ From (2.6.7) – (2.6.9) we get ki ðtþÞðwi ðt þ 1Þ 2 wi ðtþÞÞ ¼

N X

uij ðtÞðcj ðt þ 1Þ 2 cj ðtÞÞ

ð2:6:10Þ

j¼1

for i ¼ 1; …; n: Moreover, any member of the ith fund may reallocate his or her ðWÞ wealth. The members withdraw ki ðt þ 1Þwi ðt þ 1Þ from the ith fund and invest Pn ðWÞ ðIÞ kj ðt þ 1Þwj ðt þ 1Þ of the amount ðt þ 1Þwi ðt þ 1Þ in the jth fund, i¼1 ki ðWÞ

ðIÞ

where ki ðt þ 1Þ and ki ðt þ 1Þ are F tþ1 -measurable random variables for each i. At time t þ 1; the stream of liability payments also changes the balance of the ith fund. As a consequence the number of units of the ith fund changes from ki ðtþÞ to ki ðt þ 1Þ ðWÞ

ki ðt þ 1Þwi ðt þ 1Þ ¼ ki ðtþÞwi ðt þ 1Þ 2 ki

ðt þ 1Þwi ðt þ 1Þ

ðIÞ

þ ki ðt þ 1Þwi ðt þ 1Þ þ di ðt þ 1Þ; i ¼ 1; 2; …; n;

ð2:6:11Þ

where di ðt þ 1Þ is assumed to be F tþ1 -measurable for each i; t: By summing equations (2.6.11) for i ¼ 1; 2; …; n; we obtain: n X

wi ðt þ 1Þðki ðt þ 1Þ 2 ki ðtþÞÞ ¼ dðt þ 1Þ:

ð2:6:12Þ

i¼1

After allocation of participants, the management of the ith fund rebalances the portfolio wi ðt þ 1Þki ðt þ 1Þ ¼ ui1 ðt þ 1Þc1 ðt þ 1Þ þ · · · þ uiN ðt þ 1ÞcN ðt þ 1Þ for i ¼ 1; …; n:

ð2:6:13Þ

34

Chapter 2

To summarize, our mathematical model of dynamics of a group of funds consists of the following stochastic difference equations: wi ðtÞki ðtÞ ¼

N X

uij ðtÞcj ðtÞ;

ð2:6:14Þ

wi ðtþÞki ðtþÞ ¼ wi ðtÞki ðtÞ;

ð2:6:15Þ

j¼1

ki ðtþÞDwi ðtÞ ¼

N X

uij ðtÞDcj ðtÞ;

ð2:6:16Þ

j¼1 ðIÞ

ðWÞ

wi ðt þ 1ÞDki ðtÞ ¼ ðki ðt þ 1Þ 2 ki

wi ðt þ 1ÞDki ðtÞ ¼

ðt þ 1ÞÞwi ðt þ 1Þ þ di ðt þ 1Þ;

N X

cj ðt þ 1ÞDuij ðtÞ;

ð2:6:17Þ ð2:6:18Þ

j¼1 ðIÞ

ðWÞ

where t ¼ 0; 1; 2; … and i ¼ 1; 2; …; n: In this model, functions uij ; ki and ki play a role of control variables. The model described above is also a base for a continuous-time model presented in [31]. 2.6.3 Deriving the formula for the average rate of return According to (2.6.3), the average rate of return in discrete-time model in the time period ½s; u is as follows: ! uY 21 n X p rðs; uÞ ¼ 1þ Ai ðtÞri ðt; t þ 1Þ 2 1; ð2:6:19Þ t¼s

i¼1

where ri ðt; t þ 1Þ is the return of the ith fund in the time interval ðt; t þ 1; i.e., ri ðt; t þ 1Þ ¼

wi ðt þ 1Þ 2 wi ðtþÞ : wi ðtþÞ

ð2:6:20Þ

For convenience, we put rðs; sÞ ¼ 0 for each s. We present now two arguments for establishing definition (2.6.19). The first one is based on the analysis of changes of total assets of funds. In fact it is a ‘discrete’ version of the argument used in Section 2.6.1 to derive formula (2.6.6). Observe that uY 21 AðuÞ 2 AðsÞ Aðt þ 1Þ 2 AðtÞ ¼ 1þ 2 1: AðsÞ AðtÞ t¼s

35

Fundamental concepts of theory of interest

Applying (2.6.8) and (2.6.12) we get Aðt þ 1Þ 2 AðtÞ ¼ ¼ ¼

n X i¼1 n X i¼1 n X

wi ðt þ 1Þki ðt þ 1Þ 2 wi ðt þ 1Þki ðt þ 1Þ 2

n X i¼1 n X

wi ðtÞki ðtÞ wi ðtþÞki ðtþÞ

i¼1

wi ðt þ 1Þðki ðt þ 1Þ 2 ki ðtþÞÞ

i¼1

þ

n X

ðwi ðt þ 1Þ 2 wi ðtþÞÞki ðtþÞ

i¼1

¼ dðt þ 1Þ þ

n X

Ai ðtÞ

i¼1

wi ðt þ 1Þ 2 wi ðtþÞ : wi ðtþÞ

Hence uY 21 n AðuÞ 2 AðsÞ dðt þ 1Þ X ¼ þ 1þ Api ðtÞri ðt; t þ 1Þ AðsÞ AðtÞ t¼s i¼1

! 2 1:

If we remove the influence of the amount of contribution of fund’s liabilities (by setting dðt þ 1Þ ¼ 0), then AðuÞ 2 AðsÞ ¼ rðs; uÞ; AðsÞ by (2.6.19). The second argument is as follows. Clearly rðt; t þ 1Þ ¼

n X

Apj ðtÞrj ðt; t þ 1Þ;

j¼1

Pn p where Apj ðtÞ ¼ Aj ðtÞ=AðtÞ: Since Apj ðtÞ $ 0 and j¼1 Aj ðtÞ ¼ 1; we have the following interpretation of the average return: rðt; t þ 1Þ is equal to the expected return of one monetary unit chosen at random from the assets of the group of funds at time t, i.e., rðt; t þ 1Þ ¼ E rJ ðt; t þ 1Þ ; where J is a random variable with the distribution PrðJ ¼ jÞ ¼ Apj ðtÞ; j ¼ 1; …; n: Moreover, for all nonnegative u; s; u . s; uY uY 21 21 rðs; uÞ ¼ ð1 þ rJt ðt; t þ 1ÞÞ 2 1; 1 þ E rJt ðt; t þ 1Þ 2 1 ¼ E t¼s

t¼s

where Js ; …; Ju21 are independent random variables such that PrðJt ¼ jÞ ¼ Apj ðtÞ; j ¼ 1; …; n; t ¼ 0; 1; 2; … This means that if we repeat at time s; s þ 1; …; u 2 1 the

36

Chapter 2

procedure of choosing independently and sequentially for one period one fund in order to reinvest one monetary unit plus interest earned so far, then our capital at time u will be a random variable K with EðKÞ ¼ ð1 þ rðs; uÞÞ: So, the expected rate of return in this scheme is equal to the average rate of return rðs; uÞ; i.e.,: EðK 2 1Þ ¼ rðs; uÞ: Definition (2.6.19) appears in Barber et al. [8] in a quite different context; the above analysis justifying the use of it as a definition of the average rate of return is taken from [30]. 2.6.4 Martingale-fairness of the average rate of return r Our main result states that r, defined by (2.6.3), is martingale-fair. Theorem 2.6.1 The index r is martingale-fair, i.e., if {ci ðtÞ; t ¼ 0; 1; 2; …} is an F-martingale for each i; then {rð0; tÞ; t ¼ 0; 1; 2; …} is an F-martingale. Moreover, if {ci ðtÞ; t ¼ 0; 1; 2; …} is an F-submartingale (respectively, F-supermartingale) for each i; then {rð0; tÞ; t ¼ 0; 1; 2; …} is an F-submartingale (respectively, F-supermartingale). Proof By definition, the random variable rð0; tÞ is F t -measurable for each t because, by assumption, both ki ðtÞ and wi ðtÞ are F t -measurable. Moreover, Api ðtÞ is F t -measurable, since wi ðtÞki ðtÞ Api ðtÞ ¼ X : n wi ðtÞki ðtÞ i¼1

Of course

Pn

p j¼1 Aj ðtÞ

¼ 1 and we have

! wi ðt þ 1Þ Eðrð0; t þ 1ÞlF t Þ ¼ ðrð0; tÞ þ 1ÞE F 2 1 wi ðtþÞ t i¼1 ! n X wi ðt þ 1Þ p F ¼ ðrð0; tÞ þ 1Þ Ai ðtÞE 2 1: wi ðtþÞ t i¼1 n X

Api ðtÞ

We show that wi ðt þ 1Þ F ¼1 E wi ðtþÞ t

37

Fundamental concepts of theory of interest

for each i: Recall that wi ðt þ 1Þki ðtþÞ ¼ ui1 ðtÞc1 ðt þ 1Þ þ · · · þ uiN ðtÞcN ðt þ 1Þ; wi ðtþÞki ðtþÞ ¼ wi ðtÞki ðtÞ ¼ ui1 ðtÞc1 ðtÞ þ · · · þ uiN ðtÞcN ðtÞ; for i ¼ 1; …; n: Since cðtÞ is an F-martingale, we get ! N X wi ðt þ 1Þ 1 E E uij ðtÞcj ðt þ 1ÞlF t F ¼ wi ðtþÞ t ki ðtþÞwi ðtþÞ j¼1

¼

N N X X 1 1 uij ðtÞEðcj ðt þ 1ÞlF t Þ ¼ u ðtÞcj ðtÞ ki ðtþÞwi ðtþÞ j¼1 ki ðtþÞwi ðtþÞ j¼1 ij

¼

ki ðtÞwi ðtÞ ¼ 1: ki ðtþÞwi ðtþÞ

The proof of the first part of the theorem is completed. The proof of the second part is analogous, so it will be omitted. A We will now give an example of an investment efficiency index, which is not martingale-fair. Example 2.6.1 In the Polish law regulations (see [73]) the following definition of the average return of a group of pension funds can be found 1

0 rPL ðs; tÞ ¼

C B n X B Ai ðsÞ 1 Ai ðtÞ C C; ri ðs; tÞB þ n n C BX X 2 @ i¼1 Aj ðsÞ Aj ðtÞ A j¼1

ð2:6:21Þ

j¼1

where by ri ðs; tÞ we denote the rate of return of the ith fund during a given time period ½s; t; i.e., ri ðs; tÞ ¼

wi ðtÞ 2 wi ðsÞ : wi ðsÞ

The average rate of return defined in Polish law in general is not a martingale provided the prices of assets are martingales. In fact, assume that s ¼ 0; ki ð0Þ ¼ ki ðtÞ ¼ k; wi ð0Þ ¼ 1; and uij ð0Þ ¼ uij ðtÞ ¼ uij with k; uij [ R for each i; j: After an

38

Chapter 2

elementary algebra we get n X n 1 X 1 rPL ð0; tÞ ¼ wi ðtÞ 2 1 þ 2n i¼1 2

ðwi ðtÞÞ2

i¼1 n X

; wi ðtÞ

i¼1

where wi ðtÞ ¼ ðui1 c1 ðtÞ þ · · · þ uiN cN ðtÞÞ=k: Since ðw1 þ · · · þ wn Þ2 # nðw21 þ · · · þ w2n Þ for all real wi ; we have 0X 1 n 2 ðwi ðtÞÞ C n 1 X 1 B B C Eðwi ðtÞÞ 2 1 þ EB i¼1 EðrPL ð0; tÞÞ ¼ C n A 2n i¼1 2 @ X wi ðtÞ i¼1

$2

n 1 1 X þ E wi ðtÞ ¼ 0 ¼ EðrPL ð0; 0ÞÞ; 2 2n i¼1

ð2:6:22Þ

and equality holds in (2.6.22) if and only if Prðw1 ðtÞ ¼ · · · ¼ wn ðtÞÞ ¼ 1: Suppose that uik – ujk for some i; j; k; and suppose c1 ðtÞ; …; cN ðtÞ are not linearly dependent, i.e., for any real a1 ; …; aN such that la1 l þ · · · þ laN l – 0; Prða1 c1 ðtÞ þ · · · þ aN cN ðtÞ ¼ 0Þ , 1: Then Prðw1 ðtÞ ¼ · · · ¼ wn ðtÞÞ , 1 and EðrPL ð0; tÞÞ . EðrPL ð0; 0ÞÞ: This means that {rPL ð0; tÞ; t ¼ 0; 1; 2; …} is not martingale-fair.

A

2.6.5 Axiomatic approach to the investment efficiency index In this section we formulate a list of properties which any properly defined average rate of return should possess. Some of them can be found in [40]. The average rate of return r meets all the demands. The proofs are straightforward so they will be omitted. Property 2.6.1 If the group consists of only the ith fund, then rðs; uÞ ¼

wi ðuÞ 2 wi ðsþÞ : wi ðsþÞ

The next property is a multiplication rule when time interval is divided into subintervals.

39

Fundamental concepts of theory of interest

Property 2.6.2 For every s , t , u 1 þ rðs; uÞ ¼ ð1 þ rðs; tÞÞð1 þ rðt; uÞÞ: The next property says that the average rate of return r is consistent with respect to aggregation of funds. Property 2.6.3 If funds are grouped, and if the average rates of return are calculated according to (2.6.3) over the time interval ½t; t þ 1Þ; then the average rate of return over the averages within the groups equals the average rate of return of all funds. Example 2.6.1 (continuation) It is easy to check that rPL does not possess Properties 2.6.2 and 2.6.3. For instance, we show that Property 2.6.3 is not satisfied. Suppose there are three funds with the rate of returns r1 ; r2 and r3 ; respectively, on a given time period ½t; t þ 1: Then 3 X 1 Ai ðtÞ A ðt þ 1Þ rPL ðt; t þ 1Þ ¼ ri þ i ; 2 AðtÞ Aðt þ 1Þ i¼1 where 3 X Aðt þ 1Þ ¼ Ai ðt þ 1Þ: i¼1

Let us merge the first fund with the second one. In this way there are two funds: the first one has the return 1 0 r1g ¼

C B 2 X 1 B Ai ðtÞ Ai ðt þ 1Þ C C; ri B þ C 2 2 X 2 B A @X i¼1 Ai ðtÞ Ai ðt þ 1Þ i¼1

i¼1

r2g

and the return of the second one equals ¼ r3 : By definition (2.6.21), 0 2 1 2 X X Ai ðtÞ Ai ðt þ 1Þ C B C 1 g A3 ðtÞ 1 gB A3 ðt þ 1Þ i¼1 i¼1 g Cþ r þ þ rPL ¼ r1 B : 2 Aðt þ 1Þ C 2 B AðtÞ Aðt þ 1Þ @ AðtÞ A 2 g After an easy algebra we get a conclusion that rPL – rPL if and only if r1 – r2 : A

In our opinion, the fact that a given interest rate index does not have Property 2.6.3 renders it to be of no practical value, and its use in evaluation of investment funds performance is potentially harmful to the markets and individual investors. A detailed consideration on grouping funds is placed in Section 2.6.6. Here we would like to

40

Chapter 2

Table 2.1 Two-year average rates of return of the group of open pension funds in Poland Time interval Since 30-06-1999 30-09-1999 31-12-1999 31-03-2000 30-06-2000 29-09-2000 31-12-2000 30-03-2001 29-06-2001 28-09-2001 31-12-2001 29-03-2002

Average rate of return To

rPL (%)

r (%)

Difference (%)

All pension funds (%)

29-06-2001 28-09-2001 31-12-2001 29-03-2002 28-06-2002 30-09-2002 31-12-2002 31-03-2003 30-06-2003 30-09-2003 31-12-2003 31-03-2004

22.097 18.091 21.464 15.614 19.779 23.131 21.965 26.290 32.953 38.829 26.046 25.966

21.040 18.008 21.331 15.531 19.701 23.050 21.831 26.293 32.928 38.794 26.021 25.929

1.057 0.083 0.133 0.083 0.078 0.081 0.134 20.003a 0.025 0.035 0.025 0.037

20.785 16.430 19.580 13.907 18.014 21.277 20.086 22.661 29.315 36.665 24.229 24.136

a

On 13th January 2003, OFE Ego (1.4% of the whole pension funds market) merged with OFE Skarbiec Emerytura (2.4% of the whole market).

show the influence of aggregation of funds on the index value on the example of rPL ; defined by (2.6.21), using real data. A general rule is that rPL typically overestimates the true rate of return for all pension funds treated as one, while the average rate of return r; defined by (2.6.3), shows exactly this true rate. An interesting illustration of this rule is presented in Table 2.1 where 2-year average rates of return of the group of open pension funds in Poland are calculated.1 The rates rPL ; defined by (2.6.21), and r; calculated via (2.6.3) in a day-to-day manner, are compared for each quarter from the end of June 2001 to the end of March 2004. It is easily seen that rPL always overestimates r except for the one period (April, 2001– April, 2003) when the opposite relationship occurs. The explanation of this phenomenon is the merger of two funds, which happened on the 13th of January 2003. This caused a rapid decrease of the interest rate rPL : As discussed in detail in Example 2.6.1 above, rPL is not consistent with respect to grouping funds, as well as to appearance of new ones. For that reason, it was officially calculated not for all but only for some funds operating on the market in Poland. In the fourth column of Table 2.1 we present r calculated for the same group of funds that rPL was calculated for. The last column contains r calculated for all pension funds operating in Poland. Since r is consistent with respect to grouping of funds (or to appearing of new ones), this does not become a problem. Observe that the official measurement of the pension funds investment results, contained in the third column, overestimates significantly the true rate of return achieved by the market of all funds (sixth column). 1

The authors thank Jan Pelc for collecting the data and providing the calculation results presented in Table 2.1.

41

Fundamental concepts of theory of interest

Property 2.6.4 If on a subset of a probability space the accounting units of all funds have the same values over ½s; u; i.e., w1 ðtÞ ¼ w2 ðtÞ ¼ · · · ¼ wn ðtÞ for every t [ ½s; u; then rðs; uÞ ¼

w1 ðuÞ 2 w1 ðsþÞ w1 ðsþÞ

holds on the same subset. If the relative number of units (in relation to the number of all units) possessed by each fund does not change in time, the basic formula takes the following additive form. Property P 2.6.5 Suppose that ki ðtÞ ¼ ai fðtÞ for all t [ ½s; u; i ¼ 1; 2; …; n; where ai $ 0; ni¼1 ai ¼ 1 and f : ½s; u ! ð0; 1Þ: Then rðs; uÞ ¼

n X

Api ðsÞri ðs; uÞ;

i¼1

where ri ðs; uÞ ¼ ðwi ðuÞ 2 wi ðsþÞÞ=wi ðsþÞ: Observe that the above formula is very simple but it holds true only if the funds have constant proportion of units within the group. Further comments on that can be found in [29]. If the number of units of every fund is constant over the time interval ½s; u i.e., fðtÞ ¼ 1 for all t; then rðs; uÞ ¼

AðuÞ 2 AðsÞ : AðsÞ

Property 2.6.6 For every pair s; u such that s , u min

1#i#n;s#t,u

ri ðt; t þ 1Þ # rðs; uÞ #

max

1#i#n;s#t,u

ri ðt; t þ 1Þ:

The next property says that the influence of small funds on the average return of a group of funds is asymptotically negligible. Property 2.6.7 Assume that for some k and some elementary events the following inequality holds: maxi–k Ai ðtÞ # uAk ðtÞ; for every t [ ½s; uÞ and some u [ ð0; 1Þ: Then on that subset of the probability space rðs; uÞ ¼ where 1ðuÞ ! 0 as u ! 0:

wk ðuÞ 2 wk ðsÞ þ 1ðuÞ; wk ðsÞ

42

Chapter 2

Note that the following formula of average rate of return does not possess Property 2.6.7 rV ðs; uÞ ¼ ½ð1 þ r1 ðs; uÞÞ· · ·ð1 þ rn ðs; uÞÞ1=n 2 1; where ri ðs; uÞ is defined by (2.6.20). The average return rV ðs; uÞ is a theoretical analogue of the well-known Value Line Composite Index (VLIC index) since

rV ðs; uÞ ¼

n Y wi ðuÞ wi ðsÞ i¼1

!1=n 21:

The next property says that if we move capital from a less effective fund to a more effective one, then the average return increases. Property 2.6.8 Let s , t , u: Suppose that wi ðsÞ ¼ wi ðtÞ ¼ wi ðuÞ for every i ¼ 3; 4; …; n and suppose r1 ðs; tÞ , r2 ðs; tÞ: Moreover, suppose that some participants transfer their assets from the first fund to the second one at time t: Then the average return increases over the time interval ½s; u if and only if r1 ðt; uÞ , r2 ðt; uÞ:

2.6.6 Merger of funds Consider n pension funds at time t ¼ 0; 1; 2; …; t: Assume that at time t the nth fund and the (n 2 1)th one merge with each other forming a new fund, say (n 2 1)th. The assets of the new fund are equal to kn21 ðtÞwn21 ðtÞ þ kn ðtÞwn ðtÞ: At time t; we establish a number of units of the new fund, say kn21 ðtþÞ: The value of one unit of the new fund, wn21 ðtþÞ; is calculated according to the formula kn21 ðtþÞwn21 ðtþÞ ¼ kn21 ðtÞwn21 ðtÞ þ kn ðtÞwn ðtÞ: Suppose that afterwards the number of funds does not change up to the moment u . t: How should the average rate of return over the time interval ½0; u be calculated? Observe that this merger of the two funds identified as the nth and the (n 2 1)th can be treated as an allocation of all assets of the nth fund to the (n 2 1)th one. After allocation, the units of the (n 2 1)th fund split so that the new value of the unit of the (n 2 1)th fund is equal to wn21 ðtþÞ and the number of units is equal to kn21 ðtþÞ:

43

Fundamental concepts of theory of interest

By (2.6.19), rð0; uÞ ¼

t21 X n Y t¼0

£

j¼1 n22 X j¼1

£

Apj ðtÞ

wj ðt þ 1Þ wj ðtÞ

!

wj ðt þ 1Þ w ðt þ 1Þ þ Apn21 ðtÞ n21 Apj ðtÞ wj ðtÞ wn21 ðtþÞ

u21 Y

n22 X

t¼tþ1

j¼1

Apj ðtÞ

!

wj ðt þ 1Þ w ðt þ 1Þ þ Apn21 ðtÞ n21 wj ðtÞ wn21 ðtÞ

! 2 1;

ð2:6:23Þ

provided there is no split of units of other funds up to the moment u. The rate of return r 0n21 ð0; tÞ of the new (n 2 1)th fund at the moment t equals r 0n21 ð0; tÞ

t21 Y An21 ðtÞ wn21 ðt þ 1Þ An ðtÞ wn ðt þ 1Þ þ ¼ 2 1; An21;n ðtÞ wn21 ðtÞ An21;n ðtÞ wn ðtÞ t¼0

ð2:6:24Þ

where by An;n21 ðtÞ ¼ kn21 ðtÞwn21 ðtÞ þ kn ðtÞwn ðtÞ we denote the combined assets of funds numbered n and n 2 1 at time t. Example 2.6.2 Suppose at time t ¼ 0 there are five funds with the following number of units and values of units: k1 ð0Þ ¼ 106 ; k2 ð0Þ ¼ 9 · 105 ; k3 ð0Þ ¼ 4 · 105 ; k4 ð0Þ ¼ 3 · 105 ; k5 ð0Þ ¼ 2 · 105 ; and w1 ð0Þ ¼ 10:5; w2 ð0Þ ¼ 9:4; w3 ð0Þ ¼ 4:3; w4 ð0Þ ¼ 5; w5 ð0Þ ¼ 8:5 (in monetary units). At the moment t ¼ 1; k1 ð1Þ ¼ 106 ; k2 ð1Þ ¼ 9:2·105 ; k3 ð1Þ ¼ 4:3·105 ; k4 ð1Þ ¼ 3·105 ; k5 ð1Þ ¼ 2:2·105 ; and w1 ð1Þ ¼ 10:8; w2 ð1Þ ¼ 9:7; w3 ð1Þ ¼ 4:4; w4 ð1Þ ¼ 5:5; w5 ð1Þ ¼ 8:6: At time t ¼ 1 fund No. 4 merges with fund No. 5. The new fund has k4 ð1þÞ ¼ 6·105 units so that the value of one unit equals w4 ð1þÞ ¼

k4 ð1Þw4 ð1Þ þ k5 ð1Þw5 ð1Þ ¼ 5:9: k4 ð1þÞ

44

Chapter 2

Assume that at time t ¼ 2 k1 ð2Þ ¼ 1:2·106 ; k2 ð2Þ ¼ 9:4·105 ; k3 ð2Þ ¼ 4:3·105 ; k4 ð2Þ ¼ 6:1·105 ; w1 ð2Þ ¼ 10:9; w2 ð2Þ ¼ 9:6; w3 ð2Þ ¼ 4:4; w4 ð2Þ ¼ 6:2: Then the average return on the time period ½0;2 equals ! ! 5 3 X X w ð1Þ w ð2Þ w ð2Þ j j rð0;2Þ ¼ þ Ap4 ð1Þ 4 Apj ð0Þ Apj ð1Þ 2 1; w4 ð1þÞ wj ð0Þ wj ð1Þ j¼1 j¼1 (see (2.6.23)) and rð0;2Þ ¼ 0:040384 ¼ 4%: By (2.6.24) the rate of return of the new fund at t ¼ 1 equals 5.312%. Observe that the arithmetic mean of the rate of return of fund No. 4 and fund No. 5 at time t ¼ 1 is equal to (10% þ 1.117%)/2 ¼ 5.58% and is greater than the rate of return of the new fund at the same time. A

Exercises Exercise 2.6.1 Consider two funds: the first attaining 25% rate of return during a given period of time ðs; uÞ while the second loses 2 20% during the same time. The initial assets of the first fund were 100 monetary units, and of the second one were 125 monetary units. The number of accounting units does not change at each fund during the considered time. What is the true rate of return for both funds treated together? Calculate the value of rPL ðs; uÞ defined by (2.6.21) and rðs; uÞ defined by (2.6.3). Solution The initial assets of both the funds together were 225 monetary units, and their terminal assets have the same value. Since no contributions/ withdrawals have happened, the true rate of return when both funds are treated together is 0%. According to Property 2.6.5, rðs; uÞ reduces now to rðs; uÞ ¼

100 125 25% 2 20% ¼ 0% 225 225

which is the true rate of return. However, (2.6.21) gives 1 100 125 1 125 100 rPL ðs; uÞ ¼ 25% þ þ þ ð220%Þ ¼ 2:5%; 2 225 225 2 225 225 which means overestimation of the true rate of interest.

A

Exercise 2.6.2 Consider a sequence of independent and identically distributed cash flows C0 ; C1 ; C2 ; … with a finite P expectation m ¼ EðC0 Þ: Show that the accumulated value of the account, SðtÞ ¼ ti¼0 Ci ; is

45

Fundamental concepts of theory of interest

† a martingale, if m ¼ 0; † a submartingale, if m $ 0; † a supermartingale, if m # 0; with respect to a stream of s-algebras F ¼ {F t ; t ¼ 0; 1; 2; …}; where F t ; sðC0 ; C1 ; …; Ct Þ denotes the smallest s-algebra such that all C0 ; C1 ; …; Ct are F t -measurable. Solution Obviously each SðtÞ is F t -measurable. Since Ctþ1 is independent of C0 ; C1 ; …; Ct ; EðCtþ1 lF t Þ ¼ E Ctþ1 ¼ m and EðSðt þ 1ÞlF t Þ ¼ E Ctþ1 þ

t X

! Ci lF t

t X Ci ¼ m þ SðtÞ ¼ E Ctþ1 þ

i¼0

i¼0

hold almost surely. Hence, if m ¼ 0; {SðtÞ; t ¼ 1; 2; …} is a martingale. If m $ 0; the above implies EðSðt þ 1ÞlF t Þ $ SðtÞ;

a:s:

and {SðtÞ; t ¼ 1; 2; …} is a submartingale. Finally, if m # 0; EðSðt þ 1ÞlF t Þ # SðtÞ; and {SðtÞ; t ¼ 1; 2; …} is a supermartingale.

a:s: A

P Exercise 2.6.3 In Exercise 2.6.2 put S0 ðtÞ ¼ ti¼0 Ci 2 ðt þ 1Þm: Show that {S0 ðtÞ; t ¼ 0; 1; 2; …} is a martingale with respect to F no matter what the value of m is. Solution Define C 0t ¼ Ct 2 m and note that EðC 0t Þ ¼ 0: Now apply the result of Exercise 2.6.2. A

This page is intentionally left blank

47

Chapter 3 Life insurance and annuities

3.1 Annuities Finite (and, occasionally, infinite) stream of payments paid at regular time intervals is called an annuity. The most common frequency of payment used by us will be annual, as the basic unit of time used for our calculations is a year. We will start analyzing annuities by looking at those that have level payments. An annuity making a payment at the end of each payment period is called an annuity immediate. A basic n-year annuity immediate makes unit payments for exactly n years and has the present value: a n i ¼ v þ v2 þ · · · þ vn ¼ v

1 2 vn 1 2 vn 1 2 vn ¼ ¼ : 12v ð1 þ iÞd i

ð3:1:1Þ

Note that when it is clear what the interest rate is, we usually drop it in the notation: a n i ¼ a n : The accumulated value at the end of n years is s n i ¼ ð1 þ iÞn21 þ ð1 þ iÞn22 þ · · · þ 1 ¼

ð1 þ iÞn 2 1 : i

ð3:1:2Þ

If the payment is P then the formulas need to be multiplied by P: Denote by a€ n the present value of a level annual payment of a monetary unit paid at the beginning of each year for n years. Such an annuity is called an annuity due. It has the present value a€ n ¼ 1 þ v þ · · · þ vn21 ¼

1 2 vn : d

ð3:1:3Þ

Its accumulated value at the end of n years is s€ n i ¼ ð1 þ iÞn þ ð1 þ iÞn21 þ · · · þ ð1 þ iÞ ¼

ð1 þ iÞn 2 1 : d

ð3:1:4Þ

48

Chapter 3

Example 3.1.1 Let i ¼ 0:25 and n ¼ 4: Then v ¼ 0:8, a€ n ¼ 1 þ 0:8 þ ð0:8Þ2 þ ð0:8Þ3 ¼ 2:952; and a n ¼ 0:8 þ ð0:8Þ2 þ ð0:8Þ3 þ ð0:8Þ4 ¼ 2:3616:

A

Note that a€ n ¼ 1 þ a n21 ; 1 1 ¼ þ i; an sn

s€ n ¼ s n21 2 1;

a€ n ¼ ð1 þ iÞa n ;

s€ n ¼ ð1 þ iÞs n ;

1 1 ¼ þ d: a€ n s€ n

ð3:1:5Þ

The last two identities in the above list illustrate that a loan of one can be repaid by level payments throughout the n-year period of the loan or by paying interest every year and making payments that accumulate to the loan amount at the end of the n-year period of the loan. A deferred annuity starts payments at a time in the future. If the first payment is made at the time n þ 1; and the last one at the time n þ m; then its present value is nla m

¼ nþ1l€a m ¼ a mþn 2 a n :

ð3:1:6Þ

Note the following vn 2 vm ¼ iða m 2 a n Þ:

ð3:1:7Þ

We also have the following, which we will term the David Ricardo Identity: 1 ¼ vn þ ia n : While other texts do not use this name, it refers to the fact that a loan of one can be repaid by paying interest on it until a time in the future when the loan is paid in full. This resembles the idea put forth by the great classical economist David Ricardo in the management of public debt in Great Britain following the Napoleonic Wars: as long as interest was paid dutifully on the debt, there was no need to pay it off, and instead the debt could be rolled over, or even issued as a perpetuity, i.e., an annuity that lasts forever. A perpetuity-immediate makes payments at the end of each period and its present value is a 1 ¼ lim a n ¼ n!1

1 : i

ð3:1:8Þ

49

Life insurance and annuities

A perpetuity due makes payments at the beginning of each period and its present value is a€ 1 ¼ lim a€ n ¼ n!1

1 : d

ð3:1:9Þ

If the interest rate is not constant, the present and accumulated values of an annuity have to be calculated from the basic principles, and they are an ¼

1 1 1 þ þ ··· þ ; að1Þ að2Þ aðnÞ

ð3:1:10Þ

sn ¼

aðnÞ aðnÞ aðnÞ þ þ ··· þ : að1Þ að2Þ aðnÞ

ð3:1:11Þ

and

If the compounding frequency of the interest exceeds the payment frequency, we need to adjust the annuity formulas. This can be done in two ways: † by using equivalent payments made with the same frequency as compounding, † by finding an equivalent interest rate over the payment period. If a unit payment is made every k years at the end of each k-year period, with an annual interest rate of i; then using the two approaches we can either: † use an annual payment of 1=s k paid at the end of each year, or † use an equivalent interest rate over k years, denoted by j; given by j ¼ ð1 þ iÞk 2 1: In practice, it is quite common that payments are not made annually, but rather spread over smaller regular payments throughout the year. For example, mortgage payments are typically made monthly in the United States, and the pension annuities in the US are also paid with the same monthly frequency. To model that common occurrence, assume that the annual unit payment is divided into m installments, each equal to 1=m of a monetary unit. A unit mthly annuity immediate pays a total of a monetary unit over a year, paid each mth of the year, each payment made at the end of the mth of the year (if the payment is made at the beginning of the mth of the year, this is the unit mthly annuity due), for n years and its present value is ðmÞ

1 2 vn i ¼ ðmÞ a n ; ðmÞ i i

ð3:1:12Þ

ð1 þ iÞn 2 1 i ¼ ðmÞ s n : ðmÞ i i

ð3:1:13Þ

an ¼ while its accumulated value is ðmÞ

sn ¼

50

Chapter 3

The corresponding formulas for the annuity due are 1 2 vn d ¼ ðmÞ a€ n ; dðmÞ d ð1 þ iÞn 2 1 d ¼ ¼ ðmÞ s€ n : ðmÞ d d

ðmÞ

a€ n ¼ ðmÞ

s€ n

ð3:1:14Þ ð3:1:15Þ

An alternative approach is also possible: find an equivalent interest rate j effective over the payment period, given by j ¼ ð1 þ iÞ1=m 2 1: A continuous annuity pays a monetary unit per year in a continuous fashion and if it lasts for n years, its present value is a n ¼

1 2 vn i ¼ an ; d d

ð3:1:16Þ

while its accumulated value at the end of n years is s n ¼

ð1 þ iÞn 2 1 i ¼ an : d d

An alternative approach to the valuation of a continuous annuity is ðn a n ¼ vt dt:

ð3:1:17Þ

ð3:1:18Þ

0

In general, if a payment is made continuously at the annual rate Pt ; then the present value of such a stream of payments is ðn vt Pt dt: ð3:1:19Þ 0

Varying annuities introduce a new level of complication: varying amounts of payments. If payments increase geometrically, i.e., if each consecutive payment is ð1 þ rÞ times the previous payment, then the present value of such a stream of payments is 1þr n 12 1þi 1 · v þ ð1 þ rÞ · v2 þ · · · þ ð1 þ rÞn21 · vn ¼ : ð3:1:20Þ i2r A basic increasing annuity immediate makes payments of 1 at time 1, 2 at time 2, etc., with the last payment of n made at time n: The present value of such an annuity is ðIaÞ n ¼

a€ n 2 nvn ; i

ð3:1:21Þ

s€ n 2 n : i

ð3:1:22Þ

while its accumulated value is ðIsÞ n ¼

51

Life insurance and annuities

Note also that a basic increasing perpetuity-immediate is worth: 1 2 vn 2 nvn a€ 2 nv 1 d : lim ðIaÞ n ¼ lim n ¼ lim ¼ n!1 n!1 n!1 di i i n

ð3:1:23Þ

If we consider an increasing n-year annuity due, its present value is ðI a€ Þ n ¼ ð1 þ iÞ

a€ n 2 nvn a€ 2 nvn a€ 2 nvn ¼ n ¼ n ; i iv d

ð3:1:24Þ

and the present value of an increasing perpetuity-due is 1 2 vn 2 nvn a€ n 2 nv 1 d lim ðI a€ Þ n ¼ lim ¼ lim ¼ 2: n!1 n!1 n!1 d d d n

ð3:1:25Þ

A basic decreasing annuity immediate makes payments of n at time 1, n 2 1 at time 2, etc. with the last payment of 1 made at time n: The present value of such an annuity is n 2 an ; i

ð3:1:26Þ

nð1 þ iÞn 2 s n : i

ð3:1:27Þ

ðDaÞ n ¼ while the accumulated value is ðDsÞ n ¼

We also have the following for the decreasing annuity due and the continuously decreasing continuous annuity: ðD€aÞ n ¼

n 2 an ; d

aÞ n ¼ ðD

n 2 a n : d

ð3:1:28Þ

Note that ðDaÞ n þðIaÞ n ¼ ðn þ 1Þa n :

ð3:1:29Þ

If the payments of 1=m are made at the end of each mth of the first year, 2=m at the end of each mth of the second year, etc. for n years, then the present value of such an increasing annuity is ðmÞ

ðIaÞ n ¼

1 1=m 1 n a€ 2 nvn v þ v2=m þ · · · þ vnm=m ¼ n ðmÞ : m m m i

ð3:1:30Þ

If the payments at the end of consecutive mths are 1=m2 ; 2=m2 ; 3=m2 ; etc. then the present value of such an increasing annuity is ðmÞ

ðI ðmÞ aÞ n ¼

ðmÞ

a€ n 2 nvn 1 1=m 2 2=m nm nm=m v þ v þ · · · þ v ¼ : m2 m2 m2 iðmÞ

ð3:1:31Þ

In the limit, as m ! 1; we get the following continuous increasing annuity present

52

Chapter 3

value ðIa Þ n ¼

a n 2 nvn : d

ð3:1:32Þ

Let us also note that if a single payment of 1 is made at time n; its present value at time 0 is vn : If a perpetuity of 1 is paid starting at time n; its present value at time 0 is vn =d: If an increasing perpetuity of 1; 2; 3; …; is paid starting at time n; then its present value at time 0 is vn =d2 . These three formulas may be useful for analyzing various cash flows. Exercises Exercise 3.1.1 May 2003 SOA/CAS Course 2 Examination, Problem No. 8. Kathryn deposits 100 into an account at the beginning of each 4-year period for 40 years. The account credits interest at an annual effective interest rate of i: The accumulated amount in the account at the end of 40 years is X; which is five times the accumulated amount in the account at the end of 20 years. Calculate X: Solution The effective interest rate over a 4-year period is j ¼ ð1 þ iÞ4 2 1: Using that rate, we have: X ¼ 100€s10 j ¼ 100

ð1 þ jÞ10 2 1 ð1 þ jÞ5 2 1 ¼ 5 £ 100€s 5 j ¼ 500 : dj dj

Therefore 5¼

ð1 þ jÞ10 2 1 ¼ ð1 þ jÞ5 þ 1; ð1 þ jÞ5 2 1

ð1 þ jÞ5 ¼ 4:

Hence j ¼ 31:9508%: Also (note that the subscript j refers to the interest functions involving the interest rate j), dj ¼

j ¼ 24:2142%: 1þj

Therefore: X ¼ 100€s10 j ¼ 100

ð1 þ jÞ10 2 1 42 2 1 ¼ 6194:712: ¼ 100 dj 0:242142

A

Exercise 3.1.2 May 2003 SOA/CAS Course 2 Examination, Problem No. 22. A perpetuity costs 77.1 and makes annual payments at the end of the year. This perpetuity pays 1 at the end of year 2, 2 at the end of year 3,…, n at the end of year ðn þ 1Þ: After year ðn þ 1Þ; the payments remain constant at n: The annual effective interest rate is 10.5%. Calculate n:

53

Life insurance and annuities

Solution The cost of this perpetuity is n · vnþ1 a€ n 2 n · vn n · vnþ1 a n nvnþ1 nvnþ1 a n ¼v · ¼ 2 þ ¼ : v · ðIaÞ n þ þ i i i i i i i Since i ¼ 10:5%; we have an a ¼ n ¼ 77:10: i 0:105 Hence a n ¼ 8:0955; and with i ¼ 10:5% this gives n ¼ 19:

A

Exercise 3.1.3 May 2003 SOA/CAS Course 2 Examination, Problem No. 26. 1000 is deposited into Fund X; which earns an annual effective rate of 6%. At the end of each year, the interest earned plus an additional 100 is withdrawn from the fund. At the end of the 10th year, the fund is depleted. The annual withdrawals of interest and principal are deposited into Fund Y; which earns an annual effective rate of 9%. Determine the accumulated value of Fund Y at the end of year 10. Solution The interest earned in Fund X during the first year is 60. In the second year it is reduced by 6, the interest on the 100 withdrawn. Next year it is again reduced by 6, and this continues until the end of the 10th year. Thus the withdrawals, when treated as deposits into Fund Y, amount to a decreasing 10-year annuity which is six times the unit 10-year decreasing annuity. Therefore, the total accumulated in Fund Y is ! 10ð1:09Þ10 2 s10 0:09 6ðDsÞ10 0:09 þ 100s10 0:09 ¼ 6 þ 100ð15:19293Þ 0:09 ¼ 565:38 þ 1519:29 ¼ 2084:67:

A

Exercise 3.1.4 May 2003 SOA/CAS Course 2 Examination, Problem No. 45. A perpetuity-immediate pays 100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8% greater than the preceding payment. Immediately after the 10th payment of the 25-year annuity, the annuity will be exchanged for a perpetuity-immediate paying Y per year. The annual effective rate of interest is 8%. Calculate Y: Solution The value of the perpetuity is 100/0.08 ¼ 1250. The value of the 25-year annuity immediate it is exchanged for is 1250 ¼ Xðv þ 1:08v2 þ · · · þ 1:0824 v25 Þ ¼ 25Xv:

54

Chapter 3

Since we know that the interest rate is 8%, this implies that X ¼ 54: When the second exchange happens, the equation of value is Y ¼ 54ð1:0810 v þ 1:0811 v2 þ · · · þ 1:0824 v15 Þ ¼ 54ð1:08Þ9 · 15: 0:08 The solution is Y ¼ 129:5:

A

Exercise 3.1.5 November 2001 SOA/CAS Course 2 Examination, Problem No. 5. Mike buys a perpetuity-immediate with varying annual payments. During the first 5 years, the payment is constant and equal to 10. Beginning year 6, the payments start to increase. For year 6 and all future years, the current year’s payment is K% larger than the previous year’s payment. At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50. Calculate K; given K , 9:2: Solution The present value of this perpetuity is 0 1 K t þ1 1þ X B 100 C 10a 5 9:2% þ 10v59:2% @ A < 38:70 þ 6:44 1:092 t¼1

K 100 ¼ 167:50: K 0:092 2 100 1þ

Now we just need to solve this linear equation: K 167:50 2 38:70 100 ; ¼ K 6:44 0:092 2 100 1þ

and the solution is K ¼ 4:

A

Exercise 3.1.6 November 2001 SOA/CAS Course 2 Examination, Problem No. 12. To accumulate 8000 at the end of 3n years, deposits of 98 are made at the end of each of the first n years and 196 at the end of each of the next 2n years. The annual effective rate of interest is i: You are given that ð1 þ iÞn ¼ 2: Determine i: Solution Effectively, the accumulated deposits amount to 98 paid over 3n years and another 98 paid over 2n years. Hence we have 98s 3n þ 98s 2n ¼ 8000; so that ð1 þ iÞ3n 2 1 ð1 þ iÞ2n 2 1 8000 þ ¼ : i i 98

55

Life insurance and annuities

Because ð1 þ iÞn ¼ 2; we have 23 2 1 22 2 1 8000 þ ¼ : i i 98 Therefore 10 ¼

8000 i; 98

and i¼

98 ¼ 12:25%: 800

A

Exercise 3.1.7 November 2001 SOA/CAS Course 2 Examination, Problem No. 16 Olga buys a 5-year increasing annuity for X: Olga will receive 2 at the end of the first month, 4 at the end of the second month, and for each month thereafter the payment increases by 2. The nominal interest rate is 9% convertible quarterly. Calculate X: Solution Let j be the effective monthly interest rate equivalent to the nominal rate of 9% convertible quarterly. Then ð1 þ jÞ3 ¼ 1 þ 0:09=4 so that j ¼ 1:02251=3 2 1: The desired value of X is X ¼ 2ðIaÞ 60 j ¼ 2

a€ 60 j 2 60v60 j j

< 2729:70:

A

Exercise 3.1.8 November 2001 SOA/CAS Course 2 Examination, Problem No. 27. A man turns 40 today and wishes to provide supplemental retirement income of 3000 at the beginning of each month starting on his 65th birthday. Starting today, he makes monthly contributions of X to a fund for 25 years. The fund earns a nominal rate of 8% compounded monthly. Each 1000 will provide for 9.65 of income at the beginning of each month starting on his 65th birthday until the end of his life. Calculate X: Solution Since each 1000 will buy 9.65 of monthly income, in order to provide 3000 of monthly income, the amount of capital required is 3000 ¼ 310 881: 9:65

56

Chapter 3

In order to accumulate that amount by making 300 monthly payments at 8% compounded monthly (which is equivalent to 2/3% monthly interest rate), the monthly contribution must be X¼

310 881 < 324:73: a€ 300 2 %

A

3

Exercise 3.1.9 November 2001 SOA/CAS Course 2 Examination, Problem No. 35. At time t ¼ 0; Sebastian invests 2000 in a fund earning 8% convertible quarterly, but payable annually. He reinvests each interest payment in individual separate funds each earning 9% convertible quarterly, but payable annually. The interest payments from the separate funds are accumulated in a side fund that guarantees an annual effective rate of 7%. Determine the total value of all funds at t ¼ 10: Solution The annual interest amount earned on the principal is 2000ð1:024 2 1Þ ¼ 164:86: Over 10 years, the total of such interest is 1648.60. The interest earned on the annual interest is 164:86ð1:02254 2 1Þ ¼ 15:35: The total accumulation of these payments of 15.35 is s€ 9 2 9 15:35ðIsÞ 9 ¼ 15:35 ¼ 836:80: 0:07 Altogether, Sebastian will have after 10 years: 2000 þ 1648:60 þ 836:89 ¼ 4485:49:

A

3.2 Loan amortization Let us assume that a hypothetical borrower repays a lender by a series of payments at regular intervals, at a fixed interest rate, with the following notation: L is the amount of the loan, n the number of payments, P the amount of the level payment, paid at the end of each period, and i the effective interest rate per payment period (typically a year, although for mortgages in the United States, payments are made monthly, while for bonds issued in the US, semi-annual payments are a norm, and in all other countries around the world annual payments on business loans and bonds are a norm).

57

Life insurance and annuities

With the above notation, we have L ¼ Pa n i ; and P¼

L : an i

The balance of the loan immediately after the kth payment is Bk ¼ Pan2k i ; prospectively, while retrospectively it is Lð1 þ iÞk 2 Ps k i :

ð3:2:1Þ

Also note that Bkþt ¼ ð1 þ iÞt Bk ; for fractional t: Each payment can be split into a portion paying the interest Ik and the portion paying the principal Pk ; so that P ¼ Pk þ Ik for every k; and Ik ¼ iBk21 ¼ Pi

1 2 vðn21Þ2k ¼ Pð1 2 vn2kþ1 Þ ¼ P 2 Pk ; i

ð3:2:2Þ

Pk ¼ P 2 Ik ¼ Pvn2kþ1 :

ð3:2:3Þ

and

One important observation is that the ratio of principal portions of two consecutive payments is always v or 1 þ i (depending on which way we divide them). If the loan payments are not level, amortization has to be done step by step. Suppose that the payments are Pð1Þ ; Pð2Þ ; …; PðnÞ ; at times 1; 2; …; n: Then we have L ¼ Pð1Þ v þ Pð2Þ v2 þ · · · þ PðnÞ vn ; Bk ¼ Pðkþ1Þ v þ Pðkþ2Þ v2 þ · · · þ PðnÞ vn2k ¼ Lð1 þ iÞk 2 Pð1Þ ð1 þ iÞk21 2 · · · 2 PðkÞ ¼ Bk21 ð1 þ iÞ 2 PðkÞ ;

ð3:2:4Þ

where the last version of the balance formula, the recursive one, is especially useful for such nonstandard loans. Furthermore, Ik ¼ iBk21 ;

58

Chapter 3

and Pk ¼ PðkÞ 2 Ik : The above amortization procedure is standard for mortgage loans in the United States, although payments are typically made monthly. For corporate bonds issued in the US, it is standard to only make interest payments (at fixed coupon rates) between issue date and maturity date, with full repayment of principal at maturity date. However, in case of some bonds, especially issued by local governments (cities, counties and states) in the US it is common to accumulate money for the principal repayment in a separate fund, called a sinking fund, by making a payment, in addition to the regular interest payment, every period. Let us use the following notation: L is the amount of the loan, n the number of payments, i the effective interest rate per payment period paid by the borrower to the lender, j the effective interest rate earned by the borrower on the sinking fund, D the periodic sinking fund deposit (assumed level), and P the total periodic outlay by the borrower (it is equal to the interest payment to the lender plus the sinking fund deposit). Then we have the following: L ¼ Ds n j ; and D¼

L ; sn j

as well as P ¼ Li þ D ¼ Li þ

L ¼ sn j

L anj 1þði2jÞa n j

:

ð3:2:5Þ

|ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ}

Often denoted by a n i&j

The net loan balance at time k (after consideration for the accumulation in the sinking fund) is L 2 Ds k j :

ð3:2:6Þ

Therefore, the principal and interest portions of the kth payment are the following two: the net principal repaid, equal to Ds k j 2 Ds k21 j ¼ Dð1 þ jÞk21 ;

ð3:2:7Þ

Li 2 jDs k21 j :

ð3:2:8Þ

and the net interest paid:

59

Life insurance and annuities

In the special case when i ¼ j we have L 1 L P ¼ Li þ ¼L iþ ; ¼ sn j sni ani i.e., the payment is exactly the same as in the case of level amortization discussed previously. Also, net interest is the same as interest payment in that previous method, and principal repaid (which is the increase in the sinking fund balance in this method, and principal repaid in the previous one) is the same as before. Note that in all the discussions above, we always observed the same phenomenon that the ratio of two consecutive principal repayments, in any methodology, is always either the discount factor, or one plus the interest rate, at the interest rate applicable to the principal. Exercises Exercise 3.2.1 May 2003 SOA/CAS Course 2 Examination, Problem No. 15. John borrows 1000 for 10 years at an annual effective interest rate of 10%. He can repay this loan using the amortization method with payments of P at the end of each year. Instead, John repays the 1000 using a sinking fund that pays an annual effective rate of 14%. The deposits to the sinking fund are equal to P minus the interest on the loan and are made at the end of each year for 10 years. Determine the balance in the sinking fund immediately after repayment of the loan. Solution We have a 10 ¼ 6:14457: Therefore, the payment using the amortization method is 1000/6.14457 ¼ 162.745. The interest on the loan amount of 1000 is 1000 · 10% ¼ 100. Hence, the deposits into the sinking fund equals 162.745 2 100 ¼ 62.745. At 14% we have s 10 0:14 ¼ 19:3373: The accumulated value of the sinking fund at the end of the 10-year term of the loan is 62:745 · 19:3373 ¼ 1213:319: The balance in the sinking fund after repayment of the loan will be: 1213.319 2 1000 ¼ 213.319. A Exercise 3.2.2 May 2003 SOA/CAS Course 2 Examination, Problem No. 33. At an annual effective interest rate of i; i . 0; both of the following annuities have a present value of X:

60

Chapter 3

(i) a 20-year annuity immediate with annual payments of 55, (ii) a 30-year annuity immediate with annual payments that pay 30 per year for the first 10 years, 60 per year for the second 10 years, and 90 per year for the final 10 years. Calculate X: Solution Note that 30 · a 10 þ 60 · v10 · a 10 þ 90 · v20 · a 10 ¼ a 10 · ð30 þ 60v10 þ 90v20 Þ ¼ 55a 20 ¼ 55 · a 10 ð1 þ v10 Þ: Therefore 90v20 þ 5v10 2 25 ¼ 0; and this quadratic equation has two solutions: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 25 þ 9000 v ¼ 25 ^ ; 180 10

but only one of them is positive: v10 ¼

90 ¼ 0:5: 180

From this we can find the interest rate: i ¼ 7:18%; and hence X ¼ 55 · a 20 ¼ 574:60:

A

Exercise 3.2.3 May 2003 SOA/CAS Course 2 Examination, Problem No. 39. A 30-year loan of 1000 is repaid with payments at the end of each year. Each of the first 10 payments equals the amount of interest due. Each of the next 10 payments equals 150% of the amount of interest due. Each of the last 10 payments is X: The lender charges interest at an annual effective rate of 10%. Calculate X: Solution Since the first 10 payments equal the annual interest due, the amount of principal outstanding at the end of 10 years is the amount of loan: 1000. For the next 10 years, each payment equals 150% of interest due. The lender charges 10%, therefore 5% of the principal outstanding will be used to reduce the principal. At the end of 20 years, the amount outstanding is 1000ð1 2 0:05Þ10 ¼ 598:74:

61

Life insurance and annuities

Each of the last 10 payments is 598:74 ¼ 97:4417: a 10 10%

A

Exercise 3.2.4 November 2001 SOA/CAS Course 2 Examination, Problem No. 6. A 10-year loan of 2000 is to be repaid with payments at the end of each year. It can be repaid under the following two options: (i) Equal annual payments at an annual effective rate of 8.07%. (ii) Installments of 200 each year plus interest on the unpaid balance at an annual effective rate of i: The sum of the payments under option (i) equals the sum of the payments under option (ii). Determine i: Solution If P is the payment under option (i) then 2000 ¼ Pa 10 8:07% ; and hence P ¼ 299: Therefore, the total of the payments under this option is 2990. Under option (ii) the total of all payments is also 2990, and since principal payments total 2000, interest paid will be 990. The interest paid is equal to the sum of interest charged on all outstanding principal amounts throughout loans existence, giving: 990 ¼ i · ð2000 þ 1800 þ 1600 þ · · · þ 200Þ ¼ i · 11 000: Therefore i ¼ 0:09:

A

Exercise 3.2.5 November 2001 SOA/CAS Course 2 Examination, Problem No. 9. A loan is amortized over 5 years with monthly payments at a nominal interest rate of 9% compounded monthly. The first payment is 1000 and is to be paid 1 month from the date of the loan. Each succeeding monthly payment will be 2% lower than the prior payment. Calculate the outstanding loan balance immediately after the 40th payment is made. Solution The key issue is to determine the outstanding balance of a loan when the payments are not level. The monthly payment at time t is 1000 · 0:98t21 : Since the actual amount of the loan is not given, the outstanding balance must be calculated prospectively. The balance at time 40 is the present value of the future payments made

62

Chapter 3

at times 41 to 60. This equals: 0:9840 0:9841 0:9859 þ1000 þ···þ1000 1:00751 1:00752 1:007520 0:9840 0:98 0:98 2 0:98 19 þ ¼ 1000 1þ þ···þ ¼ 6889:11: 1:0075 1:0075 1:0075 1:00751

1000

A

3.3 Bonds A loan is a liability to the borrower, but an asset to the lender. Loans can be given for the purpose of consumption or investment (economic investment, i.e., creation of productive capacity). Individuals generally borrow for consumption (even a purchase of a home is, effectively, consumption), while businesses borrow for economic investment purposes. Bank loans are one form of borrowing by businesses, very popular outside the United States. In the United States, however, businesses often borrow by issuing bonds. Bonds represent borrowing by businesses or governments, with a prescribed structure, described in bond covenants, offered as financial investments (securities) for purchase by various investors. Outside the United States, the market for bonds is dominated by securities issued by governments, but in the United States, there is a large, thriving private sector borrowing through the bond markets. If bonds are issues for sale to general public in the United States, they generally must be registered with the Securities Exchange Commission (SEC). This regulatory body of the Federal Government of the United States is the main supervisory entity for the securities markets. Bonds may be secured by real assets (e.g., real estate), which become property of the lender in case of default, or unsecured. Bonds secured by property are called mortgages or mortgage bonds. Unsecured bonds are called debentures. Bonds issued in the United States offer mostly a fixed payment throughout the life of the bonds, and repayment of the principal loan amount at maturity. The fixed payments are termed coupons and are usually paid semi-annually. In other countries fixed coupon bonds are relatively rare, and those that do offer that feature usually pay annually. In what follows we will analyze standard ‘American-style’ fixed coupon bonds and understand their pricing. We will use the following notation: F ¼ the par value of the bond, i.e., the amount given on the bond certificate. That amount may not be equal to the actual amount borrowed or repaid, but it is most often equal to the principal repaid at maturity. r ¼ the coupon rate of the bond, stated in its covenants. The actual coupon amount is Fr: C ¼ the redemption value of the bond, paid at maturity, and commonly equal to F:

63

Life insurance and annuities

n ¼ the number of coupon payment periods (assumed equal) till maturity. In the United States, a coupon payment period is usually 6 months. P ¼ the market price of the bond. i ¼ the bond yield, calculated as the internal rate of return on bond’s cash flows, given its market price, and expressed over the coupon payment period (so that it is not necessarily the annual effective rate, but rather it is commonly the effective semi-annual rate). Given those concepts, we also introduce the following: g¼

Fr ; C

is termed the modified coupon rate, and K ¼ Cvn ; is the present value of the redemption value, where, as always, v ¼ 1=ð1 þ iÞ: If P . C then we say that the bond sells at a premium and P 2 C is its premium. If P , C; then the bond sells at a discount and C 2 P is the amount of its discount. We also write k¼

P2C : C

If the bond is selling at the price P ¼ C we say that it sells at par. We have the following standard formulas for the price of a bond in relation to its yield: ‘Frank’ Formula : Makeham’s Formula : Premium-Discount Formula :

P ¼ Fra n þ K; g P ¼ K þ ðC 2 KÞ; i P 2 1 ¼ k ¼ ðg 2 iÞa n : C

The first of the three above is a direct application of the calculation of the present value of an annuity (as the coupon payments form an annuity), and the other two follow from the first one. If a bond is bought at a discount, it will still be redeemed at par. Thus the investor who bought this bond will enjoy price appreciation between the time of purchase and redemption. But the bond is selling at a discount because its coupon is below the current market rate. Thus the investor is getting a coupon, which is too low in relation to the current market level of interest rates, but he/she is properly compensated for it by having the bond increase in value until it reaches full redemption value at maturity. This process of marking up the price of a bond bought at a discount is called the amortization of discount. If a bond is bought at a premium, it will still pay only the

64

Chapter 3

regular redemption value at maturity, thus it will gradually decline in value between the time of purchase and maturity. This will, however, be accompanied by payments of coupons in excess of current market level of interest rates, which was the reason why the bond was sold at a premium in the first place. Overall, any kind of a bond, sold at a discount, sold at par, or sold at a premium, will offer the same yield as the current market level of appropriate yields, and the whole process balances itself. We should make sure that we understand the process of premium/discount amortization. The book value of a bond at time t is the value at time t of the remaining future cash flows at the yield rate. Let us denote it by BVt : The terminology is related to the fact that this used to be the value of a bond shown in accounting statements of financial institutions, and still is the value shown in the statutory financial statement of US insurance companies (also in the Generally Accepted Accounting Principles, i.e., GAAP, statement, if the bond is designated as held-to-maturity). In particular, BVk is the book value immediately after the kth coupon is paid. We have BVk ¼ Fra n2k þ Cvn2k ¼ v · Fr þ v · BVkþ1 : Note that the second part of the above equation gives us the recursive formula relating the book values of a bond between two consecutive coupon payments. We also have BVk ð1 þ iÞ ¼ Fr þ BVkþ1 : We also see that the change in book value between two consecutive coupon payments is DBVk ¼ BVkþ1 2 BVk ¼ i BVk 2 Fr: This change, if positive, is the amount of the ‘write-up’ in the bond value during the kth period, if the bond was sold as a discount, and if negative, its opposite is the amount of the ‘write-down’ in the kth period, if the bond was sold at a premium. Note also that BV0 ¼ P and BVn ¼ C: We can either use the general formula for finding the book values in between or use the two boundary values and the recursive relationship to arrive at the same intermediate book value. Exercises Exercise 3.3.1 May 2003 SOA/CAS Course 2 Examination, Problem No. 42. A 10 000 par value 10-year bond with 8% annual coupons is bought at a premium to yield an annual effective rate of 6%. Calculate the interest portion of the seventh coupon. Solution The interest rate is i ¼ 6%: Using it we have BV6 ¼ 10 000v4 þ 800a 4 0:06 ¼ 7920:94 þ 2772:08 ¼ 10 693:

Life insurance and annuities

65

Thus, the interest portion of the seventh coupon is I7 ¼ i · BV6 ¼ 0:06 · 10 693 ¼ 641:58:

A

3.4 Term structure of interest rates Until now, we have assumed that we can use the same interest rate for discounting cash flows for all maturities. In reality, the rate used for discounting cash flows for various maturities differs. This can be observed most directly by comparing interest rates for the pure discount bonds, also known as zero coupon bonds, i.e., bonds that make only one payment at maturity and no intermediate coupon payments. Term structure of interest rates (TSIR) is the pattern of interest rates for discounting cash flows of different maturities. The specific functional relationship between the time of maturity and the corresponding interest rate is also called the yield curve, especially when represented graphically. Longer term bonds usually offer higher yields. Such a pattern of interest rates is termed an upward sloping yield curve. If interest rates are the same for all maturities, we call this pattern a flat yield curve. Finally, a rare but sometimes occurring situation when longer term maturity interest rates are lower than shorter term rates is termed an inverted yield curve. When the yield curve is constructed, the interest rates used in it are typically derived from bonds perceived as risk free. In the United States, it is most common to utilize bonds issued by the Federal Government, i.e., United States Treasury Bills (those with maturities up to a year), Notes (those with maturities between 1 and 10 years), and Bonds (those with maturities of 10 years or more). But this explanation does not make it clear what interest rate is used in the yield curve for each maturity. In fact, there are three ways to define the yield curve (and TSIR). The first one assigns to each maturity the coupon rate of a bond of that maturity trading at par (typically a newly issued bond). This is termed the bond yield curve. The second approach assigns to each maturity the interest rate on a zero-coupon bond of that maturity. This construct is called the spot yield curve, and the interest rates given by it are called spot rates. Finally, the third approach uses short-term interest rates in the future periods of time implied by current bond rates or spot rates. Let us explain this concept. First, we must clarify what we mean by short-term interest rates. Short-term interest rate or short rate refers to an interest rate applicable for a short period of time, up to 1 year and possibly for less, including the possibility of an instantaneous rate over the next infinitesimal period of time. If we use the 1-year rate as the short rate for the purpose of deriving forward rates, we have the following relationship: ð1 þ yn Þn ¼ ð1 þ r1 Þð1 þ r2 Þ· · ·ð1 þ rn Þ;

66

Chapter 3

where yn is the spot rate for maturity n; n ¼ 1; 2; …; and ri is the forward rate from time i 2 1 and i; i ¼ 1; 2; …; n: We also have 1 þ rn ¼

ð1 þ yn Þn : ð1 þ yn21 Þn21

It should be noted that forward rates should not be misconstrued as predictors of future interest rates. They represent current cost of transactions in future interest rate instruments. A contract, which is entered into now, but the exchange of goods/services/money takes place in the future, is called a forward. Forward interest rates are in fact interest rates on bonds purchased on the forward, and this is where the name originates. The theory that current forward rates are unbiased predictors of future uncertain short rates is called the expectations hypothesis. It is not a commonly accepted theory, and instead the following are offered as explanations for the typical upward sloping structure of yield curve: Liquidity preference theory: Liquidity refers to the ability to sell the asset without having to cut its price in relation to the market. Liquidity premium is the risk premium that must be paid by a less liquid investment when compared with otherwise identical fully liquid investment. This theory claims that less liquid longer term bonds must offer premium yield when compared with more liquid shorter term bonds. This implies that forward rates have a positive bias in relation to expected future short rates. Preferred habitat theory: This theory claims that investors in the market have preferences for certain maturities, and will only buy other maturities when compensated for it. Since short maturities are commonly preferred, short-term interest rates should be lower than long-term interest rates, at least most of the time. Market segmentation theory: In this theory, investors in various maturities actually form separate markets. The yield curve can also be studied for the continuously compounded interest rate, i.e., for the force of interest. In this case the force of interest d is a function of time d ¼ dðtÞ; that function representing the TSIR. Again, we must carefully distinguish between the following concepts: The spot rate for time t, representing the effective annual interest rate for a zerocoupon bond issued now and maturing at time t: The bond coupon yield, representing the annual coupon rate for a par bond issued now and maturing at time t: In this case, we would have to consider a continuously paid coupon, and, of course, such entities do not exist in reality. The forward rate, (instantaneous force of interest for forward purchases of bonds), which, for the force of interest, is best explained by presenting its mathematical relationship to the spot rate as follows: if dt is the spot force of interest for time t; and

67

Life insurance and annuities

ft the forward force of interest at time t; then dt t

ðe Þ ¼ e

tdt

¼ exp

ð t 0

fs ds ;

or, equivalently 1 dt ¼ t

ðt 0

fs ds:

This effectively means that the spot rate for time t is the mean value of the forward rates between times 0 and t: The set of spot forces of interest {dt } is termed the spot yield curve, while the corresponding set of forward rates {ft } is called the forward yield curve. An yield curve is termed arbitrage free if the forward rates given by it are nonnegative. This terminology is explained by the fact that negative values of forward rates would lead to riskless arbitrage opportunities. Note also that

ft ¼ t

ddt þ dt ; dt

which shows us that if ddt =dt is positive then ft . dt ; while if ddt =dt is negative then ft , dt : Exercises Exercise 3.4.1 You are given the following information about risk-free zero coupon bonds: the price of a 1-year bond maturing at 1000 is now 962, the price of a 2-year zero-coupon bond also maturing at 1000 is 916, and the price of a 3-year zero coupon bond maturing at 1000 is 864. Calculate the 1-year, 2-year, and 3-year spot rates, and 1-year forward rates at times 0, 1, and 2. Solution The 1-year spot rate is 1000 2 1 ¼ 3:9501%; 962 as we must have 1000 ¼ 962 · ð1 þ y1 Þ: The 2-year spot rate is 1000 0:5 21 ¼ 4:48459%; 916 because 1000 ¼ 916 · ð1 þ y2 Þ2 : The 3-year spot rate is 1000 1=3 21 ¼ 4:99342%; 864 as 1000 ¼ 864 · ð1 þ y3 Þ3 :

68

Chapter 3

To calculate the first year forward rate, note that ð1 þ y1 Þ ¼ ð1 þ r1 Þ so that r1 ¼ 3:9501%: For the second year forward rate, we have ð1 þ y2 Þ2 ¼

1000 1000 ¼ ð1 þ r1 Þð1 þ r2 Þ ¼ ð1 þ r2 Þ; 916 962

and therefore ð1 þ r2 Þ ¼

962 ; 916

r2 ¼ 5:02183%:

For the third year forward rate, we calculate: ð1 þ y3 Þ3 ¼

1000 1000 962 ¼ ð1 þ r1 Þð1 þ r2 Þð1 þ r2 Þ ¼ ð1 þ r3 Þ; 864 962 916

and therefore ð1 þ r3 Þ ¼

916 ; 864

r3 ¼ 6:01852%:

A

Exercise 3.4.2 You are given that the current forward rate from time 0 to time 1 is 5%, from time 1 to time 2 it is 6%, and from time 2 to time 3 it is 7%. You purchase a 3-year coupon bond paying an annual coupon of 60, with the payment of 1000 principal at maturity. If in 1 year the yield curve is flat at 7% for all maturities, what is the total return of this bond for the year? Solution The current price of this bond is 60 60 1060 þ þ ¼ 57:14 þ 53:91 þ 890:08 ¼ 1001:13: 1:05 1:05 · 1:06 1:05 · 1:06 · 1:07 At the end of the year, the bond will pay 60, and its price will be 60 1060 þ ¼ 56:07 þ 925:85 ¼ 981:92: 1:07 1:072 Thus, the 1001.13 investment will provide 1041.92 ¼ 981.92 þ 60.00, for a rate of return of 1041:92 2 1 ¼ 4:07%: 1001:13

the

total

payoff

of

A

3.5 Depreciation of fixed assets When sizable purchases of fixed assets are made by businesses, such businesses generally do not assign the cost of such a purchase to the period of time when they purchased the asset, but instead spread the cost of purchase over time. This is done for two reasons:

Life insurance and annuities

69

† Accounting principles call for matching of revenues and expenses used to generate such revenues. Since the asset is a fixed asset, i.e., purchased with the intention of using it for an extended period of time in the process of production and generation of revenues, costs should be in some way associated with those revenues. One could also argue that the fixed asset has value, and showing the entire expense as a cost to the company is misleading, as the company received value for it. Instead, the company should only show those expenses that correspond to the wear and use of the fixed asset over its productive life. This, in fact, is the reason why the expenses of purchase of a fixed asset are generally spread over the useful life of the asset, not over the infinite, or some other arbitrary, future horizon. † Tax authorities would not welcome such a large one-time expense, resulting in significant reduction of taxable income of the business. There are several methodologies used for depreciation of assets. Sometimes, a business might use one methodology for its regular accounting statements, and a different one for the tax statement (not necessarily because of desire to lower taxes, in fact, most of the time, it is the tax authority that calls for a specific, generally slow, depreciation for tax purposes, often different from regular accounting methodology). Sometimes a business might not depreciate (amortize) the cost of a fixed asset for its own accounting, but must depreciate (amortize) the cost for tax purposes. In the United States, for example, the first year expenses of issuing a life insurance policy are generally very large in relation to the premium paid, if such premium is paid annually. Thus the insurance company may spend $500 to issue a policy which pays the first year premium of $600 in the case of whole life insurance, or even spend $350 to issue a term life insurance for a policy that pays the first year premium of $300. If the cost is immediately expensed, the company shows a loss for the first year of policy existence (as the increase in the policy reserve is also an expense). Insurance solvency regulators, who in the United States are located at the state level (as opposed to the federal government of the United States), are concerned with life insurance company’s solvency, and generally want a realistic picture of company’s expenses in the financial statement. The financial statement produced for them, based on statutory accounting principles (SAP), generally treats the first year expense as a cost of doing business. But the federal tax authorities, which are concerned with efficient collection of income taxes, prescribe that the insurance firm must treat the first year expenses as a cost of acquiring a long-term fixed asset (the life insurance policy) and depreciate (amortize) those expenses over an appropriate useful life of the asset. Thus, US-based insurance companies generally produce three sets of accounting statements: † Statutory (SAP) statement for insurance regulators, † GAAP statement for firm owners (required if the company’s owners are stockholders and the company stock is traded publicly on an organized stock

70

Chapter 3

exchange supervised by the SEC, a branch of the federal government of the United States), † tax statement, prepared based on rules prescribed by the Internal Revenue Code, and supervised by the Internal Revenue Service, the federal governments tax authority, which is a part of the Department of Treasury. In what follows, we will present four basic methodologies of depreciation. They are † Sinking Fund Method, also known as Compound Interest Method. † Straight-Line Method. † Constant Percentage Method, also known as Compound Discount Method or Declining Balance Method. † Sum of the Year’s Digit Method. The most commonly used method in the United States is the straight-line method, but there are some assets for which the tax authority specifically prescribes different methodologies. The sinking fund method is most commonly used by actuaries, but rarely used by other professionals. The constant percentage method is more commonly used in Canada. Let us begin with the presentation of the Sinking Fund Method. For all depreciation methods we will use the following notation: A will be the asset value at the time of purchase, at the beginning of n accounting periods over which this asset will be depreciated, S will be the salvage value (this is what we can get for the asset by selling it at the end of its useful life), R will be the level periodic return produced by the asset, net of expenses, i will be the yield rate of the asset, j will be the sinking fund yield, Bt will be the book (accounting) value of the asset based on the depreciation method being used, calculated at time t; and Dt ¼ Bt21 2 Bt will be the depreciation expense produced by the methodology utilized, in the period ½t 2 1; t: From the economic point of view, periodic return from the asset is a compensation for the capital used to purchase it, A; plus the sinking fund deposit needed to accumulate funds to replace the asset at the end of its useful life. Therefore R ¼ Ai þ

A2S : sn j

ð3:5:1Þ

Note that there is no need to accumulate the entire purchase price of the asset A; because the amount S will be recovered by selling the old asset at the end of its useful life. One can assume for simplification that I ¼ j but that, of course, need not be the case. In the Sinking Fund Method it is assumed that for each accounting period ½t 2 1; t the depreciation charge is exactly offset by the increase in the sinking fund balance. This way the total of the asset book value and the sinking fund balance is

71

Life insurance and annuities

always the asset purchase price, i.e., A ¼ Bt þ

A2S s : sn j n j

ð3:5:2Þ

The actual depreciation charge in period t is Dt ¼

A2S ð1 þ jÞt21 ; sn j

ð3:5:3Þ

again producing the familiar pattern of two consecutive ‘principal adjustments’ with ratio equal to the accumulation factor ð1 þ jÞ or the discount factor vj : This method of depreciation produces a geometrically decreasing pattern of depreciation charges. Such a pattern is often undesirable, however, because we know that many real assets decline in value significantly more in their early years of use than in later years. We will now present the Straight-Line Depreciation Method. In this method, the periodic depreciation charge and the book value are given by the following formulas: A2S ; n t t t Bt ¼ 1 2 A þ S ¼ A 2 ðA 2 SÞ: n n n Dt ¼

ð3:5:4Þ ð3:5:5Þ

While unrealistic economically, this method is very simple and commonly used. The next method we will consider is the Constant Percentage, also called Compound Discount, or Declining Balance Method. In this method, both the depreciation charge and the next period book value are a constant proportion, fixed in advance, of the preceding book value. We have Dt ¼ dBt21 ;

ð3:5:6Þ

Bt ¼ Bt21 ð1 2 dÞ;

ð3:5:7Þ

Að1 2 dÞn ¼ S;

ð3:5:8Þ

and

d ¼12

S A

1=n :

ð3:5:9Þ

A variation of the declining balance method uses a factor equal to the multiple of the straight-line depreciation, given as percentage. Thus in that method, the factor is d0 ¼ k=n; where k is the multiple used, typically given as a percentage. The last depreciation charge is then set at the amount needed to bring the final book value to the level of the salvage value S: The last method we will consider is the Sum of the Year’s Digit Method. This method produces a desirable pattern of higher early depreciation charges.

72

Chapter 3

Let Sr ¼

r X

i¼

i¼1

1 rðr þ 1Þ: 2

ð3:5:10Þ

The formula (3.5.10) should be properly called the Gauss Formula, as Gauss discovered it at age 7, when instructed by his elementary school teacher to add all numbers from 1 to 100. In this method: D1 ¼

n ðA 2 SÞ; Sn

D2 ¼

Bt ¼ A 2

n21 n 2 ðt 2 1Þ ðA 2 SÞ;…; Dt ¼ ðA 2 SÞ; …; Sn Sn t X r¼1

Dr ¼ S þ

n X

Dr ¼ S þ

r¼tþ1

Sn2t ðA 2 SÞ: Sn

ð3:5:11Þ

ð3:5:12Þ

Exercises Exercise 3.5.1 May 2003 SOA/CAS Course 2 Examination, Problem No. 5. A machine is purchased for 5000 and has a salvage value of S at the end of 10 years. The machine is depreciated using the sum-of-the-years-digits method. At the end of year 4, the machine has a book value of 2218. At that time, the depreciation method is changed to the straight-line method for the remaining years. Determine the new depreciation charge for year 8. Solution The sum of the digits from 1 to 10 is 55. Depreciation charges at times 1, 2, 3 and 4 add up to

10 9 8 7 34 þ þ þ ð5000 2 SÞ; ð5000 2 SÞ ¼ 55 55 55 55 55

so that the book value of the machine at time 4 is 5000 2

34 55 000 34 000 34 21 000 34 ð5000 2 SÞ ¼ 2 þ S¼ þ S ¼ 2218 55 11 11 55 11 55 24 398 ¼ : 11

This is a linear equation, and it can be solved to S¼

8495 : 17

73

Life insurance and annuities

Starting at time 4, the machine is depreciated linearly from book value of 2218 to the salvage value of 8495/17 at time 10. Each annual depreciation charge equals: 8495 17 ¼ 29 211 ¼ 9737 < 286:38: 6 6 · 17 34

2218 2

A

Exercise 3.5.2 November 2001 SOA/CAS Course 2 Examination, Problem No. 42. A copier costs X and will have a salvage value of Y after n years. (i) Using the straight-line method, the annual depreciation expense is 1000. (ii) Using the sum of the years digits method, the depreciation expense in year 3 is 800. (iii) Using the declining balance method, the depreciation expense is 33.125% of the book value in the beginning of the year. Calculate X: Solution We have X2Y ¼ 1000; n so that 1000n ¼ X 2 Y: On the other hand, n22 ðX 2 YÞ ¼ 800: Sn Therefore

n22 Sn

· 1000n ¼ 800;

so that n2 2 2n ¼ 0:8: Sn By trial and error, we can then determine that n is not 1, 2, or 3, but it is 4. This implies that 12

Y X

0:25 ¼ 0:33125;

74

Chapter 3

and Y ¼ 0:2X: Therefore, X 2 0:2X ¼ 1000; 4 and X ¼ 5000:

A

Exercise 3.5.3 November 2001 SOA/CAS Course 2 Examination, Problem No. 45. A manufacturer buys a machine for 20 000. The manufacturer estimates that the machine will last 15 years. It will be depreciated using the constant percentage method with an annual depreciation rate of 20%. At the end of each year, the manufacturer deposits an amount into a fund that pays 6% annually. Each deposit is equal to the depreciation expense for that year. How much money will the manufacturer have accumulated in the fund at the end of 15 years? Solution Under the constant percentage method, the annual depreciation charge for year k is Dk ¼ dBk21 ¼ dCð1 2 dÞk21 ¼ 0:2 · 20 000 · 0:8k21 ¼ 4000 · 0:8k21 : The accumulated value of 15 values of Dk is D15 þ D14 · 1:06 þ · · · þ D1 · 1:0614 ¼ 4000ð0:814 · 1:060 þ 0:813 · 1:06 þ · · · þ 0:80 · 1:0614 Þ 1:06 15 12 0:8 ¼ 36 328:83: ¼ 4000 · 0:814 1:06 12 0:8

A

3.6 Capitalized cost From the economic point of view, the question of depreciation of a fixed asset is related to the issue of a cost of owning a fixed asset and its capitalized cost (i.e., cost of owning such an asset forever). After all, the main purpose of calculation of depreciation is to show it as an expense of owning the fixed asset in the accounting income statement of the business. The following are costs involved in owning a fixed asset (including economic opportunity cost of the money spent on the purchase of it): † loss of interest income on the purchase price A; i.e., opportunity cost of money, † depreciation cost, and † maintenance expenses.

75

Life insurance and annuities

The periodic charge for the asset can therefore be expressed as þ

Ai |{z}

Interest opportunity cost

A2S

sn j |ﬄ{zﬄ}

þ

M |{z}

:

ð3:6:1Þ

Maintenance expense

Sinking fund deposit

The capitalized cost of the asset is, therefore, the present value of a perpetuity making the payment of the periodic charge stated above: Capitalized cost ¼ A þ

A2S M : þ is n j i

ð3:6:2Þ

A complication arises when we try to compare two fixed assets that in addition to costs also produce some outputs, and those outputs are produced at different rates per unit of time. In that case, it is necessary to divide by the number of items produced per unit of time in order to produce a meaningful comparison. If machine 1 produces U1 items per unit of time, and machine 2 produces U2 items per unit of time, then those two machines would be equivalent if: A1 i þ

A 1 2 S1 A 2 S2 þ M1 A2 i þ 2 þ M2 sn1 j sn2 j ¼ : U1 U2

ð3:6:3Þ

If we assume that i ¼ j then this simplifies to A1 S A2 S 2 1 þ M1 2 2 þ M2 an1 sn1 an2 sn2 ¼ : U1 U2

ð3:6:4Þ

3.7 Random future lifetime In the mathematics of life insurance and pensions, the key random variable is the length of time that the person insured, or covered by the pension plan, will remain alive or employed as a pension plan participant, or covered by the plan. When analyzing pension plans, the future lifetime need not refer only to actual physical life. Instead, it may refer to the future length of service (while working) in the plan itself or, the future lifetime of an already retired plan beneficiary, receiving a life annuity (an annuity lasting till the end of the recipient’s life), or a temporary annuity, such as an annuity paid for 10 years certain, or annuity paid for 10 years as long as the beneficiary is alive. In order to analyze this random variable, we start by introducing the following random variable: X is the age at death of a newborn, or the future lifetime of a newborn, i.e., a random number of years (a real number, not necessarily an integer) a newborn will live. We assume that X is a continuous (or, more precisely, absolutely continuous) random variable. The probability density function (PDF) of X

76

Chapter 3

is written as fX ðxÞ ¼ sðxÞmðxÞ ¼ x p0 mðxÞ ¼ 2s0 ðxÞ: Here, t px is the probability that a person aged x will survive additional t years (and similarly t qx is the probability of dying within t years for the same person), sðxÞ ¼ PrðX . xÞ;

ð3:7:1Þ

and

mðxÞ ¼ 2

s0 ðxÞ d f ðxÞ fX ðxÞ ¼ 2 ðln sðxÞÞ ¼ X ¼ ; sðxÞ dx sðxÞ 1 2 FX ðxÞ

ð3:7:2Þ

is the force of mortality (also called the hazard rate in statistics). Note that (3.7.2) implies an equivalent identity: ðt p ¼ exp 2 m ðx þ sÞ ds : ð3:7:3Þ t x 0

For small values of s we arrive at the following approximation: s qxþt

¼

Fðt þ sÞ 2 FðtÞ sf ðtÞ < ¼ smðx þ tÞ: 1 2 FðtÞ 1 2 FðtÞ

ð3:7:4Þ

In the t px and t qx notation, it is common to drop the first subscript t if t ¼ 1: Also, if the survival probability or the probability of death is considered after a deferral period of length s; then the notation is s lt px and s lt qx ; respectively. The cumulative distribution function (CDF) of X is ðx ðx FX ðxÞ ¼ 1 2 sðxÞ ¼ fX ðsÞ ds ¼ x q0 ¼ 1 2 exp 2 mðsÞ ds 0

0

¼ 1 2 x p0 :

ð3:7:5Þ

The survival function sðxÞ ¼ PrðX . xÞ is always assumed to have these properties: † sðxÞ is decreasing (because people die), † sð0Þ ¼ 1 (because everyone is alive at birth), † sðxÞ is absolutely continuous, i.e., its derivative exists everywhere but on a set of the Lebesgue measure zero, and the function sðxÞ can be recovered from the derivative as the integral of the said derivative, † limx!1 sðxÞ ¼ 0 (everyone dies eventually). It should be noted that ð1 ð1 x!1 d ðln sðxÞÞ dx ¼ 2ln sðxÞlx¼0 ¼ 1: mðxÞ dx ¼ 2 0 0 dx

ð3:7:6Þ

77

Life insurance and annuities

In what follows, we will often use the following important property of any random variable, which is nonnegative almost surely (i.e., with probability one), and whose expected value exists: ð1 EðXÞ ¼ sðxÞ dx: ð3:7:7Þ 0

This can be proved by simple integration by parts. Similarly, for a random variable K; which is nonnegative almost surely and assumes only integer values, we have the following analogous formula, also quite useful in calculations of expected values: EðKÞ ¼

1 X

ð3:7:8Þ

PrðK $ kÞ:

k¼1

For a person already aged x; or, as commonly referred to, life aged x; often written as ðxÞ; we introduce a concept analogous to X: TðxÞ is the future lifetime (length of life remaining, calculated in years, as a continuous random variable). Usually we just write TðxÞ as T: For a pension plan participant or beneficiary, this need not be just the future lifetime, it may also be the length of work service with the plan provider, or time until retirement, or length of life while receiving plan benefits. Typically, we do not assume that we know the gender of the life ðxÞ (also referred to as status ðxÞ in case when we are considering life until some other event than just death of a person), and since there are significant differences in mortality between genders, we may sometimes need to be more specific about this assumption. If we do ignore the gender, and study the entire population, then the probability distribution of future lifetime is effectively a mixed distribution of the distributions applicable to males and females, with weights in the distribution weighing equal to appropriate gender’s population share. The PDF of TðxÞ is fT ðtÞ ¼ t px · mðx þ tÞ:

ð3:7:9Þ

In what follows we generally assume that the force of mortality depends only on attained age of the person, so that the force of mortality for TðxÞ at duration t is mðx þ tÞ: The CDF of TðxÞ is ðt ðt FT ðtÞ ¼ t qx ¼ s px · mðx þ sÞ ds ¼ 1 2 exp 2 mðx þ sÞ ds : ð3:7:10Þ 0

0

Observe as well that ¼ 1 2 FT ðtÞ:

ð3:7:11Þ

¼ Prðt , TðxÞ , t þ sÞ ¼ s px · t qxþs :

ð3:7:12Þ

t px

Note also that s t qx

l

78

Chapter 3

We also study the random variable KðxÞ ¼ bTðxÞc; i.e., the greatest integer function evaluated at TðxÞ; which is the number of whole years lived for a life age ðxÞ: Usually, we just write KðxÞ as K: The probability function of K is PrðK ¼ kÞ ¼ k lqx ¼ k px · qxþk :

ð3:7:13Þ

The CDF of K is FK ðxÞ ¼ PrðK # kÞ ¼ PrðT , k þ 1Þ ¼ qx þ 1 px · qxþ1 þ · · · þ k px · qxþk ¼ 1 2 px ð1 2 qxþ1 Þ þ 2 px · qxþ2 þ · · · þ k px · qxþk ¼ 2 qx þ 2 px · qxþ2 þ · · · þ k px · qxþk ¼ · · · ¼ kþ1qx :

ð3:7:14Þ

In general, FK ðyÞ ¼ kþ1qx ; where k ¼ b y c: The expected value of the random variable T written as EðTÞ is denoted by e x ; and commonly referred to as complete life expectancy. We will only consider random variables for which EðTÞ , 1 and EðT 2 Þ , 1: By definition ð1 t · fT ðtÞ dt: ð3:7:15Þ e x ¼ 0

We note that the survival function of T ¼ TðxÞ is t px : By integrating (3.7.15) by parts, or simply referring to (3.7.7), we obtain: ð1 ð3:7:16Þ EðTÞ ¼ e x ¼ t px dt: 0

Again by definition EðT 2 Þ ¼

ð1 0

t2 fT ðtÞ dt:

Because EðT 2 Þ , 1; we also have lim t2 ð1 2 FðtÞÞ ¼ 0;

t!1

and 2

EðT Þ ¼ 2

ð1 0

tt px dt:

ð3:7:17Þ

Therefore 2

2

VarðTÞ ¼ EðT Þ 2 ðEðTÞÞ ¼ 2

ð1 0

tt px dt 2 ðex Þ2 :

ð3:7:18Þ

79

Life insurance and annuities

The expected value of KðxÞ is called the curtate life expectancy of ðxÞ and EðKÞ ¼ ex ¼

1 X

ð3:7:19Þ

k px ;

k¼1

EðK 2 Þ ¼

1 X

ð2k þ 1Þkþ1 px ¼

k¼0

1 X

ð2k 2 1Þk px ;

ð3:7:20Þ

k¼1

and VarðKÞ ¼ EðK 2 Þ 2 ðex Þ2 :

ð3:7:21Þ

Note that sT ðnÞ ¼ n px ¼ 1 2 FT ðnÞ ¼

ð1 n

fT ðtÞ dt ¼

ð1 n

t px mðx

þ tÞ dt;

ð3:7:22Þ

i.e., you live n years if and only if you die later than that. In general, we do not know the complete nature of the probability distribution of X and T: Two simplistic assumptions are used commonly, without any illusions of their applicability to real life. The first one assumes that X has uniform distribution on ½0; v; where v is the limiting age, i.e., an age at which the lifespan of this population ends. This assumption is called the De Moivre’s Law. Under this assumption: v EðXÞ ¼ ; 2 VarðXÞ ¼ EðTÞ ¼

v2 ; 12

v2x ; 2

and VarðTÞ ¼

ðv 2 xÞ2 : 12

We will use DML to denote De Moivre’s Law assumption. The second simplifying assumption is that of a constant force of mortality m; which we will denote by CF. Under this assumption, both X and T have the same exponential distribution with mean 1=m and variance 1=m2 : Also, the exponential distribution of T implies the geometric distribution for K (although it is possible for K to have the geometric distribution without T having the exponential distribution). Several more sophisticated functional models have been proposed for describing human mortality. Gompertz (Bowers et al. [14]) proposed a model in which the force of mortality grows exponentially with age:

mðxÞ ¼ Bcx ;

ð3:7:23Þ

80

Chapter 3

where B and c are both positive parameters of the model. Based on (3.7.3), we can derive the following formula for the probability of survival under Gompertz model: n px

¼ gc

x

ðcn 21Þ

ð3:7:24Þ

:

Also, sðxÞ ¼ gc

x

21

ð3:7:25Þ

;

and

mðx þ tÞ ¼ mðxÞct :

ð3:7:26Þ

A modification of the Gompertz model, created by Makeham (Bowers et al. [14]), postulates:

mðxÞ ¼ A þ Bcx ;

ð3:7:27Þ

where n px

x

¼ sn gc ðc

n

21Þ

;

g ¼ e2B=ln c ;

s ¼ e2A ;

and

x21

sðxÞ ¼ sx gc :

Weibull (Bowers et al. [14]) proposed an alternative in which the force of mortality increases with polynomial, not exponential, order with

mðxÞ ¼ kxn ;

ð3:7:28Þ

and nþ1

sðxÞ ¼ e2ðk=nþ1Þx :

ð3:7:29Þ

Also (3.7.29) implies that t px

nþ1

¼ e2ðk=nþ1ÞððxþtÞ

2xnþ1 Þ

:

ð3:7:30Þ

Exercises Exercise 3.7.1 Calculate the life expectancy in the Weibull model with parameter n ¼ 1: Solution By (3.7.16) and (3.7.30), after a change of variables in the integral, we get: rﬃﬃﬃﬃﬃﬃﬃﬃ ð ð1 ð1 k 2p kx2 =2 1 2z2 =2 2 2 ½ðx þ tÞ p dt ¼ exp 2 2 x dt ¼ dz e e x ¼ t x pﬃﬃ e 2 k 0 0 x k rﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2p kx2 =2 ¼ ð1 2 Fð kxÞÞ; e k where F is the CDF of the standard normal distribution Nð0; 1Þ:

A

81

Life insurance and annuities

Exercise 3.7.2 November 2002 SOA/CAS Course 3 Examination, Problem No. 1. Given the survival function sðxÞ; where 8 1; 0 # x , 1; > > > < x e sðxÞ ¼ 1 2 ; 1 # x , 4:5; > 100 > > : 0; 4:5 # x: Calculate mð4Þ: Solution Recall the basic definition of the force of mortality: e4 0 2s ð4Þ e4 100 ¼ ¼ mð4Þ ¼ ¼ 1:202553422: 4 sð4Þ 100 2 e4 e 12 100

A

Exercise 3.7.3 November 2001 SOA/CAS Course 3 Examination, Problem No. 1. You are given ( 0:04; 0 , x , 40; mðxÞ ¼ 0:05; x . 40: Calculate e 25: 25 : Solution Because of the peculiar form of the force of mortality formula, it is best to break the calculation at the age where the force changes. Thus: ð 15 ð 10 e 25: 25 ¼ e 25: 15 þ 15 p25 · e 40: 10 ¼ p dt þ p · t 25 t p40 dt 15 25 0 0 ð 10 ð 15 e20:04t dt þ e20:04£15 · e20:05t dt ¼ a 15 d¼0:04 þ e20:60 · a 10 ¼ 0

20:60

0 20:50

d¼0:05

12e 12e þ e20:6 ¼ 25 2 25e20:60 þ 20e20:60 2 20e21:1 0:04 0:05 ¼ 25 2 5e20:60 2 20e21:1 ¼ 15:5985: ¼

A

Exercise 3.7.4 November 2000 SOA/CAS Course 3 Examination, Problem No. 36. Given (i) mðxÞ ¼ F þ e2x ; x $ 0 (ii) 0:4 p0 ¼ 0:50: Calculate F:

82

Chapter 3

Solution ð 0:4 ð 0:4 2x 0:50 ¼ 0:4 p0 ¼ exp 2 mðxÞ dx ¼ exp 2 ðF þ e Þ dx 0

0

e0:8 2 1 ¼ exp 2 0:4F þ : 2 Taking natural logarithms of both the sides and rearranging we obtain F ¼ 0:2009: Exercise 3.7.5 May 2000 SOA/CAS Course 3 Examination, Problem No. 1. Given (i) e 0 ¼ 25; (ii) lx ¼ v 2 x; 0 # x # v; (iii) TðxÞ is the future lifetime random variable. Calculate VarðTð10ÞÞ: Solution Since we are working with De Moivre’s Law, v is twice the life expectancy of a newborn. Hence v ¼ 50: For the 10 year old, the distribution of the remaining lifetime is uniform on ½0; 40; and its variance is 402 400 ¼ 133:333… ¼ 3 12

A

3.8 Mortality tables In the early models of life insurance, it was common to work with a deterministic cohort, i.e., a group of survivors from a group of initial l0 newborns. The size of such a group at age x is lx ; and the number of people who die between ages x and x þ 1 is denoted by dx : The collection of data describing the size of the cohort and its mortality is called a mortality table. Mortality tables are constructed from data collected by governments and insurance companies. In the United States, national census is performed every 10 years, and the data so collected is used to create tables adopted by the insurance regulators. The tables are, in turn, the source for estimation of survival and death probabilities. We have, for example: lxþ1 : lx

ð3:8:1Þ

lxþk ¼ px · pxþ1 · pxþ2 · · ·pxþk21 : lx

ð3:8:2Þ

px ¼ Similarly, k px

¼

83

Life insurance and annuities

Valuation of pension plans often requires the use of commutation functions. We will now define two functions that will be used often later in this chapter: D x ¼ vx · l x ; Nx ¼

þ1 X

ð3:8:3Þ ð3:8:4Þ

Dxþk :

k¼0

Additional discussion of commutation functions is given by Aitken [1] and Bowers et al. [14]. In the insurance practice, a group underwritten for coverage usually has different mortality than the general population. A cohort for which some additional information about mortality is obtained is called a select cohort. If the effect of underwriting wears off after some period of time, we say that the group ultimately becomes like the general population, and the table so obtained is called select and ultimate. We use the following notation for a select table: t p½x ; t q½x ; where ½x represents select mortality, and q½xþr ¼ q½x2jþrþj ¼ qxþr ;

ð3:8:5Þ

where r is the select period. Mortality tables created in practice show data only for the integer ages. This creates some degree of difficulty for insurance firms and pension plans, as they have to work with the insured people and plan participants who may die, retire, or terminate contracts at noninteger ages. In order to better model those situations, we introduce the random variable: S ¼ T 2 K:

ð3:8:6Þ

We have 0 # S , 1 and S is the fraction of a year lived in the year of death. Because the standard assumption is that T has an absolutely continuous distribution, so does S: The CDF of the random variable S conditional on K ¼ k is given by the formula PrðS # slK ¼ kÞ ¼

s qxþk

qxþk

:

ð3:8:7Þ

Assume now that the random variables S and K are independent. Then PrðS # slK ¼ kÞ does not depend on k and for a certain function FS ðsÞ; for each s [ ½0; 1; s qxþk

¼ FS ðsÞ · qxþk ;

ð3:8:8Þ

for every k ¼ 0; 1; …: Of course, FS in (3.8.8) is the CDF of S: If we assume that S is uniformly distributed, then (3.8.10) becomes an equality. Furthermore, the independence of K and S implies in this case (uniform distribution of S and we will return to this special case in detail later): VarðTÞ ¼ VarðKÞ þ

1 : 12

84

Chapter 3

Also, the expected value of the conditional distribution of S given that K ¼ k will be termed the fractional life expectancy. In what follows, we will use the following discrete version of the random variable S: For an m [ : (the set of positive integers) we define a new random variable 1 bmS þ 1c: m

SðmÞ ¼

ð3:8:9Þ

This variable is obtained by rounding S down to the smallest multiple of the fraction 1=m that already exceeds S: The random variable SðmÞ is discrete, with a jump in its CDF on the atoms 1=m; 2=m; …; 1: If K and S are independent, then so are K and SðmÞ : If S is uniformly distributed, then SðmÞ has a uniform distribution on the set of its atoms. While we do not generally know the exact probability distribution of S; several approximation assumptions are commonly used. They are called the fractional age assumptions. The first such assumption is called the Uniform Distribution of Deaths (UDD) assumption, which states that S has the uniform distribution on the unit interval ½0; 1: Let us note that De Moivre’s Law also assumes uniform distribution, but UDD is different, as it gives a different uniform distribution in each year of age. Under UDD, K and S are independent, and EðTÞ ¼ EðKÞ þ UDD

1 ; 2

VarðTÞ ¼ VarðKÞ þ UDD

1 : 12

ð3:8:10Þ

ð3:8:11Þ

We also have the following UDD identities for 0 # s þ t # 1; s [ ½0; 1; t [ ½0; 1; t qx

¼ t · qx ;

ð3:8:12Þ

UDD

mðx þ tÞ ¼

UDD

qx ; 1 2 tqx

qx ¼ fT ðtÞ ¼ t px mðx þ tÞ ¼ ð1 2 t qx Þmðx þ tÞ ¼ ð1 2 tqx Þmðx þ tÞ; UDD

UDD

lxþt ¼ lx 2 tdx ; UDD

ð3:8:13Þ ð3:8:14Þ ð3:8:15Þ

and s qxþt

¼

UDD

sqx : 1 2 tqx

ð3:8:16Þ

85

Life insurance and annuities

The second partial age assumption assumes constant force (CF) of mortality in the year of death. We have then: smxþ1=2 ¼ 2 ln s px ;

ð3:8:17Þ

¼ e2smxþ1=2 ¼ psx :

ð3:8:18Þ

CF

as well as s px

Furthermore, the conditional distribution of S; given that K ¼ k; is PrðS # slK ¼ kÞ ¼

1 2 psxþk ; 1 2 pxþk

0 # s # 1:

ð3:8:19Þ

This is the truncated exponential distribution. Note that in this case the random variables S and K are not, in general, independent, but in the special case when pxþk ¼ px (e.g., when the force of mortality is constant throughout all the ages) is independent of k; K and S are independent. The last commonly used partial age assumption is the one of Balducci, denoted as BAL. Under this model, the probability of death by the end of the year for a person who has survived a fraction s of the year between ages x and x þ 1 is a linear function of the form: 12s qxþs

¼ ð1 2 sÞqx :

BAL

ð3:8:20Þ

As px ¼ ð12s pxþs Þs px (3.8.20) implies that s px

¼

BAL

1 2 qx : 1 2 ð1 2 sÞqx

ð3:8:21Þ

Recalling the definition of the force of mortality, we obtain

mðx þ sÞ ¼

BAL

qx ; 1 2 ð1 2 sÞqx

ð3:8:22Þ

and PrðS # slK ¼ kÞ ¼

1 2s pxþk s ¼ : 1 2 ð1 2 sÞqxþk 1 2 pxþk

ð3:8:23Þ

Note that (3.8.23) implies that the random variables S and K are not independent. The Balducci assumption is also commonly called the hyperbolic assumption, because the graph of lxþs for 0 # s # 1 is a hyperbola in this case. Exercises Exercise 3.8.1 Using the constant force assumption, calculate the fractional life expectancy of S given that K ¼ k; under the constant force assumption for women,

86

Chapter 3

and men. You have the following data: qxþk ¼ 0:00576 for women and qxþk ¼ 0:01652 for men (this data comes from the mortality table for Poland, age x ¼ 50; k ¼ 5). Solution We have EðSlK ¼ kÞ ¼

ð1

ð1

s d PrðS # slK ¼ kÞ ¼ ð1 2 PrðS # slK ¼ kÞÞds 0 ð1 1 1 qxþk s ¼12 1 2 pxþk ds ¼ 1 2 1þ : qxþk qxþk ln pxþk 0 0

For the data given, we obtain: EðSlK ¼ 5Þ ¼ 0:499952 for women and EðSlK ¼ 5Þ ¼ 0:49861 for men. Let us note that these fractional life expectancies differ very little from the approximation of 0.50 given by the UDD assumption. A Exercise 3.8.2 November 2003 SOA/CAS Course 3 Examination, Problem No. 16. For a space probe to Mars: (i) The probe has three radios, whose future lifetimes are independent, each with mortality following k lq0

¼ 0:1ðk þ 1Þ;

k ¼ 0; 1; 2; 3;

where time 0 is the moment the probe lands on Mars. (ii) The failure time of each radio follows the hyperbolic assumption within each year. (iii) The probe will transmit until all three radios have failed. Calculate the probability that the probe is no longer transmitting 2.25 years after landing on Mars. Solution We have 0 lq0 ¼ 0:1; 1 lq0 ¼ 0:2; 2 lq0 ¼ 0:3; 3 lq0 ¼ 0:4; and therefore, q0 ¼ 1=10; q1 ¼ 2=9, q2 ¼ 3=7; q4 ¼ 1: We want to find the probability 3 7 p2 · ð2:5 q0 Þ ¼ ð1 2 2:25 p0 Þ ¼ ð1 2 2 p0 · 0:25 p2 Þ ¼ 1 2 BAL 10 1 2 ð1 2 0:25Þq2 3

3

3

0

13 4 3 7 39 B C 7 · ¼ @1 2 ¼ 0:06918676: A¼ 3 3 10 95 12 £ 4 7

A

87

Life insurance and annuities

Exercise 3.8.3 November 2001 SOA/CAS Course 3 Examination, Problem No. 2. For a select-and-ultimate mortality table with a 3-year ultimate period: q½60 q½61 q½62 q½63 q½64

¼ 0:09; ¼ 0:10; ¼ 0:11; ¼ 0:12; ¼ 0:13;

q½60þ1 q½61þ1 q½62þ1 q½63þ1 q½64þ1

¼ 0:11; ¼ 0:12; ¼ 0:13; ¼ 0:14; ¼ 0:15;

q½60þ2 q½61þ2 q½62þ2 q½63þ2 q½64þ2

¼ 0:13; ¼ 0:14; ¼ 0:15; ¼ 0:16; ¼ 0:17;

q½60þ3 q63 q½61þ3 q64 q½62þ3 q65 q½63þ3 q66 q½64þ3 q67

¼ 0:15; ¼ 0:16; ¼ 0:17; ¼ 0:18; ¼ 0:19;

(i) White was a newly selected life on 01/01/2000. (ii) White’s age on January 1, 2001 is 61. (iii) P is the probability on January 1, 2001 that White will be alive on January 1, 2006. Calculate P: Solution The probability sought is 5 p½60þ1

¼ p½60þ1 · p½60þ2 · p63 · p64 · p65 ¼ ð1 2 0:11Þð1 2 0:13Þð1 2 0:15Þ ð1 2 0:16Þð1 2 0:17Þ ¼ 0:4589:

A

Exercise 3.8.4 May 2001 SOA/CAS Course 3 Examination, Problem No. 1. For a given life age 30, it is estimated that an impact of a medical breakthrough will be an increase of 4 years in e 30 ; the complete expectation of life. Prior to the medical breakthrough, sðxÞ followed De Moivre’s Law with v ¼ 100 as the limiting age. Assuming De Moivre’s Law still applies after the medical breakthrough, calculate the new limiting age. Solution With De Moivre’s Law, TðxÞ is uniformly distributed on ½0; v 2 x; and e x ¼

ðv 2 xÞ : 2

Prior to the medical breakthrough, v ¼ 100 and e 30 ¼

70 ¼ 35: 2

After the breakthrough the limiting age value increases by 4 and therefore: e 30 ¼ 39 ¼ 35 þ 4 ¼ Hence, the new limiting age value is 108.

ðv 2 30Þ : 2

88

Chapter 3

Exercise 3.8.5 May 2001 SOA/CAS Course 3 Examination, Problem No. 13. Mr. Ucci has only three hairs left on his head and he will not be growing any more. (i) The future mortality of each hair is k lqx

¼ 0:1ðk þ 1Þ;

where k ¼ 0; 1; 2; 3; and x is Mr. Uccis age. (ii) Hair loss follows the hyperbolic assumption at fractional ages. (iii) The future lifetimes of the three hairs are independent. Calculate the probability that Mr. Ucci is bald (has no hair left) at age x þ 2:5: Solution Since three independent hairs must ‘die’ by age x þ 2:5; the probability sought is ð2:5 qx Þ3 : We are given that qx ¼ 0:1; 1lqx ¼ 0:2; and 2lqx ¼ 0:3: From the first two of the probabilities, it follows that 2 qx ¼ 0:1 þ 0:2 ¼ 0:3 and hence 2 px ¼ 0:7: From this, combined with 2lqx ¼ 0:3; it follows that 0:3 ¼2lqx ¼ 2 px · qxþ2 ¼ 0:7qxþ2 so that qxþ2 ¼ 3=7: With Balducci’s Law t pxþ2

¼

pxþ2 : pxþ2 þ t · qxþ2

Therefore

0:5 qxþ2

¼ 1 2 0:5 pxþ2 ¼ 1 2

4 7 4 3 þ 0:5 · 7 7

¼

1:5 3 ¼ : 5:5 11

Finally, 2:5 qx

¼ 2 qx þ 2 px ·

0:5 qxþ2

¼ 0:3 þ 0:7 ·

3 ¼ 0:490909; 11

and ð2:5 qx Þ3 ¼ 0:1183:

A

Exercise 3.8.6 November 2000 SOA/CAS Course 3 Examination, Problem No. 4. Mortality for Audra, age 25, follows De Moivre’s Law with v ¼ 100: If she takes up hot air ballooning for the coming year, her assumed mortality will be adjusted so that for the coming year only, she will have a constant force of mortality of 0.1. Calculate the decrease in the 11-year temporary complete life expectancy for Audra if she takes up hot ballooning.

89

Life insurance and annuities

Solution From De Moivre’s Law, t p25

¼12

t : 75

Before the mortality adjustment ð 11 ð 11 t 112 e 25: 11 ¼ ¼ 10:1933: p dt ¼ 1 2 dt ¼ 11 2 t 25 75 2 · 75 0 0 To analyze the effect of 1-year mortality adjustment on the temporary life expectancy, we use the recursive relation: e 25: 11 ¼ e 25:1 þ p25 · e 26: 10 : Only e 25: 1 and p25 are affected by the change in mortality. With the change, we have p25 ¼ e20:10 and e 25:1 ¼

ð1

e20:10t dt ¼

0

1 2 e20:10 ¼ 0:951626: 0:10

Also e 26: 10 ¼

ð 10 0

t p26

dt ¼

ð 10 t 102 ¼ 9:324324: 12 dt ¼ 10 2 74 2 · 74 0

Substituting into the recursion relation we get the adjusted life expectancy: e 25: 11 ¼ e 25:1 þ p25 · e 26: 10 ¼ 0:951626 þ e20:10 · 9:324324 ¼ 9:388624: The decrease is 10:1933 2 9:3886 ¼ 0:8047:

A

Exercise 3.8.7 May 2000 SOA/CAS Course 3 Examination, Problem No. 12. For a certain mortality table, you are given: (i) mð80:5Þ ¼ 0:0202; (ii) mð81:5Þ ¼ 0:0408; (iii) mð82:5Þ ¼ 0:0619; (iv) deaths are uniformly distributed between integral ages. Calculate the probability that a person aged 80.5 will die within 2 years. Solution Under UDD 0:0408 ¼ mð81:5Þ ¼

UDD

q81 ; 1 2 0:5q81

90

Chapter 3

and therefore q81 ¼ 0:04: Similarly, q80 ¼ 0:02 and q82 ¼ 0:06: Hence 2 q80:5

¼ 0:5 q80:5 þ 0:5 p80:5 · ðq81 þ p81 · ¼

0:5 q82 Þ

0:01 0:98 þ · ð0:04 þ 0:96 · 0:03Þ ¼ 0:0782: 0:99 0:99

A

3.9 Life insurance We will now consider insurance contracts, which make a one-time payment of a specified amount of insurance in the case of death of the insured, such payment made to a named policy beneficiary. In order to calculate the present value of the benefit, one should use the actual future interest rates between now and the death of the insured. But these interest rates are not known ex ante, and in practice actuaries choose an appropriate valuation rate for valuation (calculation of reserve) purposes, or for premium calculation. Similarly, for pension plans, a valuation rate is used for establishing actuarial liabilities and normal cost of a given plan. Such an interest rate chosen by the actuary will be denoted by i; and the corresponding discount factor will be written as v: As in the previous sections, the age of a person under consideration (i.e., person entering into the life insurance contract) will be denoted by x: The actual benefit amount and the time of payment of the benefit for the life insurance policy will depend on the random variable TðxÞ; the future lifetime of a life age ðxÞ; starting from the policy effective date. The present value of the said benefit will be denoted by Z; and Z is, just as TðxÞ; a random variable. The expected value of Z; EðZÞ; is called the actuarial present value of the benefit, or the single benefit premium, or net single premium, of the policy. The first form of a life insurance contract we will consider is traditionally called the whole life insurance. In this contract, the insurance amount (which, for simplification, we will assume to be a monetary unit) is paid upon death of the insured. The payment can be made either immediately (the policy is then termed continuous) or at the end of the year of death (the policy is then termed discrete). Using the notation introduced earlier in this chapter, we assume that the insured was aged when the policy was issued, T is the future lifetime of the insured, and K ¼ bTc; the number of whole years lived by the insured. The present value of the benefit payment random variable is denoted by Z: For the discrete policy, the death benefit is assumed to be paid at the end of the year of death, and we have Z ¼ vKþ1 :

ð3:9:1Þ

Furthermore EðZÞ ¼

1 X k¼0

vkþ1 PrðK ¼ kÞ ¼

1 X k¼0

vkþ1 · k px · qxþk :

ð3:9:2Þ

91

Life insurance and annuities

Recall that this expected value is also termed single benefit premium for the policy. We will denote it for the discrete whole life insurance by Ax : Therefore Ax ¼

1 X

vkþ1 · k px · qxþk :

ð3:9:3Þ

k¼0

Note that Ax ; Ax ðvÞ is a function of the discount factor v: In all practical applications v is a fraction, i.e., v [ ð0; 1Þ: Let us also observe that Ax ð·Þ is the probability generating function of the random variable K þ 1 as expressed by the function GKþ1 ðvÞ ¼

1 X

vkþ1 PrðK þ 1 ¼ k þ 1Þ:

ð3:9:4Þ

kþ1¼1

It is not hard to see that Ax ð·Þ is an increasing function of v [ ð0; 1Þ: In fact, if EðKÞ , 1; then 1 X d Ax ðvÞ ¼ ðk þ 1Þvk PrðK ¼ kÞ; dv k¼0

v [ ð0; 1Þ:

ð3:9:5Þ

Using (3.9.5) and the definition of curtate life expectancy, we infer that d A ð1Þ ¼ EðK þ 1Þ ¼ ex þ 1; dv x

v [ ð0; 1Þ:

ð3:9:6Þ

If EðK 2 Þ , 1 then the second derivative of the function Ax ð·Þ equals 1 X d2 A ðvÞ ¼ ðk þ 1Þkvk21 PrðK ¼ kÞ . 0; x dv2 k¼0

v [ ð0; 1Þ:

ð3:9:7Þ

It is easy to see, based on the above, that for any v [ ð0; 1Þ d2 Ax ðvÞ . 0; dv2

ð3:9:8Þ

as long as PrðK ¼ 0Þ , 1: Thus, if the random variable K is not degenerate at zero (i.e., there is at least one person alive at age x who survives for another year), then Ax ð·Þ is strictly convex. From (3.9.7) we also conclude that d2 Ax ð1Þ ¼ EðK 2 Þ þ EðKÞ: dv2

ð3:9:9Þ

Therefore, by combining the above with (3.9.6) and (3.9.8) we get the formula for the variance s 2 of the random variable K d2 d d 2 A ð1Þ A ð1Þ 2 1 : s ¼ 2 Ax ð1Þ 2 ð3:9:10Þ dv x dv x dv

92

Chapter 3

The properties of the function Ax ð·Þ help us to calculate all higher moments of the random variable Z: For example, the second moment is EðZ 2 Þ ¼

1 X

v2ðkþ1Þ · k px · qxþk ¼ Ax ðv2 Þ:

ð3:9:11Þ

vnðkþ1Þ · k px · qxþk ¼ Ax ðvn Þ ¼ Ax ðe2nd Þ:

ð3:9:12Þ

k¼0

In fact, it is quite easy to see that EðZ n Þ ¼

1 X k¼0

The formula (3.9.12) assumes such a simple form for this type of life insurance, because the benefit amount is 1, and any power of 1 is equal to 1 itself. There is only one other number that can replace 1 and still allow for this property, that number is 0. This gives rise to the popular Rule of Moments for the calculation of the moments of the random variable Z: as long as the amount of insurance benefit can be only 0 or 1, the nth moment of Z is calculated by replacing the force of interest d; in the formula for EðZÞ; by its multiple nd: In particular, the single benefit premium calculated with the force of interest multiplied by n is denoted by a superscript of n placed in front of the regular notation for the single benefit notation, e.g., 2 Ax for discrete whole life. From (3.9.12) we also conclude that VarðZÞ ¼ Ax ðv2 Þ 2 ðAx ðvÞÞ2 :

ð3:9:13Þ

The above considerations were concerned with the case of a discrete whole life insurance, with benefit payable at the end of the year of death. Of course, in the insurance practice, benefit is paid very soon after the death of the insured. To model the reality of the insurance marketplace, it is far more natural to use the random variable T; the future lifetime of life age x: The random variable Z; the present value of the insurance benefit for the continuous whole life insurance, is then Z ¼ vT ;

ð3:9:14Þ

and the single benefit premium for this policy is ð1 ð1 vt fT ðtÞ dt ¼ vt t · px · qxþt · dt: A x ¼ EðZÞ ¼ 0

ð3:9:15Þ

0

Recall that T ¼ K þ S; where K ¼ bTc: Therefore A x ¼ EðZÞ ¼ EðvKþ1 vS21 Þ:

ð3:9:16Þ

If we assume UDD in the year of death, then the random variables K þ 1 and 1 2 S are independent, and 1 2 S has uniform distribution on ð0; 1Þ: Therefore, ð1 Ax ¼ EðZÞ ¼ EðvKþ1 Þ · EðvS21 Þ ¼ Ax e2dðs21Þ ds ¼ i Ax : ð3:9:17Þ UDD d 0

93

Life insurance and annuities

Let us now assume that a year is divided into m equal parts (e.g., 12 months, with m ¼ 12; or four quarters, with m ¼ 4) and the benefit of the policy is paid at the end of the part of the year in which death occurs. The single benefit premium for such a plan ðmÞ of insurance will be denoted by Ax : We have ðmÞ

¼ EðvKþS Þ; AðmÞ x

ð3:9:18Þ

where SðmÞ is defined by (3.8.9). Using the UDD assumption, in a manner analogous to the one for A x ; we can derive AðmÞ ¼ x

i

UDD iðmÞ

ð3:9:19Þ

Ax :

One more important observation we should make is that the value of the single benefit premium for the unit whole life insurance can be calculated quite easily under the simple assumptions of the constant force of mortality or De Moivre’s Law. Indeed, if m is the constant force, then A x ¼

ð1

CF

e2td e2tm · m · dt ¼ 2

0

m 2tðmþdÞ t!1 m e ¼ : mþd m þd t¼0

ð3:9:20Þ

For the discrete whole life policy, if we write p ¼ px ¼ e2d ; and q ¼ 1 2 p; then we have Ax ¼

CF

1 X

vkþ1 · k px · qxþk ¼

k¼0

1 X

vkþ1 · pk · ð1 2 pÞ ¼ vð1 2 pÞ

k¼0

¼ vð1 2 pÞ

1 X

v k · pk

k¼0

1 q q ¼ ¼ : 1 2 vp ð1 þ iÞ 2 p qþi

ð3:9:21Þ

We can see that (3.9.20) and (3.9.21) are direct analogues of each other. On the other hand, under De Moivre’s Law, we have ð v2x a 1 · dt ¼ v2x ; ð3:9:22Þ A x ¼ vt · DML 0 v2x v2x and Ax ¼

vX 2x21

vkþ1 · k px · qxþk ¼

DML

k¼0

¼

vX 2x21 k¼0

vkþ1 ·

vX 2x21 k¼0

1 a ¼ v2x : v2x v2x

vkþ1 ·

v2x2k 1 · v2x v2x2k ð3:9:23Þ

94

Chapter 3

Term life insurance is a simple modification of the whole life insurance policy. Unit term life contract calls for payment of death benefit of one monetary unit if the death of the insured occurs within a prescribed term, n years, from the policy effective date. If the insured survives n years, the policy ends without a benefit payment. First, let us consider the discrete model in which the payment is made at the end of the year of death. Then the present value of the death benefit equals ( Kþ1 v ; K # n 2 1; Z¼ ð3:9:24Þ 0; K $ n: The single benefit premium for this contract is denoted by A1x: n : By definition A1x: n ¼

n21 X

vkþ1 PrðK ¼ kÞ ¼

k¼0

n21 X

vkþ1 · k px · qxþk :

ð3:9:25Þ

k¼0

Just as for Ax ; A1x: n ; A1x: n ðvÞ is also a function of v: The second moment of the present value of benefit random variable Z is given by the formula: EðZ 2 Þ ¼

n21 X k¼0

v2ðkþ1Þ PrðK ¼ kÞ ¼

n21 X

v2ðkþ1Þ · k px · qxþk ¼ A1x: n ðv2 Þ:

ð3:9:26Þ

k¼0

The Rule of Moments applies to this unit term life insurance. Also, VarðZÞ ¼ A1x: n ðv2 Þ 2 ðA1x: n ðvÞÞ2 :

ð3:9:27Þ

Unit term life insurance is also considered in the continuous case, for which ( T v ; T # n; Z¼ ð3:9:28Þ 0; T . n: For the continuous case, the single benefit premium is denoted by A 1x: n : The Rule of Moments also applies. A pure endowment insurance unit contract pays a monetary unit after a prescribed number of years, e.g., n; if and only if the insured survives that number of years. While in the definition there is no mention of the length of future lifetime of the insured, the present value of the benefit does depend on it. For this policy, there is no difference between the discrete or continuous model, and present value of benefit random variable is ( ( 0; K # n 2 1; 0; T , n; Z¼ ¼ ð3:9:29Þ n v ; K $ n; vn ; T $ n:

95

Life insurance and annuities

Therefore, Z is the product of vn and a zero-one Bernoulli Trial random variable with the probability of success n px : The single benefit premium for this pure endowment policy is denoted by A1x: n or by n Ex ; and it equals EðZÞ ¼ A1x: n ¼ vn · n px :

ð3:9:30Þ

The variance of Z is given by the formula VarðZÞ ¼ v2n · n px · n qx :

ð3:9:31Þ

The Rule of Moments applies to the unit pure endowment. A similar type of life insurance policy is an endowment, which pays a monetary unit at death of the insured if such death occurs before the n years of policy term and also pays a monetary unit if the insured survives those n years. Effectively, this insurance is a sum of a unit pure endowment and a unit year term insurance. The present value of benefit random variable for the discrete version of this policy is ( Kþ1 v ; K # n 2 1; Z¼ ð3:9:32Þ vn ; K $ n: Of course, the present value of benefit for this plan of insurance is the sum of the present value of benefit for a pure n-year term insurance (let us denote that random variable by Z1 ) and the present value of benefit for pure n-year endowment (let us denote that random variable by Z2 ). Using that observation, we establish the formula for the single benefit premium Ax: n for endowment insurance: Ax: n ¼ EðZ1 Þ þ EðZ2 Þ ¼ A1x: n þ Ax: n 1 :

ð3:9:33Þ

This insurance also has a continuous version. Since the product of the benefit payment for Z1 and Z2 is always zero, CovðZ1 ; Z2 Þ ¼ EðZ1 · Z2 Þ 2 A1x: n · Ax: n 1 ¼ 2A1x: n · Ax: n 1 ;

ð3:9:34Þ

and therefore VarðZÞ ¼ VarðZ1 Þ þ VarðZ2 Þ 2 2 · A1x: n · Ax: n 1 :

ð3:9:35Þ

The above equation implies that the risk (as measured by the variance) of an endowment policy is less than the sum of risks of its two building blocks: term insurance and pure endowment insurance. The endowment insurance can also be considered for the continuous model.

96

Chapter 3

The last type of life insurance we will consider is a deferred whole life insurance, which does not pay any benefit for a beginning period, m years, and then becomes a regular whole life insurance. The present value of benefit random variable for the discrete model is ( Z¼

K # m 2 1;

0; v

Kþ1

; K $ m:

ð3:9:36Þ

The single benefit premium for this plan of insurance is denoted by m lAx and it equals

m lAx

¼

1 X

vkþ1 PrðK ¼ kÞ ¼

k¼m

1 X

vkþ1 · k px · qxþk :

ð3:9:37Þ

k¼m

By using a substitution k ¼ i þ m in (3.9.36) we also obtain:

m lAx

¼

1 X

viþmþ1 ·

iþm px

· qxþiþm ¼ vm · m px ·

i¼0

1 X

viþ1 · i pxþm · qxþmþi

i¼0

¼ vm · m px · Axþm :

ð3:9:38Þ

Exercises Exercise 3.9.1 November 2003 SOA/CAS Course 3 Examination, Problem No. 2. For a whole life insurance of 1000 on ðxÞ with benefits payable at the moment of death: ( ðiÞ

dt ¼

( ðiiÞ

mðx þ tÞ ¼

0:04; 0 , t # 10; 0:05;

t . 10:

0:06;

0 , t # 10;

0:07;

t . 10:

Calculate the single benefit premium for this insurance.

97

Life insurance and annuities

Solution The single benefit premium is ðt ðt ð þ1 1000 exp 2 ds ds exp 2 mx ðsÞ ds mx ðtÞ dt 0 0 0 ðt ðt ð 10 exp 2 ds ds exp 2 mx ðsÞ ds mx ðtÞ dt ¼ 1000 0 0 0 ðt ðt ð þ1 exp 2 ds ds exp 2 mx ðsÞ ds mx ðtÞ dt þ 1000 10 0 0 ð þ1 ðt ð 10 20:04t 20:06t e e 0:06 dt þ 1000 exp 20:4 2 0:05 ds ¼ 1000 0

10

10

20:1t 10 ð þ1 e þ70 e21 £ exp 2 0:6 þ 0:07 ds 0:07 dt ¼ 60 e20:12ðt210Þ dt 2 0:1 0 10 10 ð þ1 1 e21 0:2 2 ¼ 60 e20:12t dt þ 70 e 0:1 0:1 10 21:2 0 70 21 21 0:2 e e ¼ 593:87: 2 ¼ 600 2 600 e þ 70 e ¼ 600 2 600 e21 þ 0:12 0:12 0:12 ðt

A Exercise 3.9.2 November 2003 SOA/CAS Course 3 Examination, Problem No. 22. For a whole life insurance of 1 on (41) with death benefit payable at the end of year of death, you are given: (i) i ¼ 0:05; (ii) p40 ¼ 0:9972; (iii) A41 2 A40 ¼ 0:00822; (iv) 2 A41 2 2 A40 ¼ 0:00433; (v) Z is the present-value random variable for this insurance. Calculate VarðZÞ: Solution Note that A40 ¼ vq40 þ vp40 · A41 ; and this implies that 0:9972 0:0028 A41 2 A40 ¼ 2 : 1:05 1:05 Combining this with A41 2 A40 ¼ 0:00822;

98

Chapter 3

we get 0:9972 0:0028 ; 12 A41 ¼ 0:00822 þ 1:05 1:05 and this means that A41 ¼ 0:21649621: Similarly 2

A40 ¼ v2 q40 þ v2 p40 · 2 A41 ;

and we infer that 0:9972 2 0:0028 A41 2 2 A40 ¼ 2 : 2 1:05 1:052 Combining this with 2

A41 2 2 A40 ¼ 0:00433;

we get 0:9972 0:0028 ; 12 · 2 A41 ¼ 0:00433 þ 2 1:05 1:052 and 2

A41 ¼ 0:07192616:

Finally VarðZÞ ¼ 2 A41 2 A241 ¼ 0:07192616 2 0:216496212 ¼ 0:02505555:

A

Exercise 3.9.3 November 2001 SOA/CAS Course 3 Examination, Problem No. 8. Each of 100 independent lives purchases a single premium 5-year deferred whole life insurance of 10 payable at the moment of death. You are given: (i) m ¼ 0:004: (ii) d ¼ 0:006: (iii) F is the aggregate amount the insurer receives from the 100 lives. Using the normal approximation, calculate F such that the probability the insurer has sufficient funds to pay all claims is 0.95. You are given that the 95th percentile of the standard normal distribution is 1.645.

99

Life insurance and annuities

Solution We have ( Z¼

T , 5;

0; T

10 · v ;

T $ 5;

and ( 2

Z ¼

T , 5;

0; 2T

100 · v ;

T $ 5; 0:04 ¼ 2:426123; 0:04 þ 0:06 0:04 ¼ 11:233224; · e25£0:04 Þ 0:04þ0:12

EðZÞ ¼ v5 · 5 px · 10 · Axþ5 ¼ 10ðe25ð0:06þ0:04Þ Þ EðZ 2 Þ ¼ v10 · 5 px · 100 · 2 Axþ5 ¼ 100ðe210

· 0:06

and hence VarðZÞ ¼ EðZ 2 Þ2ðEðZÞÞ2 ¼ 5:347153: The aggregate present value for the 100 lives is ZAGG ¼ Z1 þ···þZ100 : The 95th percentile given by the normal approximation is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 100 EðZÞ þ 1:645 100 VarðZÞ ¼ 100ð2:426123Þ þ 1:645 100ð5:347153Þ ¼ 280:65: A Exercise 3.9.4 May 2000 SOA/CAS Course 3 Examination, Problem No. 13. An investment fund is established to provide benefits on 400 independent lives age x: (i) On January 1, 2001, each life is issued a 10-year deferred whole life insurance of 1000, payable at the moment of death. (ii) Each life is subject to a constant force of mortality of 0.05. (iii) The force of interest is 0.07. Calculate the amount needed in the investment fund on January 1, 2001, so that the probability, as determined by the normal approximation, is 0.95 that the fund will be sufficient to provide these benefits. You are given that the 95th percentile of the standard normal distribution is 1.645. Solution We have m 210ðmþdÞ 5 21:2 EðZÞ ¼ 1000 e ¼ 1000 ¼ 125:5; e mþd 12

100

Chapter 3

and VarðZÞ ¼ 10002

m e210ðmþ2dÞ 2 m þ 2d

5 12

2

e22:4

¼ 23 610:16:

As S¼

400 X

Z;

i¼1

we have EðSÞ ¼ 400 EðZÞ ¼ 50 200; and VarðSÞ ¼ 400 VarðZÞ ¼ 9 444 064: The amount needed is the 95th percentile of the distribution of S pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1:645 · 9 444 064 þ 50 200 ¼ 55 255:

A

Exercise 3.9.5 May 2000 SOA/CAS Course 3 Examination, Problem No. 17. The future lifetimes of a certain population can be modeled as follows: (i) Each individual’s future lifetime is exponentially distributed with constant hazard rate u: (ii) Over the population, u is uniformly distributed over ð1; 11Þ: Calculate the probability of surviving to time 0.5 for an individual randomly selected at time 0. Solution We use the Fubini Theorem for the appropriate double integral: ð1 ð 1 ð 11 PrðT . 0:5Þ ¼ fT ðtÞ dt ¼ fT ðtlQ ¼ uÞ · fQ ðuÞ du dt 0:5 0:5 u¼1 ð 11 ð 1 ð 1 ð 11 1 · du dt ¼ u e2ut · u e2ut dt · 0:10 du ¼ 10 0:5 t¼0:5 u¼1 u¼1 0 1 ð 11 25:5 20:5 2e Be C A ¼ ðe20:5u Þ · 0:10 du ¼ 0:10 · @ A ¼ 0:1205: 1 u¼1 2 2 Exercies 3.9.6 May 2000 SOA/CAS Course 3 Examination, Problem No. 21. A risky investment with a constant rate of default will pay principal and accumulated interest at 16% compounded annually at the end of 20 years if it does not

101

Life insurance and annuities

default, and zero if it defaults. A risk-free investment will pay principal and accumulated interest at 10% compounded annually at the end of 20 years. The principal amounts of the two investments are equal. The actuarial present values of the two investments are equal at time zero. Calculate the median time until default or maturity of the risky investment. Solution Let p be the constant annual rate of survival from default for the risky investment. Then its probability of survival over 20 years is p ¼ p20 : Therefore, the actuarial present value of a unit invested in the risky investment is

p · 1:1620 · 1:10220 þ ð1 2 pÞ · 0 ¼

1:16 1:10

20

p20 :

The actuarial present value of a unit investment in the risk-free investment is 1 and the two quantities are equal; therefore p ¼ 1:10=1:16: The problem effectively assumes constant force of default, i.e., exponential distribution of time to default for times between 0 and 20 and then the residual probability of paying the maturity amount at time 20. What is the median of that distribution? If the cumulative probability of default is more than 0.50, it is somewhere between 0 and 20, otherwise it is at 20. The constant force of default is m ¼ lnð1:16=1:10Þ and the median m; if it falls below 20, satisfies 1:10 m 1 e2mm ¼ ¼ : 1:16 2 Luckily, this equation has a nice solution: m¼

ln 2 ¼ 13:0512043: ln 1:16 2 ln 1:10

A

3.10 Life annuities We will consider series of payments lasting till the end of life of the insured or in some form dependent on the survival of the insured. Such streams of payments are termed life annuities. Annuities may be paid not only over the physical life of the insured, but also as long as the insured is a participant in an insured group of some type (e.g., a pension plan). For a unit annuity, the amount of payment is a monetary unit. They play a double role in the study of insurance. First, annuity policies pay benefits in this form. Secondly, life insurance premiums are paid by the insured in a form of such an annuity. The actuarial present value or single benefit premium (also called net premium) of an annuity is the expected present value of the stream of payments. If an annuity is payable as long as the insured

102

Chapter 3

lives, but no longer than a prescribed term (e.g., n years) we term it a temporary life annuity. Let T; as always, denote the future lifetime of an insured aged x (this future lifetime could be his/her length of future employment service towards pension plan benefits accrual), K ¼ bTc; and let Y be the present value of a unit life annuity payable as long as the insured survives. This could be a continuous, or discrete, annuity. For the discrete model, we will usually assume that this is an annuity due. For such a model we have Y ¼ 1 þ v þ · · · þ vK ¼ a€ Kþ1 :

ð3:10:1Þ

Recall that K is a random variable with probability distribution: PrðK ¼ kÞ ¼ k px · qxþk ;

k ¼ 0; 1; 2; …

ð3:10:2Þ

The actuarial present value of this life annuity due is denoted by a€ x and it equals: a€ x ¼ EðYÞ ¼

þ1 X

a€ kþ1 · k px · qxþk :

ð3:10:3Þ

vk I{k#K } ;

ð3:10:4Þ

k¼0

Since Y¼

1 X k¼0

where ( I{k#K } ¼

1;

k # K;

0;

k . K;

ð3:10:5Þ

then a€ x ¼ EðYÞ ¼

þ1 X

vk · PrðK $ kÞ ¼

k¼0

þ1 X

v k · k px :

ð3:10:6Þ

k¼0

Recalling the definition of the commutation functions (3.8.3) and (3.8.4) we arrive at the following formula: þ1 X

a€ x ¼

þ1 X k¼0

v

k lxþk

lx

þ1 xþk X v lxþk ¼ ¼ vx lx k¼0

þ1 X

vxþk lxþk

k¼0 x

v lx

¼

Dxþk

k¼0

Dx

¼

Nx : Dx

ð3:10:7Þ

Example 3.10.1 Consider a life annuity due payable once a year for a Polish man aged x ¼ 65 in the amount of 12 000 per annum. Let the valuation rate be i ¼ 0:05:

103

Life insurance and annuities

Mortality table for Poland gives Nx ¼ 24 896:14 and Dx ¼ 2676:52 for male aged 65 (see [68]). Calculate the single benefit premium for this annuity for the Polish man. From Eq. (3.10.7) we get the following net single premium for the annuity for the man: 12 000

Nx ¼ 111 620:20: Dx

Assume also that for a Polish woman aged x ¼ 60; with the same valuation interest rate, Nx ¼ 60 742:84 and Dx ¼ 4774:29 based on the same table. The actuarial present value of a life annuity, in the same amount as the one for the man, for the woman is

12 000

Nx ¼ 152 674:86: Dx

A

Recall that 1 2 vKþ1 : d

a€ Kþ1 ¼ Therefore a€ x ¼

1 2 Ax : d

ð3:10:8Þ

Furthermore, we also have the following variance relationship:

Varð€a Kþ1

1 2 vKþ1 Þ ¼ Var d

2

¼

Ax 2 A2x : d2

ð3:10:9Þ

Similar variance relationships between life annuities and the corresponding life insurances also hold for continuous and modal (i.e., paid m times a year) annuities. We also have the following recursion formula a€ xþ1 ¼ To prove (3.10.10) note that a€ xþ1 ¼

1 X

kþ1 px

vk k pxþ1 ¼

k¼0

¼

1 ð€a 2 p Þ: vpx x 0 x

a€ x 2 1 : vpx

ð3:10:10Þ

¼ px · k pxþ1 and 1 1 1 X 1 X vk kþ1 px ¼ vi p px k¼0 vpx i¼1 i x

ð3:10:11Þ

104

Chapter 3

Consider now a life annuity due payable m times a year, in the amount of 1=m: The present value of the such a stream of payments is written as Y ðmÞ ¼

1 v1=m v2=m vKþL=m þ þ þ ··· þ ; m m m m

ð3:10:12Þ

where K is the number of whole years lived, and L the number of mth of the K þ 1st year in which a payment is made. Therefore Y ðmÞ ¼

1 2 ðv1=m ÞKmþLþ1 : mð1 2 v1=m Þ

ð3:10:13Þ

Since mSðmÞ ¼ L þ 1;

ð3:10:14Þ

d ðmÞ ¼ mð1 2 v1=m Þ;

ð3:10:15Þ

and

then ðmÞ

Y ðmÞ ¼ ðmÞ

Thus the single benefit premium a€ x

1 2 vKþS d ðmÞ

for this life annuity equals ðmÞ

a€ ðmÞ ¼ EðY ðmÞ Þ ¼ x

ð3:10:16Þ

:

ðmÞ

1 2 EðvKþS Þ 1 2 Ax ¼ dðmÞ d ðmÞ

;

ð3:10:17Þ

where the last step uses (3.9.18). Now let us turn our attention to the UDD assumption about the year of death. We ðmÞ will use it to find the relationship between a€ x and a€ x : From (3.10.17) it follows that ðmÞ

a€ ðmÞ x

1 2 Ax ¼ dðmÞ ¼ a€ x

i 1 i ¼ 1 2 ðmÞ Ax ¼ ðmÞ 1 2 ðmÞ ð1 2 da€ x Þ UDD d ðmÞ i d i 1

id i 2 iðmÞ 2 : iðmÞ dðmÞ iðmÞ d ðmÞ

ð3:10:18Þ

Let us denote the coefficient of a€ x by aðmÞ in (3.10.18) and the free term by 2bðmÞ: It can be shown the Taylor series expansions of the coefficients a and b with respect to d are of the following form:

aðmÞ ¼ 1 þ

m2 2 1 2 d þ ··· 12m2

ð3:10:19Þ

105

Life insurance and annuities

and

bðmÞ ¼

m21 m2 2 1 þ d þ ··· 2m 6m2

ð3:10:20Þ

From the above and (3.10.18) we get the following approximation a€ ðmÞ < a€ x 2 x

m21 : 2m

ð3:10:21Þ

We will now turn our attention to the continuous model. Let us now consider a series of payments made continuously and uniformly, with a total of one monetary unit paid over Ð the whole year. The nominal value of all payments made from time 0 till time t is t0 dt ¼ t; and the present value, denoted by a t ; equals ðt ðt a t ¼ exp 2 dðsÞ ds dt; ð3:10:22Þ 0

0

where dðsÞ is the force of interest equivalent to the discount factor used. Let T be the random future lifetime of a life age ðyÞ: Our standard assumption is that T has an absolutely continuous distribution with density fT ðtÞ: The expected present value of a life annuity paid continuously to this person is denoted by a y and it equals: ðt ð1 ð 1 ð t a y ; Eða T Þ ¼ a t fT ðtÞ dt þ exp 2 dðsÞ ds dt fT ðtÞ dt: ð3:10:23Þ 0

0

0

0

If the force of interest dð·Þ is a constant function of time and it equals d; then ð1 ð1 1 1 a y ¼ ð1 2 e2dt Þf ðtÞ dt ¼ ð1 2 e2dt Þt py · mðy þ tÞ dt; ð3:10:24Þ 0 d 0 d where the last step uses (3.7.9). Recalling the formula for the single benefit premium for the continuous whole life insurance (3.9.15) we obtain ð ð 1 1 1 2dt 1 1 1 t a y ¼ 2 e t py · mðy þ tÞ dt ¼ 2 v p · mðy þ tÞ dt d d 0 d d 0 t y ¼

1 1 2 A: d d y

ð3:10:25Þ

Let us now assume that the annuity payment is made continuously with intensity hðxÞ; such intensity being dependent on the age of the insured. The present value of such a life annuity, calculated at the moment when the person reaches retirement age y; will be denoted by a hy : We then have ðt ð 1 ð t h a y ¼ exp 2 dðsÞ ds hðy þ tÞ dt fT ðtÞ dt; ð3:10:26Þ 0

0

0

where fT ð·Þ is the density of the random variable T; future lifetime of a random person aged y: By integrating the right-hand side of (3.10.26) by parts, we get the following

106

Chapter 3

equivalent formula: ðt !1 ðt h a y ¼ 2 exp 2 dðsÞ ds hðy þ tÞ dt · ð1 2 FT ðtÞÞ 0 0 0 ðt ð1 þ exp 2 dðsÞ ds hðy þ tÞð1 2 FðtÞÞ dt 0 0 ðt ð1 ¼ exp 2 dðsÞ ds hðy þ tÞð1 2 FðtÞÞ dt: 0

ð3:10:27Þ

0

We should note that the formulas (3.10.26) and (3.10.27) are valid only under the assumption that the integral on the right-hand side of (3.10.26) is finite. Since 1 2 FT ðtÞ ¼

lyþt ; ly

(3.10.27) can be written in the following form: ðt ð1 lyþt h a y ¼ exp 2 dðsÞ ds hðy þ tÞ dt: ly 0 0

ð3:10:28Þ

ð3:10:29Þ

If we denote by w the age at which a given person joins a pension plan and by sðxÞ ¼x2w pw the survival function within the pension plan, then ðt ð1 sðy þ tÞ dt: ð3:10:30Þ a hy ¼ exp 2 dðsÞ ds hðy þ tÞ sðyÞ 0 0 In the case when the force of interest is constant and equal to d we get: ð1 ð1 sðy þ tÞ sðxÞ h 2dt dt ¼ e2dðx2yÞ hðxÞ dx; a y ¼ e hðy þ tÞ sðyÞ sðyÞ 0 y where x ¼ y þ t is the age of the person t years after retiring. A temporary life annuity is an annuity, which is payable only for a specified contract term, as long as the insured is still alive. The frequency of the payment, just as with life annuities, may be annual, m times a year (monthly, quarterly, etc.), or continuous. Temporary life annuities can be treated as regular life annuities with a modified mortality table—in the modified table the probability of dying would be equal to one at the end of the term of the annuity. For the purposes of this work, however, it is more appropriate to consider separately the ending of plan participation due to death, or withdrawal, or disability, or some other reason. Thus, without modifying the mortality tables in any form, let us denote by a€ x: n the single benefit premium for an n-year temporary unit life annuity due for a life age ðxÞ: The random present value of this annuity’s benefit payments, Y; equals: ( a€ Kþ1 ; for K # n 2 1 Y¼ ð3:10:31Þ a€ n ; for K $ n:

107

Life insurance and annuities

Therefore a€ x: n ¼ EðYÞ ¼

n21 X

a€ kþ1 k px · qxþk þ a€ n > n px :

ð3:10:32Þ

k¼0

By introducing a new random variable K 0 ¼ KI{K,n} þ ðn 2 1ÞI{K,n} ;

ð3:10:33Þ

we obtain Y¼

1 X

vk I{k#K 0 } :

ð3:10:34Þ

k¼0

From the above and from (3.10.32) we infer that a€ x: n ¼

1 X

vk PrðK 0 $ kÞ ¼

k¼0

n21 X

vk PrðK $ kÞ ¼

k¼0

n21 X

vk k px :

ð3:10:35Þ

k¼0

Recalling the definitions of the commutation functions (3.8.3) and (3.8.4), we can also write: a€ x: n ¼

1 X

v k k px 2

k¼0

¼

1 X

v k k px ¼

k¼n

1 1 X X vxþk lxþk vxþk lxþk 2 x v lx vx l x k¼0 k¼n

Nx 2 Nxþn : Dx

ð3:10:36Þ

Before we proceed to the exercises, let us note the values of the annuities single benefit premium in the two simple mortality cases: constant force and De Moivre’s Law. If the force of mortality is constant and equal to m (and the force of interest is constant and equal to d) then 1 2 A x a x ¼ ¼ d

12

m 1 mþd ¼ : mþd d

ð3:10:37Þ

For the discrete life annuity due, we have 1 2 Ax ¼ a€ x ¼ d

12

q 1þi qþi : ¼ qþi d

ð3:10:38Þ

When De Moivre’s Law holds, we have a 1 2 v2x 1 2 A x v 2 x ¼ ðv 2 xÞ 2 a v2x ¼ ðDaÞ v2x ; a x ¼ ¼ d d dðv 2 xÞ v2x

ð3:10:39Þ

108

Chapter 3

and in the discrete case: a 1 2 v2x 1 2 Ax v 2 x ¼ ðv 2 xÞ 2 a v2x ¼ ðD€aÞ v2x : ¼ a€ x ¼ d d dðv 2 xÞ v2x

ð3:10:40Þ

Exercises Exercise 3.10.1 November 2003 SOA/CAS Course 3 Examination, Problem No. 9. For an annuity payable semi-annually, you are given: (i) Deaths are uniformly distributed over each year of age. (ii) q69 ¼ 0:03: (iii) i ¼ 0:06: (iv) 1000A 70 ¼ 530: ð2Þ

Calculate a€ 69 : Solution We know that ð2Þ

a€ 69 ¼ að2Þ€a69 2 bð2Þ; UDD

where i i id 1 þ i pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1:0002163; að2Þ ¼ ð2Þ ð2Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i d 2ð 1 þ i 2 1Þ · 2ð1 2 1=ð1 þ iÞÞ and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i 2 ið2Þ i 2 2ð 1 þ i 2 1Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2573991: bð2Þ ¼ ð2Þ ð2Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i d 2ð 1 þ i 2 1Þ · 2ð1 2 1=ð1 þ iÞÞ We also know that a€ 69 ¼

1 2 A69 : d

Observe that A69 ¼ vq69 þ vp69 A70 ¼

q69 p 0:03 0:97 þ A : þ 69 A70 ¼ 1:06 1:06 70 1þi 1þi

Furthermore A70 ¼

UDD

d ln 1:06 0:53 ¼ 0:5147087: A70 ¼ 0:06 i

109

Life insurance and annuities

This gives A69 ¼ 0:4993089; and a€ 69 ¼

1 2 0:4993089 ¼ 8:8455075: 0:06 1:06

Finally ð2Þ

a€ 69 ¼ að2Þ€a69 2 bð2Þ ¼ 1:0002163 · 8:8455075 2 0:2573991 < 8:59: UDD

A

Exercise 3.10.2 November 2003 SOA/CAS Course 3 Examination, Problem No. 15. For a group of individuals all aged x; of which 30% are smokers and 70% are nonsmokers, you are given (i) d ¼ 0:10: (ii) A Smoker ¼ 0:444: x (iii) A Nonsmoker ¼ 0:286: x (iv) T is the future lifetime of ðxÞ: (v) VarðaSmoker Þ ¼ 8:818: T (vi) VarðaNonsmoker Þ ¼ 8:503: T Calculate Varða T Þ for an individual chosen at random from this group. Solution Let T be the future lifetime of an individual randomly chosen from this population. We have Ax ¼ EðvT Þ ¼ 0:30A Smoker þ 0:70A Nonsmoker ¼ 0:30 · 0:444 þ 0:70 · 0:286 ¼ 0:3334: x x Also VarðaSmoker Þ ¼ 8:818 ¼ T

2 Smoker Ax 2 ðA Smoker Þ x 2

d

and hence 2 Smoker Ax

¼ 0:285316:

¼

2 Smoker Ax 2 0:4442 ; 0:102

110

Chapter 3

Furthermore, VarðaNonsmoker Þ ¼ 8:503 ¼ T

2 Nonsmoker 2 ðA Nonsmoker Þ2 Ax x 2

d

¼

2 Nonsmoker 2 0:2862 Ax ; 0:102

and 2 Nonsmoker Ax

¼ 0:166826:

Hence 2

Ax ¼ 0:30 · 0:285316 þ 0:70 · 0:166826 ¼ 0:202373:

Finally Varða T Þ ¼

Ax 2 ðA x Þ2 ¼ 9:121744: d2

2

A

Exercise 3.10.3 November 2003 SOA/CAS Course 3 Examination, Problem No. 32. Your company currently offers a whole life annuity product that pays the annuitant 12 000 at the beginning of each year. A member of your product development team suggests enhancing the product by adding a death benefit that will be paid at the end of the year of death. Using a discount rate, d; of 8%, calculate the death benefit that minimizes the variance of the present value random variable of the new product. Solution Let B be the death benefit amount. Then the random present value of the total payment made by this product is

12 000

1 2 vKþ1 þ BvKþ1 ¼ 150 000 þ ðB 2 150 000ÞvKþ1 : 0:08

Thus, the variance of the present value random variable is ðB 2 150 000Þ2 VarðvKþ1 Þ: This is minimized (it is zero) when B ¼ 150 000: Why do we get this strange answer of zero variance? Because 150 000 is the present value of a perpetuity of 12 000 at d ¼ 8%; and this new product becomes effectively such a perpetuity, but paid till death, with perpetuity value being paid upon death to the heirs. In a sense, the product is no longer random, it becomes deterministic. A

111

Life insurance and annuities

Exercise 3.10.4 November 2002 SOA/CAS Course 3 Examination, Problem No. 4. You are given that (i) mðx þ tÞ ¼ 0:01; 0 # t , 5; (ii) mðx þ tÞ ¼ 0:02; t $ 5; (iii) d ¼ 0:06: Calculate a x : Solution We have a x ¼

ð1 0

e2dt t px dt:

Thus we need to use the information about the force to find t px : We have 8 20:01t ; 0#t,5 ðt > <e ðt p ¼ exp 2 m ðsÞ ds ¼ t x x > 0 : e20:05 · exp 2 0:02 ds ; t $ 5 5 8 20:01t <e ; 0#t,5 ¼ : e20:05 · e20:02ðt25Þ ; t$5 8 < e20:01t ; 0#t,5 ¼ : e0:05 · e20:02t ; t$5 and a x ¼

ð1 0

e2dt t px dt ¼

¼

2

ð5

e20:06t · e20:01t dt þ

ð1

0

1 20:35 1 e þ 0:07 0:07

e20:06t · e0:05 · e20:02t dt

5

þ e0:05 20 þ

1 20:40 e 0:08

¼ 13:0273426969309: A

Exercise 3.10.5 November 2002 SOA/CAS Course 3 Examination, Problem No. 31. For a 20-year deferred whole life annuity due of 1 per year on (45), you are given: (i) Mortality follows De Moivre’s Law with v ¼ 105: (ii) i ¼ 0%: Calculate the probability that the sum of the annuity payments actually made will exceed the actuarial present value at issue of the annuity.

112

Chapter 3

Solution Note that v 2 45 2 1 ¼ 59; and vk ¼ 1 for any k: Therefore a45 20 l€

¼

59 X k¼20

k px

¼

59 59 59 X X 60 2 k 1 X 41 ¼ ¼ 13:666… 12 k¼ 60 60 3 k¼20 k¼20 k¼20

The sum of the annuity payments will exceed the actuarial present value at issue of the annuity if there are 14 payments, i.e., if the (45) insured survives the 20-year deferral plus 13 more years (as the annuity payments are made at the beginning of each year). What is the probability of that? It is 60 2 33 ¼ 0:45: A 33 p45 ¼ 60 Exercise 3.10.6 November 2001 SOA/CAS Course 3 Examination, Problem No. 3. For a continuous whole life annuity of 1 on ðxÞ: (i) TðxÞ is the future lifetime random variable for ðxÞ: (ii) The force of interest and force of mortality are constant and equal. (iii) a x ¼ 12:50: Calculate the standard deviation of a TðxÞ . Solution Since m ¼ d and 12:50 ¼ a x ¼

1 ; mþd

then m ¼ d ¼ 0:04: From the standard annuity variance formula: 2 2 2 1 1 m m 2 2 Varða TðxÞ Þ ¼ ð Ax 2 ðAx Þ Þ ¼ 2 m þ 2d mþd d d 2 1 0:04 0:04 2 2 ¼ ¼ 52:08: 0:04 0:12 0:08 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Hence, the standard deviation is 52:08 ¼ 7:22:

A

Exercise 3.10.7 November 2001 SOA/CAS Course 3 Examination, Problem No. 17. For a group of individuals all aged x; you are given: (i) 30% are smokers and 70% are nonsmokers. (ii) The constant force of mortality for smokers in 0.06. (iii) The constant force of mortality for nonsmokers is 0.03. (iv) d ¼ 0:08:

113

Life insurance and annuities

Calculate Varða TðxÞ Þ for an individual chosen at random from this group. Solution Let us use the superscript n for nonsmokers and s for smokers. Because of the mixed population and constant force of mortality, moments for the related whole life insurance random variable Z ¼ vT can be calculated as follows: EðZ k Þ ¼ k A x ¼ 0:30 ·

ms mn þ 0:70 · n ; m þ kd m þ kd s

where ms ¼ 0:06; mn ¼ 0:03; and d ¼ 0:08: For k ¼ 1; 2 this results in A x ¼ 0:319481; and 2 A x ¼ 0:192344: From a standard annuity variance formula Varða TðxÞ Þ ¼

1 2 1 ð Ax 2ðA x Þ2 Þ ¼ · ð0:19234420:3194812 Þ ¼ 14:11: 2 0:082 d

A

Exercise 3.10.8 November 2001 SOA/CAS Course 3 Examination, Problem No. 26. You are given: Ax ¼ 0:28; Axþ20 ¼ 0:40; A1x: 20 ¼ 0:25; and i ¼ 0:05: Calculate ax: 20 : Solution We have A1x: 20 ¼ Ax 220lAx ¼ Ax 2 A1x: 20 · Axþ20 ¼ 0:28 2 0:25 · 0:40 ¼ 0:18: and a€ x: 20 ¼

1 2 Ax: 20 ¼ d

1 2 A1x: 20 þ A1x: 20 i 1þi

¼

1:05 · ð1 2 0:18 2 0:25Þ 0:05

¼ 21 · 0:57 ¼ 11:97: Finally, ax: 20 ¼ a€ x: 20 þ A1x: 20 2 1 ¼ 11:97 þ 0:25 2 1 ¼ 11:22:

A

Exercise 3.10.9 May 2001 SOA/CAS Course 3 Examination, Problem No. 24. For a disability insurance claim (i) The claimant will receive payments at the rate of 20 000 per year, payable continuously as long as she remains disabled. (ii) The length of the payment period in years is a random variable with the gamma distribution with parameters a ¼ 2 and u ¼ 1: (iii) Payments begin immediately. (iv) d ¼ 0:05:

114

Chapter 3

Calculate the actuarial present value of the disability payments at the time of disability. Solution The actuarial present value of this disability annuity is ð1 ð1 ð 1 2 e20:05t t e2t 20000 1 20000 a t fT ðtÞ dt ¼ 20000 ðt e2t 2 t e21:05t Þ dt dt ¼ 0:05 0 0:05 Gð2Þ 0 0 ð 20000 1 20000 1:05t þ 1 21:05t 1 2t 21:05t 2t 2ðt þ 1Þe þ ¼ ðt e 2 t e Þdt ¼ e 0:05 0 0:05 1:052 0 20000 1 12 ¼ ¼ 37188: A 0:05 1:052 Exercise 3.10.10 November 2000 SOA/CAS Course 3 Examination, Problem No. 3. A person aged 40 wins 10 000 in the actuarial lottery. Rather than receiving the money at once, the winner is offered the actuarially equivalent option of receiving an annual payment of K (at the beginning of each year) guaranteed for 10 years and continuing thereafter for life. You are also given that i ¼ 0:04; A40 ¼ 0:30; A50 ¼ 0:35; and A140: 10 ¼ 0:09: Calculate K: Solution Since the actuarial present value (APV) of an annuity due to ðxÞ of one per year with 10 payments guaranteed and the rest contingent on survival is a€ 10 þ 10 l€ax ; K is determined from the APV relation We have

10 000 ¼ Kð€a 10 þ a€ 10 ¼

Furthermore,

ax Þ: 10 l€

ð1 2 1:04210 Þ ¼ 8:4353: 0:04 1:04

0:21 ¼ 0:30 2 0:09 ¼ A40 2 A140: 10 ¼

10 lA40

¼ v10 · 10 p40 · A50 ¼ v10 ·

Therefore v10 ·

10 p40

¼

0:21 : 0:35

Since a€ 50 ¼

1 2 A50 1 2 0:35 ¼ ¼ 16:90; 0:04 d 1:04

10 p40

· 0:35:

115

Life insurance and annuities

it follows that a40 10 l€

¼ v10 ·

10 p40

· a€ 50 ¼

0:21 · 16:90 ¼ 10:14: 0:35

Substituting these results into the first equation gives 10 000 ¼ Kð€a 10 þ10 l€ax Þ ¼ Kð8:4353 þ 10:14Þ;

K ¼ 538:35:

A

Exercise 3.10.11 November 2000 SOA/CAS Course 3 Examination, Problem No. 20. Y is the present value random variable for a special 3-year temporary life annuity due on ðxÞ: You are given that t px ¼ 0:9t ; K is the curtate-future lifetime of ðxÞ; and that 8 1:00; K ¼ 0; > > < Y ¼ 1:87; K ¼ 1; > > : 2:72; K ¼ 2; 3; … Calculate VarðYÞ: Solution We have PrðK ¼ 0Þ ¼ 1 2 px ¼ 0:10; PrðK ¼ 1Þ ¼ 1 px 2 2 px ¼ 0:90 2 0:81 ¼ 0:09; PrðK . 1Þ ¼ 2 px ¼ 0:81; EðYÞ ¼ 0:10 · 1 þ 0:09 · 1:87 þ 0:81 · 2:72 ¼ 2:4715; EðY 2 Þ ¼ 0:10 · 12 þ 0:09 · 1:872 þ 0:81 · 2:722 ¼ 6:407; VarðYÞ ¼ 6:407 2 2:47152 ¼ 0:299:

A

Exercise 3.10.12 November 2000 SOA/CAS Course 3 Examination, Problem No. 26. A fund is established to pay annuities to 100 independent lives age x: Each annuitant will receive 10 000 per year continuously until death. You are given d ¼ 0:06; A x ¼ 0:40; 2 A x ¼ 0:25: Calculate the amount needed in the fund so that the probability, using the normal approximation, is 0.90 that the fund will be sufficient to provide the payments.

116

Chapter 3

Solution For each person Y ¼ 10 000a TðxÞ is the random present value of that person’s annuity. If Y1 ; Y2 ; …; Y100 are independent and identically distributed as Y; then YAGG ¼ Y1 þ · · · þ Y100 ; is the aggregate present value. Assuming that YAGG is approximately normal in distribution, we can compute the fund needed as the 90th percentile as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ EðYAGG Þ þ 1:282 VarðYAGG Þ: We have EðYAGG Þ ¼ 100 EðYÞ; VarðYAGG Þ ¼ 100 VarðYÞ; 1 2 0:40 ¼ 100 000; EðYÞ ¼ 10 000ax ¼ 10 000 · 0:06 2 0:25 2 0:16 A x 2 A 2x VarðYÞ ¼ 10 0002 ¼ 25 · 108 : ¼ 108 · 2 0:062 d The fund required is EðYAGG Þ þ 1:282

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VarðYAGG Þ ¼ 100 · 100 000 þ 1:282 100 · 25 · 108

¼ 10 000 000 þ 1:282 · 500 000 ¼ 10 641 000:

A

Exercise 3.10.13 November 2000 SOA/CAS Course 3 Examination, Problem No. 35. For a special 30-year deferred annual whole life annuity due of 1 on (35): (i) If death occurs during the deferral period, the single benefit premium is refunded without interest at the end of the year of death, (ii) a€ 65 ¼ 9:90; (iii) A35: 30 ¼ 0:21; (iv) A135: 30 ¼ 0:07: Calculate the single benefit premium for this special deferred annuity. Solution If P is the single benefit premium, then the actuarial present value of the refund of P without interest if death occurs during the deferral period is PA135: 30 : Hence P is the solution of the equation: P ¼ P · A135: 30 þ v30 ·

30 p35

· a€ 65 :

117

Life insurance and annuities

From the information given, v30 · 30 p35 ¼ A135: 30 ¼ A35: 30 2 A135: 30 ¼ 0:21 2 0:07 ¼ 0:14: Hence P¼

v30 · 30 p35 · a€ 65 0:14 · 9:90 ¼ 1:49: ¼ 1 2 0:07 1 2 A135: 30

A

Exercise 3.10.14 May 2000 SOA/CAS Course 3 Examination, Problem No. 29. For a whole life annuity due of 1 on ðxÞ; payable annually qx ¼ 0:01; qxþ1 ¼ 0:05; i ¼ 0:05; and a€ xþ1 ¼ 6:951: Calculate the change in the actuarial present value of this annuity due if pxþ1 is increased by 0.03. Solution We have 6:951 ¼ a€ xþ1 ¼ 1 þ v · pxþ1 · a€ xþ2 ¼ 1 þ

0:95 a€ ; 1:05 xþ2

and therefore a€ xþ2 ¼ 6:57742: Using this we calculate a€ x ¼ a€ x: 2 þ v2 · px · pxþ1 · a€ xþ2 ; ðnewÞ

¼ a€ x: 2 þ v2 · px · pxþ1 · a€ xþ2 : a€ ðnewÞ x The difference is equal to v2 · px · 0:03 · a€ xþ2 ¼

1 1:05

2 · 0:99 · 0:03 · 6:57742 ¼ 0:1772:

A

Exercise 3.10.15 May 2000 SOA/CAS Course 3 Examination, Problem No. 39. For a continuous whole life annuity of 1 on ðxÞ: (i) TðxÞ; the future lifetime of ðxÞ; has a constant force of mortality 0.06. (ii) The force of interest is 0.04. Calculate Prða TðxÞ . a x Þ:

118

Chapter 3

Solution We have 1 2 vT 1 2 vT 1 2 A x Prða T . a x Þ ¼ Pr . a x ¼ Pr . ¼ Prð2vT . 2A x Þ d d d m 21 m ln ¼ Pr e2dT , ¼ Pr T . ¼ t 0 px ; mþd dþm d where 21 m t0 ¼ ln : dþm d Therefore m=d 6=4 m 6 2 mt 0 ¼ ¼ ¼ 0:4648: A t0 px ¼ e mþd 10 Exercise 3.10.16 November 2002 SOA/CAS Course 3 Examination, Problem No. 39. For a whole life insurance of 1 on ðxÞ; you are given: (i) mðx þ tÞ is the force of mortality. (ii) The benefit is payable at the moment of death. (iii) d ¼ 0:06: (iv) A x ¼ 0:60: Calculate the revised actuarial present value of the benefit of this insurance if mðx þ tÞ is increased by 0.03 for all t and d is decreased by 0.03. Solution ðt ð1 2tð0:0620:03Þ ¼ e exp 2 ð m ðsÞþ0:03Þ ds ðmx ðtÞþ0:03Þ dt A New x x 0

¼

0

ð1 e 0

2t · 0:06

ð t exp 0

ð1 mx ðsÞ ds mx ðtÞ dtþ0:03 e2t

¼ A Old aOld x þ0:03 x ¼0:60þ0:03

0

120:6 ¼0:80: 0:06

· 0:06

ðt exp 2 mx ðsÞ ds dt 0

A

3.11 Life insurance premiums Life insurance and annuities are typically not purchased with one lump sum, but rather with a series of premiums. These premiums are paid only as long as the insured is alive, and therefore are life annuities. What principles can be used for setting

119

Life insurance and annuities

premiums? The following three principles have been put forth as possible premium calculation principles [14]: † Percentile premiums. Charge premium large enough to ensure that the company only suffers financial loss with sufficiently low probability. † Benefit premiums. Base the premium level on the condition of the insurer’s loss having the expected value of zero. This is also referred to as the Equivalence Principle (EP). † Exponential premiums. Premium is set as the number P such that using the utility of wealth function uðxÞ ¼ 2e20:1x ; the insurer is indifferent between accepting and not accepting the risk. The most important random variable studied in derivation of the premium is the loss-at-issue of the insurer. The random present value of the excess of outflows of the insurer over the inflows of the insurer for a unit policy, if the premium is paid as an annuity, is L ¼ Z 2 PY; where P is the premium charged by the insurer, Z the unit benefit random present value, and Y the unit annuity random present value. The premium P is set by the insurer, and, as explained above, there may be various principles for setting it. The simplest and the most fundamental principle is the EP: P is chosen so that EðLÞ ¼ 0: This means, of course, that P¼

EðZÞ : EðYÞ

The models for premium combine the mode of death benefit payment (continuous, discrete, or modal) and the mode of premium payment (continuous, discrete, modal). Also note that it is possible to pay the premium for a period shorter than the one for which the coverage is offered. For a fully continuous model, death benefit is paid at the moment of death and premium is paid continuously. For a fully discrete model, death benefit is paid at the end of the year of death and the premium is paid as a life annuity due. For a semi-continuous model, death benefit is paid at the moment of death and premiums are paid as a life annuity due. Premiums are generally assumed to be paid as a level annuity. For a fully continuous model, whole life insurance, we have the following form of the loss at issue function: T

aT ¼ L ¼ v 2 P

P P 1þ vT 2 ; d d

ð3:11:1Þ

where P is the annual level of the premium paid as a continuous life annuity. We have ax ; EðLÞ ¼ A x 2 P

ð3:11:2Þ

120

Chapter 3

and VarðLÞ ¼

P 2 2 1þ ð Ax 2 A 2x Þ: d

ð3:11:3Þ

For this continuous model, the EP premium is A x Þ ¼ Ax ¼ 1 2 d ¼ dAx : Pð a x a x 1 2 A x

ð3:11:4Þ

Furthermore, the variance of the loss at issue under the EP premium is VarðLÞ ¼

2 Ax 2 A 2x Ax 2 A 2x ¼ : 2 ðda x Þ ð1 2 A x Þ2

2

ð3:11:5Þ

We have analogous formulas for the fully discrete whole life insurance. The loss at issue random variable is P Kþ1 P ð3:11:6Þ L ¼ vKþ1 2 P€a Kþ1 ¼ 1 þ 2 : v d d The expected value of the loss is EðLÞ ¼ Ax 2 P€ax ; and the variance of the loss at issue is P 2 2 ð Ax 2 A2x Þ: VarðLÞ ¼ 1 þ d

ð3:11:7Þ

ð3:11:8Þ

The EP premium (i.e., benefit premium) is Px ¼

Ax 1 dAx ¼ 2d ¼ ; a€ x a€ x 1 2 Ax

ð3:11:9Þ

and the variance of the loss at issue under the EP premium is 2

VarðLÞ ¼

2 Ax 2 A2x Ax 2 A2x ¼ : ðda€ x Þ2 ð1 2 Ax Þ2

ð3:11:10Þ

The constant force model has some interesting implications for the life insurance premium calculation. We have

m x A m þd A x Þ ¼ ¼ ¼ m; Pð 1 a x CF mþd

ð3:11:11Þ

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Life insurance and annuities

and q Ax qþi ¼ vq ¼ A1x: 1 : Px ¼ ¼ a€ x CF 1 þ i qþi

ð3:11:12Þ

Also, under the constant force assumption, with the EP premiums, the variance of the loss at issue function has a very simple form. For the fully continuous model it is 2 m m m ð m þ dÞ 2 2 2 m2 2 2 Ax 2 A x m þ 2d mþd m þ 2d ¼ ¼ VarðLÞ ¼ 2 d2 ð1 2 A x Þ2 CF m 12 mþd m þ 2d md2 m3 þ 2m2 d þ md2 m2 2 m2 þ 2 m2 m þ 2d m þ 2d m þ 2d ¼ ¼ d2 d2 m ¼ ¼ 2 A x ; ð3:11:13Þ m þ 2d and for the fully discrete model it is VarðLÞ ¼ p · 2Ax : CF

ð3:11:14Þ

Exercises Exercise 3.11.1 November 2003 SOA/CAS Course 3 Examination, Problem No. 31. For a block of fully discrete whole life insurances of one on independent lives age x; you are given i ¼ 0:06; Ax ¼ 0:24905; 2 Ax ¼ 0:09476; and p ¼ 0:025; where p is the contract premium for each policy. You are also given that losses are based on the contract premium. Using the normal approximation, calculate the minimum number of policies the insurer must issue so that the probability of a positive total loss on the policies issued is less than or equal to 0.05. Solution The individual loss at issue is L ¼ vKþ1 2 pa€ Kþ1 ¼

173 Kþ1 53 v : 2 120 120

Therefore VarðLÞ ¼

173 120

2 ·ð

2

Ax 2 A2x Þ ¼

173 120

2

ð0:09476 2 0:279052 Þ < 0:06803464;

122

Chapter 3

and

sL < 0:26083451; where sL is the standard deviation of the individual loss at issue. The total loss on n policies is

L ¼ L1 þ ·· · þ Ln ; with EðLÞ ¼ 20:0826196n; and pﬃﬃﬃ VarðLÞ ¼ 0:26083451 n: Using the normal approximation, we estimate the probability of a positive total loss as

L 2 EðLÞ 0 þ 0:0826196n pﬃﬃﬃ . sL 0:26083451 n ¼ 0:05:

PrðL . 0Þ ¼ Pr

<12F

0:0826196n pﬃﬃﬃ 0:26083451 n

This implies that 0:0826196n pﬃﬃﬃ ¼ z95 ¼ 1:645: 0:26083451 n

Therefore, n ¼ 27:

pﬃﬃﬃ n ¼ 5:1933532594 and n ¼ 26:9709181: But n must be an integer, thus A

Exercise 3.11.2 November 2002 SOA/CAS Course 3 Examination, Problem No. 11. For a fully discrete whole life insurance of 1000 on (60), the annual benefit premium was calculated using the following: i ¼ 0:06; q60 ¼ 0:01376; 1000A60 ¼ 369:33; and 1000A61 ¼ 383:00: A particular insured is expected to experience a firstyear mortality rate 10 times the rate used to calculate the annual benefit premium. The expected mortality rates for all other years are the ones originally used. Calculate the expected loss at issue for this insured based on the original benefit premium. Solution The original benefit premium is

P60

0:06 0:36933 A60 dA60 1:06 ¼ 0:0331480357: ¼ ¼ ¼ 1 2 0:36933 a€ 60 1 2 A60

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Life insurance and annuities

The new single benefit premium is New New ANew 60 ¼ vq60 þvp60 A61 ¼

1 1 0:1376þ 0:8624£0:383 ¼ 0:4414143396: 1:06 1:06

Thus, the expected value of the loss is 12ANew New New New 60 1000ðA60 2P60 · a€ 60 Þ ¼ 1000 A60 2P60 · d 0 1 120:4414143396 C B ¼ 1000@0:441414339620:0331480357 A ¼ 114:2983843: 0:06 1:06 A Exercise 3.11.3 November 2001 SOA/CAS Course 3 Examination, Problem No. 5. For a fully discrete 2-year term insurance of 1 on ðxÞ; you are given that px ¼ 0:75; pxþ1 ¼ 0:80; and that 0.95 is the lowest premium such that there is a 0% chance of loss in the first policy year. Z is the random variable for the present value at issue of future benefits. Calculate VarðZÞ: Solution The present value of the loss in the first policy year for this insurance in year one is either v 2 P if death occurs or 0 2 P if death does not occur. The lowest premium so that there is a 0% chance of loss in this year is P ¼ v: Therefore, v ¼ 0:95: Furthermore A1x: 2 ¼ v · qx þ v2 · 1 lqx ¼ 0:95 · 0:25 þ ð0:95Þ2 · 0:75 · 0:20 ¼ 0:372875: 2 1 Ax: 2

¼ v2 · qx þ v4 · 1 lqx ¼ ð0:95Þ2 · 0:25 þ ð0:95Þ4 · 0:75 · 0:25 ¼ 0:347801:

Finally VarðZÞ ¼ 2 A1x: 2 2 ðA1x: 2 Þ2 ¼ 0:347801 2 0:3728752 ¼ 0:208765:

A

Exercise 3.11.4 November 2001 SOA/CAS Course 3 Examination, Problem No. 34. For a fully continuous whole life insurance of one, you are given that m ¼ 0:04; d ¼ 0:08; and L is the loss-at-issue random variable based on the benefit premium. Calculate VarðLÞ: Solution Because of the constant force assumption, we just use the variance of the loss at issue formula: VarðLÞ ¼ 2 A x ¼

0:04 ¼ 0:20: 0:04 þ 0:16

A

124

Chapter 3

Exercise 3.11.5 November 2000 SOA/CAS Course 3 Examination, Problem No. 22. For a fully continuous whole life insurance of one on ðxÞ we are given that (i) p is the benefit premium, (ii) L is the loss-at-issue random variable with the premium equal to p; (iii) Lp is the loss-at-issue random variable with the premium 1:25p; (iv) a x ¼ 5:0; (v) VarðLÞ ¼ 0:5625: Calculate the sum of the expected value and the standard deviation of Lp : Solution In this case we have A x Þ ¼ 1 2 d ¼ 1 2 0:08 ¼ 0:12; Pð a x 5 2 0:12 VarðLÞ ¼ 1 þ ð2 A x 2 A 2x Þ ¼ 0:5625; 0:08 1:25 · 0:12 2 2 p ð Ax 2 A 2x Þ: VarðL Þ ¼ 1 þ 0:08 Therefore

15 2 1þ 8 VarðLp Þ ¼ · 0:5625 ¼ 0:744; 12 2 1þ 8

EðLp Þ ¼ A x 2 ð1:25 · 0:12Þax ¼ 1 2 da x 2 0:15ax ¼ 1 2 0:08ax 2 0:15ax ¼ 1 2 5 £ 0:23 ¼ 20:15; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ EðLp Þ þ VarðLp Þ ¼ 0:7125:

A

Exercise 3.11.6 November 2000 SOA/CAS Course 3 Examination, Problem No. 35. For a special 30-year deferred annual whole life annuity due of 1 on (35): (i) If death occurs during the deferral period, the single benefit premium is refunded without interest at the end of the year of death. (ii) a€ 65 ¼ 9:90: (iii) A 35: 30 ¼ 0:21: (iv) A135: 30 ¼ 0:07: Calculate the single benefit premium for this special deferred annuity.

125

Life insurance and annuities

Solution If P is the single benefit premium, then the actuarial present value of the refund of P without interest if death occurs during the deferral period is PA135: 30 : Hence P is the solution of the equation P ¼ PA135: 30 þ v30 · 30 p35 · a€ 65 : From the information given v30 · 30 p35 ¼ A135: 30 ¼ A35: 30 2 A135: 30 ¼ 0:21 2 0:07 ¼ 0:14: Finally P¼

v30 · 30 p35 · a€ 65 0:14 · 9:90 ¼ 1:49: ¼ 1 2 0:07 1 2 A135: 30

A

3.12 Life insurance reserves Life insurance reserves are an analogue of the accrued liabilities (also called actuarial liabilities) of pension plans. The basic methodology behind the calculations of the two is also analogous, nearly identical, and we will present this more fundamental life insurance methodology first before introducing the corresponding pension funding concepts. Assume that the insured was aged x when the life insurance policy under consideration was issued and that we are now at duration t into the policy existence. Denote by t L the prospective loss function at duration t; defined as the random variable equal to present value at time T of the future payments made by the insurance company minus the present value at time t of the future premium receipts by the insurance company. The expected value of this random variable t L is called the tth year terminal benefit reserve. In general, reserve means: † Prospectively. The excess of future benefits over future premiums, we nearly always take this to be the actuarial present value of future benefits minus the actuarial present value of future premiums (this effectively says that reserves are set by the EP). † Retrospectively. The accumulation of past premiums paid and not used yet to pay benefits. Effectively, reserve is that part of insurance company’s assets, which is really customers money, as it is set aside to pay future benefits (in life insurance, annuities, pensions, etc.) or claims (in property, casualty, liability insurance, etc.). We will analyze the loss function for the two key forms of life insurance: fully continuous whole life and fully discrete whole life. For the fully continuous model,

126

Chapter 3

we have TðxÞ2t a 2 P tL ¼ v TðxÞ2t ¼

P P 1þ vTðxÞ2t 2 ; d d

ð3:12:1Þ

where P is the premium paid (continuously) for this policy (we do not assume that it must be the benefit premium). The reserve for this policy is P P Eðt LlTðxÞ . tÞ ¼ 1 þ EðvTðxÞ2t lTðxÞ . tÞ 2 d d P P ¼ 1þ (3.12.12) Axþt 2 : d d While technically, the reserve calculation should use the conditional distribution, we will assume, for simplicity, that we can work with unconditional distribution. Also note that P 2 P 2 2 TðxÞ2t Varðt LÞ ¼ 1 þ Varðv Þ¼ 1þ ð Axþt 2 A 2xþt Þ: d d

ð3:12:3Þ

Now suppose that for this fully continuous policy the premium is set by the EP: A x Þ ¼ Ax : P ¼ Pð a x Then the benefit reserve (the use of the term benefit reserve always indicates that the EP premium is assumed) is

t VðAx Þ

A x Þaxþt ¼ ðPð A xþt Þ 2 Pð A x ÞÞaxþt ¼ A xþt 2 Pð A x Þ Pð xþt ¼ 1 2 a xþt ¼ Axþt 2 Ax ¼ 12 A A xþt Þ a x Pð 1 2 A x ¼

A x Þ A xþt Þ 2 Pð Pð : PðA xþt Þ þ d

ð3:12:4Þ

All the formulas in (3.12.4) are prospective, utilizing future benefits and premiums. But the reserve is also equal to the actuarial accumulated value of past premiums. In order to better explain that, we introduce a new concept, the accumulated cost of insurance: ¼

t kx

A 1x: t ; t Ex

ð3:12:5Þ

127

Life insurance and annuities

for the continuous model, and t kx

¼

A1x: t ; t Ex

ð3:12:6Þ

for the discrete one. Then we also have (for t , n):

t VðAx Þ

A x Þsx: t 2t k x : ¼ Pð

ð3:12:7Þ

Analogous formulas hold for other forms of continuous insurance. For the fully discrete whole life, we have the following benefit premium reserve: Px a€ € xþt ¼ ðPxþt 2 Px Þ€axþt ¼ 1 2 Axþt ¼ 1 2 xþt t Vx ¼ Axþt 2 Px a Pxþt a€ x ¼

Axþt 2 Ax P 2 Px ¼ Px s€ x: t 2t kx : ¼ xþt 1 2 Ax Pxþt þ d

ð3:12:8Þ

In order to better understand the risk exposure of the insurance firm, we also study the variance of the loss function t L: If the premium is set by the EP, then for the fully continuous whole life policy we have Varðt LÞ ¼

2

2 Axþt 2 A 2xþt Axþt 2 A 2xþt ¼ ; 2 ðda x Þ ð1 2 A x Þ2

ð3:12:9Þ

while for the fully discrete whole life: 2

Varðt LÞ ¼

2 Axþt 2 A2xþt Axþt 2 A2xþt ¼ : ðda€ x Þ2 ð1 2 Ax Þ2

ð3:12:10Þ

It is interesting to note that under the constant force assumption, the rationale for the life insurance reserve does not hold: mortality in the later years of the policy does not increase, and therefore future benefits and future premiums are in balance and there is no need to hold a reserve to account for any imbalance between them. It is not surprising then that under constant force, for the fully continuous whole life policy:

t VðAx Þ

A x Þaxþt ¼ ¼ A xþt 2 Pð

CF

m 1 2m ¼ 0: mþd mþd

ð3:12:11Þ

Under constant force, and the EP premium, we also have Varðt LÞ ¼ 2 A x ¼ CF

CF

m : m þ 2d

ð3:12:12Þ

For the fully discrete whole life we have t Vx

¼ 0;

CF

ð3:12:13Þ

128

Chapter 3

and with the EP premium: Varðt LÞ ¼ p · 2 Ax ¼ CF

pq : q þ 2i þ i2

ð3:12:14Þ

Before we proceed to the exercises, let us add one more definition. A contract premium is the premium actually paid by the customer, which generally exceeds the benefit premium, as it must also include provision for expenses, risk, and profits of the insurer. A contract premium is also commonly called a gross premium. Exercises Exercise 3.12.1 November 2003 SOA/CAS Course 3 Examination, Problem No. 14. For a fully discrete whole life insurance of 1000 on (40), the contract premium is the level annual benefit premium based on the mortality assumption at issue. At time 10, the actuary decides to increase the mortality rates for ages 50 and higher. You are given: (i) d ¼ 0:05: (ii) Mortality assumptions are: at issue k lq40

¼ 0:02;

k ¼ 0; 1; 2; …; 49;

and revised prospectively at time 10 k lq50

¼ 0:04;

k ¼ 0; 1; 2; …; 24:

(iii) 10 L is the prospective loss random variable at time 10 using the contract premium. Calculate Eð10 LlKð40Þ $ 10Þ using the revised mortality assumption. Solution The original assumption is the uniform distribution on ½0; 50. Therefore, A40 ¼

1 1 2 1 50 a vþ v þ ··· þ v ¼ 50 < 0:350760909; 50 50 50 50

(this is, of course, just like De Moivre’s Law) and a€ 40 ¼

1 2 A40 < 12:9848: d

Hence, the original premium is P¼

1000A40 < 27:013: a€ 40

129

Life insurance and annuities

We know that 2 P€aRevised : Eð10 LlKð40Þ $ 10Þ ¼ 1000ARevised 50 50 The revised assumption is uniform on ½0; 25: From this we have ¼ ARevised 50

1 1 2 1 25 a vþ v þ ··· þ v ¼ 25 < 0:549183924: 25 25 25 25

Also a€ Revised ¼ 50

1 2 ARevised 50 < 9:01632152: d

Therefore Eð10 LlKð40Þ $ 10Þ ¼ 549:183924 2 27:013 · 9:01632152 < 305:62:

A

Exercise 3.12.2 November 2003 SOA/CAS Course 3 Examination, Problem No. 30. Michael, age 45, is expected to experience higher than standard mortality only at age 64. For a special fully discrete whole life insurance of 1 on Michael, you are given: (i) The benefit premiums are not level. (ii) The benefit premium for year 20, p19 ; exceeds P45 for a standard risk by 0.010. (iii) Benefit reserves on his insurance are the same as benefit reserves for a fully discrete whole life insurance of 1 on (45) with standard mortality and level benefit premiums. (iv) i ¼ 0:03: (v) 20 V45 ¼ 0:427: Calculate the excess of q64 for Michael over the standard q64 : Solution Let us use the superscript (M) for all items concerning Michael. We have the following standard recursive relations: ð19 V45 þ P45 Þð1 þ iÞ ¼ q64 þ p64 · 20V45 ; M M M þ p19 Þð1 þ iÞ ¼ qM ð19 V45 64 þ p64 · 20V45 : M M ¼ 0:427; i ¼ 0:03; 19 V45 ¼ 19 V45 ; p19 ¼ The following are given: 20 V45 ¼ 20 V45 P45 þ 0:01: Substituting these into the reserves recursive relations, we get

1:03ð19 V45 þ P45 Þ ¼ 0:427 þ 0:573q64 ; 1:03ð19 V45 þ P45 þ 0:01Þ ¼ 0:427 þ 0:573qM 64 :

130

Chapter 3

Now we can treat 1:03ð19 V45 þ P45 Þ as one variable, whose value from the first equation can be substituted to the second equation, and this results in 0:427 þ 0:573q64 þ 0:0103 ¼ 0:427 þ 0:573qM 64 : Hence qM 64 2 q64 ¼

0:0103 ¼ 0:01797556719: 0:573

A

Exercise 3.12.3 November 2003 SOA/CAS Course 3 Examination, Problem No. 38. For a fully discrete 10-payment whole life insurance of 100 000 on ðxÞ; you are given: i ¼ 0:05; qxþ9 ¼ 0:011; qxþ10 ¼ 0:012; qxþ11 ¼ 0:014: The level annual benefit premium is 2078. The benefit reserve at the end of year 9 is 32 535. Calculate 100 000Axþ11 : Solution Let k V be the benefit reserve for this policy at the end of year k for this policy. This reserve is calculated exactly at time k; just before the premium for year k þ 1 is paid (that premium is also paid at time k). We have 100 000Axþ11 ¼

11 V;

so we need to calculate 11 V: We have 9 V ¼ 32 535: Time 9 is the last time premium is paid for this policy. Using this recursive reserve formula: ð32 535 þ 2078Þ · 1:05 ¼ 0:011 · 100 000 þ 0:989 ·

10 V;

we conclude that 10 V

¼ 35 635:642063:

The next recursive relationship is 35 635:642063 · 1:05 ¼ 0:012 · 100 000 þ 0:988 ·

11 V;

so that 100 000Axþ11 ¼

11 V

¼ 36 657:311909:

A

Exercise 3.12.4 November 2002 SOA/CAS Course 3 Examination, Problem No. 25. For a fully discrete whole life insurance of 1000 on (20), you are given: 1000P20 ¼ 10; 1000 · 20 V20 ¼ 490; 1000 · 21 V20 ¼ 545; 1000 · 22 V20 ¼ 605; and q40 ¼ 0:022: Calculate q41 : Solution Using the information about the reserves we get: 0:49 ¼

20 V20

¼ A40 2 P20 a€ 40 ¼ 1 2 d a€ 40 2 0:01€a40 ¼ 1 2 ðd þ 0:01Þ€a40 ;

131

Life insurance and annuities

0:545 ¼

21 V20

¼ 1 2 ðd þ 0:01Þ€a41 ;

and 0:605 ¼ 22V20 ¼ 1 2 ðd þ 0:01Þ€a42 : Hence a€ 40 ¼

0:51 ; d þ 0:01

a€ 41 ¼

0:455 ; d þ 0:01

a€ 42 ¼

0:345 : d þ 0:01

Furthermore, a€ 40 ¼ 1 þ vp40 a€ 41 ¼ 1 þ ð1 2 dÞp40 a€ 41 ; thus 0:51 0:455 0:455 ¼ 1 þ ð1 2 dÞð1 2 q40 Þ ¼ 1 þ 0:978ð1 2 dÞ : d þ 0:01 d þ 0:01 d þ 0:01 This gives us 0:51 ¼ d þ 0:01 þ 0:978 · 0:455 · ð1 2 dÞ; and d ¼ 0:0991153313: From this we get a€ 41 ¼ 4:169899819;

a€ 42 ¼ 3:620022918:

Thus a€ 41 ¼ 1 þ ð1 2 dÞð1 2 q41 Þ€a42 ; or 4:169899819 ¼ 1 þ 0:9008846687 · ð1 2 q41 Þ · 3:620022918; and we conclude that q41 ¼ 0:0280027842:

A

Exercise 3.12.5 November 2002 SOA/CAS Course 3 Examination, Problem No. 38. A large machine in the ABC Paper Mill is 25 years old when ABC purchases a 5-year term insurance paying a benefit in the event the machine breaks down. You are given the following: (i) Annual benefit premiums of 6643 are payable at the beginning of the year. (ii) A benefit of 500 000 is payable at the moment of breakdown. (iii) Once a benefit is paid, the insurance contract is terminated.

132

Chapter 3

(iv) Machine breakdowns follow De Moivre’s Law with lx ¼ 100 2 x: (v) i ¼ 0:06: Calculate the benefit reserve for this insurance at the end of the third year. Solution Note that because of (iii), this coverage is just like a regular term life insurance. After 3 years, the machine is 28 years old and there are 2 years of coverage remaining. Under De Moivre’s Law: a 1 1 2 1:0622 A 128: 2 ¼ 2 ¼ ¼ 0:026220282: 72 72 ln 1:06 We also have a€ 28: 2 ¼ 1 þ

1 72 2 1 ¼ 1:930293501: 1:06 72

Therefore, the reserve is 500 000 · A 128: 2 2 6643 · a€ 28: 2 ¼ 500 000 · 0:026220282 2 6643 · 1:930293501 ¼ 287:2: A Exercise 3.12.6 November 2001 SOA/CAS Course 3 Examination, Problem No. 4. For a special fully discrete whole life insurance on ðxÞ: (i) The death benefit is 0 in the first year and 5000 thereafter. (ii) Level benefit premiums are payable for life. (iii) qx ¼ 0:05: (iv) v ¼ 0:90: (v) a€ x ¼ 5:00: (vi) 10 Vx ¼ 0:20: 10 V is the benefit reserve at the end of year 10 for this insurance. Calculate 10 V:

Solution The first step is to determine the benefit premium. Note that we have v ¼ 0:9; d ¼ 1 2 v ¼ 0:10: From the EP the annual benefit premium, denoted as P; satisfies the relationship P€ax ¼ 5000 · 1 lAx ¼ 5000ðv · px · Axþ1 Þ: We know that a€ x ¼ 5 thus Ax ¼ 1 2 d a€ x ¼ 1 2 0:10 · 5 ¼ 0:5: Furthermore 0:5 ¼ Ax ¼ v · qx þ v · px · Axþ1 ¼ 0:9 · 0:05 þ v · px · Axþ1 ;

133

Life insurance and annuities

and this results in v · px · Axþ1 ¼ 0:455: Substituting these into the premium equation leads to P¼

5000 · 0:455 ¼ 455: 5

Now we turn to the reserve. We know that 10 V

¼ 5000Axþ10 2 455€axþ10 :

Information needed now is provided by what we know about standard whole life reserve 10 Vx : a€ a€ 0:20 ¼ 10 Vx ¼ 1 2 xþ10 ¼ 1 2 xþ10 ; a€ x 5 and a€ xþ10 ¼ 5 · ð1 2 0:20Þ ¼ 4: Also Axþ10 ¼ 1 2 d a€ xþ10 ¼ 1 2 0:10 · 4 ¼ 0:60: Substituting these results into the reserve relation results in a 10th reserve of 5000 · 0:6 2 455 · 4 ¼ 1180:

A

Exercise 3.12.7 November 2001 SOA/CAS Course 3 Examination, Problem No. 16. You are given: (i) Mortality follows De Moivre’s Law with v ¼ 100: (ii) i ¼ 0:05: (iii) The following annuity-certain values: a 40 ¼ 17:58; a 50 ¼ 18:71; a 60 ¼ 19:40: Calculate

10 VðA40 Þ:

Solution Under De Moivre’s Law 1 a : v 2 x v2x The given annuity values could be calculated from i ¼ 0:05: Luckily, they are given and this speeds up the calculation. Using the standard reserve formula involving A x ¼

134

Chapter 3

life insurance net single premiums: A 50 2 A 40 ¼ 10 VðA40 Þ ¼ 1 2 A 40

18:71 19:40 2 50 60 ¼ 0:0752: 19:40 12 60

A

Exercise 3.12.8 November 2001 SOA/CAS Course 3 Examination, Problem No. 25. For a special fully continuous whole life insurance on (65): (i) The death benefit at time t is bt ¼ 1000 e0:04t ; t $ 0: (ii) Level benefit premiums are payable for life. (iii) mð65 þ tÞ ¼ 0:02; t $ 0: (iv) d ¼ 0:04: the benefit reserve at the end of year 2. Calculate 2 V; Solution The random present value of benefit is not really random: Z ¼ bT vT ¼ ð1000 · e0:04T Þ · e20:04T ; and therefore EðZÞ ¼ 1000: Because of the constant force assumption, the level benefit premium P is determined from the following: a65 ¼ P 1000 ¼ EðZÞ ¼ P

1 1 ; ¼ P 0:02 þ 0:04 mþd

and hence P ¼ 60: The second year reserve is ¼ Eð2 ZÞ 2 60a67 ;

2V

where a 67 ¼

1 1 ; ¼ 0:06 mþd

and 2 Z is the present value of the future benefit at duration 2, given by 2Z

¼ b2þU vU ¼ 1000 e0:04ð2þUÞ · e20:04U < 1083:29;

where U is the future lifetime at age 67, i.e., U ¼ T 2 2lT . 2: Substituting into the reserve relation results in ¼ 1083:29 2 60 ·

2V

1 ¼ 83:29: 0:06

A

135

Life insurance and annuities

Exercise 3.12.9 May 2001 SOA/CAS Course 3 Examination, Problem No. 5. For a fully discrete 20-payment whole life insurance of 1000 on ðxÞ; you are given: (i) i ¼ 0:06; (ii) qxþ19 ¼ 0:01254; (iii) The level annual benefit premium is 13.72. (iv) The benefit reserve at the end of year 19 is 342.03. Calculate 1000Pxþ20 ; the level annual benefit premium for a fully discrete whole life insurance of 1000 on ðx þ 20Þ: Solution The idea is to use the given 19th reserve together with premium, interest, and mortality for this year to calculate the 20th reserve. Since this is a 20-pay insurance, the 20th reserve is 1000Axþ20 : The funds available per in-force policy at the start of year 20 equal the 19th reserve plus the premium for the 20th year, i.e., 342:03 þ 13:72 ¼ 355:75: The actuarial present value of year-end fund needs for such a policy is v · qxþ19 · 1000 þ v · pxþ19 · ð1000 · 20 20 Vx Þ ¼ 355:75: |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} death in year 20

survival year 20

Note the use of the left superscript of 20, which indicates the policy is a 20-payment one. Equating the funds available with the actuarial present value of year-end needs (a version of the recursive reserve relation, or, equivalently, assets equal liabilities with benefit premium accounting) and substituting n ¼ 1=1:06; qxþ19 ¼ 0:01254; results in the 20th reserve of 369.18. As stated above this is the same as 1000Axþ20 : Therefore 1000Pxþ20 ¼

1000Axþ20 1000dAxþ20 ¼ ¼ 33:13: a€ xþ20 1 2 Axþ20

A

Exercise 3.12.10 November 2000 SOA/CAS Course 3 Examination, Problem No. 11. For a special single premium 20-year term insurance on (70), the death benefit, payable at the end of the year of death, is equal to 1000 plus the benefit reserve. We also have q70þt ¼ 0:03; t $ 0; and i ¼ 0:07: Calculate the single benefit premium for this insurance. Solution Because this policy’s benefit includes the reserve, we must work with the recursive relation between consecutive reserves. Let P be the single premium. Then in year one: 0V

þ P ¼ v · q70 · ð1000 þ 1 VÞ þ v · p70 · 1 V;

and in years 2 –20 kV

þ 0 ¼ v · q70þk · ð1000 þkþ1 VÞ þ v · p70þk · kþ1 V;

136

Chapter 3

where k ¼ 1; 2; …; 19: Replacing p70þk by 1 2 q70þk ; multiplying both the sides by vk and moving both the reserve terms to the left-hand side results in the relations 0V

2 v · 1V þ P ¼ v · q70 · 1000;

kþ1 V

þ vk · 0 ¼ vkþ1 · q70þk · 1000;

and vk · k V 2 vkþ1 ·

Summing all these relations and using 0 V ¼ P¼

19 X

20 V

k ¼ 1; 2; …; 19:

¼ 0 we obtain, after cancellation

vkþ1 · q70þk · 1000:

k¼0

Substituting q70þk ¼ 0:03 results in P ¼ 30a 20

0:07

¼ 317:82:

A

Exercise 3.12.11 November 2000 SOA/CAS Course 3 Examination, Problem No. 33. For a fully discrete 3-year endowment insurance of 10 000 on (50), you are given: i ¼ 0:03; 1000q50 ¼ 8:32; 1000q51 ¼ 9:11; 10 000 · 1 V50: 3 ¼ 3209; 10 000 · 2 V50: 3 ¼ 6539: 0L

is the prospective loss random variable at issue, based on the benefit premium. Calculate the variance of 0 L: Solution The level benefit premium (per 10 000) for this policy is p ¼ 10 000v 2 10 000 · 2 V50: 3 ¼ 9708:74 2 6539 ¼ 3169:74:

We have 8 10 000v2 p€a 1 ¼ 6539; > > < 10 000v2 2 p€a 2 ¼ 3178:80; 0L ¼ > > : 10 000v3 2 p€a 3 ¼ 283:52;

for K ¼ 0; for K ¼ 1; for K $ 2:

Furthermore PrðK ¼ 0Þ ¼ q50 ¼ 0:00832; PrðK ¼ 1Þ ¼ p50 · q51 ¼ 0:99168 · 0:00911 ¼ 0:0090342; PrðK $ 2Þ ¼ 1 2 PrðK ¼ 0Þ 2 PrðK ¼ 1Þ ¼ 0:98265;

137

Life insurance and annuities

and Eð0 LÞ ¼ 0; since p is the benefit premium. Therefore, we can calculate the variance of 0 L as Varð0 LÞ ¼ Eð0 L2 Þ ¼ 0:00832 · 65392 þ 0:00903 · 3178:802 þ 0:98265 · ð283:52Þ2 ¼ 453 895:

A

Exercise 3.12.12 May 2000 SOA/CAS Course 3 Examination, Problem No. 3. For a fully discrete 2-year term insurance of 400 on ðxÞ : i ¼ 10%; 400P1x: 2 ¼ 74:33; 400 · 1 Vx:1 2 ¼ 16:58: The contract premium equals the benefit premium. Calculate the variance of the loss at issue. Solution Let us start by finding qx and qxþ1 : We have 400 · 1 Vx:1 2 ¼ 16:58 ¼ 400vqxþ1 2 400P1x: 2 ¼

400 q 2 74:33: 1:10 xþ1

Therefore, qxþ1 ¼ 0:25: Also, 400 · 0 Vx:1 2 ¼ 0 ¼ 400vqx 2 400P1x: 2 þ 400vpx · 1 Vx:1 2 : Hence qx ¼

400P1x: 2 ð1 þ iÞ 2 400 · 1Vx:1 2 ¼ 0:17: 400 2 400 · 1Vx:1 2

The loss random variable is described as follows: 8 400 > > 2 74:33 ¼ 289:30; K ¼ 0; > > 1:10 > > > > < 400 1 2 74:33 1 þ ¼ 188:68; K ¼ 1; L ¼ 2 0 1:10 1:10 > > > > > > > > 274:33 1 þ 1 ¼ 2141:90 K $ 2: : 1:10 Since the contract premium is the benefit premium, the premium is set by the EP, and the variance of the loss at issue is the second moment of the loss. It equals: Varð0 LÞ ¼ ð289:30Þ2 · 0:17 þ ð188:68Þ2 · 0:25 · ð1 2 0:17Þ þ ð141:90Þ2 · ð1 2 0:25Þ · ð1 2 0:17Þ ¼ 31 450:

A

138

Chapter 3

Exercise 3.12.13 May 2000 SOA/CAS Course 3 Examination, Problem No. 24. For a fully discrete whole life insurance with nonlevel benefits on (70): (i) The level benefit premium for this insurance is equal to P50 : (ii) q70þk ¼ q50þk þ 0:01; k ¼ 0; 1; …; 19: (iii) q60 ¼ 0:01368: (iv) k V ¼ k V50 ; k ¼ 0; 1; …; 19: (v) 11 V50 ¼ 0:16637: Calculate b11 ; the death benefit in year 11. Solution For the fully discrete unit whole life insurance issued to (50) the relation between the 10th and 11th reserves is 10 V50

þ P50 ¼ v · q60 · 1 þ v · p60 · 11V50 :

For the variable benefit plan on (70) in this problem, the relation between the 10th and 11th reserve is 10 V

þ P50 ¼ v · q80 · b11 þ v · p80 · 11V;

where b11 is the unknown death benefit in the 11th year. Since 10 V50 ¼ 10 V and 11 V50 ¼ 11 V; we see from the above equations that the left-hand sides are equal. Setting the right-hand sides equal and substituting q60 ¼ 0:01368; q80 ¼ q60 þ 0:01 and 11 V50 ¼ 11 V ¼ 0:16637 results in a linear equation in the unknown b11 : The solution is b11 ¼ 0:648: A Exercise 3.12.14 May 2000 SOA/CAS Course 3 Examination, Problem No. 26 For a fully discrete 3-year endowment insurance of 1000 on ðxÞ : qx ¼ qxþ1 ¼ 0:20; i ¼ 0:06; and 1000Px: 3 ¼ 373:63: Calculate 1000ð2 Vx: 3 2 1Vx: 3 Þ: Solution ð10000 Vx: 3 þ 1000Px: 3 Þð1 þ iÞ ¼ px · 1000 · 1Vx: 3 þ qx · 1000; and ð10001 Vx: 3 þ 1000Px: 3 Þð1 þ iÞ ¼ px · 1000 · 2Vx: 3 þ qx · 1000: The first formula gives us ð1000 · 0 þ 373:63Þ · 1:06 ¼ 0:80 · 1000 · 1Vx: 3 þ 200; so that 10001 Vx: 3 ¼ 245:06:

139

Life insurance and annuities

For the second year we have ð245:06 þ 373:63Þ · 1:06 ¼ 0:80 · 1000 · 2Vx: 3 þ 200; and therefore 10002 Vx: 3 ¼ 569:76: The difference is 569:76 2 245:06 ¼ 324:70:

A

3.13 Multiple lives models In the United States, pensions and pension rights are generally considered a common property if one of the spouses of a married couple is a pension plan participant. Pension benefit payment default is a joint-life-and-survivor annuity: such an annuity is paid as long as at least one of the spouses is alive, and pays at least 50% of the original benefit to the surviving spouse if the participant dies. Because of these requirements, it is sometimes important to model annuities and life insurance related to the survival of multiple lives. These models generally consider a new artificial entity, called a status, which is alive as long as one, or both, or all, of several real entities are alive. One artificial entity that is also quite useful, which we have considered already is n : this status is alive for exactly n years, and dies instantly at the end of the n-year period. If x is a person aged x; and y is a person aged y; then x : y denotes the status alive as long as both x and y are alive, while x : y is the status, which is alive as long as at least one of the two lives x or y is alive. In relation to these statuses, we also consider their future lifetime random variables, e.g., 4

Tðx : yÞ ¼ minðTðxÞ; TðyÞÞ;

Tðx : yÞ ¼ maxðTðxÞ; TðyÞÞ:

Let us also note that the semicolon symbol is often dropped in the notation for joint life statuses, as long as the meaning is clear. The most commonly used property of multiple lives models relates to an important literary work of Western Civilization, which we will quote now:

Then he dropped two (fir-cones) in (the river) at once, and leant over the bridge to see which one of them would come out first; and one of them did; but as they were both the same size, he didn’t know if it was the one which he wanted to win, or the other one. So the next time he dropped one big one and one little one, and the big one came out first, which was what he said it would do, and the little one came out last, which was what he had said it would do, so he had won twice … (Milne, [51]).

140

Chapter 3

This led to the creation of the Poohsticks game, and for that reason, we will term the following the Poohsticks Formulas: TðxyÞ þ TðxyÞ ¼ TðxÞ þ TðyÞ;

ð3:13:1Þ

TðxyÞ · TðxyÞ ¼ TðxÞ · TðyÞ;

ð3:13:2Þ

aTðxyÞ þ aTðxyÞ ¼ aTðxÞ þ aTðyÞ ;

ð3:13:3Þ

for any parameter a; t pxy

þ t pxy ¼ t px þ t py ;

ð3:13:4Þ

t qxy

þ t qxy ¼ t qx þ tqy ;

ð3:13:5Þ

and fTðxyÞ ðtÞ þ fTðxyÞ ðtÞ ¼ fTðxÞ ðtÞ þ fTðyÞ ðtÞ:

ð3:13:6Þ

The life expectancies for joint lives are defined as follows: EðTðxyÞÞ ¼ e xy ;

ð3:13:7Þ

EðTðxyÞÞ ¼ e xy ;

ð3:13:8Þ

and for KðxyÞ ¼ bTðxyÞc; KðxyÞ ¼ bTðxyÞc; EðKðxyÞÞ ¼ exy ; EðKðxyÞÞ ¼ exy : We have the following Poohsticks formulas for life expectancies:

ð3:13:9Þ ð3:13:10Þ

e xy þ e xy ¼ e x þ e y ;

ð3:13:11Þ

exy þ exy ¼ ex þ ey :

ð3:13:12Þ

It should be noted that Poohsticks can only be used when the relationship between TðxÞ and TðyÞ is not relevant. For example, the covariance of TðxÞ and TðyÞ is an expression of their statistical dependence, and the Poohsticks approach cannot be used for it CovðTðxyÞ; TðxyÞÞ " #" # ¼ CovðTðxÞ; TðyÞÞ þ EðTðxÞÞ 2 EðTðxyÞÞ EðTðyÞÞ 2 EðTðxyÞÞ ; ð3:13:13Þ and CovðvTðxyÞ ; vTðxyÞ Þ ¼ ðA x 2 A xy ÞðA y 2 A xy Þ:

ð3:13:14Þ

Furthermore, if TðxÞ and TðyÞ are uncorrelated then CovðTðxyÞ; TðxyÞÞ ¼ ðex 2 e xy Þðey þ e xy Þ:

ð3:13:15Þ

141

Life insurance and annuities

Exercises Exercise 3.13.1 November 2003 SOA/CAS Course 3 Examination, Problem No. 39. You are given: (i) Mortality follows De Moivre’s Law with v ¼ 105: (ii) (45) and (65) have independent future lifetimes. Calculate e 45:65 : Solution The appropriate Poohsticks formula says e 45:65 ¼ e 45 þ e 65 2 e 45:65 : We have e 45 ¼

105 2 45 ¼ 30; 2

e 65 ¼

105 2 65 ¼ 20; 2

and

as well as e 45:65 ¼

ð 40 0

ð 40 t t 5t t2 þ 12 12 · 12 dt ¼ dt 60 40 120 2400 0

40 5t2 t3 ¼ 40 2 5 · 1600 þ 64 000 ; þ ¼ t2 240 7200 240 7200 0 ¼ 40 2

100 80 360 2 300 þ 80 140 þ ¼ ¼ : 3 9 9 9

Thus e 45:65 ¼ 30 þ 20 2

140 < 34:44: 9

A

Exercise 3.13.2 November 2002 SOA/CAS Course 3 Examination, Problem No. 33. XYZ Co. has just purchased two new tools with independent future lifetimes. Each tool has its own distinct De Moivre survival pattern. One tool has a 10-year maximum lifetime and the other one has a 7-year maximum lifetime. Calculate the expected time until both the tools have failed.

142

Chapter 3

Solution We have, because of what we know about De Moivre’s Law, e x ¼ 5; e y ¼ 3:5: Also ð ð1 ð7 7 2 t 10 2 t 1 7 · dt ¼ e xy ¼ ð70 2 17t þ t2 Þ dt ¼ 2:683: t px · t py dt ¼ 7 10 70 0 0 0 Therefore e xy ¼ e x þ e y 2 e xy ¼ 5 þ 3:5 2 2:683 < 5:81:

A

Exercise 3.13.3 May 2001 SOA/CAS Course 3 Examination, Problem No. 9. ðxÞ and ðyÞ are two lives with identical expected mortality. You are given (i) Px ¼ Py ¼ 0:1: (ii) Pxy ¼ 0:06; where Pxy is the annual benefit premium for a fully discrete insurance of one on ðxyÞ: (iii) d ¼ 0:06: Calculate the premium Pxy ; the annual benefit premium for a fully discrete insurance of one on ðxyÞ: Solution We use the relation 1 ¼ d þ Px ; a€ x and its analogues 1 ¼ d þ Pxy ; a€ xy and 1 ¼ d þ Pxy : a€ xy First 1 ¼ d þ Px ¼ 0:06 þ 0:10 ¼ 0:16; a€ x so a€ x ¼ a€ y ¼

1 : 0:16

Finally a€ xy is determined from a€ xy ¼ a€ x þ a€ y 2 a€ xy ¼ 4:167:

143

Life insurance and annuities

Hence Pxy ¼

1 1 2 0:06 ¼ 0:18: 2d ¼ a€ xy 4:167

A

Exercise 3.13.4 May 2001 SOA/CAS Course 3 Examination, Problem No. 23. A continuous two-life annuity pays: † 100 while both (30) and (40) are alive, † 70 while (30) is alive but (40) is dead, and † 50 while (40) is alive but (30) is dead. The actuarial present value of this annuity is 1180. Continuous single life annuities paying 100 per year are available for (30) and (40) with actuarial present values of 1200 and 1000, respectively. Calculate the actuarial present value of a two-life continuous annuity that pays 100 while at least one of them is alive. Solution We are given that 1180 ¼ 70a30 þ 50a40 2 20a30:40 ¼ 70 · 12 þ 50 · 10 2 20a30:40 : Therefore, a 30:40 ¼ 8: Now we can use the Poohsticks formula to obtain a 30:40 ¼ a 30 þ a 40 2 a 30:40 ¼ 12 þ 10 2 8 ¼ 14: We conclude that 100a30:40 ¼ 1400:

A

Exercise 3.13.5 November 2000 SOA/CAS Course 3 Examination, Problem No. 1. For independent lives ðxÞ and ðyÞ; qx ¼ 0:05; qy ¼ 0:10: Deaths are uniformly distributed over each year of age. Calculate 0:75 qxy : Solution Thanks to independence, 0:75 qxy

¼ 1 20:75 pxy ¼ 1 20:75 px ·

0:75 py

¼ 1 2 ð1 20:75 qx Þ · ð1 20:75 qy Þ ¼ 1 2 ð1 2 0:75qx Þ · ð1 2 0:75qy Þ UDD

¼ 1 2 0:9625 · 0:925 ¼ 0:10969:

A

144

Chapter 3

Exercise 3.13.6 November 2000 SOA/CAS Course 3 Examination, Problem No. 30. For independent lives (50) and (60):

mðxÞ ¼

1 ; 100 2 x

0 # x , 100:

Calculate e 50:60 : Solution To calculate the last survivor life expectancy we use the Poohsticks identity: e x:y þ e x:y ¼ e x þ e y : Under de Moivre’s Law with v ¼ 100; TðxÞ is uniformly distributed on ½0; v and e x ¼

v2x 100 2 x ¼ : 2 2

Hence e 50 ¼ 25; e 60 ¼ 20; and e 50:60 ¼ ¼

ð1 0

t p50 t p60

dt ¼

ð 40 0

ð 40 t t 12 12 dt ¼ 50 40

ð2000 2 90t þ t2 Þdt

0

2000

29 333:33 ¼ 14:667: 2000

Finally e 50:60 ¼ 25 þ 20 2 14:667 ¼ 30:333:

A

3.14 Multiple decrements In practical actuarial work on pension plans, one must consider various modes of participants leaving the plan, such as retirement, death, disability, or withdrawal. We do that by considering multiple decrements models for survival, with decrements being those modes of leaving the group. We have, as before, T being the future lifetime of ðxÞ; a continuous random variable. As always, it has a discrete cousin: K ¼ bTc: We introduce a new random variable J; describing the mode of decrement. This is a discrete random variable. Effectively, we are now studying the joint distribution of T and J (or K and J). We use the upper right superscript to denote the mode of decrement, and t denotes the effect of all modes of decrement combined. We have the following notation for the multiple decrement tables: ð tÞ

† lx is the (expected) number of people in the group (cohort) at age x: ð jÞ † dx is the (expected) number of people departing the group (cohort) between ages x and x þ 1 due to cause j:

145

Life insurance and annuities

ð jÞ

ð jÞ

ð tÞ

† qx ¼ dx =lx is the probability that ðxÞ departs the group in the next year due to cause j; with other decrements also after ðxÞ (i.e., with competition from other causes).P ðtÞ ð jÞ † qx ¼ m j¼1 qx is the probability of departure, regardless of cause, when all causes compete. We also have fT;J ðt; jÞ ¼ t pðxtÞ · mð jÞ ðx þ tÞ;

ð3:14:1Þ

fT ðtÞ ¼ t pðxtÞ mðtÞ ðx þ tÞ;

ð3:14:2Þ ð jÞ

fK;J ðk; jÞ ¼ PrðK ¼ k; J ¼ jÞ ¼ k lqðx jÞ ¼ k pðxtÞ · qxþk ; ðt ð jÞ ðtÞ ð jÞ q ¼ PrðT # t; J ¼ jÞ ¼ s px · m ðx þ sÞ ds; t x 0

FT ðtÞ ¼ t qðxtÞ ¼ fJ ð jÞ ¼

ð1 0

m X j¼1

ð tÞ t px

· mð jÞ ðx þ tÞ dt ¼

ð jÞ t qx ; 1 X k

ð3:14:3Þ ð3:14:4Þ ð3:14:5Þ

lqxð jÞ ¼

ð jÞ 1 qx ;

ð3:14:6Þ

k¼0

and fJ ð jlT ¼ tÞ ¼

ðtÞ fT;J ðt; jÞ px · mð jÞ ðx þ tÞ mð jÞ ðx þ tÞ ¼ ðtÞ ¼ t ð tÞ : ðtÞ fT ðtÞ m ðx þ tÞ t px · m ðx þ tÞ

ð3:14:7Þ

An important consideration in multiple decrement models is the interaction between them. Models of that nature are commonly termed competing risks in statistics. If we want to understand the nature of the individual decrements, i.e., individual risks, we are interested in modeling a population, which is subjected only to one of the risks. This single-risk population is termed the associated single decrement table, and its actuarial functions are denoted with a prime. In order to study these models, we consider these alternative random variables: Tj is the waiting time until decrement ð jÞ removes ðxÞ from the population, if other decrements do not compete with decrement ð jÞ: We assume that the force of mortality (hazard rate) for individual decrements is the same in this model as in the multiple decrement model. We use the following notation: d sTj ðtÞ ; mð jÞ ðx þ tÞ ¼ 2 dt sTj ðtÞ ðt 0 ð jÞ ð jÞ sTj ðtÞ ¼ t px ¼ exp 2 m ðx þ sÞ ds ;

ð3:14:8Þ ð3:14:9Þ

0

0 ð jÞ t qx

¼ 1 2 t p0x ð jÞ ¼ FTj ðtÞ;

fTj ðtÞ ¼ t p0x ð jÞ mð jÞ ðx þ tÞ:

ð3:14:10Þ ð3:14:11Þ

146

Chapter 3

We also have the following relationships between the multiple decrement model and the associated single decrement tables (all these follow from simple probability considerations):

mðtÞ ðx þ tÞ ¼

m X

mð jÞ ðx þ tÞ;

ð3:14:12Þ

j¼1

ð tÞ t px

fT ðtÞ ¼ t pxðtÞ mðtÞ ðx þ tÞ; ðt m Y 0 ð tÞ ¼ PrðT . tÞ ¼ m ðx þ sÞ ds ; t px ð jÞ ¼ exp 2

ð3:14:13Þ ð3:14:14Þ

0

j¼1 ð tÞ t qx

¼

m X

ð jÞ t qx ;

ð3:14:15Þ

j¼1

and 12

m X

ð jÞ ðtÞ ð tÞ t qx ¼ 1 2 t q x ¼ t px ¼

j¼1

m Y

ð1 2 t q0x ð jÞÞ:

ð3:14:16Þ

j¼1

In the simplest case of only two decrements ð1Þ t qx

ð tÞ 0 ð1Þ þ t qð2Þ þ t q0x ð2Þ 2 t q0x ð1Þ t q0x ð2Þ : x ¼ t qx ¼ t qx

ð3:14:17Þ

If we assume the constant force over each year of age in each decrement mð jÞ ðx þ tÞ ¼ ð jÞ mx then ð jÞ

q0x ð jÞ ¼ 1 2 p0x ð jÞ ¼ 1 2 ð pðxtÞ Þqx

ðt Þ

=qx

;

ð jÞ

p0x ð jÞ ¼ ð pðxtÞ Þqx

ðtÞ

=qx

;

ð3:14:18Þ

and ð jÞ

ln p0x ð jÞ ¼

qx

ð tÞ

qx

ð jÞ

· ln pðxtÞ ;

qx

ðtÞ

ð jÞ

¼

qx

mx

ð tÞ

:

mx

To show this, observe that ð1 ð1 ðt Þ ðtÞ ð jÞ e2mx mxð jÞ dt ð jÞ ð jÞ t px mx ðtÞ dt qx mx 0 0 ¼ ¼ ¼ : ð ð 1 1 ðtÞ ð tÞ ðtÞ qx mx ðtÞ ðtÞ 2 mx ð tÞ p m ðtÞ dt e m dt x x t x 0

ð3:14:19Þ

ð3:14:20Þ

0

If we make the assumption of the uniform distribution of each decrement ð jÞ ð jÞ within each year of age in the multiple decrement table (i.e., t qx ¼ t · qx ; ð jÞ ð jÞ t px ¼ 1 2 t · qx ) then we have ð jÞ

q0x ð jÞ ¼ 1 2 p0x ð jÞ ¼ 1 2 ðpðxtÞ Þqx

ðtÞ

=qx

;

ð3:14:21Þ

147

Life insurance and annuities

and this turns out to be the same as (3.14.18). To show it in this case, note that ðt ð jÞ tqx ¼ s pðxtÞ mð jÞ ðx þ sÞ ds: 0

Differentiate this with respect to t and obtain qxð jÞ ¼ t pðxtÞ mð jÞ ðx þ tÞ: Therefore 2

ð jÞ

qx

ð jÞ

m ðx þ tÞ ¼

ðtÞ

1 2 t qx

¼

d ð tÞ ð1 2 tqx Þ qð jÞ x dt : ð tÞ ðtÞ qx 1 2 tqx

By integrating both sides we get ðt ð jÞ ð jÞ qx qx 2 mð jÞ ðx þ sÞ ds ¼ ðtÞ lnð1 2 tqðxtÞ Þ ¼ ðtÞ lnðt pðxtÞ Þ: 0 qx qx

ð3:14:22Þ

ð3:14:23Þ

We also know that lnð t p0x ð jÞ Þ ¼ 2

ðt

mð jÞ ðx þ sÞ ds:

0

Therefore 0 ð jÞ t px

ð jÞ

¼ ð t pðxtÞ Þqx

ðt Þ

=qx

:

ð3:14:24Þ

The formula (3.14.21) is the special case of (3.14.24) when t ¼ 1: The uniform distribution of each decrement within each year of age can be also assumed in each single decrement table t q0x ðiÞ ¼ tq0x ðiÞ ; and this assumption is significantly different from uniform distribution of each decrement in the observed combined table. In this new case, we have ð1 ðtÞ ðiÞ qðiÞ ¼ t px m ðx þ tÞ dt x 0 1 0 ! ð1 Y B C ð1 2 t · q0x ð jÞ Þ · @t p0x ðiÞ mðiÞ ðx þ tÞ Adt ¼ |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ ﬄ {zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ ﬄ } 0 j–i ! PDF of Ti ¼q0x ðiÞ ð1 Y ¼ q0x ðiÞ ð1 2 t · q0x ð jÞ Þ dt: ð3:14:25Þ 0

j–i

For example, when there are only two decrements, under uniform distribution in the associated decrement table: 1 0 ð2Þ ð1Þ 0 ð1Þ 1 2 qx ; ð3:14:26Þ qx ¼ qx 2

148

Chapter 3

and with three decrements:

qð1Þ x

¼

q0x ð1Þ

1 0 ð2Þ 1 0 ð2Þ 0 ð3Þ 0 ð3Þ 1 2 ðqx þ qx Þ þ ðqx qx Þ ; 2 3

0 ðiÞ qðiÞ x ¼ qx

ð1 0

Y

ð3:14:27Þ

! 0 ð jÞ t px

fTi ðtlTi # 1Þ dt:

ð3:14:28Þ

j–i

In pension and insurance practice, it is common to have benefits depending on the mode of decrement. An important and simple principle for working with such plans is that their single benefit premium can be calculated by adding the single benefit premiums for all decrements that can be calculated separately first, i.e.,

t

EðBðt; jÞ · v Þ ¼

m ð1 X j¼1

Bðt; jÞ · vt · t pðxtÞ · mxð jÞ ðtÞ dt : |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

ð3:14:29Þ

0

APV of benefit payable for decrement j

Exercises Exercise 3.14.1 November 2003 SOA/CAS Course 3 Examination, Problem No. 7. A whole life policy provides that upon accidental death as a passenger on an airplane a benefit of 1 000 000 will be paid. If death occurs from other accidental causes, a death benefit of 500 000 will be paid. If death occurs from a cause other than an accident, a death benefit of 250 000 will be paid. You are given (i) Death benefits are payable at the moment of death. (ii) mð1Þ ¼ 1=2 000 000 where (1) indicates accidental death as a passenger on an airplane. (iii) mð2Þ ¼ 1=250 000 where (2) indicates death from other accidental causes. (iv) mð3Þ ¼ 1=10 000 where (3) indicates nonaccidental death. (v) d ¼ 0:06: Calculate the single benefit premium for this insurance. Solution Note that

mðtÞ ¼

1 1 1 þ þ ¼ 0:0001405: 2 000 000 250 000 10 000

149

Life insurance and annuities

Therefore, the single benefit premium is ð þ1 ð þ1 1 20:06t 20:0001405t dtþ 1000000 e e · 2000000 0 0 ð þ1 1 1 dt þ dt 500000 e20:06t e20:0001405t · 250000 e20:06t e20:0001405t · 250000 10000 0 ¼

1 1 1 1 27:5 þ2 · þ 25 · ¼ ¼457:262577: 2 0:0601405 0:0601405 0:0601405 0:0601405 A

Exercise 3.14.2 November 2003 SOA/CAS Course 3 Examination, Problem No. 27. For a double decrement table, you are given ð1Þ

ðtÞ

(i) mx ðtÞ ¼ 0:2mx ðtÞ; t . 0; ðtÞ (ii) mx ðtÞ ¼ kt2 ; t . 0; (iii) q0x ð1Þ ¼ 0:04: ð2Þ

Calculate 2 qx : Solution We know that ðtÞ mð1Þ x ðtÞ ¼ 0:2mx ðtÞ ¼

k 2 t : 5

But ðt 3 1 0:04 ¼ q0x ð1Þ ¼ 1 2 p0x ð1Þ ¼ 1 2 exp 2 ðk þ 5Þt2 dt ¼ 1 2 e2ðk=5Þðð1=3Þt Þl0 0

¼ 1 2 e2ðk=15Þ : Therefore e2ðk=15Þ ¼ 0:96;

k ¼ ð215Þ · ln 0:96 ¼ lnð0:96215 Þ:

Also

ðtÞ t px

mðxtÞ ðtÞ ¼ t2 lnð0:96215 Þ; t . 0; ðt 3 3 2 215 ¼ exp 2 r lnð0:96 Þ dr ¼ e15 ln 0:96ð1=3Þt ¼ 0:965t : 0

Finally ð tÞ 2 212 mð2Þ Þ: x ðtÞ ¼ 0:8mx ðtÞ ¼ t lnð0:96

150

Chapter 3

Therefore ð2Þ 2 qx ¼

ð2 0

ðtÞ ð2Þ t px mx ðtÞ

1 ¼ lnð0:96212 Þ 3 ¼ 24 ln 0:96

dt ¼

ð2

0:965t

3

0

ð8

& & & & u ¼ t3 t ¼ 0 ) u ¼ 0 & & · t2 lnð0:96212 Þ dt ¼ & 2 & & t dt ¼ 1 du t ¼ 2 ) u ¼ 8 & 3

0:965u du ¼ lnð0:9624 Þ

0

ð8

e5ðln

0:96Þu

du

0

1 4 40 ð0:965u Þlu¼8 u¼0 ¼ 2 ð0:96 2 1Þ < 0:64370708: 5 ln 0:96 5

A

Exercise 3.14.3 November 2002 SOA/CAS Course 3 Examination, Problem No. 2. For a triple decrement table, you are given ð1Þ

(i) mx ðtÞ ¼ 0:3; t . 0; ð2Þ (ii) mx ðtÞ ¼ 0:5; t . 0; ð3Þ (iii) mx ðtÞ ¼ 0:7; t . 0; ð2Þ

Calculate qx : Solution Note that ð2Þ ð3Þ mðxtÞ ðtÞ ¼ mð1Þ x ðtÞ þ mx ðtÞ þ mx ðtÞ ¼ 0:3 þ 0:5 þ 0:7 ¼ 1:5:

Under individual constant forces, we have ð tÞ qðiÞ x ¼ qx

lnðp0x ðiÞ Þ ðtÞ

lnðpx Þ

¼ qxðtÞ

ln e2m ln e

ðiÞ

2mðtÞ

¼ qðxtÞ

mðiÞ 0:5 ¼ 0:258956613: ¼ ð1 2 e21:5 Þ 1:5 mðtÞ A

Exercise 3.14.4 November 2002 SOA/CAS Course 3 Examination, Problem No. 13. For a double-decrement table where cause 1 is death and cause 2 is withdrawal, you are given (i) Deaths are uniformly distributed over each year of age in the singledecrement table. (ii) Withdrawals occur only at the end of each year of age. ð tÞ (iii) lx ¼ 1000: ð2Þ (iv) qx ¼ 0:40: ð1Þ ð2Þ (v) dx ¼ 0:45dx : Calculate p0x ð2Þ :

151

Life insurance and annuities

Solution The key observation is that since withdrawals occur at the end of the year, while deaths have the continuous uniform distribution, the probability of withdrawal does not affect the probability of death, only the probability of death affects the probability of withdrawal. In other words, withdrawals at the end of the year do not change the exposure to death during the year (they change it at one point only and this has no affect on probabilities in a continuous distribution). ð1Þ Thus p0x ð1Þ ¼ px : We can see quickly that ð2Þ qð1Þ x ¼ 0:45qx ¼ 0:18; ð2Þ pðxtÞ ¼ 1 2 qð1Þ x 2 qx ¼ 1 2 0:18 2 0:40 ¼ 0:42; ð tÞ

p0x ð2Þ ¼

ð tÞ

px px 0:42 ¼ 0:512195: ¼ ð1Þ ¼ 0 ð1Þ 0:82 px px

A

Exercise 3.14.5 November 2002 SOA/CAS Course 3 Examination, Problem No. 14. You intend to hire 200 employees for a new management training program. To predict the number of people who will complete the program, you build a multiple decrement table. You decide that the following associated single decrement assumptions are appropriate: (i) Of 40 hires, the number who fail to make adequate progress in each of the first three years is 10, 6, and 8, respectively. (ii) Of 30 hires, the number who resign from the company in each of the first three years is 6, 8, and 2, respectively. (iii) Of 20 hires, the number who leave the program for other reasons in each of the first three years is 2, 2, and 4, respectively. (iv) You use the uniform distribution of decrement assumption in each year in the multiple decrement table. Calculate the expected number of those employees who fail to make adequate progress in the third year. Solution From the data given we get the following: 10 1 ¼ ; 40 4

q01 ð1Þ ¼

q00 ð2Þ ¼ q00 ð3Þ

6 1 ¼ ; 30 5 2 1 ¼ ; ¼ 20 10

q00 ð1Þ ¼

6 1 ¼ ; 30 5

q02 ð1Þ ¼

q01 ð2Þ ¼

q02 ð2Þ ¼

q01 ð3Þ

8 1 ¼ ; 24 3 2 1 ¼ ; ¼ 18 9

q02 ð3Þ

8 1 ¼ ; 24 3

2 1 ¼ ; 16 8 4 1 ¼ : ¼ 16 4

152

Chapter 3

Therefore ðtÞ p0

ðtÞ

1 1 1 ¼ 12 12 12 ¼ 0:54; 4 5 10

p1 ¼

1 1 1 12 12 12 ¼ 0:47407; 5 3 9

and ðtÞ p2

1 1 1 ¼ 12 12 12 ¼ 0:4375: 3 8 4

It follows that ð tÞ

ð tÞ

ðtÞ

ð tÞ

l2 ¼ l0 · p0 · p1 ¼ 200 · 0:54 · 0:47407 ¼ 51:20; ð1Þ

q2 ¼

ln p02 ð1Þ ln

ð tÞ p2

ð1Þ

ð tÞ

· q2 ¼ 0:257590;

ð1Þ

ð tÞ

d2 ¼ q2 · l2 ¼ 14:

A

Exercise 3.14.6 November 2002 SOA/CAS Course 3 Examination, Problem No. 34. XYZ Paper Mill purchases a 5-year special insurance paying a benefit in the event its machine breaks down. If the cause is ‘minor’ (1), only a repair is needed. If the cause is ‘major’ (2), the machine must be replaced. Given (i) The benefit for cause (1) is 2000 payable at the moment of breakdown. (ii) The benefit for cause (2) is 500 000 payable at the moment of breakdown. (iii) Once a benefit is paid, the insurance contract is terminated. ð1Þ ð2Þ (iv) mx ðtÞ ¼ 0:100 and mx ðtÞ ¼ 0:004 for t . 0: (v) d ¼ 0:04: Calculate the actuarial present value of this insurance. Solution We have

mðxtÞ ðtÞ ¼ 0:104; for all t; so that ð tÞ t px

¼ e20:104t :

153

Life insurance and annuities

Therefore, the cost of the first benefit is ð5 2000 e20:04t e20:104t · 0:1 dt ¼ 200 0

5 1 e20:144t ð20:144Þ 0

¼ 1388:8889 · ð1 2 e20:72 Þ ¼ 712:84409: The value of the second benefit is ð5 500000

e

20:04t

e

20:104t

0

5 1 20:144t e · 0:004 dt ¼ 2000 ð20:144Þ 0 ¼ 13888:889 · ð1 2 e20:72 Þ ¼ 7128:4409:

The total of the two is 7841.28.

A

Exercise 3.14.7 November 2001 SOA/CAS Course 3 Examination, Problem No. 20. Don, age 50, is an actuarial science professor. His survival in his career is subject to two decrements (i) Decrement 1 is mortality. The associated single decrement table follows De Moivre’s Law with v ¼ 100: (ii) Decrement 2 is leaving academic employment with mð2Þ ð50 þ tÞ ¼ 0:05; t $ 0: Calculate the probability that Don remains an actuarial science professor for at least five but less than 10 years. Solution From (i) and De Moivre’s Law, it follows that 0 ð1Þ t p50

¼12

t 50

for t # 50:

From (ii), t p050 ð2Þ ¼ e20:05t for t . 0: Hence t 20:05t ðtÞ p ¼ 1 2 e t 50 50

for t # 50:

The probability sought is ðtÞ

ð tÞ

Prð5 # T , 10Þ ¼ sT ð5Þ 2 sT ð10Þ ¼ 5 p50 2 10 p50 45 20:25 40 20:5 e e ¼ 2 ¼ 0:216: 50 50

A

154

Chapter 3

Exercise 3.14.8 May 2001 SOA/CAS Course 3 Examination, Problem No. 10. For students entering a college, you are given the following from a multiple decrement model: (i) 1000 students enter the college at t ¼ 0: (ii) Students leave the college for failure (1) or all other reasons (2). (iii) ( ð1Þ m ðtÞ ¼ m ¼ constant; 0 # t # 4;

mð2Þ ðtÞ ¼ 0:04;

0 # t # 4:

(iv) 48 students are expected to leave the college during their first year due to all causes. Calculate the expected number of students who will leave because of failure during their fourth year. ð tÞ

Solution Since mðtÞ ðtÞ ¼ m þ 0:04 is constant, we have t p0 ¼ e2ðmþ0:04Þt : The expected decrement of 48 in the first year is the same as ðtÞ

ð tÞ

q0 · 1000 ¼ ð1 2 p0 Þ · 1000 ¼ ð1 2 e2ðmþ0:04Þ1 Þ · 1000: Hence 0:048 ¼ 1 2 e2ðmþ0:04Þ1 ; and m ¼ 2ln 0:952 2 0:04: The expected number leaving due to failure (cause 1) in the fourth year is ð1Þ

1000 · 3 lq0 ¼ 1000 · ¼ 1000 ·

ð4 3

ð4

ð tÞ

ð1Þ t p0 m ðtÞ dt ¼ 1000 ·

e2ð2ln

0:95220:04þ0:04Þt

ð4

e2ðmþ0:04Þt ð2ln 0:952 2 0:04Þ dt

3

ð2ln 0:952 2 0:04Þ dt

3

¼ 1000 ·

ð4

0:952t ð2ln 0:952 2 0:04Þ dt

3

¼ 1000 · ð2ln 0:952 2 0:04Þ ·

0:9524 2 0:9523 ¼ 7:74: ln 0:952

A

Exercise 3.14.9 November 2000 SOA/CAS Course 3 Examination, Problem No. 7. For a multiple decrement table, you are given that decrement (1) is death, decrement (2) is disability, and decrement (3) is withdrawal. You also have q060 ð1Þ ¼ 0:010; q060 ð2Þ ¼ 0:050; and q060 ð3Þ ¼ 0:100: Withdrawals occur only at the end of

155

Life insurance and annuities

the year. Mortality and disability are uniformly distributed over each year of age in ð3Þ the associated single decrement tables. Calculate q60 : Solution Only decrements (1) and (2) occur during the year, and therefore the probability of survival to the moment just before end the year is 0 ð1Þ 1 p60

· 1 p060 ð2Þ ¼ ð1 2 0:01Þð1 2 0:05Þ ¼ 0:9405:

Probability of survival through the entire year is 0 ð1Þ 1 p60

· 1 p060 ð2Þ · 1 p060 ð3Þ ¼ 0:9405ð1 2 0:10Þ ¼ 0:84645:

We have, therefore ð3Þ

q60 ¼ Probability of dying of cause 3 when other causes operate ¼ Probability of surviving to the moment just before the end of the year BUT not surviving the whole year ¼ 0:9405 2 0:84645 ¼ 0:09405: A Exercise 3.14.10 November 2000 SOA/CAS Course 3 Examination, Problem No. 9. For a special whole life insurance of 100 000 on ðxÞ; you are given that (d ¼ 0:06 and the death benefit is payable at the moment of death. If death occurs by accident during the first 30 years, the death benefit is doubled. Furthermore, mðtÞ ðx þ tÞ ¼ 0:008; t $ 0; and

mð1Þ ðx þ tÞ ¼ 0:001;

t $ 0;

where mð1Þ is the force of decrement due to death by accident. Calculate the single benefit premium for this insurance. Solution The benefit here depends on both the time and mode of decrement. It can be viewed as the combination of 100 000 at death plus an extra 100 000 at death if death is accidental and within 30 years. Thus the single benefit premium is given by ðt Þ

100 000 · mmðtÞ þd |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

Constant force ðmain insuranceÞ

þ

EðZÞ |ﬄ{zﬄ}

;

Accidental Death

where ( T

Z ¼ BðT; JÞv ¼

100 000 e20:06T ; if J ¼ 1 and T # 30; 0;

otherwise:

156

Chapter 3

where Z is a function of both T and J; which have the joint density ( 0:001 e20:008t ; J ¼ 1; fT;J ðt; jÞ ¼ t pðxtÞ · mð jÞ ðx þ tÞ ¼ 0:007 e20:008t ; J ¼ 2: So EðZÞ ¼

2 ð1 X j¼1

0

t

Bðt; jÞv · fT;J ðt; jÞ dt ¼

ð 30

100 000 e20:06t · e20:008t · 0:001 dt

0

1 2 e20:068ð30Þ ¼ 1279:37: 0:068 The total single benefit premium is ¼ 100 ·

100 000 ·

mðtÞ 0:008 þ 1279:37 ¼ 13 044:08: þ EðZÞ ¼ 100 000 · 0:068 mðtÞ þ d

A

Exercise 3.14.11 May 2000 SOA/CAS Course 3 Examination, Problem No. 15. ð tÞ ð1Þ In a double decrement table: l30 ¼ 1000; q030 ð1Þ ¼ 0:100; q030 ð2Þ ¼ 0:300; 1 lq30 ¼ ðtÞ 0:075; l32 ¼ 472: ð2Þ Calculate q31 : Solution We have ðtÞ

p30 ¼ p030 ð1Þ · p030 ð2Þ ¼ 0:9 £ 0:7 ¼ 0:63; and ð tÞ

ðtÞ

ðtÞ

l31 ¼ p30 · l30 ¼ 630: Also ð1Þ

ð1Þ

d31 ¼

1

lq30

ð tÞ

· l30 ¼ 75:

Furthermore, ðtÞ

ðtÞ

ðtÞ

d31 ¼ l31 2 l32 ¼ 630 2 472 ¼ 158; and ð2Þ

ðtÞ

ð1Þ

d31 ¼ d31 2 d32 ¼ 158 2 75 ¼ 83: Therefore ð2Þ

q31 ¼

ð2Þ

d31

ð tÞ l31

¼

83 ¼ 0:1317: 630

A

Part 2 Valuation of Assets and Liabilities

This page is intentionally left blank

159

Chapter 4 General principles of pension plan valuation

4.1 Objectives and principles of pension funding The financing of future liabilities by an individual has the following two risks: the risk that the person at some point may not have sufficient funds to continue providing resources for the liabilities (due to deterioration of financial condition), and the risk of fluctuation in the value of liabilities (apart from any deterioration of provider’s financial condition). An individual enters into the insurance arrangements precisely in order to decrease or, at last, control them. Of course, the other side of the transaction, the insurer or the plan provider, increases the exposure to such risks with every new customer. Why do then insurers accept such increasing risk by taking on new customers? Because a large portfolio of risks provides a pooling diversification benefit, resulting from the Central Limit Theorem or the Law of Large Numbers. Sufficiently diversified liabilities may in fact have their random nature dramatically reduced and become nearly deterministic. This is especially true of retirement benefit liabilities for a large group. Creation of a fund designated for provision of retirement benefit payments affords additional advantages to plan participants: † increase the financial security of future benefit payments; † by creating an asset structure which appropriately matches the liabilities, make the benefits payments less exposed to market risk and to macro-economic factors. The same rewards derived from a fund accumulation are also present when an employer or another pension plan provider accumulates a fund in order to provide pension benefits. In view of these considerations, we must view accumulation of capital assets in a fund to be one of the central functions of financial intermediaries engaged in the provision of pension benefits, such as insurance firms and pension plans. In order to purchase those capital assets, such intermediaries collect premium payments from their customers. In Section 4.2 we will discuss general principles of

160

Chapter 4

calculation of a premium for a given set of liabilities. In Chapter 5 we will discuss the actuarial methods of calculation of an appropriate premium for pension plans, and present details of four basic methods of valuation. However, the existing insurance literature does not provide an answer to a very natural question—what form of accumulation of funds for a given set of future liabilities is optimal, from the point of view of the insurance firm, pension plan, or the customer? The method of accumulation of premium may turn out to be a controversial issue in practice. Actuaries are concerned with solvency of the insurance enterprise or the pension plan under consideration and generally prefer early accumulation of assets, as this provides greater financial security. But other divisions of an insurance firm or the plan may be more interested in using the financial resources provided by the premiums or contributions for physical plant investment, marketing, or other forms of growing the business. In the case of pension plans, the employer usually feels that the resources could be better used when applied towards growing the core business of the firm, not pension plan assets. Such a redirection would move the actual payment to plan participants into the future, but may be beneficial to plan participants if indeed the employer does deliver extraordinary growth in the meantime. In fact, this financing redirection has been used for quite a while in Germany, following World War II. However, in the United States funding a pension plan through internal growth is not allowed on the basis of the 1974 Employee Retirement Income Security Act (ERISA). In the case of an insurance firm, one can look at the question of proper funding as a question of maximization of profit. Chalke [17] discusses a conflict between actuarial and marketing divisions of an insurer, so common in the United States. He postulates that the conflict can be resolved by optimization of premium level, and focusing the firm on maximization of profits, with constraints set by solvency margins. Of course, maximization of profits is a standard objective in pricing of products other than insurance, but Chalke’s work was the first one to point out that this standard approach can be applied to insurance. General theory of pricing is presented by Stigler [71]. One can, of course, argue that in the case of a pension plan, maximization of profits is never a correct objective and the true goal is to provide secure delivery of future benefits. The issue of solvency of the insurance firm, and setting appropriate levels of premiums and reserves, are of course at the very core of the work of that firm’s actuary. But the problem of finding an optimal scheme of premium payments has not received any attention in actuarial literature. Level premium has been traditionally utilized in both life insurance and other forms of insurance. There has been some move away from the traditional approach with the introduction of universal life policies. But with the sole exception of the recent work of these authors [32], the existing literature on pension plans concerned with their methods of funding (see, for example Aitken [1], McGill [47]) does not even consider the question of optimality of a funding method.

General principles of pension plan valuation

161

In this book, we propose the following general approach to the problem. We will divide the premium paid for the pension fund cost into the following two parts: † planned premium, covering the planned growth of liabilities to the plan participants; and † additional premium, used to pay for amortization of the difference between the planned and actual growth of unfunded liabilities. The planned premium is called the normal cost of the plan. It is paid at the beginning of the fiscal period and established by the actuary. A pension actuary has more flexibility in the choice of valuation assumptions than a typical life actuary. Actual methods of establishing the normal cost are shown in Chapter 5. But the flexibility of the pension plan actuary is not absolute: pension valuation is subject to government regulation, and in the United States the actuary has the fiduciary duty to plan participants, as prescribed by the 1974 ERISA Law. The additional premium is an unexpected cost, typically termed supplemental cost, to the plan provider and, as such, a source of additional risk (see Section 8.4). Minimization of risk would most likely call for spreading this additional cost equally over time. This issue will be discussed in more detail in Section 4.3.

4.2 General principles of pension valuation The basis for valuation of pension liabilities is the same actuarial mathematics developed for life insurance in Chapter 2. It is not difficult to note similarities of various concepts used in the areas of pensions and life insurance, for example, ‘accrued liabilities’ is the pension analog of ‘benefit reserve’. Nevertheless, the mathematics of pension plans is treated as a separate area of actuarial science. There are several reasons for that. First, the number of plan participants is often significantly smaller than a typical number of customers of a life insurance company. Random fluctuations of mortality play a greater role in the planning of pension plan funding than in life insurance funding. We will address this issue of random fluctuations in Chapter 6. Second, pension plan valuation interest rate is not set as strictly by law as life insurance valuation rate, and this creates additional uncertainty in the pension case. Also, pension benefits come in a much greater variety than life insurance benefits, because of many ancillary benefits available for plan participants (such as survivor benefits, disability benefits, etc.). This also adds uncertainty to the cost of plan funding. And, to make things even more complicated, pension benefits are based on salary history, but there is a great variety in benefit formulas, from straight career average to various forms of final average. Third, when the plan funding experience differs from actuarial assumptions, the consequences are not a direct analog of the life insurance situation. Careful conservative valuation basis in life insurance can only bring

162

Chapter 4

eventually higher profits to the firm and those can be returned to policyholders, especially for mutual companies. Excessive normal cost level for pension funding will show a higher pension expense for the plan provider, and such higher cost will cause a private provider to show smaller profitability, while a government provider will have to increase tax burden to fund pensions. If the normal cost is too low, this is also undesirable, as it creates a possible threat to plan solvency. In the context of the issues raised above, we will now present the fundamental concepts and the methodology of pension valuation and the calculation of normal cost and plan surplus. The key concepts of pension valuation are: † † † †

accrued liabilities, normal cost, unfunded liabilities, and actuarial gain.

We will discuss the meaning of these concepts and then present a general rule of pension valuation dynamics. We will then introduce basic pension funding methods, which will be explained in detail in Chapter 5. 4.2.1 Accrued liabilities Assume that a plan participant earns a right to retire (with a life or temporary annuity) at age y: Let us also assume that this participant will receive the retirement benefit as a monthly annuity, with the total annual benefit amount BðyÞ: A properly funded pension plan should accumulate financial resources sufficient to pay these benefits. If the annuity is a life annuity, then the present value of benefits at retirement is BðyÞ€að12Þ y ; ð12Þ

where a€ y is the actuarial present value (i.e., expected present value, see (3.10.17)) of a monthly life annuity due payable beginning at age y; with monthly installments equal to 1/12 of the nominal annual amount. If upon retirement at age y the beneficiary will instead receive a temporary annuity for a term of n years, then the present value of these benefits is ð12Þ

BðyÞ€ay: n ; ð12Þ

where BðyÞ is the annual benefit amount, and a€ y: n ; given by the formula (3.10.36), is the actuarial present value of an n year temporary annuity, payable monthly, with monthly installments equal to 1/12. In what follows, we will concentrate on the case of life annuities, but of course the case of temporary annuities will require just a minor adjustment. If we replace in all ð12Þ ð12Þ formulas a€ y by a€ y: n ; we obtain the temporary annuity analogs of life annuity

163

General principles of pension plan valuation

formulas. One can also use the same formulas as for life annuities, and merely modify the laws of mortality used. For temporary n year life annuities we would simply need to assume that all plan beneficiaries do not survive beyond age y þ n; in other words, replace the single life status ðyÞ by the joint life status y : n : Typically, a pension plan participant earns benefit rights with the passage of years of service. There are two ð12Þ approaches in determining the value at retirement BðyÞ€ay of the benefit earned by a participant. The first approach used, for example, in the unit credit method (see Section 5.1) determines the accrued benefit BðxÞ directly, as a function of age x; or service years, of a participant. If the participant starts service at age w; and at current time t is aged x; with w , x , y; then 0 , BðxÞ , BðyÞ; as it is usually assumed that BðwÞ ¼ 0: Denote by ALjt the accrued liability with respect to the jth plan participant at time t; defined here as ALjt ¼ Bðxj Þ€að12Þ yj

Dyj Dxj

;

where the index j means that the value is assigned to the jth participant, and Dyj =Dxj is the actuarial discount factor used for discounting to time t; accounting for mortality (or, more generally, probability of any form of decrement, i.e., departure from the group of plan participants). Total accrued liability for the entire plan, denoted by ALt ; is then equal to X j ALt ; ALt ¼ j[At

where At is the group of active participants at time t: The method described above is especially useful when the rules of the plan clearly define the function BðxÞ: Otherwise, it is more convenient to use a second approach, which, although less natural, leads to fewer problems of the kind described in Example 5.1.1. This second approach first determines the normal cost, and then, using it, establishes accrued liabilities. This will be discussed in Section 4.2.2. 4.2.2 The normal cost versus accrued liabilities The normal cost, NCt ; is defined as the premium which needs to be paid into the plan at the beginning of every year (or, more generally, every fiscal period) in order to pay that year increment of all liabilities, assuming that all actuarial assumptions concerning mortality, salaries, interest rates, and expenses, turn out to be correct. Thus the normal cost is effectively the annual cost of the plan, if the actuarial assumptions turn out to be exact predictions. In reality, assumptions are not perfect, and differences between assumptions and reality (termed actuarial gains or

164

Chapter 4

actuarial losses) need to be regularly amortized. The total premium paid must include this amortization, and such premium is termed the plan cost. The Fundamental Principle of Pension Plan Actuarial Valuation is:

The normal costs of a pension plan must be spread over time in such a way that at the time of retirement of a given participant all normal costs to be paid on that participant’s behalf have the same accumulated actuarial value as the actuarial present value of the benefits to be paid to this participant.

The above rule is, of course, a version of the Equivalence Principle used widely in actuarial mathematics in life insurance (see Section 3.11 and e.g., Bowers et al. [14]). The following illustrates the application of this principle to valuation of a pension plan, by relating the normal cost and the accrued liabilities.

The accrued liabilities with respect to the jth participant at every year are equal to the accumulated actuarial value of the normal costs paid on behalf of this participant.

The above is the retrospective formula for the accrued liabilities, in terms of the normal costs paid up to date. By using the Equivalence Principle, we can also derive this prospective formula for accrued liabilities:

At each year the accrued liabilities with respect to the jth plan participant are equal to the actuarial present value of future benefits ðPVFBjt Þ minus the actuarial present value of future normal costs ðPVFNCjt Þ: ALjt ¼ PVFBjt 2 PVFNCjt : It is easy to see that the retrospective formula follows from the prospective formula combined with the Equivalence Principle. In fact, the two formulas are equivalent, given the Equivalence Principle.

Using the above principles, one can first choose an appropriate method of determining normal costs, and then calculate accrued liabilities. This approach is used in several methods, for example the Entry Age Normal cost method. In this method, one assumes that all participants join the plan at approximately the same age w and

165

General principles of pension plan valuation

later retire at approximately the same age y: Assume that the normal cost for the jth participant is level and equal to NCj : Then from the Fundamental Principle of Pension Plan Actuarial Valuation, we infer that NCj a€ w: yw ¼ Bj ðyÞ€að12Þ y

Dy ; Dw

from which we can calculate NCj : Using the retrospective formula we calculate accrued liabilities ALjt with respect to this participant in the year t; in which this participant was aged x; arriving at ALjt ¼ Bj ðyÞ€að12Þ y

Nw 2 Nx Dy ; Nw 2 Ny Dx

where Nx is the commutation function given by the formula (3.8.4). This calculation will be elaborated further in Chapter 5. We can also obtain its analog using the prospective approach: ALjt ¼ Bj ðyÞ€að12Þ y

Dy Nx 2 Ny 2 NCj ; Dx Dx

where the first expression is the actuarial present value of future benefits, and the second one is the actuarial present value of future normal costs. Of course, both formulas are equivalent. A similar approach is used in the Individual Level Premium method (see Section 5.3), but in that case the assumption of approximately equal age of each new participant is no longer used. 4.2.3 Unfunded actuarial liabilities Let Ft denote the assets of the plan at time t: If the plan is in ideal balance then its assets Ft equal accrued liabilities ALt : In reality there is usually a nonzero difference UALt ¼ ALt 2 Ft ; which will be termed the actuarial deficit or the unfunded accrued liability. Of course, one must establish values of ALt and Ft to calculate the unfunded actuarial liability. Chapter 5 covers valuation of liabilities, while Chapter 7 is devoted to questions related to valuation of assets. Unfunded actuarial liabilities should be amortized. This is discussed in Chapter 5. If UALt , 0; then we will term its opposite (its absolute value) the actuarial surplus (not to be confused with the actuarial gain). A large actuarial surplus should also be amortized. 4.2.4 Actuarial gain Let C be the contribution paid by the plan provider in a given year t: Let I be the amount by which the assets grow in a given year, as a result of investment portfolio

166

Chapter 4

performance, and let P be the amount paid for purchases of annuities for retiring participants. Then the assets at time t þ 1 are Ftþ1 ¼ Ft þ I þ C 2 P: Let us also denote by IC the return earned by the contribution C paid in the year t; and we will assume that the contribution is paid at the beginning of the year. If the actuarial assumptions about the interest rate turn out to be correct, and the plan provider pays normal cost and the entire unfunded actuarial liability from the previous year, then we have: ðUALt þ NCt Þð1 þ iÞ 2 ðC þ IC Þ ¼ 0:

ð4:2:1Þ

The left-hand side of (4.2.1) is called the expected unfunded actuarial liability. In the situation described by (4.2.1), this expected unfunded liability has been reduced to zero. If the actuarial assumptions turn out to be correct, then the contribution C together with return on it IC will reduce to zero the unfunded actuarial liability UALtþ1 in the year t þ 1: In reality, of course, this usually does not happen, and thus Ga ; ðUALt þ NCt Þð1 þ iÞ 2 ðC þ IC Þ 2 UALtþ1 is nonzero. Since the above expression Ga is the difference between the expected and actual unfunded actuarial liability, it is called the actuarial gain. The actuarial gain is positive if the actuarial unfunded liability is less than the expected unfunded actuarial liability. As we had pointed out already, achieving an actuarial gain is not an objective of a pension plan. Thus, for example, in the method of frozen initial liabilities, the valuation of ALt and NCt is set in such a way that Ga ; 0 by definition (see Section 5.4). 4.2.5 Some additional remarks There are two additional concepts used in the actuarial literature in relation to the accrued actuarial liabilities. The two represent methods of establishing liabilities of a pension system. † Accumulated Benefit Obligation (ABO) calculated as those liabilities, which have been already earned by the plan participants, based on their service up to date, but not taking into consideration expected future service (if we were to include future expected service, we would also have to include future expected normal costs paid). This is an analog of the retrospective definition of accrued liabilities. † Projected Benefit Obligation (PBO) calculated as the liabilities with respect to current participants, which have been earned based on their service up to date, and on their expected future service, after subtracting expected future normal costs to be paid on their behalf. This is an analog of the prospective definition of accrued liabilities.

General principles of pension plan valuation

167

The ABO method is similar to approaches used in accounting and in the economic literatures, while the PBO method is similar to the typical approach used in life insurance reserving. If the Fundamental Principle of Pension Plan Actuarial Valuation holds, then both the methods, if used at exactly the same time, should produce exactly the same result. Otherwise, there can be dramatic differences between values produced by them. For example, in social insurance (such as the Old Age, Survivors and Disability Insurance, known popularly as Social Security), the ABO method is almost never used. It is common instead to use the PBO method, and often assume an infinite or nearly infinite time horizon. Because social insurance is assumed to pay out exactly the same total amount as they receive in premiums paid, their PBO must equal zero, unless the premium is not calculated correctly. Of course, this effectively means that premiums for the benefits of current workers will be eventually paid by future workers. At the same time the ABO liabilities of the system can be quite substantial. The duty of the actuary, be it in life insurance or in pension plans, is an appropriate design of the system so that liabilities are paid when due. The unfunded liabilities of a pension plan can be quite large, or in the case of social insurance, even larger than just large. The actuary must pay careful attention to the solvency of the entire insurance company, and to economic value as well as liquidity of the assets held. But, surprisingly, the actuarial and insurance literature has largely ignored the question of what is the optimal method of paying off an existing liability. This problem appeared naturally during the process of pension reform in Poland, when the old system of special pension privilege liabilities to the coal miners had to be discharged separately from the new pension system. The first of the authors of this work had then proposed the solution presented in Section 4.3, and had performed the calculations for the system, together with K. Makarski and M. Weretka. The authors of the present book had later presented the theoretical solution in their work [32], and proved that the solution is optimal with respect to stability of the premium flow. Section 4.3 will present that result in detail.

4.3 Optimal payoff of a liability In this section we address the problem of finding an optimal funding method for a given set of liability cash flows. In the context of a pension plan, we can view this as an optimal approach of paying off existing unfunded or any other actuarial liabilities. We will address this problem for deterministic liabilities cash flows. The discount function used to produce present values is described by the force of interest {dt : t $ 0}: Assume that all liabilities cash flows are contained in the time interval ½0; T; and that those cash flows are Ct paid at times t ¼ 1; 2; …; T (some of the cashÐ flows Ct t may be equal to zero). Their present values at time t ¼ 0 are: Lt ¼ Ct expð2 P 0 ds dsÞ; t ¼ 1; 2; …; T: Define the sum of all present values up to time t; as ALt ¼ ti¼1 Li : Let AL0 ¼ 0:

168

Chapter 4

The liabilities will be paid with a stream of premiums Pt ; also indexed by t ¼ 1; 2; …; T (and we also allow some of the Pt values to equal zero). We will additionally assume that the premiums are paid at the beginning of a period corresponding to each one of them so that the premium PÐt is paid at time t 2 1: The present value of Pt at time t ¼ 0 is PVðPt Þ ¼ Pt expð2 t21 The sum of all 0 ds dsÞ:P present values of premiums up to time t will be denoted by APt ¼ ti¼1 PVðPi Þ: Our objective is to find an optimal stream of premium payments. But first we must ask ourselves: what patterns of premium payments are acceptable? One possible solution is to prepay all liabilities completely in advance by a premium P1 paid at time t ¼ 0: Another alternative is to pay level premiums throughout the period ½0; T (or premiums with level present values). Our discussion in Section 4.1 implies that acceptable funding methods should satisfy the following conditions: Constraint 4.3.1 PVðPt Þ is a nonincreasing function of time, i.e., PVðPt Þ $ PVðPtþ1 Þ

for t ¼ 1; …; T 2 1:

This condition means that delaying payments into the future is severely restrained. But a full prepayment, or level premium do meet this condition. Constraint 4.3.2 for t ¼ 1; 2; …; T 2 1;

APt $ ALt

ð4:3:1Þ

and APT ¼ ALT : Using actuarial terminology this condition means that the reserve is never negative (in fact it is equal to at least one premium), and this in practice means that there never is a need to borrow funds to pay the liability cash flows. In Chapters 9 and 10 we call (4.3.1) Axiom of Solvency and investigate its influence on the immunization of terminal surplus APT 2 ALT against the interest rate perturbations. Under the above two constraints, we seek to minimize simultaneously all of the following quantities: APt 2 ALt

for t ¼ 1; 2; …; T:

ð4:3:2Þ

It should be noted that we intend to minimize all of these quantities simultaneously, seeking the minimum among all cash flows of premiums {P1 ; …; PT } satisfying Constraints 4.3.1 and 4.3.2. Because the premium Pt ; is paid at the beginning of the year, and if APt ¼ ALt ; t ¼ 1; 2; …; T; the reserve would be minimal and equal to one premium. The problem of simultaneous minimization of the expressions (4.3.2), under Constraints 4.3.1 and 4.3.2, when applied to pension plans, means that seeking such a method of amortization of unfunded actuarial liability, which assures liquidity

169

General principles of pension plan valuation

(Constraint 4.3.2), and does not allow for payments to be inappropriately postponed (Constraint 4.3.1), at the same time puts the least financial burden of cost on the plan provider. As we will show later, the optimal funding method, which solves this problem also gives the premium flow, which is the most stable in time in terms of premiums present values. Suppose, in contrast to the normal cost, which was discussed in Section 4.2 and will be presented in detail in Chapter 5, that this premium Pt is determined for amortization of unfunded actuarial liability that appeared as a result of a difference between expectations of the growth of assets and liabilities, and the actual growth of them. This is, in fact, an unexpected cost for the provider, creating an additional risk (see also Section 8.4). Stable distribution of the premium paying for this cost lowers the risk to the provider. It should be noted that the inequality in Constraint 4.3.1 cannot be replaced by an equality, as this could result in a contradiction with the Constraint 4.3.2. Let F be a continuous, real-valued, piecewise linear function defined on ½0; T; such that Fð0Þ ¼ 0: The derivative f of this function exists everywhere but at the bentpoints of the graph of F: The function f (where the derivative exists) is piecewise constant, and can be extended to the entire interval ½0; T; by letting it have the value of 0 at 0 and making it continuous from the left-hand side on ½0; T: After performing such an extension, this function f ; which we will term the continuous from the left derivative of the function F; is uniquely defined and for each t [ ½0; T FðtÞ ¼

ðt

f ðsÞds:

0

Now consider the function FðtÞ ¼ ALt ; which is defined at each integer value t [ ½0; T: Extend it to the entire interval ½0; T by making it piecewise linear and continuous, with the smallest possible number of bent-points. Let F p be the smallest concave majorant of F; i.e., the smallest function F p on ½0; T; such that F p is concave and F p ðtÞ $ FðtÞ for t [ ½0; T: Let f p be the continuous from the left derivative of the function F p (note that f p ð0Þ ¼ 0 by definition). Theorem 4.3.1 Premiums Ppt minimize APt 2 ALt for t ¼ 1; 2; …; T among all cash flows of premiums satisfying Constraints 4.3.1 and 4.3.2 if, and only if PVðPpt Þ ¼ f p ðtÞ

for t ¼ 1; …; T;

where f p is the continuous from the left derivative of the smallest concave majorant of the function giving the accumulated through time t present values of liabilities. Proof To prove this contention, consider an arbitrary cash flow of premiums {P1 ; …; PT }; satisfying Constraints 4.3.1 and 4.3.2 and minimizing all APt 2 ALt for t ¼ 1; 2; …; T:

170

Chapter 4

Because PVðPt Þ is a nonincreasing function of t; the piecewise linear function F p ; defined by linear interpolation of the relationship t 7 ! APt for t [ ½0; T; is concave. Based on Constraint 4.3.2 we conclude that F p ðtÞ $ FðtÞ; where FðtÞ ¼ ALt for t ¼ 1; 2; …; T: Since APt 2 ALt is minimized for every t ¼ 1; 2; …; T; F p is the smallest concave majorant. On the other hand, if F p is the smallest concave majorant of the function FðtÞ ¼ ALt ; defined first for all integers t; and then extended to all t [ ½0; T; then F p satisfies the Constraint 4.3.2, because it is a majorant, and the Constraint 4.3.1, because it is concave. Additionally F p minimizes APt 2 ALt for every t ¼ 1; 2; …; T; as it is the smallest concave majorant. A The cash flow of premiums defined above has two additional optimality properties. The present values Qpt ; PVðPpt Þ minimize their ‘square error’ P distance from a premium cash flow with level present values, Q ¼ T 21 Tt¼1 Lt ; assuming Constraint 4.3.2. More precisely, the following theorem holds: Theorem 4.3.2 ðQp1 ; …; QpT Þ minimizes EðQ1 ; …; QT Þ ;

T X

ðQt 2 QÞ2 ;

t¼1

under Constraints 4.3.1 and 4.3.2. Proof Let D be the set of all present values of premium flows satisfying the Constraints 4.3.1 and 4.3.2. It is easy to see that D is a convex set. Furthermore, E attains its minimum on D for a set of present values of premiums ðQp1 ; …; QpT Þ; if for every other set of present values of premiums ðQ1 ; …; QT Þ [ D : EðQ1 ; …; QT Þ ¼

T X

ðQpt 2 QÞ2 þ

t¼1

$

T X

T X

ðQt 2 Qpt Þ2 þ 2

T X

t¼1

ðQpt 2 QÞðQt 2 Qpt Þ

t¼1

ðQpt 2 QÞ2 :

t¼1

The above inequality is satisfied if the following inequality holds: T X

ðQpt 2 QÞðQt 2 Qpt Þ $ 0:

t¼1

Let us note an identity T X t¼1

at bt ¼ aT

T X t¼1

bt 2

T 21 X t¼1

ðatþ1 2 at Þ

t X s¼1

bs ;

ð4:3:3Þ

171

General principles of pension plan valuation

true for any ða1 ; …; aT Þ and ðb1 ; …; bT Þ: Denote at ¼ Qpt 2 Q and bt ¼ Qt 2 Qpt : By using the above identity we obtain T X

ðQpt 2 QÞðQt 2 Qpt Þ

t¼1

¼

ðQpT

2 QÞ

T X

Qt 2

t¼1

¼2

T21 X t¼1

ðQptþ1 2 Qpt Þ

T X

! Qpt

t¼1 t X

2

T21 X

ðQptþ1 2 Qpt Þ

t¼1

Qs 2

s¼1

t X

t X

ðQs 2 Qps Þ

s¼1

! Qps

$ 0;

s¼1

where the last inequality is true because Qpt is a nonincreasing function of t; and Pt p; accumulated s¼1 Qs t ¼ 1; …; T; creates the smallest concave majorant for Pthe t present values of liabilities cash flows, which implies that the sum Q s¼1 s cannot be P less than ts¼1 Qps : A Let us note that the property proved above means also that the premium flow given by ðPp1 ; …; PpT Þ gives the smallest value of max PVðPt Þ;

t¼1;…;T

while satisfying Constraints 4.3.1 and 4.3.2, and even only under Constraint 4.3.2. The optimal premium flow given by the derivative of the smallest concave majorant of the function determined by present values of liabilities cash flows accumulated to time t; depends on the force of interest dt : Could we use stochastic interest rates in this procedure? The Fundamental Theorem of Asset Pricing (see Section 7.2, Ross [62], and Dybvig and Ross [20]) states that in an efficient market, lack of arbitrage is equivalent to the price of a financial instrument being equal to the expected present value (with respect to an appropriate probability measure, the so-called risk neutral measure, see Sections 7.1 and 7.2) of its future cash flows. In practice this means that we can at least estimate the price of nonmarketable instruments, e.g., insurance liabilities, by calculating the average of present values of future cash flows for a large sample of future interest rate scenarios. Seemingly, our method of optimizing premiums cannot be applied to such a situation, as the optimal premium flow is established for each scenario separately, and there is no global solution. However, if we use a premium flow which, on the average, is the smallest concave majorant of the stream of the expectations of liabilities, we will minimize EðAPt Þ 2 EðALt Þ, for all t ¼ 1; 2; …; T; as well as some immunization risk measure Re (see Section 10.5.2 for details).

172

Chapter 4

The connection between existing unfunded actuarial liabilities and the stream Ct , t ¼ 1; 2; …; T; is quite often obvious. If, for example, the plan has no assets, then one way liabilities can be settled is by buying life annuities for each participant upon retirement. If the fund has unfunded liabilities that need to be amortized over time, then the amortization payments can be treated as a stream of liabilities to be settled. If the group of plan participants is diversified enough, these liabilities do become nearly deterministic. Most importantly, the method of smallest concave majorant is optimal in combining the need for liquidity of the pension plan with the postulate of minimal funding cost to the plan provider.

Exercises Exercise 4.4.1 A pension plan provides an annual benefit equal to 2% of final year compensation for each year of service at normal retirement age of 65. A participant has accumulated 25 years of service at normal retirement age and has the final salary of 40 000. What is the annual pension benefit earned by this participant, and what is the single benefit premium for the pension annuity if D65 ¼ 210; and N65 ¼ 2201? Solution The annual benefit earned by this participant is: 0:02 · 40 000 · 25 ¼ 20 000: The benefit premium for the pension annuity is (see (3.10.7)) 20 000 · a€ 65 ¼ 20 000 ·

N65 2201 ¼ 209 619:05: ¼ 20 000 · 210 D65

A

Exercise 4.4.2 At the beginning of the year 2003, accrued liabilities of a pension plan for the employees of The Honorable Life Insurance Company were 100 000. The assets of the plan at the same time were 150 000. On January 1 2003, a contribution of 10 000 is made to the plan. At the same time, 50 000 is used for purchases of life annuities fully discharging the liabilities for retiring participants, although based on the actuarial assumptions the cost of those annuities should have been 60 000. During the year 2003, assets of the plan earn 45%, while the actuarial valuation rate is 8%. The normal cost of the plan for 2003 is established by the plan actuary as 20 000 (as of January 1, 2003). There are no deaths or terminations other than retirement in 2003. Calculate the actuarial gain for this plan for the calendar year 2003. Solution Initial unfunded actuarial liability is UAL2003 ¼ AL2003 2 F2003 ¼ 100 000 2 150 000 ¼ 250 000:

General principles of pension plan valuation

173

Assets at the end of the year are F2004 ¼ F2003 þ C 2 P þ I ¼ 150 000 2 50 000 þ 10 000 þ ð150 000 2 50 000 þ 10 000Þ · 0:45 ¼ 159 500: Accrued liabilities during 2003 increase with the valuation rate, but are decreasing by the discharge of liabilities of 60 000, and increased by the normal cost of 20 000. Therefore, the accrued liabilities as of January 1, 2004 are ð100 000 2 60 000 þ 20 000Þ · 1:08 ¼ 64 800: The unfunded actuarial liabilities on January 1, 2004, are therefore: 64 800 2 159 500 ¼ 294 700: The expected unfunded actuarial liability on January 1, 2004, on the other hand, is ðUAL2003 þ NC2004 Þð1 þ iÞ 2 ðC þ IC Þ ¼ ð250 000 þ 20 000Þ · 1:08 2 10 000 · 1:08 ¼ 240 000 · 1:08 ¼ 243 200: The actuarial gain is Ga ¼ ð243 200Þ 2 ð294 700Þ ¼ 51 500:

A

Exercise 4.4.3 Mr. Romanov, who is 25 years old, is planning for retirement, and in order to provide for it, he estimates that he will need 2 200 000 monetary units at the age of 65, adjusted for inflation, i.e., in terms of today’s purchasing power. He does not have any current savings. His daughter, Anastasia, was just born, and Mr. Romanov also plans to provide for her college education, which will cost 180 000 monetary units when she turns 18, also in terms of today’s purchasing power. Mr. Romanov invests conservatively in securities with rates of return matching the rate of inflation. Calculate the optimal annual savings amount that Mr. Romanov should invest every year at the beginning of the year, to fund his liabilities in an optimal fashion. Solution Effectively, Mr. Romanov will earn 0% real rate of return on his savings, and for all funds, present value equals future value (in purchasing power). We can use the smallest concave majorant funding with the following future liabilities: – 2 200 000 in 40 years, and – 180 000 in 18 years.

174

Chapter 4

The linear function connecting the point ð0; 0Þ and the point ð18; 180 000Þ has the slope 180 000 2 0 ¼ 10 000: 18 2 0 The linear function connecting the point ð18; 180 000Þ and the point ð40; 2 380 000Þ; where 2 380 000 ¼ 180 000 þ 2 200 000; has the slope 2 380 000 2 180 000 2 200 000 ¼ ¼ 100 000: 40 2 18 22 Since the second slope is higher, the smallest concave majorant will require the level funding of the terminal amount of 2 380 000 over 40 years, i.e., 59 500 per year or about 4958.33 per month. A Exercise 4.4.4 Assume now that Mr. Romanov of Exercise 4.4.3 has decided instead to invest in stocks, earning an annual real (i.e., after inflation) rate of return of 7%. How will this change his optimal funding structure? Solution We can use the smallest concave majorant methodology again, but in this case we should work with all liabilities and funding amounts being expressed as present values at 7%. Mr. Romanov’s retirement funding requires 2 200 000 < 146 916:84; 1:0740 in terms of present value at time 0, while Anastasia Romanov’s college education funding requires 180 000 < 53 255:50; 1:0718 in today’s purchasing power. The slope of the line connecting the point ð0; 0Þ and ð18; 53 255:50Þ is 53 255:50 2 0 < 2958:64; 18 2 0 while the slope of the line connecting the point ð18; 53 255:50Þ and the point ð40; 200 172:34Þ; where 200 172:34 ¼ 53 255:50 þ 146 916:84; is 146 916:84 < 6678:04; 22 therefore, level funding of the amount of 200 172.34 over 40 years, i.e., 5004.31 a year, in terms of today’s present value, is called for. This means that the actual optimal funding is

General principles of pension plan valuation

175

† Year 1: 5004.31, † Year 2: 5004:31 · 1:07 < 5354:61; † Year 3: 5004:31 · 1:072 < 5729:43; etc., with Year 40 funding equal to: 5004:31 · 1:0739 < 70 034:40: It should be noted that the total savings at the end of Year 18, when Anastasia heads for college, would be 18 · 5004:31 · 1:0718 < 304 456:03; significantly in excess of the college funding needs, indicating the simultaneous funding of retirement achieved while saving for college. A Exercise 4.4.5 Now assume now that Mr. Romanov of Exercise 4.4.3 is willing to take a very high level of risk and, as a result, will earn substantially higher rate of return. How high will that rate of return have to be so that Mr. Romanov does not save for retirement while saving for the cost of college education of Anastasia Romanov? Solution The situation desired will require that the smallest concave majorant should be linear, so that these three points: 180 000 180 000 2 200 000 ð0; 0Þ; 18; þ ; 40; ; ð1 þ iÞ18 ð1 þ iÞ18 ð1 þ iÞ40 lie on a straight line. This implies that 180 000 180 000 2 200 000 180 000 20 þ 2 18 18 40 ð1 þ iÞ ð1 þ iÞ ð1 þ iÞ ð1 þ iÞ18 ; ¼ 40 2 18 18 2 0 i.e., 10 000 100 000 ¼ ; 18 ð1 þ iÞ ð1 þ iÞ40 and ð1 þ iÞ22 ¼ 10; i < 11:03%:

A

Exercise 4.4.6 Mr. Louis Bourbon is a king of a Western European country plagued by high levels of poverty among its elderly population. Mr. Bourbon proposes to introduce a flat benefit, paid in gold, to the elderly aged 65 and above, paid commencing on the 65th birthday of each citizen, until death of the citizen. The benefit will be funded by general revenue budget, i.e., by all taxes levied by the kingdom, not by special taxes allocated to this program. Because of recent expenditures of support of a war of independence for major competing power’s overseas colonies, Mr. Bourbon is concerned about the cost of the program and would

176

Chapter 4

like to minimize the expenditures, while not being forced to borrow to finance the program. All payments will be made in gold, which earns 0% real rate of return, i.e., it exactly retains its purchasing power. Mr. Bourbon estimates that the total benefit payment cost will be 100 000 gold coins the first year, 200 000 gold coins the second year, 300 000 gold coins the third year, then 500 000 gold coins for seven years, and after that 400 000 every year forever. Calculate the optimal funding for this retirement scheme designed by Mr. Louis Bourbon. Solution Since all payments are in gold, and gold is assumed to neither appreciate nor depreciate in purchasing power, we can take all amounts as present values. The optimal funding is determined by the smallest concave majorant. The slope of the line connecting the two initial points: ð0; 0Þ and ð1; 100 000Þ is 100 000, while the slope of the line connecting ð0; 0Þ and ð2; 300 000Þ is 150 000, thus Year 2 payment will need to be pre-funded. The slope of the line connecting ð0; 0Þ and ð3; 600 000Þ is 200 000, so Year 3 payment will also need to be pre-funded. The slope of the line connecting the points ð0; 0Þ and ð4; 1 100 000Þ is 275 000, and as this is greater than 200 000, indicating further pre-funding need. Since the funding needs following Year 4 are 500 000 every year, they are always bigger than the funding needs in the first 3 years, and we should find the slope of the line connecting point ð0; 0Þ and ð10; 4 100 000Þ; which is 410 000, and this indicates the optimal funding level for the first 10 years. The optimal funding for the years beyond the 10th year is the constant 400 000 liability payment the same year. A Exercise 4.4.7 November 2002 SOA Course 5 Examination, Problem No. 10, Section C. John is 30 years old and has been working 5 years for Company A, which did not have a pension plan. His company is going to adopt a pension plan. He is now also considering a new position with two other companies. Company A provides a salary with 2% annual increase. The pension benefit is 2% of the employee average salary during the last 5 years of service times the number of years of service. They will recognize in the pension formula the 5 years previously worked. There is no penalty if an employee retires before age 60. Company B offers a starting salary 20% higher than Company A with 2% annual increase. The pension benefit is 2% of the employee average salary during the last 3 years of service times the number of years of service. They offer no recognition of past work. There is no penalty if an employee retires before age 60. Company C offers a starting salary 45% higher than Company A with 2% annual increase. The pension benefit is 2% of the employee final salary times the number of years of service. They offer no recognition of past work. Furthermore, there is a 2% penalty per year if an employee retires before age 60. Compare each offer based on the pension benefit John would receive if he retires at age 55.

General principles of pension plan valuation

177

Solution Let us assume for simplicity that the current salary of John in Company A is 100 000 annually. We will compare the annual benefit rates for the three choices. Choice 1: Company A. Since John started with Company A at age 25, upon retirement at age 55, John will have 30 years of service. All of the service will be recognized in the pension formula, and there will be no reduction due to retirement before age 60. John’s salary in the last 5 years of work, counting backwards, will be 100 100 100 100 100

000 · 1.0224 < 160 000 · 1.0223 < 157 000 · 1.0223 < 154 000 · 1.0222 < 151 000 · 1.0221 < 148

843.72, 689.93, 597.97, 566.63, 594.74.

The average of these quantities is 154 658.60. The annual benefit rate will be 30 (years of service) times 2% (per service year) of 154 658.60, i.e., approximately 90 795.16. Choice 2: Company B. In this company, John’s starting salary will be 120 000. During the last 3 years of service, John’s salary will be: 120 000 · 1.0224 < 193 012.47, 100 000 · 1.0223 < 189 227.91, 100 000 · 1.0223 < 185 517.56. The average of these quantities is 189 252.65. The annual benefit rate will be 25 (years of service) times 2% (per service year) of 189 252.65, i.e., approximately 94 626.32. Choice 3: Company C. In this company, John’s starting salary will be 145 000. His final annual salary will be 145 000 · 1:0224 < 233 223:40: The benefit upon retirement at age 55 will be 2% times 25 (years of service) times 233 223.40 times 0.90 (a total of 10% benefit reduction for early retirement), i.e., 104 950.53. As we see, benefit is the highest with the Company C choice, second highest with Company B, and the lowest with Company A. A Exercise 4.4.8 November 2001 SOA Course 5 Examination, Problem No. 4. You are given the following retirement plan information for an individual. Current age is 40, entry age was 30, and retirement age is 65. The social insurance (such as Social Security in the United States) annual benefit is 11 700. Current salary is 80 000. Annual salary growth is 3%. Personal savings accumulation rate is 7%. Final salary (at age 64) will be 162 624. The annuity conversion factor at age 65 is 8.1958. Retirement benefit is 1% of final 5-year average salary times the number of years of service.

178

Chapter 4

This individual begins saving for retirement at his current age. Calculate the level percentage of salary that should be allocated to personal savings each year to provide this individual with a 70% replacement ratio at retirement. Solution The 70% replacement ratio at retirement means that the initial total retirement benefit must be equal to 70% of the final salary, i.e., 0:70 · 162 624 ¼ 113 836:80: This person will receive retirement benefits from three sources: social insurance (this is known to be 11 700), personal savings and the employer-sponsored plan. Therefore, personal savings, and the pension plan must provide together the annual benefit of 113 836:80 2 11 700 ¼ 102 136:80 Let us now calculate the pension benefit that this person will receive. Because the salary is assumed to increase 3% annually, the final average salary (calculated over the last 5 years of work) is 162 624 þ

162 624 162 624 162 624 162 624 þ þ þ 1:03 1:032 1:033 1:034 < 153 422:68: 5

The benefit amount will be 1% times this final average salary times the number of years of service (equal to 65 minus 30 for this person), i.e., 0:01 · 153 422:68 · 35 < 53 697:94: Thus the annual benefit provided by the personal savings must be 102 136:80 2 53 697:94 ¼ 48 438:86: Since the annuity conversion factor at age 65 is 8.1958, the amount needed to fund such a lifetime annuity with personal savings is 48 438:86 · 8:1958 ¼ 396 995:21: This amount is to be accumulated by contributions amounting to a level percentage of salary, where we know that savings earn 7%, while salary increases 3% every year. Current salary is 80 000, and contributions will be made over 25 years. If we denote by x the fraction of salary that must be contributed to this personal savings plan, then we must have 396 995:21 ¼ x · 80 000 · ð1:0725 þ 1:0724 · 1:03 þ · · · þ 1:0324 · 1:07Þ 1:03 25 12 1:07 ¼ x · 80 000 · 1:0725 · : 1:03 12 1:07 This solves to x < 0:0556: Therefore, 5.56% of salary must be saved every year in order to achieve 70% replacement ratio upon retirement. A

179

Chapter 5 Valuation of pension plan liabilities

5.1 Unit credit method We will for now assume that retirement benefit rights are earned continuously during participant’s years of service. At the time of hire, at assumed age w; participant’s benefit earned is 0; at retirement age y; benefit earned is BðyÞ; and at any age x between w and y; it equals an intermediate value BðxÞ; where 0 , BðxÞ , BðyÞ: The quantity BðxÞ will be termed the accrued benefit at age x: If BðyÞ equals the actual annual salary, the method of valuation will be called Traditional Unit Credit, but if the benefit rate is based on salary projected to the retirement age – we will call it Projected Unit Credit. The present value of accrued benefit for the jth participant at age x is D yj Bj ðxj Þ€að12Þ ; yj D xj where Dx ¼ lx vx ; lx denotes the cohort size at age x; surviving from the original newborn cohort size l0 ; v is the discount factor, i.e., v ¼ 1=ð1 þ iÞ; i the valuation rate (assumed long-term rate of return on plan assets). Note that the ratio Dyj Dxj

¼

l yj l xj

ð1 þ iÞxj 2yj ¼ ð12yj 2xj qxj Þð1 þ iÞxj 2yj

is calculated based on the tables of pension plan participation (analogue of ordinary mortality table for survival as a pension plan participants group), with the understanding that the current value at time t (corresponding to age xj of the jth plan participant) of a monetary unit, which will be paid at age yj ; is equal to ð1 þ iÞxj 2yj : Denote by At the group of all plan participants at time t; but only including active, not retired participants. If the plan assets at time t are X j[At

Bj ðxj Þ€að12Þ yj

D yj D xj

;

180

Chapter 5

then they are sufficient, independently of the age distribution among participants, to pay the cost of purchasing life annuities for all participants at retirement age y: In what follows we will assume that a participant attaining retirement age y purchases a life annuity with annual benefit BðyÞ and thus is no longer included in the balance sheet of assets and liabilities of the plan. In practice, of course, the participant may not receive the benefit in the form of such a benefit buyout, but instead receive a life annuity from the plan. We will, however, treat such a situation as essentially equivalent to a buyout, with such a participant considered to be a part of a separate plan, devoted only to payment of benefits. 5.1.1 Accrued liabilities The unit credit valuation method treats accrued liabilities of the plan as discounted accrued benefits of all active plan participants. Therefore, accrued liabilities of the plan at time t; denoted by ALt ; are equal to ALt ¼

X

Bj ðxj Þ€að12Þ yj

j[At

D yj D xj

ð5:1:1Þ

:

An ideally balanced plan has assets equal to ALt at every time t: The accrued liabilities change over time, because the group of active plan participants changes, and the participants who remain in the plan earn additional time of service. We will assume, for simplicity, that no new participants join the plan (we can assign them to a new, separate plan). This way, the group of active participants can only decline in size over time. Let T be the set of all participants who leave the plan in the time period between t and t þ 1; without attaining any benefit rights. Let R be the set of plan participants who retire during the same time period and start receiving benefits at age y: Then Atþ1 ¼ At w ðT < RÞ: We will now consider the change in accrued liabilities during the same year. We have, by definition ALtþ1 ¼

¼

X

Bj ðxj þ 1Þ€að12Þ yj

Dyj

j[Atþ1

Dxj þ1

X

Dyj

j[At

Bj ðxj þ 1Þ€að12Þ yj

Dxj þ1

2

X j[T
Bj ðxj þ 1Þ€að12Þ yj

D yj Dxj þ1

:

Since 1þi qx ð1 þ iÞ qx l qx 1 þ ¼ þ ¼ xþ1 þ ¼ ; Dx l x vx Dxþ1 Dxþ1 Dxþ1 lx Dxþ1 Dxþ1

181

Valuation of pension plan liabilities

then ALtþ1 ¼

"

X

j

B ðxj þ

1Þ€að12Þ yj

j[At

X

2

D yj D xj

Bj ðxj þ 1Þ€að12Þ yj

j[T
¼

X

ðBj ðxj Þ þ DBj Þ€að12Þ yj

j[At

X

2

Bj ðxj þ 1Þ€að12Þ yj

j[T
ð1 þ iÞ þ qxj

D yj

#

Dxj þ1

Dyj Dxj þ1 D yj D xj

ð1 þ iÞ þ

X j[At

Dyj Dxj þ1

qxj Bj ðxj þ 1Þ€að12Þ yj

Dyj Dxj þ1

;

where DBj denotes the increase of accrued benefit earned by the jth plan participant in the time period from time t till time t þ 1: Thus ! X j ð12Þ Dyj DB a€ yj ALtþ1 ¼ ALt þ ð1 þ iÞ D xj j[At ! X j X D D y y j j 2 B ðxj þ 1Þ€að12Þ 2 qxj Bj ðxj þ 1Þ€að12Þ yj yj Dxj þ1 Dxj þ1 j[T j[A t

2

X

Bj ðxj þ 1Þ€að12Þ yj

j[R

D yj Dxj þ1

ð5:1:2Þ

:

5.1.2 Normal cost The expression in the second large set of parentheses on the right-hand side of (5.1.2) is equal to zero, if actuary’s assumptions concerning decrease in accrued liabilities because of departure of some plan participants from the plan before attaining retirement benefit rights are verified by actual experience. If the actuarial assumptions about the investment earnings and mortality turned out to be correct, and if accrued liabilities are initially, at time t; equal to assets, then the assets in the following year will be X D yj ALt ð1 þ iÞ 2 Bj ðxj þ 1Þ€að12Þ ; yj Dxj þ1 j[R where the second expression is equal to the cost of purchasing life annuities for all participants who retired during the year. As a result of the above, the amount X j ð12Þ Dyj NCt ¼ DB a€ yj ð5:1:3Þ D xj j[A t

should be paid as a premium to cover the increase in accrued liabilities to the level ALtþ1 : This amount NCt is termed the normal cost. The word ‘normal’ in this context refers to the fact that this is the cost to be paid under normal circumstances, when

182

Chapter 5

actuarial assumptions concur with actual development of interest, mortality and expenses. The normal cost for the jth plan participant, NCjt ; is equal to the increase of accrued liabilities to this participant multiplied by the probability of survival within the group of plan participants, i.e., Dyj : ð5:1:4Þ NCjt ¼ DBj a€ ð12Þ yj Dxj In total, we have NCt ¼

X

NCjt :

j[At

In practice, actuarial assumptions never concur perfectly with the actual experience, and plan assets do not always exceed plan liabilities. Thus the actual cost paid to the plan may differ from the normal cost by the amount of an adjustment necessary to account for the divergence between experience and assumptions, as well as assets and liabilities. 5.1.3 Unfunded actuarial liability Let us now consider further the relationship between the assets and the liabilities of the plan. Let Ft be the assets of the plan at time t: The difference ALt 2 Ft ; UALt is called the unfunded actuarial liability (or actuarial deficit). The assets will grow in the time interval from t to t þ 1 by an amount denoted by I; plan participant will make a contribution of C; and the assets will decline by an amount P spent on purchases of annuities for participants from the group R: Therefore ð5:1:5Þ Ftþ1 ¼ Ft þ I þ C 2 P: Subtract the formula (5.1.5) from (5.1.2), and this way we derive the formula for the unfunded actuarial liabilities in the year t þ 1; denoted by UALtþ1: UALtþ1 ¼ ðALt þ NCt Þð1 þ iÞ ! X j X Dyj D yj ð12Þ j ð12Þ 2 B ðxj þ 1Þ€ayj 2 qxj B ðxj þ 1Þ€ayj Dxj þ1 Dxj þ1 j[T j[A t

2

X

Bj ðxj þ 1Þ€að12Þ yj

j[R

D yj Dxj þ1

2 ðFt þ I þ C 2 PÞ

¼ UALt ð1 þ iÞ 2 ðI 2 iFt Þ þ ½NCt ð1 þ iÞ 2 C 2

X

j

B ðxj þ

1Þ€að12Þ yj

j[T

2

X j[R

Bj ðxj þ 1Þ€að12Þ yj

Dyj Dxj þ1 Dyj Dxj þ1

2

X

j

qxj B ðxj þ

j[At

!

2P :

1Þ€að12Þ yj

D yj Dxj þ1

!

183

Valuation of pension plan liabilities

Denote by IC ; the return from the investment of the contribution C (we assume that the rate of return is i) earned from the date of the contribution to the end of the year. For example, if the contribution was made at the beginning of the year, then IC ¼ iC: Let IP be the analog for the amount P; i.e., investment return lost due to the payment of the amount P for the cost of annuities for the group R: This cost IP is, of course, calculated from the moment of the purchase of annuities till the end of the year. We then have UALtþ1 ¼ UALt ð1 þ iÞ 2 ðI 2 iFt 2 IC þ IP Þ 2 ½C þ IC 2 NCt ð1 þ iÞ ! X j X D D y y j j 2 B ðxj þ 1Þ€að12Þ 2 qxj Bj ðxj þ 1Þ€að12Þ yj yj Dxj þ1 Dxj þ1 T A t

2

X R

j

B ðxj þ

1Þ€að12Þ yj

D yj Dxj þ1

! 2 P 2 IP :

ð5:1:6Þ

If the actual rate of return during the year was equal to the assumed rate i; then the second expression on the right-hand side of (5.1.6) equals 0. Moreover, if the accrued liabilities, which had been cancelled during the year due to departure of participants from group T; who leave without having earned pension rights, are exactly equal to the amount given by the actuarial assumptions, then the fourth expression is also equal to 0. Similarly, the last (fifth) expression is equal to 0, if the amount spent on purchases of annuities for retired participants is exactly as in the actuarial assumptions. 5.1.4 Actuarial gain The actuarial gain, denoted by Ga, is the following quantity: Ga ¼ ðUALt þ NCt Þð1 þ iÞ 2 C 2 IC 2 UALtþ1 :

ð5:1:7Þ

By comparing (5.1.6) and (5.1.7), we can note that the actuarial gain Ga is equal to the sum of the second, fourth, and fifth expressions on the right-hand side of the formula (5.1.6). This sum represents the change in unfunded actuarial liabilities caused by the divergence between the actuarial assumptions and the actual plan experience. If the actuarial gain is negative, then its absolute value is called the actuarial loss. It follows from the expression (5.1.6) that the most important method of reducing unfunded actuarial liabilities is making contributions that exceed the normal cost. When such contributions are indeed made, we say that unfunded actuarial liabilities are being amortized. The unit credit method of actuarial funding presented in this section has one serious disadvantage: its normal costs may grow much faster than salaries. Let us illustrate this with an example.

184

Chapter 5

Example 5.1.1 (see e.g. [4]) Let us analyze the case when the pension amount of a plan participant is determined by that participant’s wages. Assume that the pension benefit amount is set as 1% of annual compensation for each year of service (the level of compensation used for calculation is that of the year of service). Assume also that salaries grow at an annual effective rate r: The normal cost associated with the jth participant, who, at time t; is aged x (see formula (5.1.4)), changes with respect to x at the following rate 1 d d d 1 j j NCt ¼ ðln DB ðxÞÞ þ ln dx dx Dx NCjt dx d d 1 1 ðln DBj ðxÞÞ þ ln þ ln x ¼ dx dx lx v d d ðln DBj ðxÞÞ þ ½2lnðl0 PrðT . xÞÞ þ lnð1 þ iÞ; ¼ dx dx where l0 is the initial size of the plan group and T the random future lifetime, understood as the lifetime in the plan, not physical life. If T is absolutely continuous with density g, then d fT ðxÞ ð2ln PrðT . xÞÞ ¼ ¼ mx ; dx PrðT . xÞ where mx is the force of mortality, also termed the hazard rate of the random variable T (see Section 3.7). Therefore d d ln NCjt ¼ ðln DBj ðxÞÞ þ mx þ d; dx dx

ð5:1:8Þ

where d ¼ lnð1 þ iÞ is the (assumed constant) nominal force of interest (also the actuarial valuation rate), which corresponds to the effective annual interest rate i (see Section 2.4, formula (2.4.9)). Let us denote by Sjt the salary of the jth plan participant in year t. Then Bj ðx þ 1Þ ¼ Bj ðxÞ þ 0:01ð1 þ rÞSjt and DBj ðxÞ ¼ 0:01ð1 þ rÞSjt : Since the jth plan participant in year t is assumed to be aged x, then Sj 2 Sjt d d ln DBj ðxÞ ¼ ln Sjt ¼ lim tþh j : hd0 dx dx hSt

ð5:1:9Þ

Let us assume that the rate of growth of salaries is constant, with the continuously compounded rate of growth dr equivalent to the effective annual rate of growth r.

185

Valuation of pension plan liabilities

Utilizing (2.4.8), we obtain d edr h Sjt 2 Sjt ln DBj ðxÞ ¼ lim ¼ dr : hd0 dx hSjt

ð5:1:10Þ

Substituting in (5.1.8) we arrive at d ln NCjt ¼ dr þ mx þ d: dx

ð5:1:11Þ

It follows from (5.1.9) – (5.1.11) that the rate of change with respect to time of the ratio (Normal Cost)/(Salary) is given by this simple formula ! d NCjt ln ¼ mx þ d: dx Sjt If we assume now that mortality is ruled by the Gompertz Law (see Section 3.7, formula (3.7.23)), i.e.,

mx ¼ a ebx ; where a and b are positive parameters, then NCjt =Sjt changes with respect to x at a rate faster than the exponential (more precisely, the rate of change is equal to an exponential function multiplied by a doubly exponential function), while salaries are growing at an exponential rate. A

5.2 Entry Age Normal method of funding In Example 5.1.1 given in Section 5.1.4, we showed that individual normal cost may grow faster than salaries, if pension benefit is a constant percentage of salaries. Although a new plan entrant typically has a lower salary than other participants, and thus lowers the global normal cost, subsequently the global cost of the plan will grow at a faster rate than salaries. This is generally an undesirable development. A possible remedy lies in the use of the Entry Age Normal method, which we will present in this section. We will now assume that all persons joining the pension plan enter at the same age w. We will first define the normal cost for each participant individually, and subsequently define the normal cost and accrued liabilities for the whole plan. Note that this order of operations is exactly the opposite of the order used for unit credit method, where accrued liabilities are defined first (as the present value of benefits earned) and the normal cost follows. Thanks to this change in order, Entry Age Normal method tends to keep the funding cost for an individual participant at a constant level.

186

Chapter 5

5.2.1 Normal cost Consider the simplest case, when the benefit is set as a dollar amount, not as a function of salary. The annual normal cost NCj for the jth participant is defined as a level annual premium equal to the amount that the present value at the time the participant entered the plan (when the participant’s age was w) of all future normal costs is equal to the present value at the same time (age w) of all future benefits. Utilizing the results of Section 3.10 (formula (3.10.36)), we get the formula for the present value of all future normal costs NCj a€ w: yw ¼ NCj

Nw 2 Ny ; Dw

ð5:2:1Þ

where a€ w: yw is the actuarial present value of an annuity of unit payments from age w to age y, or till the death (departure from the plan) of the participant, depending on which event happens first. As the present value at age w of all future benefits is ð12Þ Bj ðyÞ€ay Dy =Dw (see Section 5.1), the formula for the annual normal cost is Bj ðyÞ€að12Þ y

Dy Nw 2 Ny ¼ NCj : Dw Dw

ð5:2:2Þ

From (5.2.1) we conclude that the present value of the contribution paid on behalf of the jth participant fully pays for the benefits (actuarially, i.e., averaged over all participants, with consideration for interest and mortality). From (5.2.2) we obtain the normal cost paying for the jth participants NCj ¼ Bj ðyÞ€að12Þ y

Dy Nw 2 Ny

ð5:2:3Þ

and for the total normal cost for the plan in the year t NCt ¼

X j[At

NCj ¼

X j[At

Bj ðyÞ€að12Þ y

Dy : Nw 2 Ny

ð5:2:4Þ

The factor Dy =ðNw 2 Ny Þ is calculated based on the tables describing membership in the plan, not a standard mortality table. Note that the normal cost (5.2.3) for each participant is constant over time, if the projected benefit Bj ðyÞ does not change. In practice, as benefits are in some way adjusted over time, this adjusted value can be considered in subsequent normal cost calculations. 5.2.2 Accrued liabilities Let us now examine the concept of accrued liabilities under the Entry Age Normal method of pension funding. We will define that the accrued liabilities ALjt with respect to the jth participant, who at time t is aged x, is equal to the accumulated value of all normal costs paid for this jth participant up to date. Let us notice that the present

187

Valuation of pension plan liabilities

value of all future normal costs, which must be paid for a participant aged w (the entry age), is (compare with (5.2.1)) NC j

Nw 2 Ny ; Dw

and the present value of all future normal costs, which must be paid for a participant aged x, is NC j

Nx 2 Ny : Dx

Since we want to consider these quantities at time t, when the participant is aged x, the first of the two must be multiplied by the accumulation factor Dw =Dx ¼ vw2x ðlw =lx Þ; and then the second quantity must be subtracted from the result so obtained; this way we calculate prospectively the current value of all normal costs paid up to date on behalf of the jth participant. Thus, using (5.2.3), we obtain ALjt ¼ NCj

Nw 2 Ny Dw Nx 2 Ny N 2 Nx 2 NCj ¼ NCj w Dw Dx Dx Dx

¼ Bj ðyÞ€að12Þ y

Nw 2 Nx Dy Nw 2 Ny Dx

ð5:2:5Þ

The accrued liabilities of the entire plan, ALt, are calculated as the sum of all ALjt for j [ At ; i.e., ALt ¼

X

Bj ðyÞ€að12Þ y

j[At

Nw 2 Nx Dy : Nw 2 Ny Dx

This is the retrospective formula for accrued liabilities (calculated using normal costs paid up to date). Using the first equality in the formula (5.2.5) and also utilizing (5.2.3), we obtain the prospective formula for accrued liabilities ALt ¼

X j[At

Bj ðyÞ€að12Þ y

X Dy Nx 2 Ny 2 NCj : Dx Dx j[A t

By the prospective formula, accrued liabilities are equal to the actuarial present value of all future benefits less the actuarial present value of all future normal costs. Of course the prospective and the retrospective formulas are equivalent. Recall that Atþ1 ¼ At w ðT < RÞ; where T denotes the group of all participants who left the group without attaining pension rights and R is the group of all participants who retired and left the group with retirement rights. Note that we assume that retirement rights are settled with annuities purchase upon departure, even if the

188

Chapter 5

departure precedes retirement, between times t and t þ 1: Thus X X ALjtþ1 2 ALjtþ1 ; ALtþ1 ¼

ð5:2:6Þ

j[T
j[At

where from (5.2.5) we infer that ALjtþ1 ¼ Bjtþ1 ðyÞ€að12Þ y

Nw 2 Nxþ1 Dy : Nw 2 Ny Dxþ1

ð5:2:7Þ

In the above formula, the index of the year t þ 1; put by the value Bj ðyÞ; indicates that the projected benefit at retirement may change over time. Denote by DBj the change, between times t and t þ 1; in the projected benefit Bj ðyÞ ði:e:; DBj ¼ Bjtþ1 ðyÞ 2 Bjt ðyÞÞ. It should be noted that in Section 5.1 the concept of the increase in benefits DBj had a significantly different meaning. Let us also note that Nx ¼ Dx þ Dxþ1 þ Dxþ2 þ · · · (see Section 3.8, formulas (3.8.3) and (3.8.4)), thus Nw 2 Nx N 2 Nx þ Dx N 2N þ 1 ð1 þ iÞ ¼ w ð1 þ iÞ ¼ w x xþ1 ð1 þ iÞ Dx Dx lx v Nw 2 Nxþ1 lxþ1 Nw 2 Nxþ1 ¼ ¼ ð1 2 qx Þ: Dxþ1 lx Dxþ1 Therefore

Nw 2 Nx N 2 Nxþ1 N 2 Nxþ1 þ 1 ð1 þ iÞ þ qx w ¼ w : Dx Dxþ1 Dxþ1

ð5:2:8Þ

Using the formula for DBj ; (5.2.7) and (5.2.8), we obtain Nw 2 Nxþ1 Dy Nw 2 Nxþ1 Dy þ DBj a€ ð12Þ y Nw 2 Ny Dxþ1 Nw 2 Ny Dxþ1 Dy Nw 2 Nx Nw 2 Nxþ1 j ð12Þ ¼ Bt ðyÞ€ay þ 1 ð1 þ iÞ þ qx Nw 2 Ny Dx Dxþ1

ALjtþ1 ¼ Btj ðyÞ€að12Þ y

þ DBj a€ ð12Þ y

Nw 2 Nxþ1 Dy : Nw 2 Ny Dxþ1

From the above, considering (5.2.5) and (5.2.3), we infer ALjtþ1 ¼ ðALjt þ NCjt Þð1 þ iÞ þ qx Btj ðyÞ€að12Þ y þ DBj a€ ð12Þ y

Nw 2 Nxþ1 Dy ; Nw 2 Ny Dxþ1

Nw 2 Nxþ1 Dy Nw 2 Ny Dxþ1 ð5:2:9Þ

189

Valuation of pension plan liabilities

where to indicate that NCj is calculated based on Bjt ; we used index t. Substituting (5.2.9) into (5.2.6), we obtain ALtþ1 ¼ ðALt þ NCt Þð1 þ iÞ þ

X

qx Btj ðyÞ€að12Þ y

j[At

X

þ

DB j a€ ð12Þ y

j[At

2

X

Nw 2 Nxþ1 Dy Nw 2 Ny Dxþ1

Nw 2 Nxþ1 Dy Nw 2 Ny Dxþ1

ðBtj ðyÞ þ DB j Þ€að12Þ y

j[T
Nw 2 Nxþ1 Dy ; Nw 2 Ny Dxþ1

ð5:2:10Þ

where in the last sum we utilized the formula (5.2.7). Let us note that the set T < R consists of all people who leave the plan between the times t and t þ 1: For these persons there is no need for any corrections of the projected benefit, and so we can assume that DBj ¼ 0 for j [ T < R: Then the equation (5.2.10) can be rewritten in this form X

ALtþ1 ¼ ðALt þ NCt Þð1 þ iÞ þ þ

X

DBj a€ ð12Þ y

j[Atþ1

X

fj 2 qx AL tþ1

Nw 2 Nxþ1 Dy Nw 2 Ny Dxþ1

fj ; AL tþ1

ð5:2:11Þ

j[T
j[At

f j denotes accrued liabilities with respect to the jth participant at time where AL tþ1 j t þ 1; calculated under the assumption that Btþ1 ¼ Btj : 5.2.3 Unfunded actuarial liabilities Unfunded accrued liability UALt at time t is defined as ALt 2 Ft ; where Ft is the value of assets at time t. Next year Ftþ1 ¼ Ft þ I þ C 2 P;

ð5:2:12Þ

where I is the total investment earnings from the assets in the time period from t till t þ 1; C the contribution paid during the year, and P the cost of the purchases of annuities for the participants in the group R. After subtracting (5.2.12) from (5.2.11) we obtain the formula for the unfunded actuarial liabilities in the year t þ 1: UALtþ1 ; ALtþ1 2 Ftþ1 ¼ ðALt þ NCt Þð1 þ iÞ 2 Ft 2 I 2 C X X Nw 2 Nxþ1 Dy fj DBj a€ ð12Þ þ qx AL þ y tþ1 N 2 N D w y xþ1 j[Atþ1 j[At X X fj 2 f j þ P: AL AL 2 tþ1 tþ1 j[T

j[R

ð5:2:13Þ

190

Chapter 5

Considering the return IC earned on the contribution C before the earned of the year t, and the return IP lost on the funds paid for the cost of purchase of retirement annuities, (5.2.13) can be rewritten into a recursive formula UALtþ1 ¼ UALt ð1 þ iÞ 2 ðI 2 iFt 2 IC þ IP Þ 2 ½C þ IC 2 NCt ð1 þ iÞ þ

X

DBj a€ ð12Þ y

j[Atþ1

2

X

X X Dy Nw 2 Nxþ1 fj 2 fj AL 2 qx AL tþ1 tþ1 Dxþ1 Nw 2 Ny j[T j[At !

f j 2 P 2 IP : AL tþ1

!

ð5:2:14Þ

j[R

5.2.4 Actuarial gain If the actuarial assumptions concur with the actual experience, then the right-hand side of (5.2.14) contains the following expressions which equal zero: the second one (showing returns on assets and contributions), the fourth one (concerning expected benefit), the fifth one (showing dissolution of reserves as a result of participants from group T leaving the plan without benefit rights), and the sixth one (concerning the cost of purchasing annuities for participants leaving with benefit rights). In a manner analogous to that presented in Section 5.1, the sum of these expressions under realistic experience (when they are not necessarily zero) is termed the actuarial gain (its absolute value is termed actuarial loss, if the sums turns out to be negative) and denoted by Ga. From (5.2.14) we obtain UALtþ1 ¼ ðUALt þ NCt Þð1 þ iÞ 2 ðC þ IC Þ 2 Ga:

ð5:2:15Þ

From the above we obtain the formula for the actuarial gain, as follows Ga ¼ ðUALt þ NCt Þð1 þ iÞ 2 ðC þ IC Þ 2 UALtþ1 :

ð5:2:16Þ

The expression ðUALt þ NCt Þð1 þ iÞ 2 ðC þ IC Þ is the expected unfunded actuarial liability. The quantity UALtþ1 is the actual unfunded actuarial liability in the year t þ 1: Thus the formula (5.2.16) gives an alternative definition of the actuarial gain—as the difference between the expected and actual unfunded actuarial liabilities in the year t þ 1 (see Section 4.2). Note that the formulas (5.2.16) and (5.1.7) are formally identical, but the expressions appearing in them are derived differently, and in practice they may produce different results.

191

Valuation of pension plan liabilities

5.3 Individual level premium method of funding The Individual Level Premium (ILP) method of funding resembles somewhat the Entry Age Normal method, but in the individual level premium method the ages of various plan participants may be assumed to be different. 5.3.1 Normal cost Consider the jth participant; denote by w the age of this participant upon entry into the plan. Let t be the number of years of service of this plan participant, and NCtj be the normal cost in the year in which the participant has t years of service, assumed to be year t. Clearly t ¼ x 2 w; where x is the age of the participant at time t, thus the notation NCtj is an analogue of the notation NCtj used before (although, for this method, it is more convenient to use the index t). As in the case of the Entry Age Normal method, we assume that the actuarial present value at age w of all future benefits is equal to the actuarial present value of all future normal costs, i.e., NC0j a€ w: yw ¼ B0j ðyÞ€að12Þ y

Dy : Dw

Because a€ w: yw ¼ ðNw 2 Ny Þ=Dw (see formula (3.10.36)), we infer from the above that NC0j ¼ B0j ðyÞ€að12Þ y

Dy : Nw 2 Ny

Therefore, the normal cost, if calculated at the beginning of service, is exactly the same as in the Entry Age Normal method (see (5.2.3)). In the following year ðt ¼ 1Þ the normal cost for the jth participant is modified by DNC1j in such a way that the current value at age w þ 1 of all future corrections of normal costs is made equal to the present value at age w þ 1 of all future benefit adjustments. Thus the ‘additional’ normal cost assures the financing of the change in the projected benefit. If the projected benefit in the year t ¼ 1 is denoted by B1j ðyÞ and we define DB1j ¼ B1j ðyÞ 2 B0j ðyÞ; then DNC1j

Nwþ1 2 Ny Dy ¼ DB1j a€ ð12Þ : y Dwþ1 Dwþ1

Therefore DNC1j ¼ DB1j a€ ð12Þ y

Dy Nwþ1 2 Ny

and NC1j ¼ NC0j þ DB1j a€ ð12Þ y

Dy : Nwþ1 2 Ny

ð5:3:1Þ

192

Chapter 5

If the projected benefit does not change in the years that follow, then the normal cost NC0j in the year t ¼ 0 and NC1j in the consecutive years will finance, actuarially, the payment of benefits B1j ðyÞ till the end of life of the plan participant (by purchasing a life annuity with this annual benefit). Let us, however, recall that the projected benefit in the year t ¼ 2 is B2j ðyÞ: Then the normal cost NC1j must be corrected by the amount DNC2j ; defined by this formula, analogous to (5.3.1)

DNC2j

Nwþ2 2 Ny Dy ¼ DB2j a€ ð12Þ ; y Dwþ2 Dwþ2

where DB2j ¼ B2j 2 B1j : Thus DNC2j ¼ DB2j a€ ð12Þ y

Dy Nwþ2 2 Ny

ð5:3:2Þ

and generally, for every t ¼ 1; …; y 2 w 2 1;

DNCtj ¼ DBtj a€ ð12Þ y

Dy : Nwþt 2 Ny

We derive the following similarly: NCtj ¼ NC0j þ DNC1j þ DNC2j þ · · · þ DNCtj :

5.3.2 Accrued liabilities The accrued liabilities with respect to the jth plan participant at the moment of this participant’s joining the plan, AL0j ; are equal to zero. In the year t ¼ 1 the accrued liabilities are equal, by definition, to the actuarial present value of all future benefits minus the actuarial present value of all future normal costs (all amounts established at age w þ 1: Therefore

AL1j ¼ ðB0j þ DB1j Þ€að12Þ y

Dy Nwþ1 2 Ny 2 ðNC0j þ DNC1j Þ : Dwþ1 Dwþ1

ð5:3:3Þ

193

Valuation of pension plan liabilities

It follows that Dy AL1j ¼ ðB0j þ DB1j Þ€að12Þ y Dwþ1 Dy Dy Nwþ1 2 Ny j j ð12Þ ð12Þ 2 B0 ðyÞ€ay þ DB1 a€ y Nw 2 Ny Nwþ1 2 Ny Dwþ1 D N 2 N N 2 N y wþ1 y wþ1 ¼ B0j a€ ð12Þ 12 ¼ NC0j w y Dwþ1 Nw 2 Ny Dwþ1 D ¼ NC0j w : Dwþ1 In the year t ¼ 2 the accrued liabilities are defined in an analogous way, as the difference between the actuarial present value (at age w þ 2) of the future benefits and the actuarial present value of future normal costs. Hence Dy Nwþ2 2Ny 2ðNC0j þDNC1j þDNC2j Þ Dwþ2 Dwþ2 D N 2N D D y wþ1 y wþ1 wþ1 ¼ ðB0j þDB1j Þ€að12Þ 2ðNC0j þDNC1j Þ y Dwþ1 Dwþ2 Dwþ1 Dwþ2 Dy Nwþ2 2Ny N 2Nwþ1 2ðNC0j þDNC1j Þ wþ2 þDB2j a€ ð12Þ 2DNC2j : y Dwþ2 Dwþ2 Dwþ2

AL2j ¼ ðB0j þDB1j þDB2j Þ€að12Þ y

ð5:3:4Þ

Let us note that the sum of the last two expressions in (5.3.4) is zero by (5.3.2). Also, the sum of the first two expressions is AL1j Dwþ1 =Dwþ2 (see (5.3.3)). Since Nwþ2 2 Nwþ1 ¼ 2Dwþ1 ; it follows from (5.3.4) that AL2j ¼ ðAL1j þ NC1j Þ

Dwþ1 : Dwþ2

By induction, we arrive at the following general formula for every t ¼ 1; …; y 2 w 2 2; ALtjþ1 ¼ ðALtj þ NCtj Þ

Dwþt : Dwþtþ1

ð5:3:5Þ

The accrued liabilities at the time t with respect to the cohort1 Att of the participants, 1

A cohort is defined as a subset of the plan participants population distinguished by their age and service; a cohort may also be defined by its members’ salary and age structure.

194

Chapter 5

who then have t years of service, are ALtt ¼

X

ALtj :

ð5:3:6Þ

j[Att

The accrued liabilities with respect to the group At of all plan participants in the year t are X XX ALt ; ALtt ¼ ALtj : ð5:3:7Þ t

j[Att

t

We will study now how ALtt changes in relation to t (and t). Note that (5.3.5) implies ALtþ1 tþ1 ¼

X

ALtjþ1 ¼

X

ðALtj þ NCtj Þ

Att

Atþ1 tþ1

X Dwþt 2 ALtjþ1 ; Dwþtþ1 T
ð5:3:8Þ

t

where Tt and Rt denote, respectively, the sets of those members of the cohort At , who, in the following year, will leave the plan because of retirement (Rt , we include all participants who leave with retirement benefit rights in this group) and for other reasons (Tt). Since Dwþt lwþt vwþt l 1 ¼ ¼ wþt ð1 þ iÞ ¼ ð1 þ iÞ 1 2 qwþt Dwþtþ1 lwþtþ1 lwþtþ1 vwþtþ1 D ¼ ð1 þ iÞ þ qwþt wþt ; Dwþtþ1 then it follows from (5.3.8) that X X X ALtþ1 ðALtj þ NCtj Þð1 þ iÞ þ qwþt ALtjþ1 2 ALtjþ1 : tþ1 ¼ Att

Att

Tt
Thus, considering (5.3.6), we obtain t t ALtþ1 tþ1 ¼ ðALt þ NCt Þð1 þ iÞ

2

X

ALtjþ1 2

X

P

j[Att

qwþt ALtjþ1

2

Att

Tt

where NCtt ¼

!

X

ALtjþ1 ;

ð5:3:9Þ

Rt

NCtj : Since ALtþ1 ¼

X X t

ALtjþ1 ;

j[Atþ1 tþ1

where the first sum is over all possible values of the number of service years t, then it

195

Valuation of pension plan liabilities

follows from (5.3.7) and (5.3.9) that ! X X t t ALtþ1 ¼ ALt þ NCt ð1 þ iÞ t

2

t

XX t

ALtjþ1

j[Tt

XX

! qwþt ALtjþ1

2

Att

t

XX t

ALtjþ1 : ð5:3:10Þ

Rt

Let us denote, as in the previous chapters, the set of all active plan participants at time t by At, the set of all participants who left the active group in the time period from t till t þ 1 with pension rights by R, and those leaving without pension rights by T. Since At ¼
X

j[T

j[At

ALtjþ1 ;

ð5:3:11Þ

j[R

where t is a function of jP (t has the same value for all members of the cohorts Att ; Tt and Rt), whereas NCt ¼ t NCtt is the normal cost of the plan in the year t. 5.3.3 Unfunded actuarial liabilities By subtracting the following equation Ftþ1 ¼ Ft þ C þ I 2 P from (5.3.11) and considering the investment return IC on C and IP on P, we obtain the formula for unfunded actuarial liabilities in the year t þ 1: UALtþ1 ; ALtþ1 2Ftþ1 ¼ ðALt 2Ft Þð1þiÞ2ðI 2iFt 2IC þIP Þ 2½C þIC 2NCt ð1þiÞ2 2

X

! ALtj þ1 2P2IP :

X j[T

ALtj þ1 2

X

!

qwþt ALtj þ1

j[At

ð5:3:12Þ

j[R

5.3.4 Actuarial gain The sum of the second, fourth, and the last expression in parentheses in the right-hand side of (5.3.12) defines the dispersion between the actual parameters in the plan experience and their actuarial assumptions. The opposite of the said sum is called the

196

Chapter 5

actuarial gain, and we will denote it, as before, by Ga. It follows from (5.3.12) that Ga ¼ ðUALt þ NCt Þð1 þ iÞ 2 C 2 IC 2 UALtþ1 : The formula (5.3.12) implies an important property of the actuarial gain for the pension funding method under consideration. Note that the actuarial gain is not affected by the dispersion between the projected and the actual benefit (or salary) levels. The individual level premium method automatically adjusts the costs when the salary changes in the subsequent year. Thus in this method the common component of the actuarial gain, and importantly the component, which tends to be negative (as salary growth tends to surprise on the upside more often than on the downside, and salary declines are nearly nonexistent, especially in regard to pensionable earnings), is excluded from the actuarial gain and amortized over future normal costs. In the methods we have considered previously, this component frequently causes actuarial losses, which must be amortized separately.

5.4 The method of Frozen Initial Liability The Frozen Initial Liability method of pension plan valuation starts with establishing the normal costs NCt for all active plan participants (this group is still denoted by At). Then this simplifying assumption is adopted: that the normal cost NCtj per participant j is equal for all participants j [ At : 5.4.1 Normal cost Let us denote by nt the number of plan participants at time t. Then NCtj ¼

1 NCt : nt

Let us note further that the actuarial present value at time t of all future normal costs for the jth participant is PVFNCtj ¼ NCtj a€ x: yx ¼ NCtj

Nx 2 Ny ; Dx

where s is the age of the participant j (see (3.10.36)). Then the actuarial present value of all future normal costs for the plan, PVFNCt, is equal to PVFNCt ¼

X j[At

PVFNCtj ¼ NCt

1 X Nx 2 Ny : nt j[A Dx

ð5:4:1Þ

t

The sum on the right-hand side of (5.4.1) is the ‘present value’ of all future service years for plan participants (recall that x, in general, does depend on j); we will denote

197

Valuation of pension plan liabilities

that quantity by PVFYt. Therefore PVFYy ¼

X Nx 2 Ny : Dx j[A t

Let us also note that PVFYt =nt is the mean value of the number of future service years of an average plan participant. The normal cost per plan participant will be termed unit normal cost and denoted by Ut. Clearly Ut ¼ NCt ·

1 ¼ NCtj : nt

ð5:4:2Þ

5.4.2 Accrued liabilities Let us further assume that the plan accrued liabilities, ALt, are equal to the difference between the actuarial present values of the future benefits (PVFBt) and the future normal costs (PVFNCt), i.e., ALt ¼ PVFBt 2 PVFNCt ¼

X

PVFBtj 2 NCt

j[At

1 PVFYt : nt

ð5:4:3Þ

nt :

ð5:4:4Þ

From (5.4.1) and (5.4.3) we obtain X NCt ¼

PVFNCt n ¼ PVFYt t

PVFBtj 2 ALt

j[At

PVFYt

The formula (5.4.4) states the relationship between the accrued liabilities and the normal costs in the year t. It is sufficient to give one of these two quantities to have them both, as the second one is easily derived. In the Frozen Initial Liability method one assumes that the accrued liabilities are given. In the year t þ 1; (5.4.3) also holds, and thus ALtþ1 ¼

X

j PVFBtþ1 2 PVFNCtþ1 ;

ð5:4:5Þ

j[Atþ1

where PVFNCtþ1 ¼

X j[Atþ1

PVFNCtj ¼ NCtþ1

1 ntþ1

X Nxþ1 2 Ny : Dxþ1 j[A tþ1

ð5:4:6Þ

198

Chapter 5

Since Dy vy ly Dy 1 ¼ xþ1 ¼ ð1 þ iÞ 1 2 qx Dxþ1 Dx v lxþ1 ¼

Dy Dy qx ð1 þ iÞ þ ð1 þ iÞ Dx Dx 1 2 qx

¼

Dy Dy ð1 þ iÞ þ qx ; Dx Dxþ1

therefore X

j PVFBtþ1 ¼

j[Atþ1

X

j Btþ1 a€ ð12Þ y

j[Atþ1

¼

X

X j Dy Dy ¼ ðBt þ DBtj Þ€að12Þ y Dxþ1 j[A Dxþ1 tþ1

Btj a€ ð12Þ y

j[At

X

X Dy Dy Dy 2 Btj a€ ð12Þ þ DBtj a€ ð12Þ y y Dxþ1 j[T
X

Dy ¼ Btj a€ ð12Þ ð1 þ iÞ 2 y Dx j[At 2

X j[R

Btj a€ ð12Þ y

X

Btj a€ ð12Þ y

j[T

X Dy Dy 2 qx Btj a€ ð12Þ y Dxþ1 j[A Dxþ1

!

t

X Dy Dy þ DBtj a€ ð12Þ ; y Dxþ1 j[A Dxþ1 tþ1

where, as always, the group of plan participants, who in the year t þ 1 leave the plan with pension rights, is denoted by R, while those leaving without pension rights comprise the group denoted by T. Hence

PVFBtþ1 ¼ PVFBt ð1 þ iÞ 2

X

g j 2 PVFB tþ1

j[T

2

X j[R

g j þ PVFB tþ1

X j[Atþ1

X

! g j qx PVFB tþ1

j[At

DBtj a€ ð12Þ y

Dy ; Dxþ1

ð5:4:7Þ

g j is the actuarial present value at time t þ 1 of the future benefits, where PVFB tþ1 calculated under the assumption that their expected values at times t and t þ 1 are equal.

199

Valuation of pension plan liabilities

After substituting (5.4.6) and (5.4.7) into (5.4.1), we obtain ALtþ1 ¼ PVFBt ð1 þ iÞ 2

X

g j 2 PVFB tþ1

j[T

þ

X j[Atþ1

X

DBtj a€ ð12Þ y

j[Atþ1

2

g j 2 PVFB tþ1

X

X

g j PVFB tþ1

j[R

j[At

j[T

X

! g j qx PVFB tþ1

Dy DBtj a€ ð12Þ 2 PVFNCtþ1 y Dxþ1

¼ ALt ð1 þ iÞ 2 þ

X

!

g j qx PVFB tþ1 2

X

g j PVFB tþ1

j[R

j[At

Dy 1 X Nxþ1 2 Ny 2 NCtþ1 ntþ1 j[A Dxþ1 Dxþ1 tþ1

1 X Nx 2 Ny þ ð1 þ iÞNCt : nt j[A Dx

ð5:4:8Þ

t

5.4.3 Unfunded actuarial liabilities and the actuarial gain Let us assume that the fund (assets of the plan) at time t þ 1 equals Ftþ1 ¼ Ft þ I þ C 2 P: Then the last equality in (5.4.8) can be rewritten as ALtþ1 2 Ftþ1 ¼ ðALt 2 Ft þ NCt Þð1 þ iÞ 2 C 2 IC 2 Ga;

ð5:4:9Þ

where IC is the total return achieved from the investment of the cost paid in the year t from the time of the payment of the said cost till the end of the year, while ! X X j j g g PVFB 2 qx PVFB Ga ¼ ðI 2 iFt 2 IC þ IP Þ þ tþ1

j[T

þ

X j[R

! g j PVFB tþ1

2 P 2 IP

tþ1

j[At

2

X

Dy 1 þ NCtþ1 ntþ1 Dxþ1

DBtj a€ ð12Þ y

j[Atþ1

X Nxþ1 2 Ny 1 X Nx 2 Ny þ ð1 þ iÞNCt 1 2 nt j[A Dxþ1 Dx j[A tþ1

! :

ð5:4:10Þ

t

The expression (5.4.9) describes the change in the unfunded actuarial liabilities between times t and t þ 1: Thus we can write it in the form UALtþ1 ¼ ðUALt þ NCt Þð1 þ iÞ 2 C 2 IC 2 Ga;

ð5:4:11Þ

where UALt ¼ ALt 2 Ft : Note that, as always, Ga is the change in the unfunded actuarial liabilities, which resulted from a variance between accrued liabilities as established in the normal cost calculation and the actual experience. In the method of

200

Chapter 5

Frozen Initial Liability it is assumed that the actuarial gain is, by definition, equal to zero. Given the value of accrued liabilities at time t ¼ 0; one can use (5.4.9) to calculate accrued liabilities in the year t þ 1: Based on the so-established value of accrued liabilities and the actuarial present value of future benefits, one can use (5.4.3) to calculate the normal cost in the year t þ 1; etc. 5.4.4 A recursive formula for the unit normal cost Since

Nx 2 Ny Nxþ1 2 Ny Nxþ1 2 Ny lxþ1 Nxþ1 2 Ny 2 1 ð1 þ iÞ þ qx ¼ þ qx Dx Dxþ1 Dxþ1 lx Dxþ1 ¼

Nxþ1 2 Ny ; Dxþ1

then X Nxþ1 2 Ny X Nxþ1 2 Ny X Nxþ1 2 Ny ¼ 2 Dxþ1 Dxþ1 Dxþ1 T
2

X Nxþ1 2 Ny X Nxþ1 2 Ny 2 qx Dxþ1 Dxþ1 T A

! ;

ð5:4:12Þ

t

where we used our standard assumption that y ¼ x þ 1 on the set R. From (5.4.4) Utþ1 ¼

NCtþ1 PVFNCtþ1 1 ¼ ¼ Ut 2 ðU · PVFYtþ1 2 PVFNCtþ1 Þ: PVFYtþ1 t ntþ1 PVFYtþ1

Now using (5.4.12) and (5.4.8), we obtain 1 Nxþ1 2Ny Dxþ1 Atþ1 ( " !# X Nx 2Ny X Nxþ1 2Ny X Nxþ1 2Ny £ Ut 21 ð1þiÞ2 2 qx Dx Dxþ1 Dxþ1 T At At ! X X g j 2 g j PVFB 2PVFBt ð1þiÞþ qx PVFB

Utþ1 ¼Ut 2 X

tþ1

T

þ

X R

g j 2 PVFB tþ1

X Atþ1

DBj a€ ð12Þ y

tþ1

At

)

Dy þALtþ1 : Dxþ1

ð5:4:13Þ

201

Valuation of pension plan liabilities

Furthermore, using (5.4.1) – (5.4.3), we derive

Ut

X Nx 2 Ny Dx

At

21

! X Nx 2 Ny 2 nt Dx At 1 X Nx 2 Ny ¼ NCt 2 NCt nt A Dx

NCt ¼ nt

t

¼ PVFNCt 2 NCt ¼ PVFBt 2 ALt 2 NCt :

ð5:4:14Þ

From (5.4.13) and (5.4.14) it follows that 1 Utþ1 ¼ Ut 2 X Nxþ1 2 Ny Dxþ1 Atþ1 ( £

"

ðPVFBt 2 ALt 2 NCt Þð1 þ iÞ þ

X g j PVFB

tþ1 2 Ut

T

Nxþ1 2 Ny Dxþ1

# X X Nxþ1 2 Ny Dy j g 2 qx PVFBtþ1 2 Ut DBj a€ ð12Þ 2 y Dxþ1 Dxþ1 At Atþ1 ) X g j 2 PVFBt ð1 þ iÞ þ ALtþ1 : PVFB þ tþ1

ð5:4:15Þ

R

Let us note further that from (5.4.9) we have ALtþ1 2 ðALt þ NCt Þð1 þ iÞ ¼ Ftþ1 2 Ft ð1 þ iÞ 2 C 2 IC 2 Ga ¼ I 2 iFt 2 IC 2 P 2 Ga: Considering the above formula in light of (5.4.15), we get 1 Utþ1 ¼ Ut 2 X Nxþ1 2 Ny Dxþ1 Atþ1 (

"

X Nxþ1 2 Ny j g PVFBtþ1 2 Ut £ I 2 iFt 2 IC þ IP þ Dxþ1 T # X X N 2 Ny Dy g j 2 Ut xþ1 2 qx PVFB DBj a€ ð12Þ 2 y tþ1 Dxþ1 Dxþ1 At Atþ1 ! ) X g j 2 P 2 IP 2 Ga : PVFB þ tþ1

R

ð5:4:16Þ

202

Chapter 5

After substituting in (5.4.16) Ga ¼ 0; we arrive at the relationship between unit normal cost in the year t þ 1 and the year t. Such a relationship can be obtained directly from the equation Ga ¼ 0 with the use of (5.4.10). In fact, from (5.4.10) we obtain ( " X Nx 2 Ny 1 Utþ1 ¼ X Ut 2 1 ð1 þ iÞ Nxþ1 2 Ny Dx At D xþ1 Atþ1 !# X X j j g g PVFB 2ðI 2 iFt 2 IC þ IP Þ 2 2 qx PVFB tþ1

tþ1

T

2

X

At

! g j 2 P 2 IP PVFB tþ1

þ

X

R

DBj a€ ð12Þ y

Atþ1

Dy Dxþ1

) :

Using now (5.4.12), we derive 1

(

Utþ1 ¼ X Nxþ1 2 Ny Dxþ1 A

Ut

X Nxþ1 2 Ny Dxþ1 A tþ1

tþ1

þUt

X Nxþ1 2 Ny X Nxþ1 2 Ny 2 qx Dxþ1 Dxþ1 T A

!

t

2ðI 2 iFt 2 IC þ IP Þ 2

X

g j PVFB tþ1

2

T

2

X

2 P 2 IP

R

! g j qx PVFB tþ1

At

! g j PVFB tþ1

X

þ

X

DBj a€ ð12Þ y

Atþ1

Dy Dxþ1

) :

Thus 1 Utþ1 ¼ Ut 2 X Nxþ1 2 Ny Dxþ1 Atþ1 ( £

I 2 iFt 2 IC þ IP þ

"

X g j PVFB

tþ1 2 Ut

T

# X Nxþ1 2 Ny j g 2 qx PVFBtþ1 2 Ut þ Dxþ1 At ) X j ð12Þ Dy 2 DB a€ y : Dxþ1 A tþ1

Nxþ1 2 Ny Dxþ1

X

!

g j 2 P 2 IP PVFB tþ1

R

ð5:4:17Þ

203

Valuation of pension plan liabilities

Note that if the actual experience coincides with the actuarial assumptions, then it follows from (5.4.17) that Utþ1 ¼ Ut : As we had assumed a priori that Ga ¼ 0; we moved all variances between experience and assumptions into the unit normal cost Ut. As the actuarial gain equals zero, (5.4.2) becomes UALtþ1 ¼ ðUALt þ NCt Þð1 þ iÞ 2 C 2 IC :

ð5:4:18Þ

5.4.5 Iterative method of normal cost calculation The expression (5.4.18) allows us to calculate the normal cost in an iterative fashion. At the time t ¼ 0; when the plan is started, we calculate the unfunded actuarial liabilities using the Entry Age Normal method. Then in the consecutive years, for t ¼ 1; 2; …; we abide by the following algorithm. 1. Establish the value of assets Ft. 2. Calculate PVFBt ¼

X

Btj a€ ð12Þ y

j[At

Dy Dx

and

PVFYt ¼

X Nx 2 Ny : Dx j[A t

3. Using (5.4.4) and knowing that ALt ¼ UALt þ Ft ; we calculated the normal cost as follows: NCt ¼

PVFBt 2 UALt 2 Ft nt : PVFYt

ð5:4:19Þ

4. Using the value of UALt, we calculate the unfunded actuarial liabilities UALtþ1 with the recursive formula (5.4.18). The definition of the initial unfunded actuarial liabilities is, in fact, quite arbitrary, as in the Frozen Initial Liability method the normal cost is defined as the actuarial present value of future benefits reduced by the accrued liabilities (which are equal to the sum of the assets and unfunded actuarial liabilities), and then divided by the average annuity factor for the number of future service years for plan participants. Initial unfunded actuarial liabilities, therefore, do not change as a result of paying a cost equal exactly to the normal cost (this is the reason for the name for this funding method—Frozen Initial Liability). However, this initial liability may be amortized with additional payments. Any divergences between the actuarial assumptions and the actual experience, which in other methods are the source of actuarial gains or losses, in this method are directly reflected in the normal cost, as shown in the right-hand side of the formula (5.4.19).

204

Chapter 5

The unit normal cost Ut is sufficient to fund the aggregate expected benefits earned during the expected future service years for all active plan participants in the group At. For the younger plan participants this will produce higher normal cost than they would generally need, while for the older plan participants the effect is exactly the opposite. This pension funding method is also used in the case when the initial unfunded liability is calculated with the unit credit method, not the Entry Age Normal. This variation is referred to as the Attained Age Normal method or Frozen Attained Age method.

5.5 Valuation of plans paying pension benefits Recall that we have been assuming that the participants leaving the plan in a given year with pension benefit rights (group R) join a separate group, which receives benefits as scheduled, and has those benefits paid with appropriate annuities (mostly life annuities, but annuities certain are also possible) purchased from a separate life insurance company. When a participant leaves the plan with pension benefit rights, annuity purchase price is paid for with an appropriate amount of plan assets. As before, let B j be the annual benefit amount for the jth plan participant, but this time let it be only for the participant in the group R. The accrued liabilities with respect to the jth plan participant are Bj a€ ð12Þ x ; ð12Þ

where x is the age of this participant (in general, x varies with j), and a€ x is the net single premium for a life annuity due for life aged x, payable monthly, with annual benefit of 1 (i.e., monthly benefit of 1/12 of a monetary unit, see (3.10.17)). 5.5.1 Accrued liabilities Let Pt be the set of all inactive plan participants with pension benefit rights at the time t. Some of the members of this group may still be awaiting the receipt of their first benefit payment and we will refer to them as retirees. The accrued liabilities with respect to all retirees, ALt, are given by the expression ALt ¼

X

Bj a€ ð12Þ x :

ð5:5:1Þ

j[Pt

Denote by D the set of all plan participants who die between the times t and t þ 1: Then Ptþ1 ¼ ðPt < RÞw D:

ð5:5:2Þ

205

Valuation of pension plan liabilities

In Section 3.10 we proved the following identity (see (3.10.10)) a€ xþ1 ¼

ð€ax 2 1Þð1 þ iÞ ; px

ð5:5:3Þ

where px is the probability that a retiree aged x survives the subsequent year (we do not now consider other decrements, i.e., other ways of leaving the pension plan group). Thus, in this section px has a somewhat different interpretation as before. From the above identity we obtain a€ xþ1 ¼ ð€ax 2 1Þð1 þ iÞ þ qx a€ xþ1 ; ð12Þ

where qx ¼ 1 2 px : In Section 3.10 we showed that a€ x the following formula (see (3.10.21)): a€ ð12Þ < a€ x 2 x

ð5:5:4Þ

can be approximated with

11 : 24

ð5:5:5Þ

From this and from the formula (5.5.4) we obtain ð12Þ

a€ xþ1 < ð€að12Þ 2 1Þð1 þ iÞ þ qx a€ xþ1 2 x

11 24

11 13 11 11 i2 qx þ qx a€ xþ1 2 ð1 þ iÞ 2 1 þ 24 24 24 24 13 11 ð12Þ < a€ ð12Þ i2 q þ qx a€ xþ1 : ð5:5:6Þ x ð1 þ iÞ 2 1 þ 24 24 x

¼

a€ ð12Þ 2 x

Recall now the formula (5.5.4) and rewrite it as a€ xþ1 ¼ a€ x ð1 þ iÞ 2 ½ð1 þ iÞ 2 qx a€ xþ1 : The first term is the actuarial present value of a life annuity in the preceding year, multiplied by the accumulation factor ð1 þ iÞ: The second term gives the reduction of the actuarial present value of the annuity, caused by the payment of the benefit at the beginning of the year, with a correction for mortality between ages x and x þ 1: Now let us consider the approximate identity (5.5.6) and note that the first term on the right-hand side is the accumulated actuarial present value of the annuity from the previous year. The second term is approximately equal to the future accumulated actuarial value (at the time corresponding to age x þ 1) of the already paid benefits of the annuity, with annual benefit level at a unit, but adjusted for the monthly mode of payment of 12 equal installments, and for mortality assumed to follow uniform distribution of deaths. To show that, let us assume that the annual interest rate i is split uniformly over 12 months, to become a monthly rate of i=12: If we ignored mortality, the sum of accumulated monthly benefit payments (paid at the beginning of each

206

Chapter 5

month, as this is an annuity due), each equal to 1/12, at the end of the year would be 1 1 11 1 1 13 ð1 þ iÞ þ 1þi 1þi þ ··· þ ¼1þi : 12 12 12 12 12 24 To consider mortality, let us assume that the group of retirees at age x decreases in total size each month by the fraction qx =12 of its size at the beginning of the year (this implies that the force of mortality is actually increasing throughout the year). Then the accumulated actuarial value, at the end of the year, of the benefit payment from the first month will be ð1 þ iÞ=12; the second month payment will accumulate to ð1 þ i · 11=12Þð1 2 qx =12Þ=12; the third month payment will become ð1 þ i · 10=12Þ ð1 2 qx · 2=12Þ=12; etc. and the last payment: ð1 þ i=12Þð1 2 qx · 11=12Þ=12: The sum of all such accumulated payments will be 1 1 11 q 1 1 11 ð1 þ iÞ þ 1þi 1þi 1 2 x þ ··· þ 1 2 qx 12 12 12 12 12 12 12 ¼1þi

13 11 11 · 13 13 11 2 qx 2 qx i < 1 þ i 2 qx : 24 24 24 · 36 24 24

ð5:5:7Þ

Let us note that the term omitted in the approximate formula (5.5.7), for mortality in the range of 10% and the interest rate in the range of 15%, is equal to 0.0025, which indicates that the error of the approximation proposed is typically quite small. Finally, the last term on the right-hand side of (5.5.6) is a correction of the ð12Þ ð12Þ recursive relationship between a€ xþ1 and a€ x ; considering mortality at the end of the year. Therefore, on one hand (5.5.7) can be used to interpret the middle term of the right-hand side of (5.5.6); and on the other hand, it can serve to derive the approximation (5.5.6) without using the approximation (5.5.5). Let us now consider the accrued liabilities with respect to the plan participants in the following (t þ 1)th year. Using (5.5.1), (5.5.2) and (5.5.6), we obtain X X X X ð12Þ ð12Þ ð12Þ ð12Þ ALtþ1 ¼ Bj a€ xþ1 ¼ Bj a€ xþ1 þ Bj a€ xþ1 2 Bj a€ xþ1 j[Ptþ1

j[Pt

j[R

j[D

13 11 ð12Þ j ð12Þ i2 q þ qx a€ xþ1 ¼ B a€ x ð1 þ iÞ 2 1 þ 24 24 x j[Pt X j ð12Þ X j ð12Þ B a€ xþ1 2 B a€ xþ1 þ X

j[R

j[D

13 i Bj 1 þ 24 j[Pt " # X j ð12Þ X j ð12Þ X 11 ð12Þ j þ B a€ xþ1 2 B a€ xþ1 2 qx B a€ xþ1 þ : 24 j[R j[D j[P

¼ALt ð1 þ iÞ 2

X

t

ð5:5:8Þ

207

Valuation of pension plan liabilities

5.5.2 Unfunded actuarial liabilities The assets of the plan in the year ðt þ 1Þ; Ftþ1 ; are equal to Ftþ1 ¼ Ft þ I þ P 2 W; where I is the total (dollar) return from the investment of the amount Ft, P denotes the premium received from the active participants’ portion of the plan for the purpose of purchasing life annuities with annual benefit B j for the members of the set R, and W denotes the pension benefit payments made during the year by the fund. The above identity can be expanded to include returns from various portions of the fund. Indeed, we have Ftþ1 ¼ Ft ð1 þ iÞ þ ðI 2 iFt þ IW 2 IP Þ þ ðP þ IP Þ 2 ðW þ IW Þ;

ð5:5:9Þ

where IP is the return earned on the premium received for the purchase of annuities, while IW is the hypothetical return from the amount W paid as benefits. As we have done this before, we denote by UALt the unfunded actuarial liabilities of the plan at the time t. Let us consider the benefit payments in two subsets: W 0 and W 00 . The subset W 0 is the amount paid for retirees from the group Pt (old retirees); and the subset W 00 consists of the payments to the group R (new retirees). Using (5.5.8) and (5.5.9), we obtain UALtþ1 ¼ ALtþ1 2 Ftþ1 ¼ UALt ð1 þ iÞ 2 ðI 2 iFt þ IW 2 IP Þ " # X j 13 11 0 i2 q 2 W 2 IW 0 2 B 1þ 24 24 x j[P t

2

P þ IP 2

X

! ð12Þ Bj a€ xþ1

00

2 W 2 IW 00

j[R

2

X j[D

ð12Þ Bj a€ xþ1

2

X

! ð12Þ qx Bj a€ xþ1

:

ð5:5:10Þ

j[Pt

5.5.3 Actuarial gain The second expression on the right-hand side of (5.5.10) is equal to zero if the actual return earned by the assets in the year under consideration is equal to the one assumed by the actuary. The third expression equals (with a possible approximation error) the difference between the assumed and the actual benefit payments to the ‘old’ retirees in the group Pt. The fourth expression is the difference between the assets received in consideration for the annuities payable to the ‘new’ retirees in

208

Chapter 5

the group R and the amount of the reserve at the end of the year, increased by the payments of the benefits W 00 together with the foregone interest on those benefits IW 00 : Finally, the last expression is the difference between the actual and projected reductions in the benefit payments due to deaths of some plan participants. All these expressions, from the second one through the fifth one, are equal to zero if the actuarial projections coincide with the actual experience of the plan. The sum of all these expressions is termed, as we have always done so before, the actuarial gain Ga. We infer from (5.5.10) that Ga ¼ UALt ð1 þ iÞ 2 UALtþ1 ; i.e., this is the difference between the expected and the actual unfunded actuarial liabilities. It follows immediately that if there were no unfunded actuarial liabilities assumed at the beginning, and the experience agrees with the actuarial assumptions, then no unfunded actuarial liability will develop in the later years. Since the actuarial gain for the group of retirees does not contain any normal costs, we can simply add accrued liabilities with respect to the retirees to the accrued liabilities with respect to the active participants and then, by adding the assets allocated to the two groups as well, use the methods of plan valuation defined previously for the totality of the plan, consisting of both retirees and plan participants. We can also keep the two groups formally separated, but with one caveat: that there must be a mechanism for collection of additional premium, if an actuarial loss develops, or providing an experience refund if an actuarial gain develops. For the pension plans in the United States, typically active plan participants and retirees are members of the same plan, and retirees may receive an experience refund in the form of a special dividend or pension cost of living adjustment.

5.6 New plan participants We have discussed in Sections 5.1 – 5.3 pension funding and valuation methods assuming naturally that upon joining the plan, accrued liability with respect to the joining participant was equal to zero. Accrued liability with respect to the entire plan is the sum of the accrued liabilities with respect to individual plan participants. If we were to add new participants to the plan under such assumptions (also assuming that new participants do not bring any new assets with them), then this does not change the level of the actuarial gain or the unfunded actuarial liability of the plan. The situation is similar with respect to the normal cost. If we use the unit credit method or Entry Age Normal or individual level premium, the plan normal cost is the sum of the normal costs of the individual plan participants. Therefore arrival of new plan participants is, formally, equivalent to viewing the entire plan as the sum of two separate plans: the ‘old’ one consisting of only ‘old’ participants and the ‘new’ one with only ‘new’ plan members.

209

Valuation of pension plan liabilities

But in the case of the method of Frozen Initial Liability, considered in Section 5.4, the additivity rule for the valuation of the ‘old’ and ‘new’ plans does not apply. This follows from the fact that the normal cost in this method is given by the formula (5.4.19), which is not additive with respect to adding of new participants. For this reason exactly, in the methods of Frozen Initial Liability and the method of Frozen Attained Age, new plan participants must be included in a more subtle way in the valuation process. Before we address the details of this issue, let us denote by N the set of all plan participants who joined the plan between the times t and t þ 1: Then the group Atþ1 of plan participants in the year t þ 1 can be written as Atþ1 ¼ ðAt < NÞw ðT < RÞ;

ð5:6:1Þ

where T is the group of people who leave the plan without pension rights and R is the group of people leaving the plan with pension rights between the times t and t þ 1: 5.6.1 The method of Frozen Initial Liability In order to adopt the formula (5.4.7) to the new circumstances resulting from new participants joining the plan, we must add to its right-hand side the expression P ð12Þ j € y Dy =Dxþ1 : As a result of this addition, the formula will assume this new j[N Btþ1 a form X

j PVFBtþ1 ¼

j[Atþ1

X

Btj a€ ð12Þ y

j[At

X j ð12Þ Dy Dy ð1 þ iÞ þ Btþ1 a€ y Dx Dxþ1 j[N

! X Dy j ð12Þ Dy 2 2 qx Bt a€ y Dxþ1 Dxþ1 j[T j[At X j X Dy Dy 2 Bt a€ ð12Þ þ DBtj a€ ð12Þ : y y D D xþ1 xþ1 j[R j[A >A X

Btj a€ ð12Þ y

t

tþ1

Therefore (5.4.7) becomes

PVFBtþ1 ¼ PVFBt ð1 þ iÞ þ

X

j Btþ1 a€ ð12Þ y

j[N

2

X

g j 2 PVFB tþ1

j[T

þ

X j[At >Atþ1

X

g j qx PVFB tþ1

j[At

DBtj a€ ð12Þ y

Dy Dxþ1

Dy : Dxþ1

! 2

X j[R

g j PVFB tþ1

210

Chapter 5

Subsequently, (5.4.8) becomes

ALtþ1 ¼ ALt ð1 þ iÞ þ

X

j Btþ1 a€ ð12Þ y

j[N

2

X

g j 2 PVFB tþ1

j[T

þ

X

Dy Dxþ1 !

g j qx PVFB tþ1

2

DBtj a€ ð12Þ y

j[At >Atþ1

g j PVFB tþ1

j[R

j[At

X

X

Dy 1 X Nxþ1 2 Ny 2 NCtþ1 ntþ1 j[A Dxþ1 Dxþ1 tþ1

1 X Nx 2 Ny þ ð1 þ iÞNCt : nt j[A Dx t

The formula (5.4.10), which gives us the actuarial gain, becomes

Ga ¼ ðI 2 iFt 2 IC þ IP Þ 2

X

j Btþ1 a€ ð12Þ y

j[N

þ

X

g j 2 PVFB tþ1

j[T

2

X

X

Dy Dxþ1 !

g j qx PVFB tþ1

j[At

DBtj a€ ð12Þ y

j[At >Atþ1

Dy 1 X Nxþ1 2 Ny þ NCtþ1 ntþ1 j[A Dxþ1 Dxþ1 tþ1

þ ð1 þ iÞNCt 1 2

1 X Nx 2 Ny nt j[A Dx

! :

t

(5.4.12) is transformed into X Nxþ1 2 Ny X Nx 2 Ny X Nxþ1 2 Ny ¼ 2 1 ð1 þ iÞ þ Dxþ1 Dx Dxþ1 N A A tþ1

t

2

X Nxþ1 2 Ny X Nxþ1 2 Ny 2 qx Dxþ1 Dxþ1 T A t

! ;

211

Valuation of pension plan liabilities

and (5.4.13) becomes ( "

1

Utþ1 ¼ Ut 2 X Nxþ1 2 Ny Dxþ1 A

Ut

X Nx 2 Ny

2 1 ð1 þ iÞ

Dx

At

tþ1

X Nxþ1 2 Ny þ 2 Dxþ1 N 2PVFBt ð1 þ iÞ 2

X Nxþ1 2 Ny X Nxþ1 2 Ny 2 qx Dxþ1 Dxþ1 T A t

X

j Btþ1 a€ ð12Þ y

N

X

þ

g j 2 PVFB tþ1

T

þ

X

!#

X

Dy Dxþ1 !

g j qx PVFB tþ1

At

g j 2 PVFB tþ1

X

DBj a€ ð12Þ y

At >Atþ1

R

) Dy þ ALtþ1 : Dxþ1

Furthermore, the formula (5.4.15) must be adjusted accordingly

1 Utþ1 ¼ Ut 2 X Nxþ1 2 Ny Dxþ1 A tþ1

( £

" ðPVFBt 2 ALt 2 NCt Þð1 þ iÞ þ

X g j PVFB

tþ1 2 Ut

T

# X Nxþ1 2 Ny j g qx PVFBtþ1 2 Ut 2 Dxþ1 At X Nxþ1 2 Ny j ð12Þ Dy 2 Btþ1 a€ y 2 Ut Dxþ1 Dxþ1 N 2

X At >Atþ1

þ

X R

DBj a€ ð12Þ y

Dy Dxþ1

) j g PVFBtþ1 2 PVFBt ð1 þ iÞ þ ALtþ1 :

Nxþ1 2 Ny Dxþ1

212

Chapter 5

And we finally arrive at the following analogue of the formula (5.4.16) 1

Utþ1 ¼ Ut 2 X Nxþ1 2 Ny Dxþ1 A

(

" I 2 iFt 2 IC þ IP þ

X

g j 2 Ut PVFB tþ1

T

Nxþ1 2 Ny Dxþ1

tþ1

# X Nxþ1 2 Ny j g 2 qx PVFB tþ1 2 Ut Dxþ1 At X j ð12Þ Dy X Nxþ1 2 Ny Dy 2 Btþ1 a€ y 2 Ut DBj a€ ð12Þ 2 y D D D xþ1 xþ1 xþ1 N A >A t

þ

X

!

tþ1

)

g j 2 P 2 IP 2 Ga : PVFB tþ1

ð5:6:2Þ

R

If we substitute in the above formula Ga ¼ 0; we obtain an analog of the formula (5.4.17). From (5.6.2) we infer that if based on the unit normal cost Ut, calculated at time t, expected actuarial present value at time t þ 1 of the future normal cost for the group N, Ut

X Nxþ1 2 Ny N

Dxþ1

;

is less than the actuarial present value (still at time t þ 1) of the future benefits for that group, i.e., less than X j Dy Btþ1 a€ ð12Þ ; y D xþ1 N then the new unit normal cost at time t þ 1; Utþ1 ; must be more than the unit normal cost without consideration for new plan participants. Even if the unit normal cost is less as a result of new participants joining the plan (if they are relatively young, the expected present value of their future service years is relatively large), the normal cost of the entire plan is bound to increase somewhat because of increased burden for the plan coming from the increased future benefits obligation (compare the formula (5.4.3)).

5.6.2 Unit credit method In the standard analysis of the unit credit method (Section 5.1) we assume that at the time of entering, the plan has no earned benefit rights. In practice, this is not always true, especially if plan participants can transfer pension rights when changing jobs. When an employee terminates employment before actual retirement,

213

Valuation of pension plan liabilities

such employee may have the following options for the pension rights earned up to date: † receive an equivalent lump sum payment, † transfer pension rights to a new employer, together with an appropriate plan assets transfer, † remain a participant in the current plan, if the new employer does not offer a pension plan or a transfer of pension rights is not possible. In the last case, because of the lack of future service years, pension benefits will be appropriately adjusted downward. If the current plan allows employee contributions (or, more generally, contributions from plan participants, even if they are not active employees)—such plans are termed contributory plans—then the employee may contribute plan participation by making appropriate contributions voluntarily. If, after some period of time, the employee under consideration returns to active employment with the plan sponsor, accrued liabilities for this employee may exceed plan assets set aside with respect to the said employee. Similarly, an employee entering a retirement plan with a new employer may acquire pension rights in the new plan, and receive asset transfer, which does not appropriately correspond to the pension rights acquired. In all such cases, we must modify the formulas expressing the accrued liabilities and the actuarial gain for the unit credit method. Thus, the formula (5.1.2), and the two expressions for ALtþ1 preceding it, must be corrected by adding to the right-hand side X

Bj ðxj þ 1Þ€að12Þ yj

j[N

D yj Dxj þ1

:

The formula (5.1.2) then becomes ALtþ1 ¼

ALt þ

X

DBj a€ ð12Þ yj

j[At

2

X

2

j[R

Bj ðxj þ 1Þ€að12Þ yj

! ð1 þ iÞ þ

Dxj

Bj ðxj þ 1Þ€að12Þ yj

j[T

X

Dyj

X

Bj ðxj þ 1Þ€að12Þ yj

j[N

D yj Dxj þ1

Dyj Dxj þ1

2

X

qxj Bj ðxj þ 1Þ€að12Þ yj

j[At

D yj Dxj þ1 ! Dyj

Dxj þ1

:

The formula (5.1.3), defining the normal cost, is unchanged. But the formula (5.1.5) becomes Ftþ1 ¼ Ft þ I þ C þ Tr 2 P;

214

Chapter 5

where Tr is the amount of the assets transferred, possibly zero, which is associated with the new participants entering the plan. The unfunded actuarial liability is therefore adjusted by the following quantity X

Bj ðxj þ 1Þ€að12Þ yj

j[N

D yj Dxj þ1

ð5:6:3Þ

2 Tr 2 ITr ;

where ITr is the dollar return from the investment of the transferred assets Tr, calculated from the date of the transfer till the end of the year t þ 1: In particular, the formula (5.1.6) becomes UALtþ1 ¼ UALt ð1 þ iÞ 2 ðI 2 iFt 2 IC 2 ITr þ IP Þ 2 ½C þ IC 2 NCt ð1 þ iÞ ! X Dyj j ð12Þ 2 Tr þ ITr 2 B ðxj þ 1Þ€ayj Dxj þ1 j[N 2

X

j

B ðxj þ

1Þ€að12Þ yj

T

2

X R

j

B ðxj þ

1Þ€að12Þ yj

D yj Dxj þ1 D yj Dxj þ1

2

X

j

qxj B ðxj þ

At

1Þ€að12Þ yj

Dyj

!

Dxj þ1

! 2 P 2 IP :

Because the unfunded actuarial liability changed by the amount (5.6.3), the actuarial gain Ga is reduced by exactly the same amount, but the formula (5.1.7) remains correct. 5.6.3 The Entry Age Normal method In the Entry Age Normal method all participants entering the plan at time t þ 1 are of the same age x ¼ w: From the formula (5.2.5) we conclude that the accrued liability with respect to these new entrants is equal to zero. Thus the arrival of the new participants does not change the accrued liability of the plan, and its unfunded accrued liability, if we assume that the new participants do not bring any assets with them. The only modification needed in the formulas in Section 5.2 is that we should assume that DBj ¼ 0 on the set N. Then the formulas (5.2.11), (5.2.13) and (5.2.14) are unchanged, but the sum over the set Atþ1 includes the elements of the set N also. An alternative modification would be to sum over the set At > Atþ1 instead of the set Atþ1. 5.6.4 Individual level premium method In this method the accrued liability with respect to the new plan participant is, by definition, equal to zero, in a manner analogous to that for the Entry Age Normal

215

Valuation of pension plan liabilities

method. As a result of this situation, the actuarial gain is unaffected by new plan entrants. In the subsequent year t þ 1; the plan normal cost changes, as the normal cost associated with the new plan participants must be added.

5.7 Aggregate pension funding methods The pension funding methods we have studied up to this point all started by calculating the normal cost and the actuarial liability for each plan participant and then assessing those quantities for the entire plan by adding individual participants’ quantities. There is an alternative to this approach, although less frequently used. Instead of assigning liabilities to individual participants, one can assign assets of the plan to such individuals and then assess the funding by the need to pay the liabilities not covered by the assets in an amortization process over an appropriate time interval. All methods using this approach are termed aggregate methods. There is, however, a variety of aggregate approaches. The simplest is termed individual aggregate. In this method, one starts by assigning a share of the entire fund of assets held by the plan to each individual participant. There are various approaches used in such assignment. Assets can be assigned on the pro-rata basis in proportion to the present value of accrued benefits or by actuarial liabilities as established by another funding method (e.g., Entry Age Normal or individual level premium) or by some other reasonable approach. Once the assets are assigned to each individual, that individuals’ normal cost is established as NCj ¼

APVðBj Þ 2 F j a€ n

where NCj is this jth participant’s normal cost, APVðBj Þ that participant’s actuarial present value of accrued benefits, Fi that participant’s fund share, and a€ n an appropriate annuity (assumed here payable at the beginning of each year for a period of amortization n, typically until this individual’s normal retirement age). Such individual normal costs can then be summed up for the entire plan’s normal cost. One very important property of the individual aggregate method is that it never produces any unfunded actuarial liability. All funding needs are amortized into the normal cost. Thus the normal cost is likely to be larger than the one produced by methods developing unfunded liabilities. Some employers may desire lower normal cost, if funding of the normal cost is required statutorily, while supplementary cost may be delayed. On the other hand, the aggregate methodology produces realistic results. The second aggregate method is usually termed just aggregate, but sometimes, in contrast with the individual aggregate, it is also called aggregate aggregate.

216

Chapter 5

Its approach is very similar to the individual aggregate method, with one key difference: that the normal cost pays the difference between total assets and total plan liabilities in a form of so-called average annuity. Let N be the number of plan participants and F the total assets of the plan. Thus, the normal cost, total for the entire plan, under the aggregate method is given by the formula N X

NC ¼

APVðBj Þ 2 F

j¼1

a€

where a€ is the average annuity, a concept specific to aggregate methods. In case of funding cost defined as a dollar amount (as opposed to percentage of salaries), if xj is the jth participant’s age and rj that participant’s retirement age, such average annuity is defined as N X

a€ ¼

a€ xj : rj 2xj

j¼1

N

:

This concept is also used for plans that define normal cost as a percentage of payroll (i.e., combined wages of all employees). In that case, the average annuity is N X

a€ ¼

APVðAll Future Salaries j Þ

j¼1 N X

: j

APVðCurrent Salary Þ

j¼1

In practice, it is quite common to express the salary-based calculations with the salary-based commutation functions. A salary function Sx is a function giving the salary at age x, or an appropriate multiple or a fraction of salary. Most commonly, it is assumed that the salary increases by a constant percentage every year, so that we multiply current year’s salary by a constant factor 1 þ s: Another approach is to specify the salary function with the use of another function sx such that Sx =sx ¼ Sy =sy for any two ages x; y: While the aggregate method has a desirable property of producing no unfunded actuarial liability, it produces a relatively high normal cost. Because of this, alternative forms of aggregate approaches have been developed, which produce a separate initial unfunded liability, which is amortized and paid separately, with the payment termed supplemental cost. The first such method is called Frozen Initial Liability (Entry Age Normal), also abbreviated as FIL (EAN). In this method, the initial actuarial liability at the pension plan inception is calculated for each plan participant using the Entry Age Normal method and then summed over all plan participants. This sum is treated as a separate

217

Valuation of pension plan liabilities

Frozen Initial Liability, denoted by FIL. Then the normal cost is calculated as N X

NC ¼

APVðBj Þ 2 F 2 FIL

j¼1

a€

:

Another alternative is to calculate the Frozen Initial Liability using the unit credit method, and the resulting aggregate method is then termed Frozen Initial Liability (Attained Age Normal) or FIL (AAN). The normal cost under FIL (AAN) is higher than under FIL (EAN), because the initial unfunded actuarial liability is smaller, while the supplementary cost is smaller than under FIL (EAN).

Exercises Exercise 5.8.1 November 2000 SOA Course 5 Examination, Problem No. 3. You are given the following pension plan information. Normal retirement benefit is 1% final 1-year salary for each year of service. The valuation interest rate is 6% per year. Salary increase is 3% (used for Projected Unit Credit Cost method only). There are no pre-retirement deaths and terminations. Retirement age is 65. The life annuity factor at age 65 is 10. There is only one participant as of January 1, 2001, hired at age 30, and age 45 on January 1, 2001. The year 2001 annual salary of the participant is 50 000, while the year 2002 annual salary is 52 500. Under each of the Traditional Unit Credit Cost method and the Projected United Credit Cost method, calculate the following: the actual liability as of January 1, 2001, the expected liability as of January 1, 2002, the actual liability as of January 1, 2002; and the liability-side actuarial gain or loss at January 1, 2002. Compare and explain the difference between the liability gain or loss under the Traditional Unit Credit Cost method and the Projected Unit Credit Cost method. Solution Traditional Unit Credit. Under Traditional Unit Credit, annual benefit accrued at age x is Bx ¼ 0:01 · Sx · ðx 2 eÞ; where Sx is the salary at age x and e the entry age. In the case of this one participant, as of January 1, 2001, accrued liability is ð12Þ

B45 · v20 · 20 p45 · a€ 65 ¼ ð0:01 · 50 000 · 15Þ · 1:06220 · 1 · 10 ¼ 23 385:35: The expected liability as of January 1, 2002 is based on the 2001 annual salary rate of 50 000 (the 3% projected increase is only used for the Projected Unit Credit method), and it equals · v19 · BExpected 46

19 p45

ð12Þ

· a€ 65 ¼ ð0:01 · 50 000 · 16Þ · 1:06219 · 1 · 10 ¼ 26 441:04:

218

Chapter 5

However, the actual liability as of January 1, 2002 is based on the actual annual salary in 2002, i.e., 52 500. Therefore, it equals B46 · v19 ·

19 p45

ð12Þ

· a€ 65 ¼ ð0:01 · 52 500 · 16Þ · 1:06219 · 1 · 10 ¼ 27 763:09:

The actuarial gain due to the liability side of the balance sheet as of January 1, 2002 is the excess of the expected liability on January 1, 2002 over the actual liability on January 1, 2002, i.e., 26 441:04 2 27 763:09 ¼ 21; 322:05: This, of course, is an actuarial loss, as it is negative. Projected Unit Credit. This funding method uses the same basic methodology as the Traditional Unit Credit method, but the benefit rate is based on salary projected to the retirement age. This method is actually very commonly used in the United States, because it agrees with the standard generally accepted accounting principles (GAAP) concerning pension funding and valuation (as opposed to the statutory valuation performed by the actuary, which is the subject of this chapter). Salary increases are assumed to be 3% annually for this funding method. At age 45 (after 15 years of service), the annual benefit rate accrued to the participant is B45 ¼ 0:01 · 50 000 · 1:0320 · 15 < 13 545:83: This represents the annual benefit rate payable in equal monthly installments accrued as of January 1, 2001. The corresponding accrued liability as of the valuation date on January 1, 2001 is B45 ·

20 E45

ð12Þ

· a€ 65 < 13 545:83 ·

1 · 10 < 42 236:55: 1:0620

The expected liability as of January 1, 2002 is based on the 2001 annual salary rate of 50 000 increased by 3%, and it equals BExpected · 46

19 E46

ð12Þ

· a€ 65 ¼ ð0:01 · 50 000 · 1:03 · 1:0319 · 16Þ · 1:06219 · 1 · 10 < 47 755:46:

The actual liability as of January 1, 2002 is based on the actual salary in 2002, i.e., 52 500. Therefore, it equals ð12Þ

B46 · v19 · 19 p45 · a€ 65 ¼ ð0:01 · 52 500 · 1:0319 · 16Þ · 1:06219 · 1 · 10 < 48 682:75: The actuarial gain due to the liability side of the balance sheet as of January 1, 2002 is the excess of the expected liability on January 1, 2002 over the actual liability on January 1, 2002, i.e., 47 755:46 2 48 682:75 ¼ 2927:29: Again, this is an actuarial loss, as it is negative.

219

Valuation of pension plan liabilities

Both methods produce an actuarial loss, but the loss is smaller under Projected Unit Credit. This is caused by the fact that Traditional Unit Credit does not assume any salary increases, and while Projected Unit Credit assumes salary increases, the 3% assumed here turns out to be less than the actual salary increase (actual increase from 50 000 to 52 500 represents 5%). A Exercise 5.8.2 November 2000 SOA Course 5 Examination, Problem No. 18. For a newly established plan, you are given the following data. The normal retirement benefit is 10 per month per year of service. Normal retirement age is 65. Valuation interest rate is 6%. There are no pre-retirement terminations other than death. There are no pre-retirement death benefits. You are given this selected annuity ð12Þ value: a€ 65 ¼ 10: The method for amortization of Frozen Initial Liability is: level dollar amount over 20 years. Plan effective date is January 1, 2001. Plan assets as of January 1, 2001 are 0. The following selected commutation factors are given: D32 ¼ 1468;

N32 ¼ 23 018;

D33 ¼ 1382;

N33 ¼ 21 550;

D63 ¼ 215;

N63 ¼ 2287;

D65 ¼ 171;

N65 ¼ 1689:

There is only one participant, Mr Jones, who was hired on January 1, 1970 and born on January 1, 1938. For Mr Jones, calculate the normal cost under the following methods as of January 1, 2001: Frozen Initial Liability (Attained Age Normal), Frozen Initial Liability (Entry Age Normal), and Aggregate. Solution Frozen Initial Liability (Attained Age Normal). In this method, the normal cost is given by the following formula NC ¼

APVðBenefitsÞ 2 Assets 2 UAL ; a€ 63: 2

where the UAL (unfunded actuarial liability) is calculated initially using the Traditional Unit Credit funding method, and a€ 63: 2 is the average future service annuity (the notation is caused by the assumption of age 63 now, and 65 being the normal retirement age). The projected annual benefit at retirement is: 120 times 33 years of service, i.e., 3960. Its portion accrued through January 1, 2001 is 31 · 3960 ¼ 3720: 33 Accrued liability under the Traditional Unit Credit method is ð12Þ

3720 · a€ 65 ·

D65 171 < 29 586:98: ¼ 3720 · 10 · 215 D63

220

Chapter 5

There are no assets in the plan on January 1, 2001, and the above is therefore the unfunded actuarial liability on the valuation date. The actuarial present value of projected benefits is ð12Þ

3960 · a€ 65 ·

D65 171 < 31 495:81: ¼ 3960 · 10 · 215 D63

Combining this with the information about the unfunded actuarial liability we obtain the normal cost under the Frozen Initial Liability (Attained Age Normal) method NC ¼

APVðBenefitsÞ 2 Assets 2 UAL 31495:81 2 0 2 29 586:98 < 686:29: ¼ N63 2 N65 2287 2 1689 D63 215

Recall that under this method, the unfunded actuarial liability of 29 586.98 is amortized and paid separately, in level amortization over 20 years, giving the annual supplemental cost of 29586:98 < 2433:52: a€ 20 Frozen Initial Liability (Entry Age Normal). In this method, the calculation of normal cost is the same as in the Frozen Initial Liability (Entry Age Normal), but the initial unfunded actuarial liability is based on the Entry Age Normal accrued liability calculation. The Entry Age Normal method normal cost is (note that the entry age was 32) ð12Þ D65

APV32 ðBenefitsÞ ¼ a€ 32: 33

120 · 33 · a€ 65

N32 2 N65 D32

D32

171 1468 < 317:50: 23 018 2 1689 1468

120 · 33 · 10 · ¼

The Entry Age Normal accrued liabilities at age 63 (current attained age of the participant) can now be calculated as the excess of the actuarial present value of future benefits over the actuarial present value of future normal costs, i.e., as ð12Þ

120· 33· a€ 65 ·

D65 N 2 N65 2 317:50 · 63 < 31495:81 2 883:09 < 30612:72: D63 D63

Since there are no assets, the unfunded actuarial liability is 30 612.72. The normal cost is then calculated as NC ¼

APVðBenefitsÞ 2 Assets 2 UAL 31495:81 2 0 2 30 612:72 < 317:50: ¼ N63 2 N65 2287 2 1689 D63 215

221

Valuation of pension plan liabilities

Again, the Frozen Initial Liability of 30 612.72 is amortized and paid off separately, and the annual supplemental cost is 30612:72 < 2668:96: a€ 20 Aggregate method. In this method, the normal cost is calculated as NC ¼

APV63 ðBenefitsÞ 2 Assets 31 495:81 2 0 < 11323:74: ¼ N63 2 N65 2287 2 1689 D63 215

There is no supplemental cost funding in this method.

A

Exercise 5.8.3 November 2002 SOA Course 5 Examination, Problem No. 15, Section C. You are given the following for a pension plan covering one individual. The pension formula is: 2.0% of salary times the number of years of participation in plan up to 25 years of service, plus 1.5% times the number of years of participation in plan in excess of 25 years of service. The pension benefit is payable as a life annuity. The valuation date is January 1, 2002. The valuation interest rate is 8.0%. The normal retirement age is 65. There are no terminations or deaths assumed prior to normal retirement age. The date of birth of the sole participant is January 1, 1952, the date of hire is January 1, 1977, and the effective date of participation is January 1, 1977. The 2001 annual salary of the participant is 100 000, and the future salary is expected to stay the same. You are also given that the actuarial present value at age 65 of 1 payable for life in the manner provided by this plan’s pension (assumed to be monthly), starting at age 65, is 10. The assets of the plan on January 1, 2002 are 150 000, while on December 31, 2002 they are 180 000. Calculate the normal cost and the unfunded actuarial liability as of January 1, 2002 under the following cost methods: Traditional Unit Credit, Projected Unit Credit, Entry Age Normal (level percentage of pay), and Aggregate (level percentage of pay). Solution Traditional Unit Credit. As of January 1, 2002, the participant has 25 years of service. Under this method, the benefit accrued up to date is B50 ¼ 0:02 · 25 · 100 000 ¼ 50 000: The accrued liability is AL50 ¼ B50 ·

15 E50

ð12Þ

· a€ 65 ¼ 50 000 · 1:08215 · 10 < 157 620:85:

The unfunded actuarial liability is therefore UAL ¼ AL 2 F ¼ 157 620:85 2 150 000 ¼ 7620:85:

222

Chapter 5

The normal cost is the actuarial present value of retirement benefit accrued in this year of service (note that only 1.5% of salary accrues this year, as this is the 26th year of service): NC50 ¼ 0:015 · 100 000 · 10 · 1:08215 < 4728:63: Projected Unit Credit. The only (and key) difference between this method and the Traditional Unit Credit is the assumption of salary increases in the Projected Unit Credit. In this problem, no salary increases are assumed, thus the results are the same for the Projected Unit Credit as for the Traditional Unit Credit. Entry Age Normal (level percentage of pay). Note again that no future salary increases are assumed. This also means that no past salary increases should be assumed, and a level percentage of pay is a fixed nominal monetary amount. The entry age for the participant was 25, and at retirement, the participant will have 40 years of service. The normal cost will be paid as an annuity over the entire service period, so that NC25 a€ 25: 40 ¼ APV25 ðBenefitsÞ: The actuarial present value of benefits at the entry age is APV25 ðBenefitsÞ ¼ B65 ·

40 E25

ð12Þ

· a€ 65

¼ ð0:02 · 25 þ 0:015 · 15Þ · 100 000 · 1:08240 · 10 < 33 372:43: Therefore NC25 ¼

APV25 ðBenefitsÞ 33 372:43 33 372:43 < 2591:31: < < a€ 25: 40 a€ 40 0:08 12:879

This represents approximately 2.59% of salary. Furthermore, the accrued liabilities as of age 50 are AL50 ¼ APV50 ðBenefitsÞ 2 NC25 · a€ 50: 15 < ð0:02 · 25 þ 0:015 · 15Þ · 100 000 · 10 · 1:08215 2 2591:31 · a€ 15 < 204 595:56 2 23 954:68 ¼ 228 550:24: Unfunded actuarial liability is UAL50 ¼ 228 550:24 2 150 000 ¼ 78 550:24:

Valuation of pension plan liabilities

223

Aggregate (level percentage of pay). Under this method, the normal cost calculation is performed as follows: NC50 ¼

APV50 ðBenefitsÞ 2 F a€ 50: 15

¼

ð0:02 · 25 þ 0:015 · 15Þ · 100 000 · 10 · 1:08215 2 150 000 a€ 15

<

228 550:23 2 150 000 < 8497:21: 9:2442

This represents approximately 8.50% of salary. Under the Aggregate method, there is no need to calculate accrued liability, as it is effectively equal to the assets of the plan. A Exercise 5.8.4 November 2003 SOA Course 5 Examination, Problem No. 6. For a defined benefit pension plan, you are given that the benefit formula is: 1.5% of final year’s salary for each year of service up to 10 years, plus 2.0% of final year’s salary for each year of service after 10 years. Furthermore, the valuation interest rate is 6% and the salary growth rate is 4%. There are no pre-retirement decrements. ð12Þ Assumed retirement age is 65. Retirement annuity factor is a€ 65 ¼ 12: Assets on January 1, 2003 are valued at 300 000 and on January 1, 2004 are 320 000. Contributions made on December 31, 2003 were 5000. The funding method used is the Projected Unit Credit. There are two employees. Employee A was hired at age 30, and on January 1, 2003 age is 40, with salary of 30 000. Employee B was hired at age 30, and on January 1, 2003 age is 60, with salary of 50 000. Calculate the unfunded accrued liability at January 1, 2003. The actual accrued liability on January 1, 2004 is 350 000. Calculate the actuarial gain as of that date. Solution For employee A, the projected final salary is 24 SA 65 ¼ 30 000 · 1:04 < 76 899:12:

This employee will have a total of 35 years of service at retirement, but only 10 years of service are earned, and the actuarial present value of those earned benefits, calculated at valuation date of January 1, 2003, is B40 ·

25 E40

ð12Þ

· a€ 65 < ð0:015 · 10Þ · 76 899:12 · 1:06225 · 12 < 32 251:30:

For employee B, the projected final salary is SB65 ¼ 50 000 · 1:044 < 58 492:93:

224

Chapter 5

This employee will have a total of 35 years of service at retirement, but as of valuation date, 30 is earned, and the actuarial present value of earned benefits, calculated at valuation date of January 1, 2003, is ð12Þ

B60 · 5 E60 · a€ 65 < ð0:015 · 10 þ 0:02 · 20Þ · 58 492:93 · 1:0625 · 12 < 288 481:51: Total accrued liability as of January 1, 2003 is therefore 32 251:30 þ 288 481:51 ¼ 320 732:81: Since assets are then 300 000, the unfunded actuarial liability on January 1, 2003 is 20 732.81. The normal cost for the year 2003 is the actuarial present value of the benefit accrual during that year: 2003 NC2003 ¼ NC2003 40 þ NC60

¼ 0:02 · 76 899:12 · 1:06225 · 12 þ 0:02 · 58 492:93 · 1:0625 · 12 < 4300:17 þ 10 490:24 ¼ 14 790:41: The expected unfunded actuarial liability on January 1, 2004, is UALExpected ¼ ðUALActual 2003 þ NC2003 Þð1 þ iÞ 2 ðC þ IC Þ 2004 ¼ ð20 732:81 þ 14 790:41Þ · 1:06 2 ð5000 þ 0Þ < 32 654:61 The actual accrued liability on January 1, 2004 is 350 000 and the assets are 320 000, giving the actual unfunded liability of 30 000. The actuarial gain in the year 2004 is therefore Ga ¼ UALExpected 2 UALActual ¼ 32 654:61 2 30 000 ¼ 2654:61:

A

Exercise 5.8.5 November 2003 SOA Course 5 Examination, Problem No. 30. For a defined benefit pension plan, you are given the following. The actuarial cost method is the Traditional Unit Credit. Normal retirement benefit is 30 per month per year of service. The valuation interest rate is 7.0%. There are no pre-retirement decrements other than death. Assumed retirement age is 65. There are 50 active participants, all of whom are age 55, as of January 1, 2002. You are given this selected mortality rate: normal cost for 2002 is 150 000. All participants are still active as of January 1, 2003. Calculate the normal cost for 2003.

225

Valuation of pension plan liabilities

Solution The normal cost for 2002 is the actuarial present value on January 1, 2002 of the benefit accrued during that year, i.e., ð12Þ

150 000 ¼ 50 · 30 · 12 · 10 E55 · a€ 65

ð12Þ

¼ 50 · 30 · 12 · p55 · 1:0721 · 9 E56 · a€ 65

ð12Þ

¼ 50 · 30 · 12 · 0:98 · 1:0721 · 9 E56 · a€ 65 : Therefore 9 E56

ð12Þ

150 000 · 1:07 50 · 30 · 12 · 0:98 107 < 9:0986395: ¼ 11:76

· a€ 65 ¼

The normal cost for 2003 is the actuarial present value on January 1, 2003, of the benefit accrued during that year, i.e., ð12Þ

150 000 · 1:07 50 · 30 · 12 · 0:98 150 000 · 1:07 ¼ 0:98

50 · 30 · 12 · 9 E56 · a€ 65 ¼ 50 · 30 · 12 ·

¼ 163 775:51:

A

Exercise 5.8.6 November 2002 SOA Course 5 Examination, Problem No. 13. A pension plan effective date is January 1, 2002. The actuarial cost method used is Frozen Initial Liability (Entry Age Normal). You are given the following selected valuation results on January 1, 2002: the actuarial present value of future benefit is 23 500, Entry Age Normal accrued liability is 18 400, actuarial value of assets is 450, and actuarial present value of average working life annuity is 15. Calculate the Frozen Initial Liability (Entry Age Normal) normal cost on January 1, 2002. Solution Under this method, the normal cost is calculated as NC ¼ ¼

APVðBenefitsÞ 2 F 2 Frozen Initial Liability Average working life annuity 23 500 2 450 2 18 400 ¼ 310: 15

A

Exercise 5.8.7 November 2001 SOA Course 5 Examination, Problem No. 6. You are given the following information for a defined benefit plan. Plan effective date was January 1, 2001. Plan year is the calendar year. Normal retirement benefit

226

Chapter 5

is 2% of final 3-year average pay for each year of service. The actuarial cost method is Entry Age Normal. The valuation interest rate is 8.00%. Salary scale increases 5% annually. There are no pre-retirement terminations other than death. There are no inactive participants. The following data for the 10 active participants as of January 1, 2001 is given: age of all of them is 45, past service is exactly 5 years, 2001 salary is 25 000. Plan assets as of January 1, 2001 are 0. Selected annuity values are ð12Þ a€ 65 ¼ 10 and a€ 10 ¼ 7: The following selected commutation functions are given: D40 ¼ 245; D45 ¼ 180; D65 ¼ 50; N40 2 N65 ¼ 3200; N45 2 N65 ¼ 2100; S40 ¼ 1; S45 ¼ 1:2763; S65 ¼ 3:3864; S N40 2 S N65 ¼ 7100; S N45 2 S N65 ¼ 5000: The supplemental liability is amortized over 10 years. Determine the annual cost as of December 31, 2001 for the 2001 plan year based on an initial valuation as of January 1, 2001. Solution Note that this exercise uses the left superscript S; which indicates the quantity being increased with the salary increase every year. This concept is used in establishment of normal cost as a percentage of salary, as such normal cost increases every year with salary. Furthermore, the function Sx effectively indicates the ratio of salary at age x to the salary at age 40, and it is important to note that S Dx ¼ Sx · Dx : For each participant, the entry age was 40. The projected annual benefit rate at age 65 for each one of them is 0:02 · 25 ·

25 000 · 1:0519 þ 25 000 · 1:0518 þ 25 000 · 1:0517 < 30 107: 3

This solution is based on the assumption that the normal cost is a constant percentage of salary, such approach being indicated by the use of salary-based commutation functions. The initial individual normal cost is established based on the entry age of 40: D65 ð12Þ D ð12Þ 50 · a€ 65 B65 · 65 · a€ 65 · 10 30 107 · D40 D40 245 < 2120: ¼ S < S Sa 7100 € 40: 25 N40 2 N65 SD 1 · 245 40

B65 · NC40 ¼

But at age 45 the individual normal cost is increased by the same factor in comparison to age 40 normal cost, as the salary at age 45 in comparison to the salary at age 40, so that NC45 ¼ NC40 ·

S45 ¼ 2120 · 1:055 < 2706: S40

Since there are 10 participants, the total normal cost is 27 060. The total annual cost includes both the normal cost and the supplemental cost. Thus we also need to calculate the supplemental cost. To establish it, we start with the total supplemental liability. As the plan assets are zero, that supplemental liability is

227

Valuation of pension plan liabilities

the accrued actuarial liability. This accrued liability at age 45 is AL45 ¼ APV45 ðFuture BenefitsÞ 2 APV45 ðFuture Normal CostsÞ ¼ 30 107 ·

S D65 ð12Þ N 2 S N65 · a€ 65 2 2706 · 45S < 24 736: D45 D45

The total for 10 participants is 247 360. This cost is amortized over 10 years, and a€ 10 ¼ 7; thus the total annual amortization is 247 360 < 35 337: 7 The total annual cost as of January 1, 2001 is 27 060 þ 35 337 ¼ 62 397: If we express it as of December 31, 2001, we need to increase the January 1 figure by the valuation interest rate effect on the liability, arriving at 62 397 · 1:08 < 67 388:76:

A

Exercise 5.8.8 A pension plan uses the Traditional Unit Credit funding method. The valuation interest rate is 6%. There are no pre-retirement decrements other than death. The normal retirement age is 65. The benefit is 20 per month for each year of service. On January 1, 2003, there are 10 active participants, all aged 50. The total normal cost for 2003 as of January 1, 2003 is 20 000. We also know that p50 ¼ 0:9: Calculate total normal cost for 2004 as of January 1, 2004. Assume that the assumed mortality is exactly the mortality experience in 2004. Solution The normal cost per participant at age 50 is NC50 ¼ 20 · 12 · v15 ·

ð12Þ

51 p50

· a€ 65 :

14 p51

· a€ 65 :

At age 51, the normal cost per participant is NC51 ¼ 20 · 12 · v14 ·

ð12Þ

We see therefore that NC50 ¼ v· NC51

15 p50 14 p51

¼ vp50 ;

so that NC51 ¼

ð1 þ iÞNC50 : p60

Because there are 10 participants on January 1, 2003, normal cost per participant at that time is NC50 ¼ 2000: Therefore, the normal cost on January 1, 2004 per

228

Chapter 5

participant is 1:06 · 2000 < 2355:56: 0:9 There are nine participants surviving as of January 1, 2004, thus the total normal cost is 1:06 · 2000 ¼ 21 200: A 9· 0:9 NC51 ¼

Exercise 5.8.9 A pension plan for a computer software company uses the Projected Unit Credit funding method. Normal retirement benefit is 2% of final salary per year of service. The valuation interest rate is 6%. Salaries are assumed to increase 3% per year. There are no pre-retirement deaths or other decrements. Normal retirement age is 65. You are also given that as of January 1, 2004, there are two participants: Mr Wang and Mr Kowalski. Mr Wang was born on January 1, 1954 and hired on January 1, 1984. Mr Wang’s salary as of January 1, 2004 is 60 000 per annum. Mr Kowalski was born on January 1, 1964 and hired on January 1, 1994. Mr Kowalski’s annual salary is 50 000. What is the normal cost for 2004 as of January 1, 2004? Solution As on January 1, 2004 valuation date, Mr Wang is 50, and has 15 years till retirement. His projected salary during the last year of service is 60 000 · 1:0314 < 90 755:38: The pension benefit that accrues in 2004 is 2% of projected salary, and the actuarial present value of that amount on valuation date is 0:02 · 60 000 · 1:0314 ·

15 E50

ð12Þ

· a€ 65 ¼ 1200 ·

1:0314 · 9 < 6816:43: 1:0615

This represents the portion of normal cost attributed to Mr Wang. Mr Kowalski is 40 years old on the valuation date and his projected salary during the last year of service is 50 000 · 1:0324 < 101 639:70: The pension benefit that accrues in 2004 is 2% of projected salary, and the actuarial present value of that amount on valuation date is 0:02 · 50 000 · 1:0324 ·

25 E40

ð12Þ

· a€ 65 ¼ 1200 ·

1:0324 · 9 < 5115:29: 1:0625

The total normal cost is 6816:43 þ 5115:29 ¼ 11 931:72:

A

Exercise 5.8.10 In the pension plan under consideration the Entry Age Normal (percentage of salary) method is used. Normal retirement benefit is 80% of final 2-year average salary. The valuation interest rate is 6%. Salary increases are assumed

229

Valuation of pension plan liabilities

to be 3% annually. There are no pre-retirement deaths or other decrements. All retirements are assumed to occur at the normal retirement age of 65. We are also ð12Þ given that a€ 65 ¼ 10: There is only one participant born on January 1, 1954. The date of hire for this participant was January 1, 1989. The 2004 salary of the participant is 100 000. What is the normal cost for 2004 as of January 1, 2004? Solution As of January 1, 2004, the participant is 50 years old, and has 15 years of service. The projected retirement benefit is B65 ¼ 0:80 · 100 000 ·

1:0313 þ 1:0314 < 119 244:94: 2

Since there are no deaths prior to retirement, the normal cost is paid as a deterministic salary-based annuity starting at entry age of 35 and ending at retirement age of 65, so that ð12Þ

NC35 · S a€ 65235 ¼ B65 · 1:06230 · a€ 65 : If we define j¼

1:06 2 1; 1:03

then S

a€ 65235 ¼ 1 þ 1 ·

1:03 þ ··· þ 1· 1:06

1:03 1:06

29 ¼ a€ 30 j :

Therefore NC35 ¼

119 244:94 · 10 · 1:06230 119 244:94 · 10 · 1:06230 < 10 176:79: < a€ 30 j 20:4011

This constitutes the following percentage of salary at age 35: 10 176:79 < 15:86%: 1:03215 · 100 000 At age 50, 15.86% of the salary is 15 860.

A

Exercise 5.8.11 A pension plan becomes effective on January 1, 2004. The valuation method used is individual level premium. Normal retirement benefit is 1000 per month. The valuation date is December 31, 2004. The valuation interest rate is 8%. There are no pre-retirement deaths or any other decrements. All retirements occur ð12Þ at the normal retirement age of 65. You are given that a€ 65 ¼ 8: There are two participants, both born on January 1, 1974. What is the normal cost for 2004 as on December 31, 2004?

230

Chapter 5

Solution The normal cost is calculated as the level annuity payment from the plan inception date to retirement, sufficient to fund the projected benefit at retirement. In this case, as the two participants are both 30 and have 35 years till retirement ð12Þ NC · S€ 35 ¼ 2 · 1000 · 12 · a€ 65 ;

therefore NC ¼

24 000 · 8 < 1031:70: S€ 35 0:08

A

Exercise 5.8.12 A pension plan is being valued on January 1, 2004. The actuarial funding method used is the Aggregate method. The valuation interest rate is 6%. There are no pre-retirement decrements. Salaries are assumed to increase 1% per year. ð12Þ It is known that a€ 65 ¼ 10: The normal cost is 10% of salary. There is only one participant, aged 55. Current salary of the participant is 50 000 per year. The assets of the plan are valued at 40 000 on January 1, 2004. Find the expected monthly benefit for this participant at normal retirement age of 65. Solution Under the Aggregate method, in the case of this plan, the nominal amount of the normal cost in 2004 is NC ¼

APVðFuture BenefitsÞ 2 Assets : Sa € 10

In this case, we have S

a€ 10 ¼ 1 þ

1:01 þ ··· þ 1:06

1:01 1:06

9 ;

and with 1:06 2 1 < 4:9505%; 1:01

j¼ we have S

a€ 10 ¼ a€ 10 j < 8:1235172:

Therefore APVðFuture BenefitsÞ ¼ Assets þ NC · S a€ 10 < 40 000 þ 5000 · 8:1235172 < 80 617:59: Let us write x for the monthly retirement benefit. Then ð12Þ

80 617:59 < APVðFuture BenefitsÞ ¼ 1:06210 · 12 · x · a€ 65 < 67:007373x: Therefore x<

80 617:59 < 1203:12: 67:007373

A

231

Chapter 6 Random fluctuation of pension liabilities

6.1 Formulation of the problem The three key variables studied in the process of pension plan valuation are: normal cost, accrued liability, and unfunded actuarial liability. In order to calculate them, we ð12Þ have consistently used the expression a€ y ; the actuarial present value of a monthly life annuity due with monthly benefit of 1/12, payable to an average retiree aged y. ð12Þ The value of the actuarial function a€ y is calculated based on the probability distribution of a future lifetime of y in the population studied. While its theoretical value is firmly established as the expected present value of a life annuity random variable, its practical estimation is quite a challenge. The first source of uncertainty of its estimate is the difference between the actual probability distribution of the future lifetime, and the one estimated by the actuary. To use the statistical terminology, a standard question concerning this source of uncertainty is whether the life expectancy estimate is unbiased. But even if we can answer the questions concerning the estimate of the probability distribution of future lifetime in the affirmative, and if we indeed obtain the correct probability distribution of this variable (a daunting task, as we have to continuously update and correct our mortality tables), we still have the second wave of issues to address, such as: 1. How large can the random fluctuations of the future lifetime of a retiree (this is also the duration of a life annuity paid by a pension plan)? 2. How can we estimate random fluctuations of our estimate of pension liabilities? The answer to the first question is provided in Bowers [14] as well as in [4], but, to our best knowledge, the second question remained largely ignored in the actuarial literature. But an estimate of the random fluctuation of pension liabilities (and any other insurance liabilities, we believe) is vital to our understanding of the relationship of the plan assets and liabilities, as well as to the planning of the process of investment

232

Chapter 6

and reserving. A new approach to this issue has been proposed in the work of Gajek [27], in which a partial answer to the second question is provided. We will present the ideas of that work here. Section 6.2 analyzes in detail random fluctuations of the present value of a life annuity for the continuous future lifetime model, and for a continuous form of interest accrual. Section 6.3 provides an evaluation of the expected value of the square error of the relative deviation of the plan liabilities, if those liabilities are established with the Unit Credit method (see Section 5.1), from the actual liabilities. We do not, however, establish any estimates of the precision of the actuarial assumptions concerning the withdrawal or retirement times. Deviations of the actual experience from the actuarial assumptions lead to actuarial gains or losses, as presented in Chapter 5 (see also [4]).

6.2 Fluctuation of the present value of a life annuity Some of the results presented in this section can be found in [4] and [27]; and some of the results concerning the coefficient of variation tT were presented in [27]. Let us assume that pensions are paid in a continuous manner, and that interest accrues at a constant force dðtÞ ¼ d: Let T be the future lifetime of a randomly chosen plan participant (of random, or given specifically, gender), who is alive at age x. We will assume that T is an absolutely continuous random variable with density f. Then Prðt # T # t þ dtÞ ¼ f ðtÞdt; and we know that f ðtÞ ¼t px · mxþt ;

ð6:2:1Þ

where t px is the probability of survival for another t years for a participant aged x, and mxþt is the force of mortality (i.e., the hazard rate function for the random variable T ) at age x þ t. Let us note that the expected value of the variable T, EðTÞ ; e x ; is given by the formula ð1 ð6:2:2Þ e x ¼ t px dt; 0

while the variance of T, VarðTÞ ; s 2 ; is given by: 2

s ¼2

ð1 0

tt px dt 2 ðex Þ2

ð6:2:3Þ

[see also the formula (3.7.18)]. ð12Þ We will consider in this section, instead of a€ y ; the present value a T of a continuous life annuity paying at a rate of one monetary unit per year.

233

Random fluctuation of pension liabilities

6.2.1 The expected value of the random variable a T As we already know [see the formula (3.10.25)] a x ; Eða T Þ ¼ ¼

ð1 0

1 ð1 2 e2dt Þf ðtÞdt ¼ d

ð1 0

1 ð1 2 e2dt Þ · t px · mxþt dt d

1 1 2 A x; d d

ð6:2:4Þ

where A x is the single benefit (i.e., net) premium for a unit fully continuous whole life insurance with the benefit payable at the moment of death. We will show that a x is a decreasing function of the force of interest d. To that effect, let us calculate the derivative d 1 a x ¼ 2 dd d

ð1

½ðdt þ 1Þe2dt 2 1 f ðtÞdt:

ð6:2:5Þ

0

We have, for an arbitrary d . 0 and t . 0; the following inequality:

dt þ 1 , e d t : Therefore, for every t . 0 ðdt þ 1Þe2dt 2 1 , 0: Combining the above with (6.2.5), we conclude that d a , 0 dd x for every d . 0: Therefore a x is a decreasing function of d. From this and (6.2.4) we conclude that sup a x ¼ lim a x ¼ lim d$0

dd0

dd0

ð1 0

1 2 e2dt f ðtÞdt ¼ d

ð1 0

t f ðtÞdt ¼ EðTÞ ¼ e x :

ð6:2:6Þ

6.2.2 The variance of the random variable a T The crucial issue in our analysis is the variability of the random variable a T with respect to its expected value a x : The best measure of such a variability is Varða T Þ:

234

Chapter 6

Clearly Varða T Þ ¼ ¼

ð1 0

a 2T f ðtÞdt 2 a 2x

ð1 0

2

1 2 e2dt d

f ðtÞdt 2

ð1 0

2 1 2 e2dt f ðtÞdt : d

ð6:2:7Þ

Since ð1 0

a 2T

ð1

2 1 2dt f ðtÞdt ¼ ð1 2 e Þ f ðtÞdt d 0 ð ð 1 2 1 2dt 1 1 22dt e f ðtÞdt þ 2 e f ðtÞdt; ¼ 2 2 2 d d 0 d 0

then, using (6.2.7) and (6.2.4), we get Varða T Þ ¼ d

22

ð 1 e

22dt

f ðtÞdt 2

ð 1 e

0

2dt

2 f ðtÞdt

:

ð6:2:8Þ

0

We will show now that Varða T Þ is a decreasing function of d. To that effect, let us d calculate the derivative Varða T Þ: From (6.2.8) we conclude that dd ð1 ð1 ð1 d Varða T Þ ¼ 2d23 2 dt e22dt f ðtÞdt þ e2dt f ðtÞdt · dt e2dt f ðtÞdt dd 0 0 0 ð 1 2 ð 1 22dt 2dt e f ðtÞdt 2 e f ðtÞdt 2 0

0

ð1 23 2 ðdt þ 1Þe22dt f ðtÞdt ¼ 2d 0

þ

ð1

ðdt þ 1Þe

2dt

f ðtÞdt ·

ð1

0

e

2dt

f ðtÞdt :

0

ð6:2:9Þ Let us note that ð ð ð d 23 Varða T Þ ¼ 2d 2 hgf þ hf · gf ; dd where hðtÞ ¼ ðdt þ 1Þexpð2dtÞ and gðtÞ ¼ expð2dtÞ: Since both g and h are strictly decreasing functions for d . 0; it follows from the Tchebyshev Inequality that: d Varða T Þ , 0; dd

235

Random fluctuation of pension liabilities

and, therefore, Varða T Þ is a decreasing function of the parameter d, as long as d . 0: From this we can also conclude that there exists a limit limdd0

d Varða T Þ; dd

possibly equal to þ1; and that such a limit is the greatest possible value of the variance Varða T Þ: Let us calculate that limit:

lim Varða T Þ ¼ lim dd0

ð 1 "

dd0

¼

ð1

0

1 2 e2dt d

ð 1

2

t 2 0

2 # 2 ð 1 1 2 e2ds f ðsÞds 2 f ðtÞdt d 0 2

sf ðsÞds

f ðtÞdt ¼ VarðTÞ;

ð6:2:10Þ

0

assuming that VarðTÞ , 1 (we will, in fact, always assume that). Thus the greatest possible value of the deviation of a T away from a x ; if measured by the variance, is equal to VarðTÞ: 6.2.3 The coefficient of variation of the random variable a T Let us now turn our attention to the coefficient of variation ta T of the random variable a T : By definition:

ta T

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Varða T Þ ; ¼ Eða T Þ

ð6:2:11Þ

and, as we have just showed above, the numerator and the denominator of the righthand side of (6.2.11) are decreasing functions of d. Thus we cannot make an immediate conclusion as to whether ta T is an increasing, decreasing, or constant function of the parameter d. However, we can show, using (6.2.4) and (6.2.7), that it is, in fact, a strictly decreasing function of d. Note that t2a T may be expressed in the following way: ð1

t2a T

¼

0 ð 1

ð1 2 e2dt Þ2 f ðtÞdt 2 2 1: 2dt ð1 2 e Þf ðtÞdt

0

Let us denote t2a T ; treated as a function of the parameter d, by fðdÞ:

236

Chapter 6

Then ð1 0

ð1 2 e

0

f ð dÞ ¼ 2

2dt

ð1

2dt

Þt e f ðtÞdt ð1 2 e2dt Þf ðtÞdt 0 ð 1 3 2dt ð1 2 e Þf ðtÞdt 0

ð1 0

2

ð1

2dt 2

ð1 2 e Þ f ðtÞdt t e2dt f ðtÞdt 0 ð 1 3 2dt ð1 2 e Þf ðtÞdt 0

ð1 ¼2

0

ð1 ð1 2 e2dt Þf ðtÞdt t e2dt f ðtÞdt 0 ð 1 3 2dt ð1 2 e Þf ðtÞdt 0

2 6 6 4

ð1 0

2dt

t e f ðtÞ dt 2 ð1 2 e2dt Þ ð 1 t e2dt f ðtÞdt

ð1 0

3 7 ð1 2 e Þf ðtÞ dt7 ð1 2 e2dt Þ ð 1 5: ð1 2 e2dt Þf ðtÞdt

0

2dt

0

ð6:2:12Þ Observe that the expression in the square bracket (denote it by a) has the following form: ð1 ð1 2dt a ¼ ð1 2 e Þg1 ðtÞdt 2 ð1 2 e2dt Þg2 ðtÞdt; ð6:2:13Þ 0

0

where t e2dt f ðtÞ g1 ðtÞ ¼ ð 1 t e2dt f ðtÞdt 0

and ð1 2 e2dt Þf ðtÞ g2 ðtÞ ¼ ð 1 ð1 2 e2dt Þf ðtÞdt 0

are the densities of two probability distributions on the half-axis ð0; 1Þ; whose cumulative distribution functions we will write as G1 and G2, respectively. It can be shown relatively easily that the probability distribution with the cumulative distribution function G1 is stochastically dominated by the distribution with the

237

Random fluctuation of pension liabilities

cumulative distribution function G2, and then one can utilize the corresponding inequality for the integrals with respect to these probability distributions. Below, we will show directly, without using that property of G1 and G2, that a , 0: First note that g1 and g2 are differentiable and, for some constant c (possibly dependent on d), the expression g2 ðtÞ edt 2 1 ¼c g1 ðtÞ t is, for d . 0; an increasing function of the variable t. Indeed, since for every d . 0; t . 0; the following inequality holds e2dt . 1 2 dt; then edt ð1 2 dtÞ # 1; and we conclude that d g2 edt ðdt 2 1Þ þ 1 . 0: ¼ dt g1 t2 Since g2 and g1 are differentiable, their ratio is increasing. We also know that the integrals of both g2 and g1 over the real line are equal to 1, thus there is a t0 . 0 such that for t . t0 g2 ðtÞ . 1; g1 ðtÞ while for 0 , t , t0 the inequality above holds in the opposite direction. The righthand side of (6.2.13) can be expressed as: ð1 g ðtÞ a ¼ ðe2dt0 2 e2dt Þ 1 2 2 g ðtÞdt g1 ðtÞ 1 0 ð t0 g2 ðtÞ 2dt0 2dt 2e Þ 12 ¼ ðe g ðtÞdt g1 ðtÞ 1 0 ð1 g2 ðtÞ 2dt0 2dt 2e Þ 12 ð6:2:14Þ ¼ þ ðe g ðtÞdt , 0; g1 ðtÞ 1 t0 where the last inequality follows from the fact that in both integrals the integrand is negative. It follows from (6.2.12) and (6.2.14) that f0 ðdÞ , 0 for every d . 0; and, therefore, ta T is a decreasing function of the parameter d. Thus pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Varða T Þ VarðTÞ ¼ tT ; ¼ ð6:2:15Þ sup ta T ¼ lim d d0 EðTÞ Eð a Þ d.0 T where in the second equality one uses (6.2.6) and (6.2.10), while tT is the coefficient of variation of the random variable T.

238

Chapter 6

6.2.4 The Gompertz Law of Mortality We will now estimate the expressions E(T ), Var(T ) and tT ; for a typical probability distribution of the future lifetime of (x). Of course, the meaning of the word “typical” is not clear in this context, but let us note that the actuarial literature generally considers the Gompetz Law to be a satisfactory model of the actual mortality for retirement ages. This model assumes

mxþt ¼ mx ebt ; where mx is the force of mortality at age (x) (also called the hazard rate of the random variable T ¼ TðxÞ; while b is some constant). Since ðt h m i mxþs ds ¼ exp 2 x ðebt 2 1Þ ; ð6:2:16Þ t px ¼ exp 2 b 0 then, using (6.2.1), we obtain the following expression for the probability density function of the random variable T ¼ TðxÞ under the Gompertz Law of Mortality: h i m f ðtÞ ¼ mx exp bt 2 x ðebt 2 1Þ : b

Example 6.2.1 Figure 6.1 shows the graph of the density f for the parameter mx ¼ 0:0204 and b ¼ 0:097; which fit quite adequately the actual mortality table dated 1971 for the population of men aged 65 in the United States (see [77]). Using (6.2.2) and (6.2.16), we further obtain ð1 h m i e x ¼ exp 2 x ðebt 2 1Þ dt b 0 ð m 1 mx =b 1 expð2eu Þdu ; emx =b H ln x ; ¼ e ð6:2:17Þ b b b lnðm =bÞ x

A where the function HðtÞ ¼

ð1

expð2eu Þdu

t

is defined and differentiable infinitely many times (i.e., it belongs to the class C 1) on the entire real line. Furthermore, it is easy to see that this function is strictly decreasing and strictly convex. The graph of the function H(t) is given in Fig. 6.2. We introduce a new parameter u ¼ lnðmx =bÞ: Then (6.2.17) can be rewritten as e x ¼

1 HðuÞ=expð2eu Þ: b

ð6:2:18Þ

239

Random fluctuation of pension liabilities

Fig. 6.1. The graph of the density for the Gompertz Law of Mortality.

Using a standard numerical software package one can use the expressions (6.2.17) or (6.2.18) to calculate the values of the parameters mx and b. We will now use the formulas (6.2.3) and (6.2.16) to calculate the second moment of the random variable T: ð1

ð1

h m i t exp 2 x ðebt 2 1Þ dt b 0 0 ð1 2 m ¼ 2 emx =b u 2 ln x expð2eu Þdu b b lnðmx =bÞ ð1 2 ¼ 2 ðu 2 uÞexpð2eu Þdu=expð2eu Þ: b u

EðT 2 Þ ¼ 2

tt px dt ¼ 2

Fig. 6.2. The graph of the function H(t).

ð6:2:19Þ

240

Chapter 6

For a t [ R let us define a new function ð1 ðu 2 tÞexpð2eu Þdu; GðtÞ ¼

ð6:2:20Þ

t

which is non-negative, differentiable inifinitely many times, strictly decreasing, strictly convex, and has the following additional property: G0 ðtÞ ¼ 2HðtÞ:

ð6:2:21Þ

Using the definition of the function G, we can rewrite the expression (6.2.19) as EðT 2 Þ ¼

2 GðuÞ=expð2eu Þ: b2

ð6:2:22Þ

The graph of the function G is given in Fig. 6.3. The values of the function G can be established, for given values of parameters mx and b, numerically, and then one can use (6.2.19) or (6.2.22) to calculate the second moment EðT 2 Þ: Using (6.2.18) and (6.2.22) we obtain a formula for the variance of the random future lifetime: 2GðuÞexpð2eu Þ 2 H 2 ðuÞ GðuÞexpð2eu Þ 2 VarðTÞ ¼ ¼ ðex Þ 2 2 1 : ð6:2:23Þ H 2 ð uÞ ðb expð2eu ÞÞ2 The coefficient of variation tT of the random variable T is given by the formula: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VarðTÞ tT ¼ : e x

Fig. 6.3. The graph of the function G(t).

241

Random fluctuation of pension liabilities

Thus from (6.2.23) it follows that: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ GðuÞexpð2eu Þ tT ¼ 2 2 1; H 2 ð uÞ

ð6:2:24Þ

and, therefore, tT depends on b only through their common relationship to the parameter u. We will show that the coefficient of variation tT converges to 0 as u ! 21: Note first that lim H 0 ðuÞ ¼ lim ð21Þexpð2eu Þ ¼ 21:

u!21

u!21

ð6:2:25Þ

Using de l’Hospital Rule, (6.2.21) and (6.2.25), we obtain:

lim

u!21

GðuÞ G 0 ð uÞ 21 1 ¼ lim ¼ : ¼ lim 0 0 2 u !21 u !21 2HðuÞH ðuÞ 2H ðuÞ 2 H ð uÞ

ð6:2:26Þ

It follows from (6.2.24) and (6.2.26) that under the Gompertz Law of Mortality

lim

u!21

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VarðTÞ ¼ 0; e x

i.e., for small values of the parameter u the coefficient of variation tT is approximately equal to zero. It should be noted that the coefficient of variation tT depends only on the ratio of the parameters mx and b in this case.

Example 6.2.2 For mx ¼ 0:01 and b ¼ 0:15; we have u ¼ 22:71; and the exact value of the coefficient of variation is tT ¼ 0:417: If, on the other hand, b ¼ 1:5; then for the same value of mx, we have u ¼ 25:01 and tT ¼ 0:27: A We will show now that for large values of the parameter u, the coefficient of variation tT is approximately equal to 1. Strictly speaking, we will show that: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VarðTÞ lim ¼ 1: u!1 e x

ð6:2:27Þ

Since HðuÞ ! 0; when u ! 1; then, on one hand, using de l’Hospital Rule, we get:

lim

u!1

e2u expð2eu Þ 2e2u expð2eu Þ 2 expð2eu Þ ¼ lim ¼ 1: u!1 HðuÞ 2 expð2eu Þ

ð6:2:28Þ

242

Chapter 6

This immediately gives lim

u!1

e2u H 0 ðuÞ ¼ 21: HðuÞ

On the other hand, using de l’Hospital Rule, the expression (6.2.21), and the above limit, we get: lim

u!1

GðuÞ G 0 ð uÞ ¼ lim u!1 2 e2u HðuÞ þ e2u H 0 ðuÞ e2u HðuÞ 1 ¼ lim ¼ 1: u!1 H 0 ðuÞ 2 u 2 u e 2e HðuÞ

ð6:2:29Þ

It follows from (6.2.28) and (6.2.29) that lim ¼

u!1

GðuÞexpð2e2u Þ ¼ 1: H 2 ðuÞ

If we consider this result in (6.2.24), we arrive at (6.2.27). The approximation pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VarðTÞ <1 e x becomes more precise, the more mx exceeds b. Example 6.2.3 For example, if mx ¼ 0:55 and b ¼ 0:07; then u ¼ 2:06 and tT ¼ 0:91: Also, in Example 6.2.1 considered previously, concerned with men in the United States who survive to age 65, mx ¼ 0:0204; b ¼ 0:097 and u ¼ 21:5592: The more precise value of the coefficient of variation for the random variable T; calculated directly from (6.2.23), is 0.526. A Figure 6.4 shows the graph of the coefficient of variation tT as a function of the parameter u, for the Gompretz Law.

6.3 Confidence intervals for accrued liabilities of a pension plan 6.3.1 General case Assume that all plan members retire at the same age y. For the purpose of the theoretical model discussed in this section, we will use the continuous version of the Unit Credit method. In this approach, benefits accrue continuously during the working years, and are paid continuously at retirement. Accrued liabilities ALt at time t with respect to the group At of active participants, are given by the

243

Random fluctuation of pension liabilities

Fig. 6.4. The graph of tT (u).

following formula: ALt ¼

X

Bj ðxj Þay

j[At

X j Dy Dy ¼ a y B ðxj Þ ; Dxj Dxj j[A t

where Dx ¼ vx lx for any age x. Let us denote by T 0j the future lifetime while in retirement for the jth plan participant, given that this participant survived in the plan till retirement. If the participant does not survive in the plan till retirement, set T 0j ¼ 0 by definition. Then the present value of the random retirement payments for all plan participants, calculated at time t is: X ALpt ¼ Bj ðxj ÞaT 0 v y2xj : j

j[At

The relative variability of ALpt away from ALt is given by the formula X l xj ALpt 2 ALt 21 a 0 2 a y pj ; ¼ a y ð6:3:1Þ DALt ; ALt ly T j j[A t

where Dy Dxj pj ¼ X Dy Bj ðxj Þ D xj j[A Bj ðxj Þ

ð6:3:2Þ

t

are positive weights summing up to 1. Let us define additional random variables Zj as: Zj ¼ 1; if the plan participant survives in the plan till retirement, and Zj ¼ 0 otherwise. Note that the random

244

Chapter 6

variables Zj have the Bernoulli distribution with the probability of success given by: PrðZj ¼ 1Þ ¼

ly : l xj

Observe also that: E

l xj ly

a T 0

j

lxj a 0 lZ ¼E E ¼ Eða T 0 Þ ¼ a y ; ly T j j

ð6:3:3Þ

where T is the future lifetime of a plan participant who survives till age y. The random variable T is defined on a different probability space and has a different probability distribution than the random variables T 0j : Thus DALt is a weighted average of the deviations of the coefficients ðlxj =ly Þa T 0 j from their expected value a y : We are interested in estimating the deviation DALt in the situation when the rate of return d $ 0 assumes the worst possible value. If the group of plan participants is uniform (in the sense that the equality (6.3.3)) holds, then from (6.3.1) we conclude that EðDALt Þ ¼ 0: In order to estimate the average value of the random fluctuations of DALt, we will calculate the variance of the deviation of DALt. If the random variables T 0j are independent then: 2 1 X 2 lxj VarðDALt Þ ¼ 2 pj VarðaT 0 Þ: ð6:3:4Þ j ly a y j[A t

Note that VarðaT 0 Þ ¼ VarðEðaT 0 lZj ÞÞ þ EðVarðaT 0 lZj ÞÞ: j

j

j

ð6:3:5Þ

We have ( EðaT 0 lZj Þ ¼

0;

when Zj ¼ 0

Eða T Þ; when Zj ¼ 1:

j

Since Eða T Þ ¼ a y ; then ly VarðEðaT 0 lZj ÞÞ ¼ a 2y j lxj

ly 12 l xj

! :

On the other hand ( VarðaT 0 lZj Þ ¼ j

0;

when Zj ¼ 0

Varða T Þ; when Zj ¼ 1:

ð6:3:6Þ

245

Random fluctuation of pension liabilities

Therefore, ly EðVarðaT 0 lZj ÞÞ ¼ Var a T j l xj

! ð6:3:7Þ

:

After substituting (6.3.6) and (6.3.7) into (6.3.5), we obtain ! ! ly ly 2 ly VarðaT 0 Þ ¼ a y 12 þ Var a T : j lxj l xj l xj

ð6:3:8Þ

From the above and from (6.3.4) we conclude that: VarðDALt Þ ¼

X

lx p2j j ly j[At

ly Varða T Þ 12 þ lxj a 2y

X

!

or the equivalent formula: Var ðDALt Þ ¼

j[At

p2j y2xj pxj

y2xj qxj þ

Varða T Þ : a 2y

ð6:3:9Þ

After utilizing the Schwartz Inequality we arrive at the estimate: X 2 Varða Þ X 2 y2xj qxj þ Varða T Þ=a2y 1 Varða T Þ T # p # pj : j nt a 2y a 2y y2xj pxj j[At j[At From the above and from (6.3.9) we arrive at the following estimate from below for Var(DALt): Varða T Þ # VarðDALt Þ: nt a 2y By using (6.3.9) we also arrive at the following estimate from above: ( ) a T Þ=a2y qp þ Var ða T Þ=a2y y2xj qxj þ Var ð ; Var ðDALt Þ # max pj # pp j[At pp y2xj pxj where

pp ¼ max pj ; j[At

pp ¼ min y2xj pxj j[At

and

qp ¼ 1 2 pp :

Let us recall that we are interested in estimating the variance of the deviation in the worst possible case. From (6.2.15) we conclude that supd$0 Varða T Þ=a2y is attained when d ¼ 0: In that case Varða T Þ ¼ VarðTÞ ¼ s 2 [see (6.2.10)] and a y ¼ e y ; where e y is the expected value of the future lifetime of the persons who survived till age y. Unfortunately, from (6.3.9) it follows that VarðDALt Þ depends on d not just through Varða T Þ=a2y but also through pj [see formula (6.3.2)].

246

Chapter 6

Since for every d $ 0 the weights pj sum up to 1, it is not possible that all coefficients pj are monotone with respect to d in the same fashion. However, it turns out that under some appropriate groupings of data, the form of relationship of pj and the force of interest does not significantly affect our analysis. We will study this issue in the next section.

6.3.2 Cohorts Let us assume that the set At is divided into l subsets—cohorts At,i, i ¼ 1; …; l; such that for all members of a cohort At,i the expression Bj ðxj ÞðDy =Dxj Þ is approximately constant (we will denote it by bi). Then by (6.3.1) we have:

DALt ¼

a 21 y

l X i¼1

X l xj bi a 0 2 a y ; l X ly T j j[At; i bi nt;i i¼1

where nt,i is the number of participants in the cohort At,i. The parameters bi, together with the number of participants nt,i in the cohorts At,i describe the age and pay structure of the plan under consideration. Let us note that this structure evolves over time depending on factors such as changes in employment, changes in pay, general employment policy, etc. If the age and pay structure is such that the number of members of a cohort decreases and is inversely proportional to bi ; i.e., nt;i ¼ c=bi ; for some constant c, then DALt ¼

l 1 X 1 X l xj a T 0 2 a y : a y l i¼1 nt;i j[A ly j

ð6:3:10Þ

t;i

Thus the relative precision of the estimate of accrued liabilities is an average over the cohorts of the average relative deviations inside the cohorts of the coefficients ðlxj =ly ÞaT 0 away from their expected value a y : j Let us note that the method of grouping of plan participants considered here causes all cohorts to have approximately the same total accrued liabilities. This makes the cohorts so defined roughly equal in their influence on the overall plan funding. Assuming that the probability distribution of the future lifetime of plan all plan participants is the same (outside of the obvious age differences), we obtain, by utilizing (6.3.10) and (6.3.3): EðDALt Þ ¼ 0:

247

Random fluctuation of pension liabilities

Now assuming that the future lifetimes for plan participants are mutually independent, we obtain, by using (6.3.10) and (6.3.8): VarðDALt Þ ¼

2 l 1 X 1 X l xj VarðaT 0 Þ j ly a 2y l2 i¼1 n2t;i j[A t;i

¼

l 1 X 1 X l2 i¼1 n2t;i j[A

t;i

1 ð1 2y2xj pxj þ t 2a T Þ; y2xj pxj

ð6:3:11Þ

since ly =lxj ¼ y2xj pxj : Let us note here that the random variable T denotes the future lifetime for plan participants upon attaining retirement age—and its probability distribution is assumed to be given by the tables applicable to the entire population. On the other hand, the length of time of survival in the pension plan, which may end not just in death, but also in withdrawal, disability, etc. is a random variable with the distribution given by the probabilities y2xj pxj : If all plan participants are in the same cohort ðl ¼ 1Þ; then VarðDALt Þ ¼

1 X n2t j[A t

1 ð1 2y2xj pxj þ t2a T Þ: p y2xj xj

This formula can be immediately derived from (6.3.9) if we assume that the sole cohort is obtained when pj < n21 for each j [ At : Similarly, the final formula t (6.3.11) follows directly from (6.3.9) after using the condition that pj < const dla j [ At;i : At a first glance, it might appear that since y2xj pxj are usually close to 1, the following approximation should work quite well: VarðDALt Þ <

l t2a T X 1 : 2 l i¼1 nt;i

ð6:3:12Þ

It is derived from (6.3.11) after substituting y2xj pxj by 1. But the approximation (6.3.12) may turn out to be imprecise. To analyze this, let us rearrange the formula (6.3.11) as follows: l t2a X 1 X VarðDALt Þ ¼ 2T l i¼1 n2t;i j[A

t;i

l 1 1 X 1 X þ 2 l i¼1 n2t;i j[A y2xj pxj

t;i

l l t2a X t2a þ 1 X 1 1 X ¼ 2T þ T2 l n2t;i j[A l i¼1 nt;i i¼1

t;i

1 21 y2xj pxj

! 1 21 : y2xj pxj

!

248

Chapter 6

Let 2

1 X Ht;i ¼ 4 nt;i j[A

t;i

be the harmonic mean of the probabilities VarðDALt Þ ¼

321 1 5 y2xj pxj

y2xj pxj

in the cohort At,i. Then

l l t2a T X t2a T þ 1 X 1 1 21 þ ðHt;i 2 1Þ: 2 2 n l l i¼1 nt;i t;i i¼1

ð6:3:13Þ

Let us compare the approximation (6.3.12) and the exact formula (6.3.13): the ratio 1 of the summand removed to the one substituted is expressed as: 1¼

l t2a T þ 1 X

t2a T

21 ðHt;i 2 1Þai ;

ð6:3:14Þ

i¼1

Pl 21 where ai ¼ n21 i¼1 nt;i : Therefore 1 is a weighted average of the deviations away t;i = from 1 of the reciprocals of harmonic means of probabilities y2xj pxj for all cohorts, multiplied by the coefficient

t2a T þ 1 t2a T

;

which may turn out to be very large for small coefficients of variation t2a T : Example 6.3.1 We will now estimate the order of 1 using an example of a pension plan with 144 participants grouped into l ¼ 6 cohorts. The number of participants in each cohort and the harmonic means Ht,i of the probabilities y2xj pxj ; j [ At;i ; in the cohorts, are given in Table 6.3.1. Table 6.3.1. i nt,i Ht,i

1 8 0.985

2 12 0.983

3 20 0.971

4 27 0.860

5 33 0.805

6 44 0.743

Utilizing the formula (6.3.14) we obtain 1 ; 1ðta T Þ ¼ 0:074

t2a T þ 1 t2a T

:

ð6:3:15Þ

The graph of the relationship between 1 and the coefficient of variation t is given in Fig. 6.5. We can see quite easily that 1 is never less than 0.074 (which is the limit of 1

249

Random fluctuation of pension liabilities

Fig. 6.5. Graph of the function 1(t).

for t ! 1). On the other hand 1 may be arbitrarily large, for sufficiently small t. Thus the approximation (6.3.12) may turn out to be imprecise. A We conclude from (6.2.15) that the maximum value of Varða T Þ=a2y equals ðs=ey Þ2 : From the approximation (6.3.12) we obtain a simple estimate sup VarðDALt Þ < d$0

l s2 1 X 1 : 2 2 e y l i¼1 nt;i

ð6:3:16Þ

The formula (6.3.11) implies a more precise estimate sup VarðDALt Þ ¼ d$0

2 2 l 1 X 1 X 1 2y2xj pxj þ s =e l2 i¼1 n2t;i j[A y2xj pxj

ð6:3:17Þ

t;i

It follows from (6.3.10) that if the number of participants in each cohort is sufficiently large then DALt has, approximately, a normal distribution with the mean of zero and the variance given by the formula (6.3.11). Thus for an arbitrary d $ 0 the following inequalities hold with approximate probability 0.95 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u u l l u X X 1:96s t 1 1 1:96s u 1 t1 2 # DALt # ; ð6:3:18Þ 2 2 e y e y l i¼1 nt;i l i¼1 nt;i if we use the approximation (6.3.16).

Example 6.3.2 In the case when the future lifetime is described by the Gompertz Law with parameters mx ¼ 0:0204 and b ¼ 0:097; the standard deviation equals s ¼ 0:526 e y (see Example 6.2.3). By using this in (6.3.12), we obtain the following

250

Chapter 6

inequalities for an arbitrary d $ 0 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u u l l u1 X u1 X 1 1 2 1:031t 2 # DALt # 1:031t 2 ; l i¼1 nt;i l i¼1 nt;i which are satisfied, approximately, with probability 0.95, if the formula (6.3.16) is sufficiently precise. A If the approximation (6.3.16) is not sufficiently precise, then instead of (6.3.18) we must use a better confidence interval vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 u X u X u 1 l 1 X y2xj qxj þ s 2 =e2 u 1 l 1 X y2xj qxj þ s 2 =e2 t @21:96t A; ; 1:96 l2 i¼1 n2t;i j[A l2 i¼1 n2t;i j[A y2xj pxj y2xj pxj 0

t;i

t;i

which for every d $ 0 has the confidence level of approximately 0.95.

Example 6.3.3 In this example we will consider the population of men in Poland who survived to age y ¼ 65: Using the mortality tables we can calculate their additional life expectancy (in years) as e y ¼ 12:878; with variance s 2 ¼ 59:894: Thus

s ¼ 0:601: e y Consider a pension plan with 144 participants, as described in Example 6.3.1. From the approximation (6.3.16) we obtain sup VarðDALt Þ < 0:00348:

ð6:3:19Þ

d$0

On the other hand, from (6.3.15) we have 1 ¼ 0:074

s 2 =e2y þ 1 ¼ 0:279: s 2 =e2y

Thus, despite the simplicity of the approximation (6.3.16), to calculate supd$0 VarðDALt Þ we should use the more complete formula (6.3.17). Then sup VarðDALt Þ ¼ 0:00445: d$0

A

If we calculate Var(DALt) using (6.3.12), in the case when there are many cohorts, we must divide the square of the coefficient of variation a T by the product of the number of

251

Random fluctuation of pension liabilities

cohorts and the harmonic mean of the number of participants in the cohorts. Therefore, Var(DALt) is, roughly, inversely proportional to the number of plan participants. Nevertheless, the length of the confidence intervals for DALt may be quite large even for plans with a large number of participants, in the range of several hundreds. Example 6.3.4 In practical applications, we can divide the group of all plan participants into 5 –25 cohorts. For this limited analysis, let us assume 6 cohorts. Table 6.3.2 shows the lengths of the 95%-confidence intervals for five cases of pension plans with different numbers of plan participants. Let us assume that the structure of the plans is the same as given in Example 6.3.1. Table 6.3.2.

1 2 3 4 5

nt

The length of the confidence intervals for DALt (%)

144 250 500 1000 10 000

26.2 19.9 14.1 9.9 3.1

We can see that even for plans with a quite large number of participants (250 – 1000) the relative error of the prediction of the accrued liabilities caused by the error in the estimate of population mortality may be large. A It follows from the above that:

Even if we know the actual mortality (probability distribution of future lifetime) of the population, then, with probability 0.95, there can be deviations from the overall population means of the order vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u X u1 l 1 X 1 s2 t ^ 1:96 1 2y2xj pxj þ 2 : l2 i¼1 n2t;i j[A y2xj pxj e t;i

Such deviations appear because of the natural fluctuations of population mortality.

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253

Chapter 7 Valuation of pension plan assets

7.1 Theories of capital markets From the point of view of financial economics, we divide the assets in the national economy into the real assets and financial assets (also known as capital assets, or securities). The real assets are used for current consumption or production, and if they are used for current production, they are a source of current income. On the other hand, the financial assets are claims to future production or consumption, i.e., claims to future income, in the form of stocks, bonds, derivative securities of various types, etc. Pension plans, just as other financial intermediaries, such as banks, insurance companies, or mutual funds, typically hold only financial assets on the asset side of their balance sheet. Sometimes, although rarely, pension plans may hold real assets, such as real estate, but even then, such assets are held for investment purposes, i.e., they are held in order to produce future income for plan participants, and consequently, they play the role of financial assets. 7.1.1 Optimal consumption decision versus the level of interest rates The value of financial assets is established in the capital markets, such as the stock exchanges, or the bond markets. The issue of pricing of capital assets is one of the central problems of modern financial theory. In the simplest model of a capital market, a consumer exchanges current consumption for a capital asset, which in turn is later exchanged for future consumption. The exchanges are, of course, held with the use of the monetary unit of a given national economy. In order to understand this process better, let us describe the preferences of the consumer under consideration with a utility function UðCt ; Ctþ1 Þ where Ct is the current consumption, and Ctþ1 is the future consumption. The consumer’s problem is then to find the maximum of UðCt ; Ctþ1 Þ; given the income level of Yt and Ytþ1 in the future.

254

Chapter 7

Let us assume that the economy offers a market for exchanging today’s income for future consumption, i.e., a capital market. If a unit of such a capital asset has the price of Pt and if the consumer buys x units, then we must have Yt 2 xPt ¼ Ct ; where the expression Ct is nonnegative and it denotes today’s consumption (we will ignore the unrealistic possibility of zero consumption). If the unit of the capital asset has the price of Ptþ1 in the future, then Ytþ1 þ xPtþ1 ¼ Ctþ1 : Therefore, the consumer will maximize overall utility UðYt 2 xPt ; Ytþ1 þ xPtþ1 Þ by choosing an optimal level of x: If the utility function U is concave and differentiable with respect to each of its variables then by differentiating U with respect to x we obtain the following expression for the optimal level of x: dU ›U ›U ¼2 P þ P ¼ 0: dx ›Ct t ›Ctþ1 tþ1

ð7:1:1Þ

Under certain assumptions this optimization problem may have a solution on the boundary of the domain of the problem function, resulting in zero consumption, but we have decided to ignore this unrealistic solution. In all other cases, (7.1.1) is a necessary and sufficient condition for the optimal division of income between consumption and investment. The condition (7.1.1) is actually equivalent to

›U Pt ›Ctþ1 ¼ : ›U Ptþ1 ›C t Using the economic terminology, the above equation has the following interpretation:

The marginal utility of future consumption has the same relationship to today’s consumption as today’s price of a capital asset has to the price of the said capital asset in the future.

It should be also noted that the ratio Ptþ1 =Pt is equal to the absolute value of the derivative dCtþ1 =dCt : The choices available to the consumer are presented in Fig. 7.1. The optimal point is the point of tangency of an indifference curve (a curve connecting all points with the

255

Valuation of pension plan assets

Fig. 7.1. Consumer’s consumption decision.

same level of UðCt ; Ctþ1 Þ) and the line given by the equation: Ctþ1 ¼ Ytþ1 þ ðYt 2 Ct Þ

Ptþ1 ; Pt

ð7:1:2Þ

obtained from the equations expressing division of income between consumption and investment at time t; and its consequences at time t þ 1; after eliminating the variable x: This is implied by the fact that the points lying on the line (7.1.2) represent all possible decisions of the consumer, while between two indifference curves, the one giving higher utility lies further North-East. The quantity i ¼ ðPtþ1 =Pt Þ 2 1 is the rate of return offered by the capital asset. If the price of a capital asset is deterministic, we call the rate of return the risk-free rate. In reality, however, most future prices and rates of return are quite uncertain. This uncertainty is the source of risk of capital assets. We will return to this subject later in this chapter. Global optimization for all consumers creates the supply of savings in the economy, given by the function xp ðYt ; Ytþ1 ; iÞ; which determines the value of nonconsumed income in relation to consumers’ income today and in the future, and to the interest rate.

256

Chapter 7

The other element creating the equilibrium in the economy is the demand for savings. That function is created as follows: for a given interest rate, the firms undertake those projects, which have a positive net present value, and reject those that have a negative net present value. The total demand is the sum of all funds needed for the projects with positive net present value.

The equilibrium of the supply and demand for savings determines the interest rate.

Of course, in reality this process is much more complex, as the savings plans of the consumers, as well as the projects undertaken by the firms have varying time horizons, and this causes the existence of various interest rates for various time horizons (this is called the yield curve or term structure of interest rates). Furthermore, there is a great uncertainty concerning the returns of the projects undertaken by the firms. This causes the value of capital assets, representing the right to income from the capital investment projects, to fluctuate because of the riskiness of such income, and changing risk preferences of the capital markets investors. We will attempt to address this issue now. 7.1.2 Optimization of an investment portfolio Markowitz [46] created the first theory of capital markets that included a consideration for the risk of capital assets. We will present the outline of his theory, commonly called the Modern Portfolio Theory. In the model of Markowitz it is assumed that the rate of return of a capital asset is a random variable R: We assume that the securities market consists of N elements with random rates of return R1 ; …; RN : The buyers and sellers of capital assets, called investors, allocate their resources to a portfolio of assets. For a given investor, the portions of this investor’s portfolio invested in each of the available assets are x1 ; …; xN : Thus the rate of return of the portfolio, also a random variable, is Rx ¼

N X

x j Rj ;

j¼1

where x ¼ ðx1 ; …; xN ÞT : Markowitz assumed that the preferences of the investors can be summarized with the expected value and the standard deviation of the rate of return of the portfolio. The standard deviation (or variance) is the measure of risk in this model. Let us note that EðRx Þ ¼

N X j¼1

xj EðRj Þ;

ð7:1:3Þ

257

Valuation of pension plan assets

and that VarðRx Þ ¼

N X N X

xl xj CovðRl ; Rj Þ:

ð7:1:4Þ

l¼1 j¼1

Markowitz assumed that every investor’s objective is to maximize the expected value of the rate of return of the portfolio, and to minimize the standard deviation (or the variance) of that rate of return. Only in rare situations one can find a portfolio, which indeed has both the highest expected rate of return and the lowest variance. In reality, however, most research in modern finance tends to predict that on average instruments with higher rates of return tend to have higher degree of risk, as described by the standard deviation. Thus, in practice, it takes some more work to find an optimal portfolio. A portfolio will be called efficient if there is no other portfolio with a higher expected rate of return and lower variance. Thus, efficient portfolios are maximal elements in the set of all portfolios with respect to the partial order given by simultaneous increasing of the expected return and decreasing of the variance of the rate of return. In the model studied here, such maximal elements, i.e., efficient portfolios, always exist. In order to find them, one usually assumes that the investor aims to minimize the following expression: VarðRx Þ 2 2t EðRx Þ;

ð7:1:5Þ

where t $ 0 is the risk tolerance coefficient for the investor (defined as the first derivative of the utility function divided by the absolute value of the second derivative of PNthe utility function). This minimization is performed under the constraint: j¼1 xj ¼ 1: Negative values of the xj coefficients are allowed and refer to short position in a given security. A short sale is a sale of a security, which is borrowed (from a broker), with that security replaced later by a purchase (covering of a short). Short sale requires a deposit (margin deposit) of a certain amount of money (typically 50% of the amount received from the sale of the security). This margin deposit may, or may not, earn interest with the broker. The money received from the sale generally does not earn interest, and is held with the broker until the short is covered. A short sale creates new supply of the security shorted, so if the shorted security pays income (e.g., dividends) that income must be produced by the short seller, in order for the holder of the newly created security to receive such income. An issue that arises in short sales is the calculation of the rate of return received by the short seller. In such a calculation, we should note that the rate of return is determined by the initial cash outlay of the investor, and the final cash flow received at the end, when the short is covered, as well as possible intermediate cash flows of dividends paid by the investor and interest received.

258

Chapter 7

General formula for the effective yield earned over the period of investment is i¼

PþI2D ; M

where the symbols have the following meanings: M is the margin requirement (initial cash outlay by the short seller), D the amount of dividends paid by the short seller to the newly created security’s owner, I the amount of interest earned by the short seller on the margin deposit (assuming there is no interest earned on the cash received from the initial short sale, if there is such interest, then it must be added here), and P the profit on the short sale transaction, i.e., sale proceeds short covering repurchase cost. In order to find the desired minimum of the expression (7.1.5), we create the Lagrangian Lðx; lÞ ¼ VarðRx Þ 2 2t EðRx Þ 2 lðeT x 2 1Þ; where e ¼ ð1; …; 1ÞT is the unit vector in RN ; and eT x is the scalar product (also known as the dot product) in the same space. The parameter l is a Lagrange multiplier. The minimization problem for (7.1.5) under the constraint PN x ¼ 1 may be reduced to the unconstrained minimization of Lðx; lÞ: j¼1 j In fact, assume that there is a l [ R and x1 [ RN such that eT x1 ¼ 1 and for every x [ RN Lðx; lÞ $ Lðx1 ; lÞ ¼ VarðRx1 Þ 2 2t EðRx1 Þ: If eT x ¼ 1; then from the above inequality we infer that VarðRx Þ 2 2t EðRx Þ $ VarðRx1 Þ 2 2t EðRx1 Þ: Therefore x1 also minimizes (7.1.5) on the set of x [ RN such that eT x ¼ 1: Since the Lagrangian Lðx; lÞ ¼

N X N X l¼1 j¼1

xl xj CovðRl ; Rj Þ 2 2t

N X j¼1

xj EðRj Þ 2 l

N X

! xj 2 1 ;

j¼1

is a convex function, it follows from the Kuhn– Tucker Theorem that the necessary and sufficient condition for it to have a minimum at x is that the gradient 7Lð·; lÞ for x is a zero-vector and that eT x ¼ 1: Since N X ›Lðx; lÞ ¼2 xl CovðRl ; Rk Þ 2 2t EðRk Þ 2 l; ›x k l¼1

259

Valuation of pension plan assets

the necessary and sufficient condition for L to reach its minimum at x under the constraint eT x ¼ 1; therefore, becomes a system of N þ 1 equations 8 X > x1 CovðRl ; Rk Þ 2 2t EðRk Þ 2 l ¼ 0; <2 l¼1 for k ¼ 1; 2; …; N; ð7:1:6Þ > : T e x ¼ 1; in which there are also N þ 1 unknowns: x1 ; …; xN and l: If we use the notation S ¼ ½CovðRl ; Rj Þ1#l; j#N and m ¼ ½EðR1 Þ; …; EðRN ÞT then the system of equations (7.1.6) can be written in the following vector form: 2Sx 2 2tm 2 le ¼ 0 eT x ¼ 1;

ð7:1:7Þ

where 0 ¼ ð0; …; 0ÞT is an N-dimensional vertical vector with all coordinates equal to zero. Assume now that † the matrix S ; ½CovðRl ; Rj Þ1#l; j#N is positive definite. † the vectors e and m ¼ ½EðR1 Þ; …; EðRN ÞT are linearly independent. If we substitute t ¼ 0 in (7.1.6) then we can easily get 8 < xMIN ¼ l S21 e 2 ; : T MIN ¼1 e x

ð7:1:8Þ

where x MIN is the solution of (7.1.6) under the condition that t ¼ 0: Let us multiply the left-hand side of the vector equation in (7.1.8) by eT and calculate that l=2 ¼ ðeT S21 eÞ21 : This eventually gives us xMIN ¼

S21 e : eT S21 e

We can proceed in an analogous fashion for t . 0 and obtain a general solution x p of the system of (7.1.6) (or (7.1.7)). xp ¼

S21 e eT S21 m 21 21 t S m 2 e : þ S eT S21 e eT S21 e

Let us use this notation: zp ¼ S21 m 2

eT S21 m 21 S e; eT S21 e

260

Chapter 7

and note that eT zp ¼ 0: Therefore:

The solution of the problem of portfolio optimization is the set of efficient portfolios of the following form: xp ¼ xMIN þ t zp ;

t $ 0;

ð7:1:9Þ

MIN

is the solution obtained under the constraint t ¼ 0; giving where x the portfolio of minimum variance, while z p is a vector dependent on the expected values and covariances of returns of securities available in the market such that eT zp ¼ 0: We say that the portfolio z p is self-financing, as all long positions (i.e., securities owned in the portfolio) in it are created with funds obtained from short positions in it. We also have EðRxp Þ ¼ EðRxMIN Þ þ t EðRzp Þ; and VarðRxp Þ ¼ VarðRxMIN Þ þ t2 VarðRzp Þ:

ð7:1:10Þ

The first of the above equalities is obtained from (7.1.3) and (7.1.9). To infer the second one, note that VarðRxp Þ ¼ VarðRxMIN Þ þ 2tðxMIN ÞT Szp þ t2 VarðRzp Þ: It can be shown easily, by using the definitions of z p and x MIN, that ðxMIN ÞT Szp ¼ 0; and this completes the proof of the equality (7.1.10). The set of efficient portfolios in the plane, which has the portfolio variance on the x-axis and the expected return of a portfolio on the y-axis, is called the efficient frontier. This set is a parabola (see Fig. 7.2). If, instead, we place the standard deviation of the portfolio return on the x-axis, then the efficient frontier so created is a hyperbola. A 7.1.3 Capital assets pricing methodologies The Markowitz’s theory was the first step in the development of further theories of pricing of capital assets. Sharpe [66] and Lintner [43] created the Capital Asset

261

Valuation of pension plan assets

Fig. 7.2. Efficient frontier.

Pricing Model (CAPM). The CAPM model considers the rate of return to be a random variable over a specified one-time period, and assumes existence of a riskfree asset over that time period. A Treasury Bill maturing at the end of the time period under consideration is usually taken to be such a risk-free asset. Furthermore, the model assumes a finite number K of investors in a market with a finite number N of risky securities. Also it is assumed that † The risk preferences of the investors are fully determined by the expected rate of return and the variance of the rate of return of a security or a portfolio. † The matrix ½CovðRl ; Rj Þ1#l#N;1#j#N is positive definite and at least one of the expected values EðRj Þ is greater than the risk-free rate of return i: † All investors have the same time horizon, the sole time period under consideration. † The kth investor possesses wealth Wk at the beginning of the time period considered, and has risk tolerance tk ; k ¼ 1; …; K: Under these assumptions, as we know from Markowitz’s analysis (see also Panjer [57, Chapter 8]), the kth investor ðk ¼ 1; 2; …; KÞ chooses the following portfolio: xpðkÞ ¼ xMIN þ tk zp : The global market demand for securities, xM ; is then given by the formula xM ¼

K 1 X W xpðkÞ ; W k¼1 k

where W¼

K X k¼1

Wk :

262

Chapter 7

This means that xM ¼ xMIN þ tM zp ; where

tM ¼

K 1 X Wt: W k¼1 k k

Therefore, the portfolio xM is efficient. Let RxM be its random rate of return. Optimality of the portfolio means that the following conditions are satisfied: 2 3 2 3 CovðR1 ; RxM Þ EðR1 Þ 2 i 6 7 6 7 6 7 6 7 .. .. 6 7 ¼ tM 6 7; . . 4 5 4 5 CovðRN ; RxM Þ EðRN Þ 2 i and VarðRxM Þ ¼ tM ðEðRxM Þ 2 iÞ: When we divide the above two expressions by each other, we obtain: EðRj Þ 2 i ¼

CovðRj ; RxM Þ ðEðRxM Þ 2 iÞ; VarðRxM Þ

for j ¼ 1; …; N: But in the state of market equilibrium, the portfolio xM must represent not only the global demand (as assumed above) but also the global supply. This implies that it is just the global market portfolio, containing all securities in proportion to their market values. The quantity bj ¼ CovðRj ; RxM Þ=VarðRxM Þ is called the beta coefficient of the jth security. In practice, the market portfolio with the rate of return RxM ; denoted simply by RM ; is very difficult, really impossible, to construct. In the United States it is common to use the index called Standard and Poor’s 500 (S&P 500) as its proxy. Sometimes S&P 500 is replaced by its combination with bond market index. But neither of such indices is ideal for the purpose of creating the market portfolio—a true market index must contain all risky financial assets, and this is simply impossible to hold in a real-life portfolio. The quantity EðRj Þ 2 i is called the risk premium of the jth instrument. The quantity EðRM Þ 2 i is the market risk premium. We can see that in the CAPM theory, the expected return of a capital asset is determined by the risk tolerance of the market, the risk of that individual capital asset, and the risk-free rate of return. The rate of return of a pension plan is, according to CAPM, a random variable with the mean given by CAPM. As such, it experiences random fluctuations. On the other hand, the rate of return assigned to plan liabilities is determined by the plan actuary and it represents a long-term average of the rate of

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263

return of plan assets. This creates a conflict of the methodologies of valuation of assets and liabilities of a pension plan. Such a problem is also experienced by banks, insurance and other financial intermediaries, whose liabilities usually remain quite stable while their assets fluctuate randomly with the changes in the capital assets markets. Can this conflict of methodologies be resolved? There are two basic approaches to its resolution: † find a way to establish the market value of pension plan (or other intermediary’s) liabilities, and † modify the value of assets on the balance sheet in a manner analogous to the one used for liabilities valuation. 7.1.4 The Principle of No Arbitrage It might at first appear strange to talk about the market value of pension plan liabilities, when there is no market for them. But the search for the market value of liabilities is consistent with the idea that all capital assets, including pension plan liabilities, should be valued consistently, and a pension plan liability is an instrument, which exchanges current consumption for uncertain future consumption, just as any other financial asset. Furthermore, from the point of view of such an extreme position, most publicly traded shares would have no market value. Indeed, of all shares of Microsoft in the hands of the investing public, only the shares that are traded at this moment have a market value. No other shares can have a market value—after all, they are not being traded. From this perspective, we could never establish the market value of any assets of a pension plan, because to have a market value they would have to be sold for cash at the moment of valuation. Similarly, we could not be sure that the price we would pay at the checkout of a grocery store for a tomato should be the same as the price posted in the store—after all, the tomato we are buying may not exactly the tomato for which the price is posted. Nevertheless, we automatically assume that the price posted and the price paid is the same. Similarly, the prices of two financial assets that provide exactly the same cash flows in the future must be the same. This reasoning is based on the following assumption:

An efficient market will not allow for simultaneous existence of two identical goods (or services) with different prices.

This assumption may not be fully satisfied for some consumer goods or consumer services markets (although for competitive markets it does work quite well), but in the case of capital assets it is the fundamental principle for the proper functioning of the markets. There may be slight divergences from it due to transaction costs, taxes, or

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some artificial barriers to trading, but nevertheless this assumption is reasonable for capital markets in principle. Furthermore, it need not apply only to securities traded in the capital markets. If two securities with identical cash flows coexist in the economy, while having two different prices, this creates an incentive for arbitrage. A fully developed definition of arbitrage will be given later. For now we will assume it to be the simultaneous purchase and sale of the same item, with the purchase made at a lower price, and the sale made at a higher price. This, of course, creates a riskless profit to the trader engaged in the practice. If two different prices coexist, traders engaged in arbitrage enjoy such riskless profit until the differing prices converge and arbitrage is no longer possible. The Principle of No Arbitrage in the capital markets, and its consequences, form the second, next to CAPM, leading theory of capital assets valuation. Its creator is Ross [62]. A more developed presentation of it is given by Panjer [57]. As we noted above, its simplest consequence is as follows:

The capital markets give us not only prices of those securities which are traded in those markets, but also all those securities, whose cash flows can be replicated with traded instruments.

It turns out that the absence of arbitrage has far reaching consequences—it implies a certain specific structure of determination of prices in the capital markets. In order to explain this, we will need some definitions. A capital asset is called an Arrow-Debreu security if it pays a monetary unit upon happening of a specific elementary event in the probability space of all possible scenarios of the future, and zero in all other cases. We assume that all capital assets are finite, and thus all capital assets are linear combinations of Arrow-Debreu securities. A capital market is called complete if all Arrow-Debreu securities have unique market prices. Consider now a one-time period model of a capital market. Let it consist of N securities S1 ; S2 ; …; SN : We will be studying prices of these assets at times 0 (the present) and 1 (the future). We assume that there are M possible states of the world in the future: v1 ; v2 ; …; vM : The set V ¼ {v1 ; v2 ; …; vM } is the set of all elementary events in the probability space of future scenarios of the world. The value of the jth asset in the state v at time 1 will be denoted by Sj ð1; vÞ: We assume that for these elementary securities Sj ð1; vÞ $ 0: The prices of these capital assets at time 0 are positive and represented by the vector: Sð0Þ ¼ ½ S1 ð0Þ

S2 ð0Þ

···

SN ð0Þ :

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Thus, the prices of capital assets at time 0 are deterministic and given, and the prices at time 1 are random variables. The prices of these capital assets at time 1 form the following matrix: 3 2 S1 ð1; v1 Þ S2 ð1; v1 Þ · · · SN ð1; v1 Þ 7 6 6 S1 ð1; v2 Þ S2 ð1; v2 Þ · · · SN ð1; v2 Þ 7 7 6 7: Sð1; VÞ ¼ 6 7 6 .. .. .. .. 7 6 . . . . 5 4 S1 ð1; vM Þ S2 ð1; vM Þ · · · SN ð1; vM Þ We will now consider portfolios of these elementary securities. Let uj be the number of units of the jth security held in a portfolio from time 0 to 1. This number can be negative (its negative value represents a short position in the jth security in the portfolio). A vector 2 3 u1 6 7 6 u2 7 6 7 7 u¼6 6 .. 7; 6 . 7 4 5 uN is called a trading strategy. The value of the portfolio given by a trading strategy at time 0 is Sð0Þu ¼ u1 S1 ð0Þ þ u2 S2 ð0Þ þ · · · þ uN SN ð0Þ; while at time 1 it is 2

u1 S1 ð1; v1 Þ þ u2 S2 ð1; v1 Þ þ · · · þ uN SN ð1; v1 Þ

3

7 6 6 u1 S1 ð1; v2 Þ þ u2 S2 ð1; v2 Þ þ · · · þ uN SN ð1; v2 Þ 7 7 6 7; Sð1; VÞu ¼ 6 7 6 .. 7 6 . 5 4 u1 S1 ð1; vM Þ þ u2 S2 ð1; vM Þ þ · · · þ uN SN ð1; vM Þ this, of course, is a random variable. Let us note that the price of a portfolio at time 0 is a linear functional on the vector space spanned by elementary securities. If there is a risk-free asset in this market, then we can assume, without loss of generality, that it is the first asset, i.e., S1 ð0Þ ¼ 1 and S1 ð1; vÞ ¼ 1 þ i for every v [ V: As before, i denotes the risk-free rate of return. We will now present a mathematical definition of arbitrage for this one-period model. A trading strategy u is called an arbitrage opportunity if Sð0Þu # 0 and, simultaneously, for the vector Sð1; VÞu . 0; i.e., all coordinates of this vector are nonnegative, and at least one coordinate is positive. This means that the payoff of this

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portfolio is nonnegative in all states of the future, and positive in some states of the future, while the cost of the portfolio is zero or less. Example 7.1.1 Let us consider the simplest example of a one-period capital market, consisting of two securities, a risk-free one, and a risky one. Both assets are assumed to have the price of 1. In this simplest model there are only two states of the world in the future, the ‘up’ state, in which the risky asset pays u . 1; and the ‘down’ state in which the risky asset pays w , u: Thus we have M ¼ N ¼ 2; Sð0Þ ¼ ½ 1 1 and " # 1þi u Sð1; VÞ ¼ : 1þi w We will also assume that i . 0 (this is the risk-free rate). Then we have

There exist no arbitrage opportunities in this market if and only if the risk-free rate i lies between the risky rates of return.

Indeed, let u ¼ ½ u1

u2 T ;

be an arbitrary trading strategy. Let us investigate when it constitutes an arbitrage opportunity. For this to happen we must have: Sð0Þu # 0

and

Sð1; VÞu . 0:

Then u1 þ u2 # 0; and ð1 þ iÞu1 þ uu2 $ 0

and

ð1 þ iÞu1 þ wu2 $ 0;

and one of the last two inequalities must be strict. Consider the case when 1 þ i . w: If u2 $ 0 then we have

u1 # 0

and

wu2 $ 2ð1 þ iÞu1 $ 2wu1 $ wu2 :

This, in turn, implies that u1 ¼ u2 ¼ 0: The trading strategy u cannot be an arbitrage opportunity in this case. If, on the other hand, u2 , 0; (and still 1 þ i . w) then

u1 . 0

and

ð1 þ iÞu1 $ 2uu2 $ uu1 :

Therefore, 1 þ i $ u: Since 2uu2 . 2wu2 ; we must have a strict inequality ð1 þ iÞu1 . 2wu2 ; and u can be an arbitrage opportunity.

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If 1 þ i # w , u then u1 ¼ 21 and u2 ¼ 1 gives ð1 þ iÞu1 þ uu2 ¼ u 2 ð1 þ iÞ . 0; and ð1 þ iÞu1 þ wu2 ¼ w 2 ð1 þ iÞ $ 0; offering an arbitrage opportunity. On the other hand, if 1 þ i $ u; an arbitrage opportunity is given by u1 ¼ 1 and u2 ¼ 2 (1 þ i)/u # 2 1. A

7.1.5 The Fundamental Theorem of Asset Pricing Let us define a state price vector as a vector C ¼ ½ cðv1 Þ cðv2 Þ

···

cðvM Þ ;

such that for every v [ V; cðvÞ . 0 and the following equality holds: Sð0Þ ¼ CSð1; VÞ; i.e., for every j ¼ 1; 2; …; N X Sj ð0Þ ¼ cðvÞSj ð1; vÞ: v[ V

A state price vector gives the prices of Arrow-Debreu securities, and thus allows us to express the price of a security as a linear combination of prices of Arrow-Debreu securities. Using this new concept we can now formulate the following consequence of the Principle of No-Arbitrage, which is also a simplified version of the Fundamental Theorem of Asset Pricing.

A one-period securities market is arbitrage free if and only if there exists a state price vector in it.

In order to prove the above, first assume that there is a state price vector. If, for a trading strategy u we have Sð1; V; Þu . 0; then Sð0Þu ¼ CSð1; VÞu . 0; since the components of a price vector must be positive. It follows that the trading strategy u cannot be an arbitrage opportunity. Now let us assume that the market is free of arbitrage (i.e., there are no arbitrage opportunities). Assume also that there are M linearly independent securities (i.e., securities whose random payoffs at time 1 are linearly independent vectors). If that is not the case, the market can be completed by adding new securities (this process is fully described in [57, Chapter 5]). Under these assumptions the matrix

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Sð1; VÞ has maximum rank and N $ M: We can assume without loss of generality . that the first M securities are the linearly independent ones. Let Sð0Þ ¼ ½Lð0Þ .. Rð0Þ; where Lð0Þ ¼ ½ S1 ð0Þ

S2 ð0Þ

· · · SM ð0Þ ;

and Rð0Þ ¼ ½ SMþ1 ð0Þ

SMþ2 ð0Þ · · · SN ð0Þ : .. Similarly, let us write Sð1; VÞ ¼ ½Lð1Þ . Rð1Þ; where 3 2 S1 ð1; v1 Þ S2 ð1; v1 Þ · · · SM ð1; v1 Þ 7 6 6 S1 ð1; v2 Þ S2 ð1; v2 Þ · · · SM ð1; v2 Þ 7 7 6 7; Lð1Þ ¼ 6 7 6 .. .. .. .. 7 6 . . . . 5 4 S1 ð1; vM Þ

S2 ð1; vM Þ · · ·

SM ð1; vM Þ

and 2

SMþ1 ð1; v1 Þ

SMþ2 ð1; v1 Þ

···

6 6 SMþ1 ð1; v2 Þ SMþ2 ð1; v2 Þ · · · 6 Rð1Þ ¼ 6 6 .. .. .. 6 . . . 4 SMþ1 ð1; vM Þ SMþ2 ð1; vM Þ · · ·

SN ð1; v1 Þ

3

7 SN ð1; v2 Þ 7 7 7: 7 .. 7 . 5 SN ð1; vM Þ

The matrix L(1) is invertible. Let C ¼ Lð0ÞðLð1ÞÞ21 : Let em be the column vector in the M-dimensional Euclidean space, whose mth component is 1, while its other components are 0, i.e., this vector effectively represents an Arrow-Debreu security. For that Arrow-Debreu security, let us introduce the following trading strategy: u ¼ ½ ððLð1ÞÞ21 em ÞT

0

···

T 0 :

Then we have Sð1; VÞu ¼ Lð1ÞðLð1ÞÞ21 em ¼ em : Since this is a positive vector, and the market is arbitrage free, the quantity Sð0Þu ¼ Lð0ÞðLð1ÞÞ21 em ¼ Cem ; is positive. This means that the mth component of the vector C is positive. Since this parameter m was arbitrary, the reasoning applies to any component. Moreover, we have . . CSð1; VÞ ¼ Lð0ÞðLð1ÞÞ21 ½Lð1Þ .. Rð1Þ ¼ ½Lð0Þ .. Lð0ÞðLð1ÞÞ21 Rð1Þ: ð7:1:11Þ

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If N ¼ M; then we have found a state price vector. If, on the other hand, N . M; then since we can choose a basis for the M-dimensional Euclidean space from the columns of the matrix Lð1Þ; there exists a matrix with dimensions M £ ðN 2 MÞ; let us call it K, such that Rð1Þ ¼ Lð1ÞK: Since the market is arbitrage free, the prices of the additional (redundant) N 2 M securities must be equal to prices of appropriate linear combinations of the linearly independent securities, producing the same cash flows in the future (at time 1), i.e., Rð0Þ ¼ Lð0ÞK ¼ CLð1ÞK ¼ CRð1Þ: Therefore, . . . CSð1; VÞ ¼ C½Lð1Þ .. Rð1Þ ¼ ½CLð1Þ .. CRð1Þ ¼ ½Lð0Þ .. Rð0Þ ¼ Sð0Þ; ð7:1:12Þ (recall that the price of a security is a linear function), and this completes the proof of this simple version of the Fundamental Theorem of Asset Pricing. Example 7.1.2 Let us consider the simplest market model with two securities: the risk-free one, and a risky one, which we have already described previously in Example 7.1.1. The equations describing the state price vector in this case are 1 ¼ ð1 þ iÞc1 þ ð1 þ iÞc2

1 ¼ uc1 þ wc2 :

ð7:1:13Þ

Their solutions

c1 ¼

ð1 þ iÞ 2 w ð1 þ iÞðu 2 wÞ

c2 ¼

u 2 ð1 þ iÞ ; ð1 þ iÞðu 2 wÞ

ð7:1:14Þ

exist and are positive, as long as the numerator and the denominator in both the fractions have the same sign. Under the following assumptions i $ 0; u . 1; u . w . 0; we arrive at the following conclusion.

A state price vector exists if and only if u . 1 þ i . w: This, of course, is equivalent to this market being arbitrage free.

A If there is a state price vector C, then we can define a function of the argument v[V QðvÞ ¼ ð1 þ iÞcðvÞ:

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Thus we have for every j ¼ 1; …; N Sj ð0Þ ¼

X QðvÞSj ð1; vÞ ; 1þi v[V

and, in particular, for the risk-free security 1 ¼ S1 ð0Þ ¼

X X QðvÞð1 þ iÞ ¼ QðvÞ: 1þi v[ V v[ V

This means that Q can be considered a discrete probability distribution on V; and the price of a security at time 0 is the expected present value of its cash flows from time 1 with respect to probability distribution. The probability measure so created is called the risk-neutral measure. Our analysis implies the following: Theorem 7.1.1 (The Fundamental Theorem of Asset Pricing). The following are equivalent: 1. The one-period securities market is arbitrage free. 2. There exists a state price vector. 3. There exists a risk-neutral probability measure. It has important consequences in the pricing of capital assets. In the model we are considering, a capital asset is defined by its cash flows at time 1, in all possible states of the world v [ V; 2 3 Xðv1 Þ 6 7 6 7 X ¼ 6 ... 7: 4 5 XðvM Þ A trading strategy u replicates a security X if Sð1; VÞu ¼ X: The basic idea of an arbitrage-free market is that a security and a portfolio created by a trading strategy replicating that security must have the same price. In particular, this means that a capital asset, whether it is traded in the capital markets or not, has a market price if it can be replicated with marketable assets. If the Arrow-Debreu securities for all possible future states of the world are marketable, or can be replicated by marketable securities, then all securities must have market prices. Panjer [57, Chapter 5] shows that a one-period model is arbitrage free and complete if and only if its state price vector is unique. Recall that for an efficient arbitrage-free capital market with arbitrary future scenarios there always is a probability distribution with respect to which each security

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Valuation of pension plan assets

has the same mean present value of future cash flows as its current price. With respect to such a probability distribution, the market behaves, after discounting with the riskfree rate, as a multi-dimensional stochastic process called a martingale. Therefore, the Fundamental Theorem of Asset Pricing can be generalized as follows: Let SðtÞ be the price of a capital asset at time t: This price is a continuous stochastic process. Let d be the force of interest given by the risk-free rate of return. Under certain assumptions of market efficiency, absence of arbitrage in the market is equivalent to existence of a probability distribution such that with respect to that distribution {e2dt SðtÞlt $ 0} is a martingale, i.e., for each u , t; Eðeðu2tÞd SðtÞl Fu Þ ¼ SðuÞ; where {Ft ; t $ 0} is a nondecreasing sequence of s-algebras, with respect to which the process {SðtÞ; t $ 0} is measurable (for definition of martingales—see Section 2.6). One can find a proof of this theorem in Harrison and Kreps [36] (see also Harrison and Pliska [37]), while Schachermayer [65] gives an elegant proof in the discrete model in a Hilbert space.

Exercises Exercises 7.1.1 May 2003 SOA/CAS Course 2 Examination, Problem No. 36. Eric and Jason each sell a different stock short at the beginning of the year for a price of 800. The margin requirement for each investor is 50% and each will earn an annual effective interest rate of 8% on his margin account. Each stock pays a dividend of 16 at the end of the year. Immediately thereafter, Eric buys back his stock at a price of 800 2 2X; and Jason buys back his stock at a price of 800 þ X: Eric’s annual effective yield, i; on the short sale is twice Jason’s annual effective yield. Calculate i: Solution Eric’s profit is 2X þ 32 2 16 ¼ 16 þ 2X while Jason’s profit is 2X þ 32 2 16 ¼ 16 2 X: The margin deposit (initial cash outlay) is 400 for both Eric and Jason. Thus we have, by comparing yields:

16 þ 2X 16 2 X ¼2 ; 400 400 which implies that X ¼ 4: Eric’s yield is i¼

16 þ 2 · 4 ¼ 6%: 400

A

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Exercise 7.1.2 Security X has a beta of 1.5. The risk-free rate is 3.00% and the market expected rate of return is 9.00%. According to the CAPM, what is this security’s expected rate of return? Solution Using the basic CAPM formula we obtain: EðrÞ ¼ rf þ bðEðrM Þ 2 rf Þ ¼ 3:00% þ 1:5ð9:00% 2 3:00%Þ ¼ 12:00%

A

Exercise 7.1.3 The market price of a security is 50. The security is a perpetuity of a constant dividend. Assume that the CAPM holds. The expected rate of return of this security is 11%. The risk-free rate is 3% and the market risk premium (over the riskfree rate) is 8%. What will be the market price of the security if the covariance of its returns with the market doubles, while no other parameters are changed? Solution Basic CAPM formula is EðrÞ 2 rf ¼ bðEðrM Þ 2 rf Þ; where

b¼

Covðr; rM Þ ; VarðrM Þ

and all risky rates of return are considered random variables. When the covariance doubles, beta doubles. We need to find the original beta first. It must satisfy 11% 2 3% ¼ b · 8%; so that b ¼ 1: After beta doubles, b ¼ 2: Therefore, new rate of return is rnew ¼ 3% þ 2 · 8% ¼ 19%: What is the dividend D of this security? At 11% expected rate, perpetuity of D is worth 50, and this must equal D=0:11 so that D ¼ 5:50: At 19% expected return, perpetuity of 5.50 is worth approximately 28.95. A Exercise 7.1.4 You are given securities market described by " # 1:10 1:20 Sð0Þ ¼ ½ 1 1 ; Sð1Þ ¼ : 1:10 0:90 Find the risk-neutral probabilities. Solution We can see that the risk-free rate is 10 p1 ¼

1þi2w 0:20 2 ¼ ¼ ; u2w 0:30 3

p2 ¼

u 2 ð1 þ iÞ 0:10 1 ¼ ¼ : u2w 0:30 3

A

Valuation of pension plan assets

273

7.2 General methodology of arbitrage-free valuation of capital assets In this section, we will attempt to develop a better understanding of the Fundamental Theorem of Asset Pricing, while in Section 7.3, we will review practical methodologies used in asset valuation. Consider a model of a capital market in which the number of states of the world at each time is finite, with time measured in discrete intervals, and an infinite time horizon. Assume that the market is complete and there are no external impediments to trade. Transactions can be performed only at times 0; 1; 2; …: Let the one-period risk-free rate at time t be iðtÞ; this means that a risk-free security purchased at time t will be worth 1 þ iðtÞ at time t þ 1; regardless of the state of the world. Let the value of the jth security at time t be written as Vj ðtÞ; and let Dj ðtÞ be the dividend (or other similar cash flow) paid at time t: We assume that the value Vj ðtÞ is ex-dividend Dj ðtÞ:

Absence of arbitrage is equivalent to the existence of a probability measure with respect to which

1 ðV ðt þ 1Þ þ Dj ðt þ 1ÞÞ ; Vj ðtÞ ¼ Et 1 þ iðtÞ j where Et is the expected value calculated at time t:

The said probability measure is termed the risk-neutral measure. It follows from the above that:

The present value of a security defined as a stochastic process of future cash flows {DðtÞ : t ¼ 1; 2; …}; is E

1 X t¼1

! DðtÞ ; ð1 þ ið0ÞÞð1 þ ið1ÞÞ· · ·ð1 þ iðt 2 1ÞÞ

where the expected value is taken with respect to the risk-neutral probability measure.

It should be stressed that the above statement does not apply only to marketable securities. It applies to all financial assets, which can be replicated with marketable

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securities. This means that pension plan liabilities have a market value even if such value is not shown on the balance sheet, as long as liabilities cash flows can be replicated with marketable instruments. In order to calculate such market value of liabilities, we should calculate the expected present value of future plan liabilities cash flows with respect to the risk-neutral probability measure. While this may appear simple, this is a rather complex modeling process in practice. In the United States, since 1991, the life insurance Standard Valuation Law requires that companies hold assets that are adequate to meet all liabilities cash flows. While this requirement does not appear mathematical in nature, in practice it means that the reserve is not only calculated with a standard actuarial mathematics reserve formula (e.g., prospectively as the actuarial present value of future benefits minus the actuarial present value of future premiums), but also that the valuation actuary should perform analysis of the firm’s solvency under a variety of scenarios of the future. Such an analysis is commonly referred to as cash flow testing. If the scenarios of the future tested in it amount to a sample from the probability distribution of the risk-neutral measure, then the sample average of the expected present value of the benefits cash flows is an estimator of the market value of the insurance firm’s liabilities. If this estimator is unbiased, this gives a reasonable approach to the calculation of the market value of liabilities, and creation of a market-based balance sheet of the insurance company. In practice, however, such estimation of the market value of liabilities is quite a challenge, as it requires that the sample does indeed represent the probability space of the risk-neutral measure, and all liabilities cash flows are modeled properly. Neither of these two tasks is trivial. In view of such significant modeling challenges, it seems reasonable to look for other methodologies of valuation that treat assets and liabilities consistently. On the other hand, inconsistency in valuations of assets and liabilities of a pension plan appears somewhat dangerous. Why?

If the assets are stated on the balance sheet according to their market value, while the liabilities are reported based on their actuarial valuation, then the plan surplus, i.e., the excess of assets over the liabilities, as reported in the balance sheet, will exhibit significant volatility, thus introducing additional uncertainty, and possibly insolvency fears among plan participants.

The actuarial valuation of pension plan liabilities is established based on an expected long-term rate of return for plan assets, with a possible conservative margin. If we do not use consistent market values on both sides of the balance sheet, then we can at least seek consistency by using analogous averaging

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Valuation of pension plan assets

procedures for both assets and liabilities of a pension plan. This kind of approach should significantly reduce surplus volatility.

7.3 Derivative securities The no-arbitrage pricing approach described here can be used, and is used quite widely in practice, as a method for valuation of derivative securities. A derivative security is a security whose cash flows are derived from cash flows of another security (called commonly the underlying). The simplest derivative is a forward, a contract for a purchase of an item at a predetermined price at a specified time in future. This kind of instrument, however, is not widely used in practice, because of significant credit risk faced by both parties to the transaction—when the future time for the transaction arrives, one of the parties may not show up for the transaction, leaving the other party (those two sides to the transaction are commonly called counterparties) with a significant problem. In order to address this issue, similar instruments with significant credit enhancements and contract enforcement mechanisms has been designed: a futures contract. In the futures contract, each counterparty must post a deposit (termed a margin deposit) to guarantee financial performance, and must agree to a standardized delivery, i.e., prescribed features of the item being transacted, e.g., quality and quantity. Futures are traded on organized exchanges, which also require that each counterparty position is evaluated at the end of each trading day (marked to market), and the counterparty enjoying a gain may take a portion of it from the account, still leaving a sufficient margin, while the other counterparty, suffering a loss, if such a loss results in insufficient margin account balance, must deposit additional funds. Futures price (the price at which the contract is to be delivered) is related to the price of the underlying through a formula derived from a no-arbitrage condition. An investor with a long position in a futures contract at price F; has an obligation to buy the underlying at price F at the contract expiration, which we will assume is at time T in future. Let us also assume that this investor can borrow or lend a riskfree effective annual interest rate i: This investor with the long futures has an alternative of buying the underlying at the current market price S: If the investor enters into the futures contract, the cash otherwise spent on the purchase can be invested risk free at the rate i; and at contract delivery day, the investor ends up with the underlying either way. This arbitrage argument gives us the basic futures – spot parity relationship: F ¼ Sð1 þ iÞT : Another important derivative security is an option. In general, an option is a contract in which one side acquires the right to buy (or sell, but only one of these two) the underlying at a predetermined price (might be a function of something, but the conditions are stated in advance) at a predetermined time or by

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predetermined time. In hedging with forwards and futures, upside potential must be sacrificed in order to receive downside protection. When hedging with options, this is not the case. Options have asymmetrical payoff patterns such that they only pay off when the underlying price moves in a specific direction. In an option contract, the long side of it has a right, but not an obligation, to purchase or sell (depending on the option) a security at a specified price. The short side counterparty must provide the market for the right of the long side. This may sound good for the long side, bad for the short side, but the long side must pay an up-front premium to the short side. The value that an option has if it were exercised instantly is known as its intrinsic value. Before its expiration, the option may have a price higher than intrinsic value, and the difference between the two is called the time value of the option. An American option can be exercised at any time prior to expiration. A European option can be exercised only at maturity. A call option gives the long side the right to buy the underlying at a fixed price, called the strike price or exercise price. A put option gives the long side the right to sell the underlying at a fixed price, called the strike price or exercise price. Let us now assume that the underlying is a stock. If ST is the stock price at the time of expiration of a European option and K the strike price, then the following summarizes the long side payoffs of call and put options: Call Option Payoff ¼ MaxðST 2 K; 0Þ ¼ ðST 2 KÞþ ; and Put Option Payoff ¼ MaxðK 2 ST ; 0Þ ¼ ðK 2 ST Þþ : The above are true for American options, but the stock price need not be the price at expiration of the option; it can be the price at any time until the expiration of the option is then exercised. An option whose intrinsic value is positive is said to be in-the-money. If the underlying is trading at exactly the exercise price, then the option is said to be at-themoney. An option that is at-the-money has intrinsic value of zero, but the opposite is not necessarily true. If an option has intrinsic value of zero, but it is not at-the-money, we say that the option is out-of-the-money. An important insight into the valuation of options is the formula called the put-call parity. It applies to European options. It assumes that no dividends are payable on the underlying, all investors may borrow and lend at the risk-free rate, there are no transaction fees or taxes, short selling and borrowing are allowed, fractional shares may be traded, and that the market does not allow any arbitrage opportunities. Under these assumptions, if i is the effective annual risk-free interest rate, T the time until expiration, c the call price, p the put price, and S the underlying price, then (Panjer [57]): c þ K e2rðT2tÞ ¼ S þ p:

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The most famous formula giving the value of European options under certain assumptions is the Black –Scholes Formula. It has all the same assumptions as the put-call parity, and additionally assumes that the continuously compounded rate of return on the underlying asset over the period of time of length T has normal distribution with mean mT and variance of s2 T; and returns over disjoint time intervals are independent (this is also called the random walk model). For European options, the call value c and the put value p are given by the formulas: c ¼ SFðd1 Þ 2 Kð1 þ iÞ2T Fðd2 Þ; p ¼ Kð1 þ iÞ2T Fð2d2 Þ 2 SFð2d1 Þ; where ln d1 ¼

S Kð1 þ iÞ2T pﬃﬃﬃﬃ s T

þ

s2 T 2

;

and

S ln Kð1 þ iÞ2T pﬃﬃﬃﬃ d2 ¼ s T

2

s2 T 2

;

where F denotes the cumulative distribution function of the standard normal distribution. Both put-call parity and the Black – Scholes formula can be modified for dividend-paying stocks by considering their value without the dividend. The last kind of a derivative security we would like to present here is an instrument used commonly by insurance companies for interest rate and currency risk management: swaps. A ‘plain vanilla’ (i.e., the simplest kind, as in the ‘plain vanilla’ ice cream) interest rate swap is a contract between two counterparties, requiring them to make interest payments to each other over the term of the contract, based on different types of underlying bonds or interest rate indices. The long party (fixed-rate payer) pays interest to the second at a fixed rate, while the short party (floating-rate payer) pays interest to the first at a rate that changes (‘floats’) according to a specified index. This kind of a swap is also called a pay-fixed swap, because the long party pays fixed. The actual cash payments are determined by multiplying the relevant rate of interest by a face amount, or principal, which is called the notional principal, or just the notional. In a currency swap, the notional amounts are specified in different currencies. This is handled by exchanging the notional amounts at the beginning of the swap and returning them at the end. Currency swaps are used to manage foreign exchange exposures. Consider, for example, a British company that wants to issue debt in Japan, but finds the process of registration of securities and issuing them in Japan too much to handle. Instead the company can issue floating debt

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in Great Britain, and enter into a swap to pay fixed long-term Japanese Yen rate, and receive British pound floating.

Exercises Exercise 7.3.1 Assume it is now June 30, 2002, and a risk-free bond maturing September 30, 2002 with 10 000 face value is selling for 9955.20. Current spot price for gold is 315 per ounce. What is the price of the futures contract for gold with September 30, 2002 delivery? Assume that there is neither any cost of storage for gold nor any convenience yield for owning it. Solution Since gold does not pay any dividends, all we have is this relationship between spot and futures price: F ¼ Sð1 þ iÞT ¼ 315 ·

10 000:00 ¼ 316:42: 9955:20

A

Exercise 7.3.2 The current price as of June 30, 2002 of a 325 call on September 30, 2002 gold is 12, with the spot price and interest rates data just as in the previous exercise. Find the price of a September 30, 2002 325 gold put as of June 30, 2002. Assume that all conditions for put-call parity to hold are satisfied. Solution Use put-call parity: 12 2 p ¼ 315 2

9955:20 325 ¼ 28:54; 10 000

p ¼ 20:54:

A

Exercise 7.3.3 An option market satisfies the condition for put-call parity. The current underlying security price is 100. A call option with a strike price of 105 and maturity 1 year from now has a current price of 4. A put option with a strike price of 105 and maturity 1 year from now has a current price of 6. Determine the short-term risk-free interest rate. Solution Using put-call parity we get 4 2 6 ¼ 100 2

105 ; 1þi

102 ¼

105 ; 1þi

i¼

3 ¼ 2:94%: 102

A

Exercise 7.3.4 A certain stock’s price is 100. A European call option on that stock has the exercise price of 95. The interest rate is 10% compounded continuously. Time to expiration of the option is 3 months (a quarter of a year). The standard deviation of

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Valuation of pension plan assets

the rate of return of the underlying stock is s ¼ 0:50 per annum. Find the price of this call option using the Black – Scholes formula. Solution We have ln

0:502 þ 0:25 · 2 95 e20:25 · 0:10 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ d1 ¼ < 0:43; 0:50 0:25

100 0:502 ln 2 0:25 · 20:25 · 0:10 2 95 e pﬃﬃﬃﬃﬃﬃﬃﬃﬃ < 0:18; d2 ¼ 0:50 0:25 100

Fðd1 Þ ¼ 0:6664;

Fðd2 Þ ¼ 0:5714:

Thus the value of the European call option is c ¼ 100 · 0:6664 2 95 e20:25 · 0:10 · 0:5714 ¼ $13:70:

A

Exercise 7.3.5 You are given securities market described by " # 1 1:20 Sð0Þ ¼ ½ 1 1 ; Sð1Þ ¼ : 1 0:90 Calculate the arbitrage-free price of a European put on the second asset with the strike (exercise) price of 1.20. Solution Start by finding the state price vector ½ c1 1 ¼ c1 þ c2 ;

c2 : We have

1 ¼ 1:20c1 þ 0:90c2 :

We have ½ c1

c2 ¼

1 3

2 : 3

Solving this system of equations, we can find the following state price vector: 1 2 ½ c1 c2 ¼ : 3 3 The put described in this problem has cash flows " # 0 ; 0:30

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at time 1 and its price equals 1 2 · 0 þ · 0:3 ¼ 0:2: 3 3

A

7.4 A survey of asset valuation methods The Society of Actuaries published in 1998 a survey of the methods of valuation of assets of defined benefit pension plans in the United States and Canada. It contained the following 11 methods of valuation (some of them being averaging methods) of asset values. 1. Fair Market Value Method. This method uses the market value of an asset as derived from trades in the market. In some cases, it allows using a mathematical model for calculation of the market value (this is very common to nontraded bonds or real estate). 2. Discounted Cash Flow Method. This method is equivalent to the standard actuarial valuation if the valuation interest rate is the same. It calculates the present value of expected cash flows of the assets. The interest rate used is the one determined to be adequate for the valuation purposes (in the case of life insurance valuation, the valuation rate is established by the Standard Valuation Law, in reference to Moody’s corporate bond index, while in the case of pension plans, the rate is established by the plan actuary). Then there are three versions of the Book Value Method; all of them value the assets based on established accounting rules. It should be noted that in the US insurance statutory valuation, stocks, and real estate are shown using their purchase price, while bonds are shown as the present value of their future cash flows based on the interest rate equal to the yield to maturity as of the date of purchase. 3. Cost Value Method. This method uses the price paid for an asset as its accounting value. This methodology does not comply with the concept of consistency of valuation of assets and liabilities. If used for a pension plan, it can produce significant volatility of plan surplus, when the plan needs to sell assets at prices at variance with their purchases prices. 4. Amortized Value Method. It is used for bonds that have a clearly defined maturity redemption value. It can also be used for some assets that have some form of amortization schedule. The amortization method used can be linear (common in accounting valuation) or exponential (i.e., compound interest). This is the standard methodology of valuation of bonds for US statutory valuation of insurance bond assets. 5. Contract Value Method. This method is used if assets are invested in another financial intermediary’s liabilities, e.g., bank deposit, Guaranteed Investment Contracts (GIC), Individual Participation Guarantee (IPG), or Deposit Administration.

Valuation of pension plan assets

281

The value of the security is defined by a contract with its issuer and typically guaranteed by the issuer in some form (bank deposits up to $100 000 are insured in the US by the Federal Deposit Insurance Corporation, while GICs guaranteed by insurance companies are indirectly insured by the state guaranty funds). For guaranteed securities, use of contract value method does appear reasonable. The next group of valuation methodologies consists of those that use some form of averaging. These methods are of great interest to us, as they attempt to provide consistency with the valuation of liabilities when the value of liabilities also uses some averaging procedure. However, most of the averaging asset valuation methods have been introduced with the objective of reducing assets volatility, without great attention to consistency with liabilities. 6. Blend of Cost and Market Values Method. This method uses an average (or weighted average) of the asset purchase cost and current market value. One can also study the ratio of the market value to the purchase cost over two or more years, and based on this the expected value of such a ratio at valuation time is predicted. 7. Write-up Method. This method does provide consistency of valuation of assets and liabilities. Asset value from last year is the starting point. Then one adds all current year asset additions (deposits) to asset portfolio, subtracts all subtractions (withdrawals) from the asset portfolio, and then one increases the value by a predetermined rate of return, usually some form or market index (e.g., the interest rate of a specific Treasury debt issue). The value so obtained may also be additionally recalculated, for example through partial amortization to current market value. 8. Deferred Recognition Method. In this method, only a portion of the change in the market value of the asset portfolio is recognized in its valuation. The rest of the change is amortized in some predetermined fashion. This implies, of course, that over time several years’ amortizations will coexist, with possibly contradictory effects on the value of the portfolio. 9. Average Market Value Method. In this methodology, one uses the average (possibly a weighted average) of the current market value and the market value from several previous years. The averaging may be also done with respect to the asset purchase cost, or the proceeds from the sale of the asset, or one may have to make an adjustment to the average calculated when the asset is sold. This method is effectively equivalent to the deferred recognition method. Both these methods are basic methods of averaging of assets values in the United States and Canada.There are also two more methods, used sporadically. 10. Trend-Line Method. Using this approach, one establishes (using linear regression, exponential regression, etc.) a trend of asset values over time. One can also divide the portfolio under consideration into sub-portfolios with their own trends and use the method for each sub-portfolio separately. The trend relationship so established is then applied to the historical value data about securities, portfolios, or sub-portfolios. Typically, current market value is one of the pieces of data used as an

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input in the formula used in the trending procedure, together with prices from previous periods, and this method ends up being an averaging methodology. 11. Average Unit Value Method. This method treats the asset portfolio as a set of investment units, or shares, in a real or hypothetical fund (similar to a mutual fund). The value of assets is established as a product of the number of units held and the price of a unit (calculated based on market data). Society of Actuaries’ Survey of Asset Valuation Methods for Defined Benefit Plans provides an overview of utilization of the above methodologies by North American (US and Canadian) defined benefit pension plans. The dominant method used is the Fair Market value one, used by 60.3% of firms surveyed. The second most used method is the Contract Value Method at 20.8%, used so often because of wide use of guaranteed instruments such as GICs, Stable Value Funds, etc., as invested assets. Next are: Write-Up, at 9.4% and Deferred Recognition, at 4.6%. All other methods are placed below 2% and did not seem to be used commonly. It should be noted that Internal Revenue Service (IRS, the US Federal Government tax authority) has produced some interpretations of tax laws that limit the use of averaging methods for assets value calculation for tax purposes. In the US pension plan contribution required for funding purposes is not necessarily allowed as a tax-deductible contribution. We should note that pension plan funding in the United States is under regulatory supervision of the Department of Labor of the Federal Government, while tax matters are handled by the IRS, which is a part of the Department of Treasury, another department of the Federal Government. To summarize, let us reiterate the principle, which we believe should be used in the valuation of pension plan assets and liabilities.

The Principle of Consistency of Valuation of Assets and Liabilities: when the pension plan liabilities have a clearly established market value, use of market values on the asset side of the balance sheet is appropriate and called for. If, however, the market value of liabilities is not known, then the asset valuation methodology should be consistent with the method used for liabilities valuation.

It should be also noted that the market values, i.e., true economic values, of assets and liabilities may have additional effects on the management and operations of insurance enterprises and pension plans, beyond the accounting statements. If assets and liabilities are valued consistently, but in a manner far removed from market, customers and competitors of the form may perceive this as an arbitrage opportunity, if the accounting values divergent from the market value are in some way available to them. Therefore, consistency of valuation of assets and liabilities for the internal management of the enterprise, or a pension plan, must be also

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283

viewed in the context of analogous consistency in the perception of the customers, competitors, and regulators. The principle is only the first step in the greater issue of a consistent management of assets and liabilities. This issue is the topic of Chapter 8.

Exercises Exercise 7.4.1 A pension plan holds two assets, a 1000 par value bond paying a fixed 6% annual coupon, and 100 shares of Consolidated Aggregate Manufacturers, Inc., trading under the symbol CAM. The plan bought these assets on January 1, 2003. The bond was purchased to yield 7%, and its maturity is on December 31, 2023. The shares of CAM were purchased at 12 per share. As of January 1, 2004, current market yield for the bond is 5%, and the CAM stock is trading at 10 per share. Establish the value of the asset portfolio of this pension plan in January 1, 2004, under the following methodologies: (a) Fair Market Value Method: using the actual market values of both assets. (b) Discounted Cash Flow Method: assuming that the bond and the stock are valued at discounted values of their cash flows, using the bond market yield and only for this part of the exercise, the stock is assumed to pay a perpetual dividend of 0.50. (c) Cost Value Method: using the actual cost of the purchase of the assets, including income paid held as cash. (d) Amortized Value Method: using the market value of the stock and the standard statutory valuation method for the bond, amortizing it to the redemption value using the yield as of the date of purchase. (e) Contract Value Method: assuming that the company decided at the beginning of the year to sell the bond and the stock, and instead purchase a GIC with the same book value, paying a guaranteed rate equal to the bond market yield as of the beginning of the year. (f) Blend of Cost and Market Value Method: assuming that for both the bond and the stock, this method utilizes the simple arithmetic average of the cost and the current market value of the asset. (g) Write-Up Method: assuming that the bond value increases every year at the rate equal to the yield as of the date of purchase, and coupons paid are deducted from the value, while for the stock, assuming no dividends and annual rate of return of 2% above than current bond market yield. (h) Deferred Recognition Method: recognizing half of change in the market value, and amortizing the rest, using linear amortization, over the following 5 years. (i) Average Market Value Method: using the average of the current and previous year’s market value.

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(j) Trend-Line Method: assuming the asset portfolio to be worth half of the current market value plus half of last year’s value increased by the market yield for bonds as of the end of last year. (k) Average Unit Value Method: assuming that the initial portfolio is one unit, and calculating the value of that unit at the end of the year, plus any income produced by the portfolio (as mentioned before, except for part (b), we assume that the stock does not pay any dividends). Solution Before we proceed with specific solutions of all parts, let us establish the market values of the bond and stock portions of the portfolio as of January 1, 2003 and as of January 1, 2004. On January 1, 2003, the stock portfolio consists of 100 shares valued at 12 per share for a total value of 1200. The value of the bond is given by the standard ‘Frank Formula’: 20 P ¼ Fra 20 7% þ Cv20 7% ¼ 60a 20 7% þ 1000v7% ¼ 635:64 þ 258:42 ¼ 894:06:

On January 1, 2004, the stock portfolio consists of 100 shares valued at 10 per share for a total value of 1000. The value of the bond is 20 P ¼ Fra 19 5% þ Cv20 5% ¼ 60a 19 5% þ 1000v5% ¼ 1120:85:

(a) The market value of the portfolio on January 1, 2004 is 1000 þ 1120:85 ¼ 2120:85: Since the coupon of 60 was not included in the calculation, we should also add it to the total value, as cash holdings, for a total of 2180.85. (b) The bond market yield is 5% as on the date of valuation. We have established the value of the bond as 1120.85 using that rate as the yield to maturity. The value of the stock is the value of 100 shares paying 50 each in perpetuity, or just a straight perpetuity of 50 at 5%, i.e., 1000. The value of the portfolio is again 2120.85 plus the cash holdings of 60 for a total value of 2180.85. (c) The cost of the stock purchase was 1200, and the bond purchase cost 894.06. In addition, the bond paid the coupon of 60, for a total portfolio value of 2154.06. (d) The standard amortized value of the bond is calculated based on the remaining cash flows and the yield as of the date of purchase, giving 19 P ¼ Fra 19 7% þ Cv19 7% ¼ 60a 19 7% þ 1000v7% ¼ 896:64:

The portfolio also has 60 in cash and 1000 in shares of CAM, for a total value of 1956.64. (e) Under this scenario, on January 1, 2003, the company, instead of 894.06 in bonds and 1200 in shares, purchases a GIC paying 7% with the initial value of 2094.06. The contract value at the end of the year is 2094:06 · 1:07 ¼ 2240:64:

Valuation of pension plan assets

285

(f) The market value of the stock is 1000, and the cost of it is 1200. The average of the two is 1100. The bond cost is 894.06, and the market value is 1120.85 for an average of approximately 10 007.46. Adding the cash position of 60, we arrive at the total portfolio value of 2167.46. (g) This method effectively values the bond portfolio using the standard statutory method, same as in (d), giving it the value of 896.64. The stock purchased at 1200 is assumed to increase in value by 9%, 200 basis points above the bond market yield of 7% at the beginning of the year, for the end of the year value of 1308. Adding the 60 cash position, we get the total portfolio value of 896:64 þ 1308 þ 60 ¼ 2264:64: (h) The stock portfolio was initially valued at 1200, and at the end of the year it stood at 1000. The change in the market value is 2 200, but only a half of it is recognized immediately so that the value of the stock is established as 1100 and over the future 5 years, additional 20 will be deducted from its value every year. The bond portion of the portfolio stood initially at 894.06 and now has a market value of 1120.85. The change of the value is 226.79 and half of it, i.e., 113.40, is immediately added to the statement bond value giving 894:06 þ 113:40 ¼ 1007:46: The remaining 113.39 will be amortized linearly over the future 5 years, adding approximately 22.68 to the bond portfolio value each of those years. By including the 60 cash position, we arrive at the statement value of asset of 1100 þ 1007:46 þ 60 ¼ 2167:46: (i) The average of the current and previous year stock value is 1100, and the average of the current and the previous year bond value is ð894:06 þ 1120:85Þ=2 ¼ 1007:46: With the 60 position in cash, the total statement value of the portfolio is 2167.46 (because of the simplified design of the problem we arrive at the same answer as in part (h), but different recognition and averaging methods can produce dramatically different results). (j) Half of the current market value of assets is, from part (a), 2180:85=2 ¼ 1090:43: Half of last year’s market value is 2094:06=2 ¼ 1047:03; and increased by 5%, it gives us 1099.38 for a total statement value of assets, including cash, of 1090:43 þ 1099:38 þ 60 ¼ 2249:81: (k) The initial portfolio is 100 shares plus 1000 par value bond. At the end of the year, the number of units is the same and since we are using the market value of the unit, we will end up with the market value for the statement. We will have to also include 60 cash from the coupon, for the total value is exactly equal to the total market value of 2180.85 as in part (a). A

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Part 3 Financial Risk Management

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289

Chapter 8 General rules of risk classification and measurement

8.1 Risk classification The key objective of a pension plan is a secure delivery of pension benefits to the plan participants. The plan sponsor has a choice of a variety of funding methods and risk management strategies. All actuarial funding methods have one characteristic in common: they require that at least some of the future liabilities of the plan are financed with capital assets purchased today. A pension plan has assets in the form of financial instruments, while its liabilities are set actuarially in relation to benefits granted. While holding securities as assets is designed to assure delivery of benefits, it does become a source of additional risks for the plan. What kind of risks does a pension plan face? The Society of Actuaries Committee on Valuation and Related Matters (1979) charged with, among others, this question, defined the following three key kinds of risks faced by a life or annuity insurance enterprise. † C1 risk: asset default and depreciation risk, i.e., the risk of suffering losses in equities or bond losses due to credit risk (but not due to interest rate risk). This is an asset-side risk. † C2 risk: pricing risk, i.e., the risk that the product issued by the financial intermediary has been issued at an inappropriate price (e.g., it did not provide the intermediary with proper compensation for risks that the intermediary has assumed). This is a liabilities-side risk. † C3 risk: interest rate risk or, more generally, asset –liability management risk, i.e., the risk that assets and liabilities of the intermediary may respond differently to changes in market prices or indexes (notably, changes in interest rates). † C4 risk: the general business risk, caused by the management, political and regulatory environment, or other socio-economic factors.

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The level of C1 risk is a function of the investment policy assumed by the intermediary. We have already discussed the issue of risk of an investment portfolio in Section 7.1. We will discuss some additional issues of asset risk in Chapter 10, where we talk about the shortfall constraints and the value at risk. In the case of insurance companies and pension plans, C2 risk is clearly the responsibility of an actuary. For a pension plan, plan liabilities may turn out to be higher than assumed, due to factors such as mortality, salaries, withdrawal, or retirement pattern, diverging from the actuarial assumptions used in valuation. C2 risk for a pension plan is related to the methodology of plan valuation. We discussed such methodologies in Chapters 4 – 6. But the asset – liability management risk C3 is a function of how an intermediary manages its assets and liabilities together. Such integrated comprehensive risk management is a relatively new idea in the management of financial intermediaries. We will discuss its ideas in Chapter 9 and Sections 10.2– 10.5. The key issue of asset –liability management is the question of whether assets and liabilities exhibit similar behavior when market conditions change, and if their behavior is different, whether the financial intermediary is compensated properly for assuming that risk. Furthermore, an intermediary should be able to understand and control the risk assumed. Clearly, an intermediary cannot control the level of interest rates (see Section 7.1). However, one of the main objectives of asset – liability management is to create a comprehensive and integrated policy of dealing with the changes in interest rates. Interest rate risk can cause insolvency of a financial intermediary. For a pension plan we arrive at this Principle of Asset –Liability Management:

Asset –liability management, and interest rate risk management in particular, for a pension plan, should be based on the principle that either assets and liabilities should behave in the same way under changing market conditions, or if they diverge under changing market conditions, the pension plan managers should understand that risk, be able to control it, and be compensated for assuming it.

This classification of risks was the basis for the risk-based capital (RBC) requirements for life and annuity insurance companies in the United States [9].

8.2 Total risk measurement and capital requirement The (United States) National Association of Insurance Commissioners (NAIC) instituted its RBC system for life insurance companies in 1993, followed by a property –casualty system in 1994 and a health system in 1998. The NAIC’s RBC

General rules of risk classification and measurement

291

system consists of two parts: † a formula that is used to set a regulatory minimum capital level for each insurer, based on that insurer’s mix of assets, liabilities, and risk, and † definition of ‘financial impairment’ and remedies to state insurance regulators in the event that an insurer meets that definition of impairment. Formulas were originally based on the Society of Actuaries Committee on Valuation and Related Matters risk classification (see Section 8.1), but over time they continuously evolve. We will present the current version below. NAIC publishes newsletters and guidelines for the calculation of RBC. The RBC system is meant to be a supplement, not a replacement, for the existing fixed minimum capital requirements that exist in each state. That is, the RBC formula requirements can be higher or lower than the fixed minimum capital requirements (which are typically $1 to $2 million), but each insurance company must meet both sets of standards. Many small insurance companies generate RBC requirements that are lower than the fixed dollar minimums, but for virtually all medium-sized and large insurers, the capital requirements generated by the RBC formula are higher than the state-fixed minimums. The RBC requirement (level of capital required in view of the risk undertaken) is calculated by multiplying risk factors times statement values, adding the results together, and then adjusting for covariance between major risk categories. The formula results are compared to the risk-adjusted capital of the insurer to develop the RBC ratio, which is the ratio of risk-adjusted capital to RBC. The ratio results are used to determine the degree to which an insurance company’s surplus is impaired. The model act specifies a series of increasingly stringent regulatory responses, as the RBC ratio decreases below 200%. A trend test is included to test whether insurers that were between the 200% breakpoint and 250% level were trending downward, which will trigger regulatory action, but an RBC ratio over 250% for a life company is sufficient to receive a passing grade on this pass/fail test. There are four ‘action levels’ under the NAIC RBC system. † Company Action Level (CAL). If this level is reached, insurer is required to automatically submit a written, detailed business plan within 45 days that details the causes and actions that have led up to the capital impairment as well as a plan for the restructuring of the insurer’s business to rebuild capital to acceptable levels. Alternatively, the company can detail plans to reduce its risk to a level commensurate with its actual capital level. † Regulatory Action Level (RAL). In this case, the insurer must conform to the requirements stated in the Company Action Level, and in addition is subject to an immediate regulatory audit. The regulator can then issue protective orders to force the insurer to either lower its risk profile or increase its capital to a level commensurate with its risk. A company that has reached the Company Action Level and that does not conform to the statutory requirements spelled

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out in the statute is also automatically deemed to have triggered the Regulatory Action Level. † Authorized Control Level (ACL) is triggered by having statutory capital that is less than the ACL RBC, as computed by the RBC formula or by failing to meet regulatory requirements imposed by the Regulatory Action Level. The ACL is the capital level at which the state insurance commissioner is authorized, although not required, to place the insurance company under regulatory supervision. † Mandatory Control Level. When that happens, the state regulator is required by statute to take steps to place the insurer under regulatory supervision. Originally, the life and annuity RBC formula considered the four categories of risk created by the Committee on Valuation and Related Matters (1979). These generic categories have been later refined by the US national regulatory supervisory agency, NAIC, and currently they have become: † † † † †

C0: C1: C2: C3: C4:

Affiliates Risk Asset Risk Insurance Risk Interest Rate Risk Business Risk

The values calculated for each category are then combined in what is commonly called the covariance formula. The original life RBC covariance formula was qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðC1 þ C3Þ2 þ C22 þ C4: It has been since then changed to qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C0 þ ðC1 þ C3Þ2 þ C22 þ C4: The results of the covariance formula produce the Company Action Level RBC capital requirement. The Company Action Level requirement is twice the ACL requirement. If the insurer’s Total Adjusted Capital is less than the ACL RBC requirement, the regulator is authorized to seize control of the company. The ACL RBC and the Total Adjusted Capital are both reported in the 5-year history page of the annual statement submitted to state insurance regulators. The RBC formula inputs and calculations are not made public. Total Adjusted Capital is defined as follows: Total Adjusted Capital ¼ statutory capital and surplus þ Asset Valuation Reserve (AVR) including AVR in separate accounts þ half of company’s liability for dividends þ company’s ownership share of

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General rules of risk classification and measurement

AVR of subsidiaries þ half of company’s ownership share of subsidiaries’ dividend liability. Separate RBC models apply to life companies, property/casualty companies and health organizations. The common risks identified in the NAIC models for all types of companies include Asset Risk—Affiliates, Asset Risk—Other, Credit Risk, Underwriting Risk, and Business Risk. The steps in RBC calculation are: † apply risk factors against annual statement values, † sum risk amounts and adjust for statistical independence (using the covariance formula), † calculate ACL RBC amount, † compare ACL RBC to adjusted capital. Total Adjusted Capital (actual capital) is divided by ACL RBC (hypothetical minimum capital) to get the RBC ratio † † † †

No action (98% of companies)—TAC/RBC over 200% Company Action Level—TAC/RBC is 150 –200% ACL—TAC/RBC is 70– 100% Mandatory Control Level—TAC/RBC is less than 70%

After the calculation of RBC, the company is also expected to perform sensitivity tests to indicate how sensitive the results are to certain risk factors’ changes. The calculation of the risk amounts C0, C1, C2, C3, and C4 is done in the following way. C0 is transferred from the subsidiary’s balance sheet in a pro-rata share. C1 is calculated by applying factors specified by NAIC to various asset classes, and then adding the results. Here is an example of such a calculation (note that bonds are grouped into classes defined by NAIC):

Asset NAIC Class 1, US Government NAIC Class 1, non-US Government NAIC Class 2 NAIC Class 3 NAIC Class 4 NAIC Class 5 NAIC Class 6 Total

Portfolio

Factor

RBC

$1000 $1000 $1000 $1000 $1000 $1000 $1000 $7000

0.000 0.010 0.010 0.020 0.045 0.100 0.300

$0.00 $3.00 $10.00 $20.00 $45.00 $100.00 $300.00 $478.00

The result for this calculation for bonds is then adjusted by a factor, which is determined by the number of bonds in the portfolio. There are no such adjustments for portfolio size for stocks and mortgages.

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For stocks C1 RBC is derived by multiplying the total value of all stocks by a factor provided by NAIC (30% for life companies, but that factors do change over time). There is also a separate calculation for mortgages held in life insurance companies’ portfolios. The mortgages calculation is based on AVR, a reserve liability set aside to account for drop in market value of bonds due to change in interest rates. After establishing AVR, there is a Mortgage Experience Adjustment (MEA), and then the RBC mortgage contribution is established by multiplying the results by separate risk factors by mortgage type (farm, residential, commercial, restructured) and by quality (in good standing, overdue, in foreclosure). For real estate assets there is a large number of separate RBC factors depending on the type and quality of property. The calculation is done separately for each property and then summed to obtain the total. The C2 risk RBC is calculated by using a tiered formula used for the net amount at risk (benefit payable minus the reserve set up for its payment). We present an example of a tiered C2 RBC calculation for life insurance. A company has $10 000 000 in net amount at risk. A factor of 3.5% is applied to the first $5 million and 2.0% for the next $5 million. Then the insurance risk RBC for this company is: 3.5% of $5 000 000 plus 2.0% of $5 000 000, for a total of $175 000 þ $100 000 ¼ $275 000. The C3 risk RBC calculation uses the asset – liability model used for year-end Asset Adequacy Analysis cash flow testing or a consistent model. One starts by running the scenarios (12 or 50) produced from the interest-rate scenario generator. The statutory capital and surplus, SðtÞ; should be captured for every scenario for each calendar year-end of the testing horizon. For each scenario, the C3 measure is the most negative of the series of present values SðtÞ; using 105% of the after-tax 1-year treasury rates for that scenario. Then one should rank the scenario-specific C3 measures in descending order, with scenario number’s measure being the positive capital amount needed to equal the very worst present value measure. Taking the weighted average of a subset (currently, scenarios ranked 17th through 5th are used with weights 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.16, 0.12, 0.10, 0.08, 0.06, 0.04, 0.02) of the scenario-specific C3 scores derives the final C3 factor for the 50-scenario set. For the 12-scenario set, the charge is calculated as the average of the C3 scores ranked 2 and 3, but cannot be less than half the worst scenario score. There are also cases when single scenario testing is allowed. Finally, the C4 RBC is calculated generally as a small percentage of premiums, in the range of 2%. In the case of pension plans there is one more unique risk: that the plan sponsor may not be able to provide the funding required. We will analyze this issue in the next section.

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Exercises Exercise 8.2.1 May 2003 SOA Course 6 Examination, Problem No. C-13. You are given the following for an insurance company that currently offers term insurance and fixed deferred annuities: corporate pre-tax target return on capital is 18%, and the RBC formula is: 1:5

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðAsset Default Risk Component2 þ Mortality Risk Component2 Þ:

The calculation of the asset default risk component (C1) of the RBC formula is performed as follows: † For bonds, the RBC factor is 1%. This company has 600 million monetary units invested in bonds, and estimates the expected rate of return on bonds as 7%. † For real estate, the RBC factor is 7%. This company has 300 million monetary units invested in real estate, and estimates the expected rate of return on real estate as 8%. † For common stocks, the RBC factor is 20%. This company has 100 million monetary units invested in stocks, and estimates the expected rate of return on stocks as 10%. For the mortality risk component (C2), the RBC factor is 0.10% of the net amount at risk (defined as the difference between the benefit amount on a life insurance policy and the reserve held for it), and this company has 10 000 million in monetary units of net amount of risk. The industry-wide ratio of C1 to C2 is 1.5. The risk-free rate is 6%. (a) Describe the shortcomings of this RBC formula. (b) Calculate the RBC-adjusted spread for this company’s asset portfolio. (c) Evaluate the competitive advantage of the company’s product lines from a cost of capital perspective. Solution (a) The shortcomings of this formula are: – – – – – – – –

limited asset classes, no component for asset – liability risk, no covariance of risks component, flat reference to amount at risk, no reference to asset diversification within asset class, no reference to asset credit quality within asset class, equal weights for asset and liability risks, probably not appropriate, no accounting for subsidiaries.

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– does not account for general business risks, – this is just a formula and does not reflect the actual quality of company’s performance, such as, e.g., underwriting standards. (b) For this company, we have C1 ¼ 600 · 1:0% þ 300 · 7% þ 100 · 20% ¼ 47 and C2 ¼ 10 000 · 0:1% ¼ 10: There is no information about the actual surplus held. In reality, the company will have a target RBC ratio that it will aim to keep, and the actual reasoning should be done for that target ratio, and the resulting capital held. We will assume 100% target ratio, and if the actual ratio is higher, the result will be proportional. Now consider the RBC formula in this problem qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ RBC ¼ Fðx; yÞ ¼ 1:5 x2 þ y2 where x is the C1 value and y is the C2 value. Suppose the C1 value increases by Dx. Then the new RBC is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ RBC ¼ Fðx þ Dx; yÞ ¼ 1:5 ðx þ DxÞ2 þ y2 : The resulting change in RBC is: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ›Fðx; yÞ · Dx: 1:5 ðx þ DxÞ2 þ y2 2 1:5 x2 þ y2 < ›x In the case of this RBC formula, we have

›Fðx; yÞ 1:5x ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ›x x2 þ y2 Let us assume that a company buys one unit of bonds. What is the amount of capital required for that? The C1 amount would be increased by Dx ¼ 0:01; as the RBC requirement for bonds is 1%. The increase in required capital is approximately 1:5 · 47 · 0:01 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ < 0:01467: 472 þ 102 Hence one additional unit of bonds will earn the following spread over the risk-free rate, after adjusting for the cost of capital required, of ð7% 2 6%Þ · 1 2 ð18% 2 6%Þ · 0:01467 ¼ 0:824%: Now assume that the company buys one additional unit of real estate. Then Dx ¼ 0:07 as the RBC requirement for real estate is 7%. The increase in required capital is

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approximately 1:5 · 47 · 0:07 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ < 0:1027: 472 þ 102 This one additional unit of real estate will earn the following spread over the risk-free rate, after adjusting for the cost of capital required, of ð8% 2 6%Þ · 1 2 ð18% 2 6%Þ · 0:1027 ¼ 0:7676%: Finally, assume that the company buys one additional unit of stocks. Then Dx ¼ 0:2 as the RBC requirement for stocks is 20%. The increase in required capital is approximately 1:5 · 47 · 0:20 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ < 0:2934: 472 þ 102 This one additional unit of stocks will earn the following spread over the riskfree rate, after adjusting for the cost of capital required, of ð10% 2 6%Þ · 1 2 ð18% 2 6%Þ · 0:2934 ¼ 0:4702%: The total spread weighted by asset allocation is: 0:60 · 0:824% þ 0:30 · 0:7676% þ 0:10 · 0:4702% ¼ 0:4944% þ 0:2303% þ 0:0470% ¼ 0:7717%: This is about 77 basis points. (c) A company enjoys a competitive advantage if it has a lower cost of capital. The capital required by the market is considered a sum of the ‘Face Capital’, which earns the risk-free rate, and ‘At-Risk Capital’, which earns the rate of return on stocks (equity cost of capital). The cost of capital for all the capital is the appropriate weighted average of the two. Because of the RBC formula qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1:5 · ðAsset Default Risk Component2 þ Mortality Risk Component2 Þ; it makes sense (from what we know from basic triangular geometry and the Pythagorean Theorem) for companies to equally balance the asset default risk component and the mortality risk component. Hence companies with relatively high proportion of C1/RBC would have advantage to sell more pure insurance risk type products like term insurance. Those with relatively low ratio have competitive advantage to sell accumulation products, as those are investment oriented, with relatively lower amount at risk. We should note that RBC requirement has the potential to tie down some capital and restrain company’s ability to grow and sell certain more capital-intensive products. This is

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particularly true if the RBC ratio is near the regulatory or rating agencies’ threshold level (e.g., trend test zone). In the case of this company C1 47 ¼ ¼ 4:7; C2 10 while for entire industry C1 ¼ 1:5; C2 hence this company is now overly concentrated in C1 risk, and it will be more competitive to sell purely insurance-oriented products. A

8.3 Asset-side risks and investment benchmarks As pointed out in Section 8.1, holding securities as assets becomes a source of the socalled C1 risks of asset default or depreciation. This kind of risks is recognized in a way depending on the type of pension plans: in case of defined benefit plans the asset depreciation forces the plan sponsor to provide additional contribution to the fund; in case of defined contribution plans it means a decrease of future benefits of the plan participants. This is one of the possible explanations for a rapid increase of the number of defined contribution plans in the United States and Canada since the late 1970s (see Section 1.2): plan sponsors were interested in moving the asset risk entirely to the plan participants. In order to increase responsibility of asset management companies for good investment performance of the funds, specially if private firms operate mandatory pension funds, regulatory bodies in several countries introduced minimal investment requirements. Such investment benchmarks are usually defined as minimal required rates of return, being functions of the average rate of return of all funds and, incidentally, another interest rate like inflation rate. For example, in Poland the minimal required rate of return, rmin, is defined as follows rmin ¼ minð0:5rPL ; rPL 2 0:04Þ

ð8:3:1Þ

where rPL is the 2-year average rate of return defined by (2.6.21). A minimal required rate of return in Chile was defined analogously. The fund management company must contribute from their own capital to the required level of the fund assets, if it is not attained at the comparison date. When using formulas like (8.3.1) the following natural questions arise. † What definition of average rate of return should we use? † What should be the bandwidth between the average rate of return and the minimal required rate of return?

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† How long should the time intervals used for the average rate of return calculation be? † How often should the investment performance of the fund be compared with the benchmark? To answer these questions one should first determine the aim of using formula (8.3.1) or similar ones. Is it a way to take some capital from a randomly chosen asset management company in order to increase fund’s assets? No, because if that asset increment were a purely random game all management firms would discount that risk in the management fee, simultaneously decreasing the assets assigned to the customers. We propose instead that:

A minimal required rate of return formula, such as (8.3.1), should be a tool to identify those pension fund management firms that systematically, not randomly, obtain investment results lower than the entire pension funds market.

The starting point, therefore, is that the average rate of return index should reflect the market behavior precisely. In Section 2.6 we have presented the formula (2.6.2) for the average rate of return rðs; tÞ; which is the proper candidate to be used for comparisons. This formula is convenient in applications and satisfies all axioms presented there. And, what is most important, it reflects accurately the market investment performance. If the rate of return index is not chosen properly, the benchmark may be a way to punish, or even to bankrupt, a randomly chosen management company. A badly chosen average index formula may even become harmful, as its effect may be for all management companies to copy each other’s investment strategies, or for the regulatory body to start increasing time intervals for the average rate of return calculation, and to decrease frequency of comparisons of the rates of return of the funds with the benchmark. In the first case, increased correlation of fund managers’ strategies may be a source of unnecessarily increased systematic risk of the markets. Such increased systematic risk may result in less capital being available in a given country’s financial markets, due to its higher cost. Poland may be considered a case study of such a situation. Immediately following the reform and partial privatization of the state pension system in April of 1999, the 2-year average rate of interest rPL (see formula (2.6.21)) was calculated every quarter and since the end of July 2001, every fund’s rate of return was compared with rmin. Quite early into the existence of the new system, one of the pension management companies was required to contributed approximately USD 17 million, or approximately 24% of its capital to the fund it managed. If this situation were to happen now, when assets managed by the company more than doubled (or when the pension market matures and funds assets are expected to increase further fivefold),

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required capital contribution could result in an insolvency of that pension manager. However, Table 2.6.1 shows that rPL overestimates the overall results of all funds. This overestimation had two sources: one is the wrong formula for the average rate of interest and the second is not taking into account all companies managing the open pension funds (compare the third, forth and sixth column of Table 2.6.1). Hence the contributions to client’s fund required by regulators from the fund managers are significantly higher than that implied by market reality. However, instead of correcting the formula, the supervisory body started to enlarge the length of investment activity taken into account to calculate rPL and decreasing frequency of the results inspection. Since 2004, the wrong formula rPL (with 15% as the upper bound put on the market shares) is calculated from the results of the last 3 years and the inspection is done semiannually. The three key questions we have posed above are closely related to each other and should be answered with consideration for their interdependence. If the market overall performance is described precisely, we can utilize it to test an investment fund’s performance and compare it to all funds, or a prescribed group of funds. This is an interesting statistical problem of testing inequality between means of two statistically dependent random variables (dependence is implied by the fact that the average rate of return for all funds contains the rate of return for every fund). This dependence should not be ignored, and it cannot be ignored in the case of the biggest funds, which constitute a substantial portion of the market, so a more adequate approach would be to compare the rate of return of a given fund with the minimal required rate of return calculated on the basis of the average rate r corresponding to all other funds. Finally, it is well known that statistical tests concerning hypotheses on relationships between means of random variables have precision levels depending on variances of those random variables. Therefore the formula (8.3.1) may be statistically meaningful only for the portfolios with very specific variances; otherwise, one should derive market-based model parameters instead of arbitrarily chosen multiplier of 0.5 and similarly arbitrarily picked threshold of 0.04.

8.4 Stability of total annual premium From the point of view of plan sponsor, one of the key problems in the pension plan risk management is the assurance of stability and predictability of the plan normal cost. The normal cost of a plan is a concept similar to the premium for life insurance, and it represents the expected premium needed to finance the difference between the present value of accrued liabilities in successive years. The difference between the value of the plan assets and liabilities is called the plan surplus, if it is positive, or plan deficit, if it is negative. A plan with a deficit must, generally, amortize it with a series of payments in addition to the normal cost, thus increasing the future costs to the plan sponsor. It should be noted that this concept of plan deficit is not the same as the notion of the actuarial deficit: the actuarial deficit is

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the difference between the value of plan liabilities and expected future normal cost (planned for this year), and the value of the plan assets. Lack of stability of the total annual premium can be caused by these factors: † instability of normal cost and † existence of plan deficit. In Chapter 5 we discussed methodologies for calculation of the normal cost, with consideration of normal cost stability over time. That presentation gave ways of calculating the normal cost in a way that adheres to the plan structure, in order to achieve a desired degree of control over the evolution of plan costs over time. Plan deficit appears when the difference between plan assets and liabilities turns out to be significantly different than the expected one. Such a difference can be caused by different reactions of plan assets and liabilities to the changes in the capital markets. Therefore

The stability of total annual premium over time depends on the chosen method of normal cost valuation and the stability of the relationship between assets and liabilities of the plan.

We will now further our study of the relationship between assets and liabilities.

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303

Chapter 9 Interest rate risk—nonrandom approach

9.1 The relationship between the price of capital assets and the interest rates 9.1.1 Duration of a security In what follows, we consider both assets and liabilities of a pension plan as capital assets (see Example 9.1.1). Assume at first that we know exactly the cash flow At produced by a security under consideration at time t: We will usually assume that the cash flows are discrete (as opposed to continuous). Assume also that there is only one interest rate regardless of maturity (i.e., the yield curve is flat) and that At does not depend on i (we will say that a security has deterministic cash flows when its cash flows do not depend on interest rate). Then the present value of the security, its price, is P ¼ PðiÞ ¼

X t$0

At : ð1 þ iÞt

ð9:1:1Þ

We will define now the basic measures of sensitivity of a security with respect to changes in interest rates. We will assume, unrealistically so, that there is only one interest rate regardless of maturity, i.e., that the yield curve is flat. The derivative of the price P of the security with respect to the interest rate, with a minus sign in front of it, i.e., 2

dPðiÞ ; di

is commonly called the dollar duration of the security. We propose a new name for it, monetary duration (denoted by Dnom), to separate it from the national origin of the currency used. This definition applies to all securities, with deterministic cash flows or with cash flows that depend on the interest rate. In the case of deterministic cash

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flows, as in (9.1.1), monetary duration is given by the formula X X tAt 1 Dnom ðPÞ ¼ ¼ t · PVðAt Þ; tþ1 ð1 þ iÞ t.0 t.0 ð1 þ iÞ where PVðAt Þ is the present value of the cash flow At : The ratio of the monetary duration to the price, equal to the logarithmic derivative of the price of a security with respect to the interest rate, with a minus sign in front of it, is called the duration D(P) of the security. Thus: DðPÞ ¼

Dnom ðPÞ 1 dP d ¼2 ¼ 2 ðlnðPÞÞ: P P di di

In the case of a security with deterministic cash flows, as in (9.1.1), duration is given by the formula: DðPÞ ¼

X PVðAt Þ 1 X tAt 1 : ¼ t· tþ1 P t.0 ð1 þ iÞ ð1 þ iÞ t.0 P

If we introduce the quantity wt ¼

PVðAt Þ ; P

then duration of a deterministic security turns out to be a weighted-average time to maturity, modified by the factor 1=ð1 þ iÞ: For that reason, the concept of duration as introduced here in the most general sense, is called modified duration for securities with deterministic cash flows (as the weighted average is modified by the factor 1=ð1 þ iÞ). For securities with cash flows dependent on interest rates, which causes the cash flows to be random in nature (as we assume that interest rates are random), duration is most often termed effective duration. However, in all cases, duration is the opposite of the logarithmic derivative with respect to the interest rate, and this is why we use here this most general concept. The weighted-average time to maturity concept is actually the original idea of duration. Macaulay [45] defined, for a security with deterministic cash flows, the concept, which is commonly called the Macaulay duration, as DM ; 2ð1 þ iÞ

dðln PÞ 1 X tAt ¼ : di P t$0 ð1 þ iÞt

ð9:1:2Þ

We do assume in our analysis for now that the interest rate i is the same for all maturities. This in turn implies that the force of interest d ¼ lnð1 þ iÞ is also level throughout all maturities. We can see easily that for a security with deterministic cash flows X P¼ e2dt At ð9:1:3Þ t$0

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Interest rate risk—nonrandom approach

and DM ¼ 2

1 dA dðln PÞ 1 X 2dt ¼2 ¼ t e At : P dd dd P t$0

ð9:1:4Þ

We will use the negative logarithmic derivative with respect to the force of interest as the general definition of Macaulay security for any security, not just a deterministic one. If we recall the definition of weights wt ¼

PVðAt Þ PðiÞ

and introduce a discrete probability distribution with PrðT ¼ tÞ ¼ wt ; then we can also see that the Macaulay duration is the expected value of this probability distribution. We can think of the time T to payment as a random variable described by that distribution, and the expected time to payment EðTÞ is therefore the Macaulay duration. Let us also note that DM ðPÞ ¼

d lnðPÞ : 1 d ln 1þi

This means that

Macaulay duration of a security is a measure of the elasticity of the price of this security with respect to the discount factor v ¼ ð1 þ iÞ21 :

In practice, Macaulay duration is most often identified with the weighted-average time to maturity of a security. Indeed, it follows from (9.1.1) and (9.1.2) that DM ¼

X t$0

tA ð1 þ iÞ2t Xt : At ð1 þ iÞ2t t$0

Let DurðAÞ be either the duration or Macaulay duration of any security with price A treated as a function of interest rate (or force of mortality) A ¼ AðiÞ ¼ AðdÞ: Suppose that A, A1, and A2 are securities such that AðiÞ ¼ AðdÞ ¼ A1 ðiÞ ^ A2 ðiÞ ¼ A1 ðdÞ ^ A2 ðdÞ: Then it follows directly from the definition of duration, or Macaulay

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duration that DurðAÞ ¼ ¼

A1 A DurðA1 Þ ^ 2 DurðA2 Þ A A A1 A2 DurðA1 Þ ^ DurðA2 Þ A1 ^ A 2 A1 ^ A2

ð9:1:5Þ

The nominal Macaulay duration of A, DnomM, is defined by X tAt ð1 þ iÞ2t DnomM ¼ t$0

for a deterministic security A, and for an arbitrary security, it is simply the opposite of the derivative of the price of the security with respect to the force of interest. We can quickly see that for a single payment At at a time t in the future, its Macaulay duration is t; its duration is t=ð1 þ iÞ and its nominal Macaulay duration is tAt =ð1 þ iÞt : Since duration of a portfolio can be calculated as a weighted average of the duration of individual payments, for securities whose cash flows do not depend on interest rates calculation of duration is actually quite easy. It is not common to consider the case of continuous stream of payments for calculation of duration, because such securities do not exist in reality. We will consider such securities for purely theoretical purposes. If instead of discrete cash flows At we have a continuous stream of such payments, with a constant force of interest d its present value is (assuming a finite price exists) ð1 PðdÞ ¼ e2td At dt: 0

The nominal Macaulay duration of such a security is ð ð1 ð1 d 1 2td d 2td DnomM ðPðdÞÞ ¼ 2P0 ðdÞ ¼ 2 e At dt ¼ 2 e At dt ¼ t e2td At dt dd 0 0 dd 0 Its Macaulay duration is ð1

ð1 t e2td At dt ð 1 P0 ð d Þ e2td At 0 ð ð ¼ 1 ¼ t 1 dt ¼ twt dt: DM ðPðdÞÞ ¼ 2 PðdÞ 0 0 e2td At dt e2td At dt 0

0

If we now consider a continuous probability distribution with density wt then its expected value again is the Macaulay duration of the security. We also conclude that the relationship DðPðiÞÞ ¼

1 D ðPðdÞÞ 1þi M

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Interest rate risk—nonrandom approach

holds for continuous streams of payments. However, all the above reasoning assumed that the security’s cash flows do not depend on the interest rate. We will discuss the case when there is such a dependence when we present practical approximations used for duration and convexity, at the end of this section. Example 9.1.1 Consider a discrete security paying a cash flow At ¼ etd at a single time t. Then its present value is PðdÞ ¼ 1: Therefore 2P0 ðdÞ ¼ 0; and the duration of this instrument is zero. Any linear combination of instruments like this, paying the accumulated value of a monetary unit at a given force of interest, will also have duration of zero, as it is a weighted average (even if in continuous form) of durations of individual payments. Similarly, if a discrete security pays a cash flow of At ¼ eðt21Þd at time t it has a price PðdÞ ¼ e2d and duration of 1. So do all the aggregations of this type of payment. These two examples illustrate the well-known fact that floating rate securities have durations between 0 and 1. A

9.1.2 Convexity of a financial asset Let A be the price of a capital asset considered as a function of interest rate. The quantity CM ðAÞ ¼

1 d2 A A dd2

will be termed the Macaulay convexity of this security. Note that for a security with P deterministic cash flows, if A ¼ t.0 At e2dt ; then X 1 X 2 t · At e2dt ¼ wt · t2 ; A t.0 t.0

CM ðAÞ ¼

where, as before, wt ¼ PVðAt Þ=P: Because A · DM ðAÞ is the monetary duration of the security, we also have CM ðAÞ ¼ 2

1 d dDM ðAÞ ðA · DM ðAÞÞ ¼ D2M ðAÞ 2 : A dd dd

The quantity 2 ¼2 MM

dDM ðAÞ ¼ CM ðAÞ 2 D2M ðAÞ dd

will be termed the Macaulay M-squared. Because DM ðAÞ ¼ 2

d ln A; dd

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we also see that d2 ðln AÞ : dd2 P For a security with deterministic cash flows A ¼ t.0 At e2dt ; X 1 X 2 MM ¼ ðt 2 DM Þ2 e2dt At ¼ wt ðt 2 DM Þ2 A t.0 t.0 2 ¼ MM

and 2 X dMM 1 X ¼2 ðt 2 DM Þ3 e2dt At ¼ 2 wt ðt 2 DM Þ3 : A t.0 dd t.0

The expression CðAÞ ¼

1 d2 A A di2

is called the convexity of a security, while M2 ¼

d2 ðlnðAðiÞÞÞ dDðAÞ ¼2 di di2

is called the M-squared of a security. What is the practical interpretation of the concept of convexity? We did note that for a security with deterministic cash flows the Macaulay duration is the weighted-average time to maturity. Consider such a deterministic security. Let us use the notation wt ¼

PVðAt Þ A e2dt ¼ Xt A At e2dt t.0

again and recall that based on (9.1.3) and (9.1.4) X tAt e2dt X t.0 DM ¼ X ¼ wt t: 2dt At e t.0 t.0

Now observe that 2 ¼ CM 2 D2M ¼ MM

X t.0

wt · t2 2 D2M ¼

X

wt · ðt 2 DM Þ2 :

t.0

This allows for a relatively simple and quite intuitive interpretation of duration, convexity and M-squared of a security. If we define a discrete probability distribution and a random variable T on the (assumed finite) set {t : At . 0} by PrðT ¼ tÞ ¼ wt ; then the Macaulay duration DM is the expected value of this random variable EðTÞ; and

309

Interest rate risk—nonrandom approach

Macaulay convexity is the second moment of this random variable, and Macaulay Msquared is the variance of it. This means that Macaulay duration can be interpreted intuitively as the expected time till maturity of cash flows of a security, Macaulay Msquared is the measure of dispersion of the cash flows of the said security, and Macaulay convexity is a sum of Macaulay M-squared and the square of Macaulay duration. What about the intuitive meaning of the regular duration, convexity and Msquared, i.e., the measures calculated with respect to the effective annual interest rate? Note that by the Chain Rule dP dd dP d lnð1 þ iÞ dP 1 dP ¼ ¼ ¼ : di di dd 1 þ i dd di dd Thus taking a derivative with respect to i is the same as taking a derivative with respect to d and multiplying the result by 1=ð1 þ iÞ: Also, taking a derivative with respect to d is the same as taking a derivative with respect to i and dividing the result by 1=ð1 þ iÞ; i.e., multiplying by 1 þ i: Let us now look at the second derivative d2 P d dP d dd dP d 1 dP ¼ ¼ ¼ di di di di dd di 1 þ i dd di2 1 dP 1 d dP 1 dP 1 dd d dP ¼2 þ þ ¼2 1 þ i di dd 1 þ i di dd dd ð1 þ iÞ2 dd ð1 þ iÞ2 dd ¼2

1 dP 1 d2 P þ : ð1 þ iÞ2 dd ð1 þ iÞ2 dd2

This means that D¼

1 D 1þi M

and C¼

1 1 D þ CM : 2 M ð1 þ iÞ ð1 þ iÞ2

For M 2 ¼ C 2 D2 we can calculate d2 ðln PÞ d M ¼ ¼ di di2 2

¼ C 2 D2 ¼ ¼

d ln P di

d ¼ di

1 dP P di

¼2

1 dP dP 1 d2 P þ P di2 P2 di di

1 1 1 C þ D 2 D2M 2 M 2 M ð1 þ iÞ ð1 þ iÞ ð1 þ iÞ2

1 1 2 MM þ DM : ð1 þ iÞ2 ð1 þ iÞ2

310

Chapter 9

Now let us assume that the financial instrument considered has deterministic cash P flows At (at time t), and let A ¼ t.0 PVðAt Þ: Then X 1 D Dnom ¼ tAt ð1 þ iÞ2t21 ¼ ; 1 þ i nomM t$0 dðln PÞ 1 X 1 ¼ D ; D¼2 wt¼ di 1 þ i t$0 t 1þi M X 1 Nominal convexity ¼ tðt þ 1ÞCFt ð1 þ iÞ2t ; ð1 þ iÞ2 t$0 X 1 1 1 C¼ wt tðt þ 1Þ ¼ C þ DM : 2 2 M ð1 þ iÞ t$0 ð1 þ iÞ ð1 þ iÞ2 2 We already noticed that MM increases with dispersion of cash flows. We can show it some other way. Let PVðAt Þ ¼ Pt : We have

d2 ðln AÞ 1 ¼ di2 ð1þiÞ2 2 ! ! ! ! ! !3 X X X X2 X X 6 t Pt Pt tPt tPt þ tPt Pt 7 7 6 t$0 t$0 t$0 t$0 t$0 7 6 t$0 6 2 7 !2 !2 7 6 X X 5 4 Pt Pt t$0

X 1 ¼ ð1þiÞ2

t$0

! Ps

X

s$0

!

X

2

t Pt 2

t$0

X

! sPs

s$0 !2

X

! tPt

t$0

þ

1 DM ð1þiÞ2

Pt

t$0

XX 1 ¼ ð1þiÞ2

t

t 2 Ps Pt 2

XX t

s–t

X

!2

s–t

stPs Pt þ

1 DM ð1þiÞ2

Pt

t$0

XX 1 ¼ ð1þiÞ2

t

ðt 2sÞ2 Ps Pt

s,t

X

!2

þ

1 DM : ð1þiÞ2

Pt

t$0

This formula, because of the expression ðt 2sÞ2 ; again reinforces the fact that the dispersion of the cash flows is, in addition to duration, a key consideration for

311

Interest rate risk—nonrandom approach

convexity. We also see that XX 2 MM ¼

d2 ðln AÞ ¼ dd2

t

ðt 2sÞ2 Ps Pt

s,t

X

!2

:

Pt

t$0

This last formula is effectively a re-interpretation of 2 MM ¼ VarðTÞ ¼ CM 2D2M

because for two independent identically distributed random variables S and T EððT 2SÞ2 Þ ¼ 2VarðTÞ ¼ 2VarðSÞ: Let us also note that we have a simple expression for the Macaulay convexity of a single payment at time t: it is equal to t 2. Its convexity is t2 t tðt þ1Þ þ ¼ ; 2 2 ð1þiÞ ð1þiÞ ð1þiÞ2 2 its Macaulay M-squared MM is zero, and its M 2 is t=ð1þiÞ2 : Furthermore if ConvexðPÞ is either the convexity or Macaulay convexity of any security, whose price is treated as a function of interest rate (or force of interest) P ¼ PðiÞ ¼ PðdÞ; and P, P1, P2 are prices of securities such that PðiÞ ¼ PðdÞ ¼ P1 ðiÞ^P2 ðiÞ ¼ P1 ðdÞ^P2 ðdÞ; then

ConvexðPÞ ¼ ¼

P1 P ConvexðP1 Þ^ 2 ConvexðP2 Þ P P P1 P2 ConvexðP1 Þ^ ConvexðP2 Þ: P1 ^P2 P1 ^P2

9.1.3 Nominal interest rates Can duration and convexity be calculated with respect to nominal interest rates, such as ið2Þ ; ið3Þ ; …; iðmÞ ? Yes, naturally, and the most natural choice is i (2) because semiannual yields are most common for bonds issued in the United States (although not common at all outside of the United States). Recall that iðmÞ 1þ m

!m ¼ 1 þ i ¼ ed :

312

Chapter 9

Therefore di 1 iðmÞ ¼ m· · 1 þ ðmÞ m m di

!m21 ¼

1þi iðmÞ 1þ m

and dd dd di · ¼ ¼ di diðmÞ diðmÞ

1 : iðmÞ 1þ m

Also, if j ¼ iðmÞ =m then dj 1 ¼ : ðmÞ m di Combining those identities, we get dP dP di ¼ ¼ di diðmÞ diðmÞ

1 þ i dP dP dP dd 1 dP ; ¼ ¼ ; dd diðmÞ diðmÞ iðmÞ di iðmÞ dd 1þ 1þ m m

dP dP diðmÞ 1 þ i dP dP dP diðmÞ 1 dP ¼ ðmÞ ; ¼ ¼m ¼m : ðmÞ ðmÞ ðmÞ dj di dj dd dj dj di di i i 1þ 1þ m m Based on these calculations, we can derive the duration with respect to iðmÞ DðmÞ ¼

1þi 1 D¼ DM ðmÞ i iðmÞ 1þ 1þ m m

and the duration with respect to iðmÞ =m as m · DðmÞ : What about convexity measure with respect to iðmÞ ? We have 0 1 d2 P d dP d B 1 dP C C ¼ ðmÞ ¼ ðmÞ B @ ðmÞ ðmÞ dd A ðmÞ 2 di di di dði Þ i 1þ m 1 d2 P 1 1 dP 2 ¼ þ : !2 !2 m dd dd2 iðmÞ iðmÞ 1þ 1þ m m

313

Interest rate risk—nonrandom approach

Therefore, convexity with respect to iðmÞ equals 1

C ðmÞ ¼ 1þ

iðmÞ m

1

!2 CM þ 1þ

iðmÞ m

!2

1 D : m M

9.1.4 The Babcock Equation The concepts of duration and convexity of a security are important tools for interest rate risk management. Duration is a measure of sensitivity of a security with respect to a change in interest rate, and convexity is a measure of sensitivity of duration to a change in interest rate. An important application of these concepts is presented in the Babcock Equation, which we will discuss now. Consider our standard deterministic security, which pays At at time t. Let Aðd; sÞ be its current value at time s as a function of the force of interest d. We have Aðd; sÞ ¼ eds

X

e2dt At ¼

t.0

X

edðs2tÞ At :

t.0

Now suppose that the force of interest changes slightly from d to d þ 1: Let Aðd þ 1; sÞ ¼ ers : Aðd; sÞ This means that r is the force of interest corresponding to the rate of return received from this security as a result of the change of the overall force of interest from d to d þ 1 (see Section 9.2.2). It turns out that r can be approximated with an expression involving Macaulay duration DM and Macaulay M-squared of the security under consideration. Indeed, we have Aðd þ 1; sÞ ¼

X

edðs2tÞ At e1ðs2tÞ ;

t.0

and also

e

1ðs2tÞ

¼e

1ðs2DM Þ

e

1ðDM 2tÞ

¼e

1ðs2DM Þ

1 2 2 1 þ 1ðDM 2 tÞ þ 1 ðDM 2 tÞ þ · · · ; 2

314

Chapter 9

which implies that Aðd þ 1;sÞ ¼ e

1ðs2DM Þ

X t.0

e

dðs2tÞ

1 2 2 At 1 þ 1ðDM 2 tÞ þ 1 ðDM 2 tÞ þ ·· · 2

¼ e1ðs2DM Þ Aðd;sÞ X dðs2tÞ X dðs2tÞ 0 1 e At ðDM 2 tÞ e At ðDM 2 tÞ2 1 t.0 B C t.0 þ 12 þ · · ·A @1 þ 1 Aðd;sÞ 2 Aðd;sÞ

¼e

1ðs2DM Þ

1 2 2 Aðd;sÞ 1 þ 0 þ 1 MM þ ·· · : 2 ð9:1:6Þ

This implies the following approximation Aðd þ 1;sÞ 1 2 < e1ðs2DM Þ 1 þ 12 MM : Aðd; sÞ 2 From the definition of r we conclude that 2 DM 1 1 2 2 DM 12 M M : r<1 12 þ ln 1 þ 1 MM < 1 1 2 þ s 2 s s 2s

ð9:1:7Þ

The Babcock Equation follows from the above if we ignore the term involving 1 2: DM r<1 12 : s On the other hand, if we assume that s ¼ DM ; then from (9.1.7) we infer that

r<

2 12 M M : 2s

The above expressions can also be derived by using the aforementioned probabilistic interpretation of duration and convexity. Let T be the previously defined random variable with the discrete probability distribution: e2dt At PrðT ¼ tÞ ¼ wt ¼ : Aðd; 0Þ The expected value of T will be now denoted by m, while a higher central moment of order j will be written as mj, i.e.,

mj ¼ EðT 2 mÞ j ;

315

Interest rate risk—nonrandom approach

for j ¼ 1; 2… Assume that the random variable T is bounded by a certain constant from above with probability 1. We then have ln EðeuT Þ ¼ ln EðeumþuðT2mÞ Þ ¼ muþln EðeuðT2mÞ Þ ! ! þ1 n þ1 n X X u ðT 2 mÞn u mn ¼ muþln E ¼ muþln 1þ n! n! n¼0 n¼2 ¼ muþ

n¼2

¼ muþ

þ1 n 1 X u mn 2 2 n¼2 n! n!

þ1 n X u mn

!2 þ···

m2 u2 m3 u3 ðm4 23m22 Þu4 ðm5 210m2 m3 Þu5 þ þ þ þ···: ð9:1:8Þ 2! 3! 4! 5!

Since 2 m ¼ EðTÞ ¼ DM ; EðT 2 Þ ¼ CM and m2 ¼ VarðTÞ ¼ MM ;

then from (9.1.8) we conclude that 1 2 ln EðeuT Þ ¼ uDM þ u2 MM þ···: 2

ð9:1:9Þ

Furthermore ers ¼

Aðd þ1;sÞ Aðd þ1;0Þ ¼ e1s ¼ e1s Eðe21T Þ: Aðd;sÞ Aðd;0Þ

Thus by using (9.1.9) for u ¼ 21; we obtain 1 2 rs ¼ 1sþln Eðe21T Þ ¼ 1s21DM þ 12 MM 2··· 2 and from this, for values of 1 sufficiently small in absolute value, we get the following approximation: 2 DM 12 MM : r < 1 12 þ s 2s

9.2 Classical immunization The simplest, and yet significant, example of the situation when assets and liabilities of a pension plan (or any other financial intermediary) may respond differently to changes in market prices of capital assets is the case when they respond differently to changes in interest rates. We will begin the analysis of this case with the study of a simplified world in which assets and liabilities depend on one interest rate only

316

Chapter 9

(i.e., we assume a flat yield curve). This approach may actually be used to all situations when assets and liabilities both depend on one common factor. Asset –Liability Management, i.e., the idea of managing assets and liabilities of a financial intermediary (a pension plan in our case) jointly, was created by Redington. Since the publication of Redington’s [60] work we know that ‘there should be consistency of treatment in the valuation of assets and liabilities’. Redington called it the cardinal rule of valuation. Consequently, he did use the same interest rate i for the valuation of both assets and liabilities. Assume that At is the asset cash flow at time t . 0 and Lt is the liability cash flow at the same time. We shall, for now, assume that both assets and liabilities are deterministic and their cash flows are independent of interest rates (we will return to the more complex case of dependence on interest rates in the next chapter). Then the present value of the assets is A¼

X t.0

At ð1 þ iÞt

(we can assume that the sum is finite, as no financial instrument in the real world makes an infinite number of payments), while the liabilities’ present value is L¼

X t.0

Lt : ð1 þ iÞt

We will consider the financial intermediary to be a pension plan, and the plan surplus is S ¼ SðiÞ ¼

X t.0

X Lt At : t 2 ð1 þ iÞ ð1 þ iÞt t.0

Let us note that Sði þ DiÞ < SðiÞ þ S0 ðiÞDi and if S0 ðiÞ ¼ 0; then Sði þ DiÞ < SðiÞ: If we consider a more accurate approximation Sði þ DiÞ < SðiÞ þ S0 ðiÞDi þ

1 00 S ðiÞðDiÞ2 ; 2

then in the case when S0 ðiÞ ¼ A0 ðiÞ 2 L0 ðiÞ ¼ 0

and

S00 ðiÞ ¼ A00 ðiÞ 2 L00 ðiÞ . 0;

ð9:2:1Þ

for small changes in the interest rate we have Sði þ DiÞ . SðiÞ: This means that a pension plan, which is able to create such a situation, will experience an increase in its surplus under any change in the interest rate. This methodology of interest rate risk management is termed immunization.

317

Interest rate risk—nonrandom approach

An insurance company could implement such a strategy by paying its customers a rate of return earned by an intermediate-term bond, while investing in a combination of short-term and long-term bonds of the same dollar duration as its liabilities. We will show that in such a case the mathematical assumptions of the immunization strategy are satisfied. We should, however, stress that this illustration is based on the unrealistic assumption used by Redington: the assumption of one interest rate, or a flat yield curve. 9.2.1 A critical discussion of classical immunization for a pension plan We will use the concepts, introduced in Section 9.1, to test the immunization strategy for a pension plan. Assume that the plan has assets A and liabilities L, and that we want to protect the surplus S ¼ A 2 L from taking negative values under small changes of the interest rate. Then the assets of the plan should be invested in such a way that the condition (9.2.1) is satisfied, i.e., A0 ðiÞ ¼ L0 ðiÞ

and

A00 ðiÞ . L00 ðiÞ:

If, on the other hand, we want to protect the ratio A=L; which equals one plus the ratio of the surplus to the liabilities, a measure likely to be important to regulators, and called the funding ratio (or the funded ratio) then the following conditions must be satisfied:

and

d lnðAðiÞÞ d lnðLðiÞÞ ¼ di di d2 lnðAðiÞÞ d2 lnðLðiÞÞ . : 2 di di2

If the funding ratio equals one and durations of assets and liabilities are matched, the above condition on M-squared is equivalent to the corresponding relationship of convexities. Recall that duration (or, more precisely, Macaulay duration) may be intuitively interpreted as a weighted-average time to maturity of cash flows, while M-squared is a measure of dispersion of cash flows. Therefore:

The immunization strategy is achieved when assets and liabilities have the same weighted-average time to maturity while assets have greater dispersion of cash flows than liabilities.

This statement immediately indicates how difficult interest rate risk management for pension plans is in practice. Liability cash flows are generally long

318

Chapter 9

term, thus they have relatively long duration. In order to match such long duration, one should invest in long-term assets, ideally long-term bonds with fixed coupon. But the supply of such bonds is quite limited. In most countries around the world, long-term fixed coupon bonds, beyond 2-year maturity, do not exist. In the United States, there are longer term US Government bonds and a limited supply of corporate issues (e.g., the 100-year bond issue by Disney, but even that issue is callable in 30 years). Given lower yields and limited Government bonds, and limited supply, one might want to naturally extend the search for assets to private issues. One possibility that presents itself is to invest in stocks. But stocks, while risky and properly treated as a long-term investment, do not exhibit long-term duration. Based on the no-arbitrage approach to pricing, we conclude that the stock prices should be the current market value of its future dividends. Dividends are a compensation that providers of capital receive for its use in the productive activities of the firm. For stocks to be competitive in relation to bonds, dividends and firm profits, from which dividends are generally derived, must stay in the range comparable to that for current interest rates (one could argue that there are periods of ‘bubbles’ or severe bear markets, when such relationships tend to diverge to extreme ranges, but over the longer term periods, applicable to pension plans, this general relationship does prevail). Stock investors generally should also earn a premium over what is earned by bond investors, due to higher risk of stocks (again, we can only expect such a relationship ‘in the long run’, and short-term divergences from it are exactly what the risk of stocks is about). This means that if the general level of interest rates increases then, maybe not immediately but eventually, the rate of return on stocks should also increase (firms will abandon projects with lower rates of return and only concentrate on those that provide the higher rate of return required by investors). This means that stocks do not behave like a long-term security, but rather a security with duration roughly equal to the period of adjustment needed when changes in the overall economy occur. One could reasonably argue that such adjustment takes between 1 and 5 years, but not 10 –30 years, which may be appropriate for pension plan liabilities. Empirical studies of US stocks indicate duration estimates in the range of 5– 6 years, i.e., significantly less than the duration of, e.g., 30-year long-term US Government bond, whose duration fluctuates around 10 years. Let us note here that duration is measured in years. And it is duration rather than maturity that is the most appropriate measure for determining whether a financial asset is long term, intermediate term, or short term. On top of the problem with duration matching for pension plans, their liabilities are life annuities, which exhibit high convexity (as they are both long term and have large dispersion of cash flows). High convexity means that the application of the immunization technique to pension plans is difficult. In order to succeed in immunization one must find assets with even greater convexity. This is just not

319

Interest rate risk—nonrandom approach

realistic in the capital markets. Bonds do not have evenly spread cash flows, and mortgages can be prepaid early, thus reducing their convexity, or even giving them negative convexity in the neighborhood of the interest rate at which they are about to be paid off. There are more troubling issues in the immunization methodology. Immunization claims that it is possible to create a situation in which any change in interest rates results in a gain to the financial intermediary using the strategy. This means a creation of riskless profit. If the funds used to purchase assets are derived from issuing liabilities, i.e., if A ¼ L; then the intermediary will have created a riskless profit without any capital outlet, i.e., arbitrage. If we believe the markets are at least reasonably efficient, this should not be possible. Panjer [57], in Chapter 3, points out this methodological weakness of immunization. One remedy for this weakness was proposed by Fong and Vasicek [25] (see also [39]). They proved that the change of S can be bounded from below by a product of two terms: one measuring the change in the interest rate structure and the other being M 2 of the portfolio (i.e., of assets and liabilities together). Generally speaking, the bound is usually negative and the optimal strategy relies on arranging the portfolio of assets and liabilities so that, given DA ¼ DL ; M-squared of the portfolio be minimal. As we have pointed out in this section, however, the demand that DA ¼ DL can hardly be fulfilled in practice. Therefore, one should rather try to bound DS from below by a (possibly negative) quantity without this demand. Elementary considerations below show that this idea is easily applicable. Indeed, first observe that X St ð1 þ iÞt DS ¼ Sði þ DiÞ 2 SðiÞ ¼ 2 1 ð1 þ iÞt ð1 þ i þ DiÞt t.0 where St ¼ At 2 Lt ; is the surplus at time t. Hence, by the Schwartz inequality, " DS $ 2

X t.0

S2t ð1 þ iÞ2t

#1=2

X t.0

ð1 þ iÞt 21 ð1 þ i þ DiÞt

2 !1=2 :

ð9:2:2Þ

Thus the change in the value of S is bounded from below by a product of two quantities: one, depending only on the portfolio structure and the other one, depending only on the change of the interest rate. Now it is clear that, in general, DS might be negative, however, decreasing the quantity X t.0

S2t ; ð1 þ iÞ2t

which can be treated as a measure of the immunization risk, leads to decreasing the decline in surplus value, at least in the worst case scenario. What is more, the bound (9.2.2) is sharp, i.e., it is attainable if the change in the interest rates

320

Chapter 9

replicates, at every t . 0; the structure of assets and liabilities in the following way

ð1 þ iÞt St t 2 1 ¼ 2c ð1 þ i þ DiÞ ð1 þ iÞt

for some constant c . 0: The above approach may be very useful in practice but it, as well as the approach by Fong and Vasicek, does not take into account the regulatory requirements on the plan solvency. Therefore we shall present in Section 9.3 and Chapter 10, an alternative approach to the pension plan immunization which does not have this disadvantage. 9.2.2 Multivariate immunization Another problem with the classical immunization methodology is the fact that it assumes only one interest rate, i.e., a flat yield curve. One possible response to this criticism is a generalization of immunization through a multivariate approach. In this new approach, one assumes that each security is a function of a vector of interest rates, where each interest rate in the vector is a rate for some important maturity. Such a model was proposed by Reitano [61]. In this approach, a pension plan surplus is considered to be a function of a finite number n of variables (interest rates): Sði1 ; …; in Þ: In order to achieve multivariate immunization, we seek such a portfolio of assets and liabilities, which satisfies the following set of differential equations:

› Sði ; …; in Þ ¼ 0; ›i j 1

j ¼ 1; 2; …; n:

Then for small changes in interest rates we will have Sði1 þ Di1 ; …; in þ Din Þ < Sði1 ; …; in Þ: If the matrix of second-order partial derivatives H¼

›2 S ›i k ›i j

; 1#k; j#n

is positive definitely, then every small change in interest rates will result in gains for the pension plan utilizing this multivariate immunization strategy. Alas, this approach is also rooted in a vain pursuit of arbitrage (Panjer [57, Chapter 3]).

321

Interest rate risk—nonrandom approach

9.2.3 Simulation approach to asset – liability management In our previous considerations, we always assumed that the cash flows of financial assets under consideration were independent of interest rates. In reality, this is the case only for very few securities (e.g., for completely riskless US Government bonds). Most corporate bonds are issued with some form of call option, while mortgages have an early prepayment option. Municipal bonds are often issued with a sinking fund. Nearly all bonds have some form of credit risk. And stocks, of course, offer the greatest uncertainty of future cash flows. According to the Fundamental Theorem of Asset Pricing, the value of an instrument with uncertain cash flows can be established as 1

0

C BX C B Atþ1 C; B EB C t Y A @ t.0 ð1 þ ik Þ k¼0

where the expected value is calculated with respect to an appropriate risk-neutral measure. But to use this formula practically, one must construct an appropriate probability measure and then assure oneself that the resulting model gives appropriate prices to the instruments, whose market prices are known. Such a process of verification of asset pricing model is termed model calibration. If the probability space considered is finite, or if we use a finite sample to approximate the theoretical value, then the price of a security can be expressed as X

PrðvÞ

v[V

X t.0

Atþ1 t Y

;

ð1 þ ik ðvÞÞ

k¼0

where v is an element of the probability space (or a sample) corresponding to a scenario of future short-term interest rates. Such a representation can be used to approximate the effective duration and convexity of a security, whose cash flows depend on interest rates. Recall that DðAÞ ¼ 2

1 dA ; A di

CðAÞ ¼

1 d2 A : A di2

By using the Taylor series expansion we obtain Aði þ DiÞ ¼ AðiÞ þ

dA 1 d2 A Di þ ðDiÞ2 þ · · ·: di 2 di2

322

Chapter 9

Let us now ignore nonlinear terms. By doing this we obtain 2

1 dA AðiÞ 2 Aði þ DiÞ < A di ADi

2

1 dA Aði 2 DiÞ 2 AðiÞ < : A di ADi

and

Now take the average of the above two expressions and obtain DðAÞ <

Aði 2 DiÞ 2 Aði þ DiÞ : 2ADi

We also have dA 1 d2 A Di þ ðDiÞ2 di 2 di2

ð9:2:3Þ

dA 1 d2 A ð2DiÞ þ ð2DiÞ2 : di 2 di2

ð9:2:4Þ

Aði þ DiÞ 2 AðiÞ < and Aði 2 DiÞ 2 AðiÞ <

After adding the above two formulas, we obtain Aði 2 DiÞ 2 2AðiÞ þ Aði þ DiÞ <

d2 A ðDiÞ2 di2

or CðAÞ ¼

1 d2 A Aði 2 DiÞ 2 2AðiÞ þ Aði þ DiÞ : < A di2 ðDiÞ2 A

Thus if we have a model of market value of a financial asset then in the case of a small change in the interest rate the above formulas may be used to approximate duration and convexity of such an asset. The estimates so obtained are commonly called the option-adjusted duration and option-adjusted convexity, as they account for options embedded in the asset under consideration. These estimates may be an effective tool for interest rate risk management for portfolios of assets and liabilities with embedded options, which, of course, are quite common among pension plans and insurance companies. This methodology may be used also for nontraded capital assets, such as pension or insurance liabilities, if we first create a model allowing for the calculation (or at least an approximation of their market values.

323

Interest rate risk—nonrandom approach

9.3 Axiom of Solvency and surplus immunization Let us consider a stream of nonnegative liability outflows {L1 ; …; Ln }; due at dates {1; …; n} and a stream of nonnegative net asset1 inflows {A1 ; …; An }; occurring at the same dates. We allow that some of the assets or liabilities be equal to zero by the investment horizon n. We assume that the assets and liabilities are valued by an actuary using the same discount function corresponding to a given basic term structure of interest rates (TSIR). Namely, let PðkÞ denote the present value at time t ¼ 0 of the monetary unit due at t ¼ k; then PVðAk Þ ¼ PðkÞAk and PVðLk Þ ¼ PðkÞLk denote the present values of Ak and Lk at t ¼ 0; respectively. Clearly, the present value of the portfolio surplus, S, is given by S¼

n X k¼1

PVðAk Þ 2

n X

PVðLk Þ:

k¼1

Throughout the section, S is treated as a target value of the portfolio under the basic TSIR. In practice usually S $ 0; because of regulatory requirements, although the immunization bounds given below are valid also in the case S , 0: The problem we are interested in is how the target value of the portfolio can change when the basic TSIR changes. Let S0 denote the present values of the whole portfolio under a perturbated TSIR. Similarly, let PV0 ðAk Þ ¼ P0 ðkÞAk and PV0 ðLk Þ ¼ P0 ðkÞLk denote the present values of the asset Ak and the liability Lk at time t ¼ 0; respectively, under this new TSIR. Then 0 n X P ðkÞ 21 : ð9:3:1Þ ðPVðAk Þ 2 PVðLk ÞÞ S0 2 S ¼ PðkÞ k¼1 We will call the function f ðkÞ ¼ P0 ðkÞ=PðkÞ a shift factor. In Section 9.2, we formulated the immunization problem as follows: find conditions under which S0 2 S $ 0 for any change of TSIR. However, the immunization postulate that S0 2 S $ 0 for any scenario of interest rates is contradictory (except for some special circumstances) with the postulate that the market does not allow arbitrage opportunity (for details, see Section 9.2). Therefore S0 2 S may be nonnegative only for some special types of change of TSIR. For example, Hu¨rlimann [39] and Montrucchio and Peccati [52] considered immunization of the portfolio against perturbations of the basic TSIR with convex shift factors. On the other hand, Fong and Vasicek [25] considered arbitrary interest rate scenarios and have obtained a lower bound on the change in the end-of-horizon value of the portfolio. This lower bound is a product of two terms and may be negative for some perturbations. The first term is a function of the change of TSIR only, while the second one depends solely on the structure of the 1

From a mathematical point of view, it is of minor importance if the costs are added to liabilities or subtracted from assets (or both); for the sake of simplicity we subtract them from asset cash flows. Net asset cash flows are the result.

324

Chapter 9

investment portfolio. All the above-mentioned results concern deterministic interest rates, assets and liabilities. In practice, insurance regulatory authorities will not allow to operate an insurance company with a negative net worth in a considerable time period. It means that the basic TSIR used by an actuary in a balance sheet to valuate assets and liabilities must imply that S $ 0 and the following inequalities hold t X

PVðAs Þ $

s¼1

t X

PVðLs Þ

for t ¼ 1; …; n 2 1:

ð9:3:2Þ

s¼1

Let us observe that Sk ¼

k X s¼1

PVðAs Þ 2

k X

PVðLs Þ

s¼1

is the present value of the portfolio surplus occurring at t ¼ k: So (9.3.2) really means that a present value of the surplus at any moment t ¼ 1; …; n 2 1 is nonnegative. We will call a series of inequalities (9.3.2) the Axiom of Solvency from now on. In the United States, in the property/casualty insurance business, the premiums received by an insurer usually do not cover the claims plus expenses to be paid. The difference is called underwriting loss or the cost of float that the business generates [16]. However, premiums are received before losses are paid giving the insurer time to invest the money. So, the Axiom of Solvency may still be satisfied even when liabilities nominally are higher than net premiums. The aim of this section is to evaluate how the difference S0 2 S can change under Axiom of Solvency in response to nonrandom perturbations of the basic TSIR corresponding to: (1) monotone factors and (2) factors with a bounded slope. In particular, Theorem 9.3.1 states that Axiom of Solvency is equivalent to the portfolio immunization against all perturbations which correspond to decreasing shift factors. We also consider arbitrary interest rate scenarios and provide a lower bound on the corresponding expected change in the portfolio value whenever (9.3.2) is satisfied. This lower bound is a sum of two products, each one consisting of two terms, namely S0 2 S $ Sð f ðnÞ 2 1Þ 2 KðnS 2 DnomM ðAÞ þ DnomM ðLÞÞ;

ð9:3:3Þ

where DnomM ðAÞ and DnomM ðLÞ denote the nominal Macaulay duration of A and L, respectively, and K is an upper bound on the change in the slope of f (see Theorem 9.3.2 for more details). In both products one term is a function of the interest rate change only, while the other depends only on the portfolio structure. So the second term can be regarded as a measure of immunization risk and minimized. In Section 10.5 we construct a funding method, which minimizes that measure in a more general situation when assets, liabilities and interest rates are random. We will also address there the issue of choosing the portfolio structure in response to the level of S and the

325

Interest rate risk—nonrandom approach

basic term structure of interest rates, as well as the issue of an appropriate target surplus level. 9.3.1 Monotone shift factors Let us start our investigations with the case of changes of TSIR, which correspond to monotone shift factors defined as follows. We shall call a shift factor f decreasing if f ðt þ 1Þ # f ðtÞ

for t ¼ 1; …; n 2 1:

ð9:3:4Þ

We begin with a sufficient and necessary condition for ‘immunization inequality’ (9.3.5) below. The following theorem was proven in [28]. We omit the proof because in Section 10.3 we will prove a more general result (see Theorem 10.3.1). Theorem 9.3.1 The following inequality holds S0 2 S $ Sð f ðnÞ 2 1Þ

ð9:3:5Þ

for every decreasing shift factor if, and only if, Axiom of Solvency holds. Hu¨rlimann [39] has given an interesting characterization of the immunization against convex shift factors in terms of the stop-loss order. This interpretation concerns a special case when S ¼ 0: Observe that (9.3.2) may then be interpreted in a way that a random variable A with support {1; …; n} and probabilities {q1 ; …; qn }; P where qk ¼ PVðAk Þ= ns¼1 PVðAs Þ; is stochastically smaller than a random variable L P with the same support and probabilities {p1 ; …; pn }; where pk ¼ PVðLk Þ= ns¼1 PVðLs Þ: The random variables A and L are called the asset and liability risks, respectively [39]. So when S ¼ 0; Theorem 9.3.1 states that the portfolio target value is immunized against all decreasing shift factors if and only if the asset risk is stochastically smaller than the liability risk, i.e., A # st L: Since the Macaulay duration of the asset and liability cash flows are defined by DM ðAÞ ¼ EðAÞ and DM ðLÞ ¼ EðLÞ; A # st L implies in particular that DM ðAÞ # DM ðLÞ: 9.3.2 Arbitrary shift factors This section concerns a general case when f is arbitrary. The following result can be found in [28]. The proof is omitted because in Section 10.3 we will prove Theorem 10.3.2, which concerns a more general case. Theorem 9.3.2 Assume that Axiom of Solvency holds. If for some constant K f ðt þ 1Þ 2 f ðtÞ # K

for every t ¼ 1; …; n 2 1;

ð9:3:6Þ

326

Chapter 9

then S0 2 S $ Sð f ðnÞ 2 1Þ 2 RK;

ð9:3:7Þ

where R ¼ nS 2 DnomM ðAÞ þ DnomM ðLÞ with DnomM ðAÞ and DnomM ðLÞ denoting nominal Macaulay durations of the assets and liabilities, respectively. It is clear that Theorem 9.3.2 gives the immunization bound for arbitrary f, because K may be defined then as maxð f ðt þ 1Þ 2 f ðtÞÞ: The right side of (9.3.7) is a difference of two quantities which are products of two terms: the first term (S or nS 2 DnomM ðAÞ þ DnomM ðLÞ; respectively) describes the portfolio structure, the second one ( f ðnÞ 2 1 or K, respectively) describes the change of TSIR only. Hence the inequality given in Theorem 9.3.2 has a similar structure as that by Fong and Vasicek [25] although it gives different bound values and holds under different assumptions. Such a product-like structure of the right side of (9.3.7) enables one to optimize the portfolio structure. The portfolio exposure to an arbitrary type of TSIR change is determined by the portfolio characteristic R which is treated as a measure of immunization risk. This problem will be investigated further in Section 10.5. 9.3.3 Conservative asset –liability management The class of decreasing shift factors can be easily interpreted using TSIR. Let {i1 ; …; in } be the short-term interest rates in succeeding time periods (i.e., forward rates), which are used by the actuary to calculate the present values of assets and liabilities, so that PðkÞ ¼

1 ; ð1 þ i1 Þ…ð1 þ ik Þ

for k ¼ 1; 2; …; n

ð9:3:8Þ

and let {i01 ; …; i0n } be interest rates corresponding to P0 ðkÞ: The condition that the shift factor is decreasing is equivalent to the inequalities P0 ðk þ 1Þ P0 ðkÞ 2 #0 Pðk þ 1Þ PðkÞ

for k ¼ 1; 2; …; n 2 1:

ð9:3:9Þ

Observe that inequalities (9.3.9) are satisfied, if ik # i0k

a:s:

ð9:3:10Þ

for k ¼ 2; …; n: Moreover, (9.3.10) implies that f ðnÞ 2 1 is nonpositive, if additionally i01 $ i1 : Thus Theorem 9.3.1 gives a sufficient condition for portfolio immunization against all scenarios with higher forward interest rates than those used by the actuary to value the assets and liabilities. The lower bound (9.3.5) is 0 (the case of full immunization) whenever S ¼ 0; for S . 0 the lower bound may be negative

327

Interest rate risk—nonrandom approach

(the case of subimmunization); finally, for S , 0 the bound may be positive (the case of superimmunization). This gives a strict mathematical justification for the following conservative strategy in asset – liability management.

In relation to your expectations of the interest rates in the future, use a conservative basic term structure of interest rates, i.e., interest rates lower than expected. Then arrange the portfolio of assets and liabilities so that the Axiom of Solvency is satisfied, given S $ 0: If the interest rates in the future are higher than the assumed ones, the difference S0 2 S will be automatically bounded from below by a nonnegative quantity SðP0 ðnÞ=PðnÞ 2 1Þ:

ð9:3:11Þ

This page is intentionally left blank

329

Chapter 10 Stochastic approach to financial risk management

10.1 One-period analysis of the plan surplus and funded ratio In the previous parts of this book, we have addressed the problem of consistency of treatment of assets and liabilities (Section 9.2 and Part 2). We will now assume that the Cardinal Rule of Valuation is satisfied, and plan contributions are sufficient to fund those liabilities that will be created in the future (although if this turns out not to be the case, we can generalize our analysis, for example, by including a portion of future contributions in plan assets). We will now concentrate on the analysis of the plan surplus, defined as the difference between the actuarial value of plan assets and the accrued liabilities of the plan: S ¼ A 2 L: The process of risk management for a pension plan is, to a great extent, the process of management of the quantity S ¼ A 2 L; and its changes due to the changes in the values of assets and liabilities. A one-period analysis of the plan surplus allows simplified modeling of the structure of risks. Namely, the interest rate risk C3 and the investment risk C1 can be treated together. But first we shall answer the following question: what is a good measure of the change in the level of plan surplus? Leibowitz et al. [42] propose the following quantity: S 2 S0 RS ¼ ; L0 where S0 is the value of the surplus at the beginning of the period under consideration, L0 the beginning value of plan liabilities, and S the terminal value of plan surplus (considered to be, at time 0, a random variable). Let us introduce the notation A F¼ ; L for the funded ratio (also called funding ratio), and denote the initial funded ratio by F0 : We then have RS ¼ F0 RA 2 RL ;

330

Chapter 10

where RA is the rate of return of the plan assets and RL the rate of return of the plan liabilities (both these quantities are random variables). Plan assets risk management can be summarized by the following two items: † strategic decision about plan assets allocation, and † current optimization of the portfolio within the adopted long-term strategy. Pension plan asset management in the United States has traditionally used a common asset allocation of approximately 60% stocks and 40% bonds. In what follows, we will assume that plan assets are invested in these two general asset classes: stocks, with a random rate of return RE ; and bonds, with a random rate of return RB : Let m; s; r with appropriate subscripts denote, respectively, expected values, standard deviations, and correlation coefficients for the probability distributions of the rates of return considered. We then have

mS ¼ F0 mA 2 mL ;

ð10:1:1Þ

s 2S ¼ ðF0 sA Þ2 þ s 2L 2 2rAL ðF0 sA ÞsL :

ð10:1:2Þ

and

If we assume that a fraction w of the asset portfolio is invested in stocks, and the rest in bonds, then RA ¼ wRE þ ð1 2 wÞRB : This means that

mA ¼ wmE þ ð1 2 wÞmB ; and

s 2A ¼ ðwsE Þ2 þ ð1 2 wÞ2 s 2B þ 2rEB wð1 2 wÞsE sB : Furthermore CovðRA ;RL Þ ¼ w EððRE 2 mE ÞðRL 2 mL ÞÞ þ ð1 2 wÞEððRB 2 mB ÞðRL 2 mL ÞÞ ¼ w CovðRE ;RL Þ þ ð1 2 wÞCovðRB ;RL Þ:

ð10:1:3Þ

The structure of the balance sheet of a pension plan is such that the plan surplus is very sensitive to divergences in behavior of plan assets and liabilities. The statistical parameters presented above clearly illustrate that. Let us now analyze the duration of plan surplus. Let Dur be either the Macaulay or effective duration of a capital asset under consideration. In the case of pension plan

Stochastic approach to financial risk management

331

surplus we have, based on (9.1.5), the following: DurðSÞ ¼ DurðA 2 LÞ ¼

A A DurðAÞ þ 1 2 DurðLÞ: S S

After simplifying that expression we get: DurðSÞ ¼ DurðLÞ þ

A ðDurðAÞ 2 DurðLÞÞ: S

This means that the duration of plan surplus is of the same order as durations of assets and liabilities, but if assets have substantially longer duration than liabilities, and S is small in relation of A (this is quite common not only for banks, life insurance companies, and pension plans, but also, although to a lesser degree, for property/casualty insurance companies), then the duration of the surplus may turn out to be extremely large, and thus the plan surplus may be subject to very high level of interest rate risk. For a pension plan, we typically have DurðAÞ , DurðLÞ; and this creates a risk structure opposite to that of banks:

If S is small in relation to A; then in the case of long duration liabilities with shorter duration assets, the duration of plan surplus may turn out to be negative, and interest rates declines cause the surplus of such a plan to decrease.

The situation described in the statement above is not only possible, but also very likely for pension plans. The liabilities of pension plans are mostly created by the necessity to pay plan participants for the length of their remaining lifetime upon retirement. The normal retirement age is commonly around 65, but early retirement age can be as early as 55, and at either of these ages, the remaining lifespan is substantial. In the United States life expectancy of a person aged 65 reached well into the eighties and continuously expands at rates surprizing to many experts. Pension plan providers and insurance companies, as well as governments providing pensions, assume risk of already substantial and increasing longevity of pension recipients. It is quite crucial that such a risk be understood and quantified. We believe the section of this book providing bounds for the variance of the present value of liabilities to be an important contribution in this area, but there is much further work to be done. Let us note a web-based international database http://www. humanmortality.org [72], a novel effort to expand this research effort globally. The increasing lifespan of pension recipients means that liabilities have ever increasing duration, and because of the significant spread of cash flows in the payments of life annuities, very large convexity. On the other hand, duration of bonds

332

Chapter 10

available for investments by pension plans is quite limited. Even the 30-year US Federal Government Bond, if bought with coupons (as opposed to a zero coupon bond), generally has duration in the range of 10. Its zero coupon bond version will have Macaulay duration of 30, but even that may be insufficient, and one cannot fund the entire pension plan with 30-year zero coupon bonds for these two key reasons: the yield on those bonds is insufficient to support the business, and, more importantly, the US Federal Government stopped issuing them (this, of course, is a reason enough to give up on the idea, to quote the statement: “We have a long list of reasons. The first one is that we do not have any cannons” in response to Napoleon, when he asked a mayor of a certain German town why the arrival of the Emperor was not greeted with canon fire salute). This problem faced by life insurance companies and pension plans of not being able to find assets of sufficient duration and convexity to correspond to that of their liabilities is commonly summarized by the life insurance industry in the Short Straddle Model discussed by Ostaszewski [56]. We believe that the industry response is quite similar to that described in Section 9.3.3 and in Section 10.5 (see also Section 4.3). However, the optimal funding structure given by the lowest concave majorant remains largely ignored, and we believe it should be considered more often. 10.1.1 Risk management by shortfall control The classical Markowitz methodology of designing an efficient portfolio seeks to find a portfolio with the greatest possible expected rate of return mS and the smallest possible standard deviation of the rate of return s 2S : This method, when applied to US markets and typical plan sponsors’ risk tolerance, often results in the common asset allocation of 60% stocks and 40% bonds. Leibowitz et al. [42] proposed an alternative approach to the design of an optimal portfolio. We will present it now. Consider at first a portfolio consisting of only assets and assume that the portfolio may be invested only in either stocks or riskless bonds. Instead of seeking portfolios efficient in Markowitz’ sense, we will now seek those portfolios that have the probability of 90% or better of achieving the rate of return considered to be the lowest acceptable return, i.e., acceptable threshold below which we would rather not fall. Example 10.1.1 If we assume that the rate of return of the asset portfolio is a random variable distributed normally with the mean of 11% and the standard deviation of 11.5%, then the probability of receiving the rate of return below 2 3.7% is exactly 10%. This 2 3.7% is termed the threshold rate of return. A The concept of shortfall threshold was the key idea of the alternative asset-liability management methodology presented in [42]. Assume now that the risk-free rate of return is rF : Let the fraction of the portfolio invested in assets have the expected return of mE ; standard deviation of returns of

333

Stochastic approach to financial risk management

sE and the normal distribution of returns. The random portfolio rate of return RP with fraction w invested in stocks and ð1 2 wÞ invested in risk-free bonds is then also distributed normally with the expected return of mP ð1 2 wÞrF þ wmE ; and the standard deviation of

sP ¼ w sE : We will now analyze the tenth percentile of the probability distribution of returns of the portfolio. We have PrðRP , kÞ ¼ 0:10;

ð10:1:4Þ

where k is the tenth percentile of the portfolio return distribution. We also have RP ¼ ð1 2 wÞrF þ wRE : Therefore PrðRP , kÞ ¼ Prðð1 2 wÞrF þ wRE , kÞ

ð1 2 wÞrF þ wRE 2 ð1 2 wÞrF 2 wmE k 2 ð1 2 wÞrF 2 wmE , w sE wsE RE 2 mE k 2 ð1 2 wÞrF 2 wmE ¼ Pr , ¼ 0:10: ð10:1:5Þ sE wsE

¼ Pr

We know that the tenth percentile of the standard normal distribution is 2 1.28, and therefore k 2 ð1 2 wÞrF 2 wmE ¼ 21:28; w sE and so k 2 ð1 2 wÞrF 2 wmE ¼ 21:28wsE ; hence

mP ¼ k þ 1:28sP :

ð10:1:6Þ

This is an equation describing all portfolios whose tenth percentile of returns meets the shortfall threshold constraint (10.1.4). In the two-dimensional plane with the ðmP ; sP Þ-coordinates these portfolios lie on a straight line defined by the (10.1.6). The mP -intercept of that line is the threshold level. We will consider portfolios acceptable, and write their set as P; if for P [ P PrðRP , kÞ # 0:10:

334

Chapter 10

An investor may be naturally interested in the maximization of the expected rate of return, i.e., finding such a portfolio Pp that max EðRP Þ ¼ EðRPpÞ: P[P

If we now add to our portfolio a short position in pension plan liabilities, we achieve a more realistic model of a pension plan. Let k be the percentile of assetliability returns, which is our shortfall threshold. Then the shortfall control constraint can be written as PrðRS , kÞ ¼ a; where a is the acceptable probability level corresponding to the accepted threshold return. This condition is equivalent to the following: RS 2 mS k 2 mS Pr , sS sS 0 1 RS 2 F0 m A þ m L k 2 F0 mA þ mL B C ﬃ , qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃA ¼ Pr@ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ðF0 sAÞ þ s 2L 2 2rAL ðF0 sA ÞsL ðF0 sA Þ þ s 2L 2 2rAL ðF0 sA Þs L ¼ a; ð10:1:7Þ where in the second equality we utilized (10.1.1) and (10.1.2). If we assume that assets and liabilities returns follow a normal distribution, the above can be rearranged to obtain an expression for the set of all portfolios satisfying the condition PrðRS , kÞ # a: In practice, however, simulation may be a better approach in finding the portfolios meeting the shortfall constraint. The authors of [42] present typical solutions of this problem, showing that the set of desired portfolios forms an egg-shaped curve in the sigma – mu plane, located just above the asset portfolio immunizing the liabilities. It should be noted that this analysis assumes that plan liabilities are given and only assets are rearranged to obtain the risk management objective. 10.1.2 Value-at-Risk The shortfall constraint approach is a risk management methodology that looks at a percentile of the return distribution as a key measure of risk. Such an approach is also used in another relatively new risk measure: Value-at-Risk. This is a methodology newly required for risk-based capital regulatory requirements for banks. For a given portfolio, given a time period length t; and a probability (significance) level a;

Stochastic approach to financial risk management

335

Value-at-Risk is defined as a number VAR such that PrðPortfolio loses more than VAR within time period tÞ , a: Bank regulations specify that this model should assume normal market conditions. Note that the Value-at-Risk is specified as a monetary amount, not a percentage. As an example, if we have a million dollar portfolio invested in a stock market index such that the probability that it will decline by more than 20% within a year is less than 10%, then with t ¼ 1 year, a ¼ 0:10; Value-at-Risk is $200 000. There are three basic methodologies used in practical Value-at-Risk calculations: parametric, simulation, and historical. Under parametric approach, one starts by estimating the expected returns, standard deviations, and coefficients of correlation for all asset classes in the portfolio. Then, using those parameters, and assuming joint multivariate normal distribution for asset classes in the portfolio, one estimates the parameters of the portfolio rate of return distribution (which will be, under these assumptions, normal). Finally, an appropriate percentile of the portfolio return distribution is estimated from its known parameters of the distribution, and based on the assumed normality of the return. Using the simulation approach, one specifies probability distribution of asset classes comprising the portfolio, and simulates their returns. Using the known asset allocation of the portfolio, this simulated data is then used to create a simulated sample of portfolio returns. This sample gives a sample distribution, whose appropriate percentile is chosen as the Value-at-Risk sought. Under the historical approach, one starts by assembling a database of historical rates of return of asset classes comprising the portfolio. Then one applies the data to the portfolio asset allocation, thus creating a sample distribution of historical returns of the current portfolio. This sample gives a sample distribution, and an appropriate sample distribution percentile is chosen as the Value-at-Risk. This, of course, is a completely non-parametric method, completely avoiding the unrealistic assumption of normality of returns used in the other methods. By far the most commonly used method is the parametric one. Let us illustrate it with a simple example. Assume that the asset portfolio consists of a position of $1 000 000 invested in asset X and $1 000 000 invested in asset Y: Assume that the daily volatilities of both assets are 0.1% and that the correlation coefficient between their returns is 0.30. In the most practiced parametric approach, it is assumed that for short-term Value-at-Risk calculation, the expected rate of return per day can be taken as zero, because it is generally very small in relation to volatility (i.e., standard deviation of returns). Since banks’ Value-at-Risk is required for a period of 2 weeks, expected return for such a short period is indeed sufficiently small for such an assumption to be justified. What is the 5-day 95% Value-at-Risk for this portfolio, assuming a parametric model with zero expected return? Under given assumptions, the standard deviation of the daily dollar change in the value of each asset is $1000. As mentioned before, we assume joint multivariate normal distribution of the returns of securities under

336

Chapter 10

consideration. Therefore, any linear combination of their returns (i.e., any portfolio return) will have a normal distribution. The portfolio expected daily return will be zero because such an expected return is assumed for individual assets. The variance of the portfolio’s daily change is then 10002 þ 10002 þ 2 · 0:3 · 1000 · 1000 ¼ 2 600 000: The standard deviation of the portfolios daily change in value is the square root of 2 600 000, i.e., $1612.45. For multi-day calculations, we make yet another unrealistic, but helpful assumption—that the consecutive daily returns are independent identically distributed (IID) random variables. This assumption is subject to debate and research in finance, but most empirical studies in the United States indicate it to be not true for US capital assets returns. Nevertheless, it does simplify the analysis greatly. Under the IID assumption, the 5-day portfolio return variance is the sum of five equal daily variances, and therefore, the standard deviation of the 5-day change in the portfolio value is pﬃﬃﬃ $1612:45 · 5 ¼ $3605:55: The fifth percentile of the standard normal distribution is 2 1.645, and therefore the fifth percentile of the 5-day portfolio return is 21:645 · $3605:55 ¼ 2$5931: This means that $5931 is the 5-day portfolio Value-at-Risk at 5% significance level (or, with 95% confidence). Hull [38, Chapter 16] provides a review of parametric Value-at-Risk methodologies. 10.1.3 Funded ratio return Yet another measure of risk management for pension plans has been proposed in [42]. It is termed funded ratio return (FRR). It is defined as A1 A A1 L 2 0 2 1 R 2 RL L1 L0 A0 L0 FRR ; ¼ ¼ A : L1 A0 1 þ RL L0 L0 As we see, the FRR depends on the rates of return of assets and liabilities, but does not depend on the level of the funded ratio. We can analyze the variable FRR the same way as the other random variables we have studied before. For example, the threshold level of return (i.e., lowest rate of return acceptable as some low percentile of the distribution of FRR) can be set for FRR. However, this methodology becomes somewhat more complicated for the variable FRR. In order for FRR to have normal distribution, RA and RL must have

337

Stochastic approach to financial risk management

lognormal distribution. We believe that practical applications of the FRR methodology would benefit from using the simulation approach.

10.2 Axiom of Solvency and random interest rates We assume throughout this chapter that the assets and liabilities are random variables and use the methodology introduced in [28]. Now inequalities (9.3.2) in Axiom of Solvency concern functions, not real numbers. We shall use therefore two versions of the Axiom of Solvency in this chapter: a strong one, when (9.3.2) holds almost surely, i.e., t X

PVðAs Þ $

s¼1

t X

PVðLs Þ

a:s: for t ¼ 1; …; n 2 1;

ð10:2:1Þ

s¼1

and a weak one, when (9.3.2) holds on the average, i.e., t X s¼1

EðPVðAs ÞÞ $

t X

EðPVðLs ÞÞ

for t ¼ 1; …; n 2 1:

ð10:2:2Þ

s¼1

A common approach in practice is to test the portfolio valuation in response to random perturbations of the interest rates [75]. Therefore, this chapter deals with a general case when TSIR changes in a random way. Hence, we shall consider shift factors f ðtÞ ¼

ð1 þ i1 Þ· · ·ð1 þ it Þ ; ð1 þ i01 Þ· · ·ð1 þ i0t Þ

where i01 ; …; i0t are random perturbations of fixed interest rates i1 ; …; it : Consequently, { f ðtÞ}t¼1;…; n is a stochastic process with possibly dependent random variables f ðtÞ; t ¼ 1; …; n: Section 10.3 concerns the case when the portfolio and TSIR are uncorrelated while Section 10.4 concerns the case when they may be statistically dependent. Our aim is to evaluate how the expectation of S0 2 S can change under a proper version of the Axiom of Solvency in response to random perturbations of the basic TSIR corresponding to: 1. monotone factors (modeled mathematically via e.g., supermartingale-like processes), and 2. factors with a bounded slope (modeled e.g., by processes with bounded expected increments). In particular, Theorem 10.3.1 states that a weak version of the Axiom of Solvency is equivalent to the portfolio immunization against all random perturbations which produce mean-decreasing shift factors. Analogously, Theorem 10.4.1 says that a

338

Chapter 10

strong version of the Axiom of Solvency is a necessary and sufficient condition for the portfolio immunization against all perturbations with supermartingale-like shift factors. We also consider arbitrary interest rate scenarios and provide a lower bound on the corresponding expected change in the portfolio value whenever (9.3.2) is satisfied on the average or almost surely, respectively.

10.3 Immunization when the portfolio is not correlated with term structure of interest rates In this section, we consider the assets and liabilities, which generate payments statistically independent of the future interest rates. More precisely, we assume that for every t ¼ 1; …; n 2 1 the difference f ðt þ 1Þ 2 f ðtÞ is not correlated with St ¼

t X

PVðAs Þ 2

k¼1

t X

PVðLs Þ;

k¼1

the portfolio surplus at the moment t: Due to this assumption, we are able to weaken Axiom of Solvency to a version equivalent to the immunization inequality (10.3.3). Namely, we shall consider in this section the following weak version of Axiom of Solvency k X s¼1

EðPVðAs ÞÞ $

k X

EðPVðLs ÞÞ

for k ¼ 1; …; n 2 1:

ð10:3:1Þ

s¼1

10.3.1 Monotone shift factors Let us start our investigations with the case of random perturbations of TSIR which correspond to monotone shift factors defined as follows. We will call a shift factor f mean-decreasing if Eð f ðt þ 1ÞÞ # Eð f ðtÞÞ

for t ¼ 1; …; n 2 1:

ð10:3:2Þ

Observe that if f is nonrandom and decreasing, it is mean-decreasing as well. We begin with a sufficient and necessary condition for some ‘immunization inequality’. It is a result somewhat analogous to that by Hu¨rlimann [39, Theorem 2.1], though we consider stochastic, not deterministic interest rates, assets, and liabilities. In particular, S may be nonzero and so does the lower bound in (10.3.3). And, what is most important, we are interested in satisfying Axiom of Solvency, which leads to different bounds when comparing with Hu¨rlimann [39] and others. The following theorem was proven in [28] Theorem 10.3.1 Assume that for every t ¼ 1; …; n 2 1 the increment f ðt þ 1Þ 2 f ðtÞ is not correlated with St ; the portfolio surplus at t under the basic TSIR.

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Stochastic approach to financial risk management

The following inequality holds EðS0 2 SÞ $ EðSð f ðnÞ 2 1ÞÞ;

ð10:3:3Þ

for every mean-decreasing shift factor if and only if a weak Axiom of Solvency holds. It is clear that the above theorem includes the case when some or all quantities appearing there are deterministic. The same remark concerns the other results of this chapter and will not be repeated from now on. The following proof of the above statement is taken from [28]. 0 Proof Applying Pn (4.3.3) with at ¼ P ðtÞ=PðtÞ 2 1 and bt ¼ PVðAt Þ 2 PVðLt Þ and observing that t¼1 bt ¼ S; we get by (9.3.1)

S0 2 S ¼ Sð f ðnÞ 2 1Þ 2

n21 X

ð f ðt þ 1Þ 2 f ðtÞÞ

t¼1

t X

PVðAs Þ 2

s¼1

t X

! PVðLs Þ :

ð10:3:4Þ

s¼1

From identity (10.3.4) we get 0

EðS 2SÞ¼ EðSð f ðnÞ21ÞÞ2

n21 X

E ð f ðt þ1Þ2f ðtÞÞ

t¼1

¼ EðSð f ðnÞ21ÞÞ2

n21 X

t X s¼1

ðEð f ðt þ1ÞÞ2Eð f ðtÞÞÞ

t¼1

PVðAs Þ2

t X

!! PVðLs Þ

s¼1

t X

! t X EðPVðAs ÞÞ2 E PVðLs Þ ;

s¼1

s¼1

ð10:3:5Þ where the last equality follows from the uncorrelation assumption. Now, if (10.3.1) and (10.3.2) are satisfied, inequality (10.3.3) follows. Conversely, assume that (10.3.3) holds for every shift factor f satisfying (10.3.2). Fix arbitrary k ¼ 1; 2; …; n 2 1 and define ( 1 2 1; for s . k; fk ðsÞ ¼ 1; for s # k; where 1 is some small positive number. Then fk is positive, decreasing and, by (10.3.3) and (10.3.5), ! k k X X 1 EðPVðAs ÞÞ 2 EðPVðLs ÞÞ $ 0: s¼1

Since k is arbitrary, (10.3.1) follows.

s¼1

A

340

Chapter 10

10.3.2 Arbitrary shift factors This subsection concerns a general case: it is allowed that f be neither meandecreasing nor supermartingale-like (the latter case will be treated in Section 10.4). The only mathematical restriction is that the expectation of f is finite during the investment horizon. The following result was proven in [28]. Theorem 10.3.2 Assume that a weak Axiom of Solvency holds. If for some constant K Eð f ðt þ 1ÞÞ 2 Eð f ðtÞÞ # K

for every t ¼ 1; …; n 2 1;

ð10:3:6Þ

then EðS0 2 SÞ $ EðSð f ðnÞ 2 1ÞÞ 2 Re K;

ð10:3:7Þ

where Re ¼ n EðSÞ 2 EðDnomM ðAÞÞ þ EðDnomM ðLÞÞ with DnomM ðAÞ and DnomM ðLÞ denoting nominal Macaulay durations of the assets and liabilities, respectively. Proof To prove the above theorem, let us observe that (10.3.5) and (10.3.6) and the weak Axiom of Solvency, altogether imply ! n21 X t t X X 0 EðS 2 SÞ # EðSð f ðnÞ 2 1ÞÞ 2 K EðPVðAs ÞÞ 2 EðPVðLs ÞÞ : t¼1

s¼1

s¼1

Now the result follows from the following identity ! n21 X t t n21 X n21 X X X PVðAs Þ 2 PVðLs Þ ¼ ðPVðAs Þ 2 PVðLs ÞÞ t¼1

¼

s¼1 n X s¼1

s¼1

s¼1 t¼s

ðn 2 sÞðPVðAs Þ 2 PVðLs ÞÞ ¼ nS 2 DnomM ðAÞ þ DnomM ðLÞ:

ð10:3:8Þ A

It is clear that Theorem 10.3.2 gives the immunization bound for arbitrary f with Eð f ðtÞÞ , 1; t ¼ 1; …; n; because K may be defined then as maxðEð f ðt þ 1ÞÞ 2 Eð f ðtÞÞÞ: The right-hand side of (10.3.7) is a difference of two quantities which are expected products of two terms: the first term (S or nS 2 DnomM ðAÞ þ DnomM ðLÞ; respectively) describes the portfolio structure, while the second one (f ðnÞ 2 1 or K; respectively) describes the change of TSIR only. Such a product-like structure of the right-hand side of (10.3.7) enables us to optimize the portfolio structure. Namely, the portfolio exposure to an arbitrary type of TSIR change is described by the portfolio characteristic Re which may be treated as a measure of immunization risk. This problem will be investigated further in Section 10.5.

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Stochastic approach to financial risk management

10.4 Immunizing arbitrary portfolios In this section, the portfolio may depend in an arbitrary way on the term structure of interest rates. In particular, it may contain derivatives based on TSIR as well as it may replicate the securities market. Let F k be an increasing stream of s-algebras such that for each k ¼ 1; …; n; all random variables PVðAt Þ and PVðLt Þ for t ¼ 1; …; k are F k -measurable. A stream {F t }t¼1;…;n forms a ‘history’ known to the actuary at each t ¼ 1; …; n: In particular, since PVðAt Þ and PVðLt Þ are F t -measurable, the actuary can determine their probability distributions at the moment t not knowing their future values. 10.4.1 Monotone shift factors We shall call {f ðtÞ}t¼1;…;n supermartingale-like with respect to {F t }t¼1;…;n ; if Eð f ðt þ 1Þ 2 f ðtÞlF t Þ # 0 a:s: for every t ¼ 1; …; n 2 1: Whenever each f ðtÞ is F t -measurable, this means that {f ðtÞ}t¼1;…;n is a supermartingale, but the definition also works without such a measurability restriction. In particular, any almost surely decreasing sequence {f ðtÞ}t¼1;…;n is supermartingale-like. It is clear, however, that the class of supermartingale-like random sequences contains plenty of sequences which are not monotone in this narrow, almost sure, sense. In this subsection, the notion of supermartingale-like random sequences will be applied for modeling monotonicity of shift factors. Theorem 10.4.1 The following inequality holds EðS0 2 SÞ $ EðSð f ðnÞ 2 1ÞÞ;

ð10:4:1Þ

for every supermartingale-like { f ðtÞ}t¼1;…;n if and only if a strong version of Axiom of Solvency is satisfied. The above theorem was proven in [28]. The arguments are as follows. Proof By identity (10.3.4), we get EðS0 2SÞ¼EðSðf ðnÞ21ÞÞ2E

n21 X t¼1

¼EðSðf ðnÞ21ÞÞ2

n21 X t¼1

ðf ðtþ1Þ2f ðtÞÞ

t X

PVðAs Þ2

s¼1

E Eðf ðtþ1Þ2f ðtÞlF t Þ

t X

!! PVðLs Þ

s¼1 t X s¼1

PVðAs Þ2

t X

!! PVðLs Þ

:

s¼1

ð10:4:2Þ

342

Chapter 10

If the strong Axiom of Solvency holds and {f ðtÞ}t¼1;…;n is supermartingale-like, then t X s¼1

PVðAs Þ2

t X

PVðLs Þ$0

Eðf ðtþ1Þ2f ðtÞlF t Þ#0 a:s:

and

s¼1

for every t¼1;…;n21: Hence EðS0 2SÞ$EðSð f ðnÞ21ÞÞ: Assume now that (10.4.1) holds for every supermartingale-like {f ðtÞ}t¼1;…;n : Suppose that for some k¼1;…;n21 the set Bk of all elementary events for which k X

PVðAs Þ,

s¼1

k X

PVðLs Þ;

s¼1

has a positive probability. Now define a random sequence {fk ðtÞ}t¼1;…;n by fk ðtÞ;1; for every t¼1;…;k; and ( fk ðtÞ¼

121; on Bk

1;

otherwise;

for every t¼kþ1;…;n; where 1 is a sufficiently small positive number. Then { fk ðtÞ}t¼1;…;n is a positive supermartingale-like random sequence. For this very sequence (10.4.1) and (10.4.2) together imply 1 E IBk

k X s¼1

PVðAs Þ2

k X

!! PVðLs Þ

$ 0:

s¼1

Since 1.0 and PðBk Þ.0; this contradicts the definition of Bk : So PðBk Þ¼0 for every k¼1;…;n21 and the strong Axiom of Solvency holds. A Since the set of all mean-decreasing shift factors contains the set of all supermartingale-like ones, the immunization inequality (10.3.3) in Theorem 10.3.1 holds for a larger class of functions than the same inequality in Theorem 10.4.1. Simultaneously this is equivalent to a weaker Axiom of Solvency than the strong version of Axiom of Solvency used in Theorem 10.4.1. This paradox is possible because we have assumed in Section 10.3 a special stochastic structure of the interest rates and the portfolio that f ðt þ 1Þ 2 f ðtÞ is uncorrelated with St for every t ¼ 1; …; n 2 1: The class of supermartingale-like shift factors can be easily interpreted using TSIR. Let {i1 ; …; in } be the (nonrandom) forward interest rates in succeeding time periods, which are used by the actuary to calculate the present values of assets and

343

Stochastic approach to financial risk management

liabilities, i.e., PðkÞ ¼

1 ; ð1 þ i1 Þ· · ·ð1 þ ik Þ

for k ¼ 1; 2; …; n;

ð10:4:3Þ

and let {i01 ; …; i0n } be random interest rates corresponding to P0 ðkÞ: The condition that the shift factor is supermartingale-like is equivalent to the almost sure inequalities 0 P ðk þ 1Þ P0 ðkÞ 2 E F # 0 for k ¼ 1; 2; …; n 2 1: ð10:4:4Þ Pðk þ 1Þ PðkÞ k Observe that inequalities (10.4.4) are satisfied, if ik # i0k a:s:

ð10:4:5Þ

for k ¼ 2; …; n: Moreover, (10.4.5) implies that f ðnÞ 2 1 is nonpositive, if additionally i01 $ i1 almost surely. But a careful verification of the proof of Theorem 10.4.1 shows that if for some elementary event the sequence of reals {f ðtÞ}t¼1 ; …;n is decreasing and inequalities (9.3.2) are satisfied, then S0 2 S $ Sð f ðnÞ 2 1Þ:

ð10:4:6Þ

Thus Theorem 10.4.1 gives a sufficient condition for portfolio immunization against all scenarios with higher interest rates than those used by the actuary to value the assets and liabilities. The lower bound (10.4.6) is 0 (the case of full immunization) whenever S ¼ 0; for S . 0 the lower bound might be negative (the case of subimmunization); finally, for S , 0 the bound might be positive (the case of superimmunization). 10.4.2 Arbitrary shift factors In this subsection, we consider immunization under a class of shift factors with bounded slope. Theorem 10.4.2 below provides a lower bound on the change in a conditional expectation of the target value of the immunized portfolio. Its formulation and proof is taken from [28]. Theorem 10.4.2 Assume that the strong Axiom of Solvency holds. If for some constant K Eð f ðt þ 1Þ 2 f ðtÞlF t Þ # K

a:s: for every t ¼ 1; …; n 2 1;

then EðS0 2 SÞ $ EðSð f ðnÞ 2 1ÞÞ 2 Re K;

ð10:4:7Þ

where Re ¼ n EðSÞ 2 EðDnomM ðAÞÞ þ EðDnomM ðLÞÞ with DnomM ðAÞ and DnomM ðLÞ denoting nominal Macaulay durations of the assets and liabilities, respectively.

344

Chapter 10

Proof To prove Theorem 10.4.2 let us observe that by (10.4.2) and the assumptions we get EðS0 2 SÞ ¼ EðSð f ðnÞ 2 1ÞÞ 2E

n21 X

Eð f ðt þ 1Þ 2 f ðtÞlF t Þ

t¼1

$ EðSð f ðnÞ 2 1ÞÞ 2

t X

PVðAs Þ 2

s¼1

t X

!! PVðLs Þ

s¼1

n21 X t X K EðPVðAs Þ 2 PVðLs ÞÞ: t¼1

s¼1

Taking into account (10.3.8), we get EðS0 2 SÞ $ EðSð f ðnÞ 2 1ÞÞ 2 K EðnS 2 DnomM ðAÞ þ DnomM ðLÞÞ:

A

Let us notice that inequality (10.4.7) is an extension of inequality (10.4.1). Indeed, setting K ¼ 0 in Theorem 10.4.2 defines the class of all supermartingale-type shift factors and then (10.4.7) reduces to (10.4.1).

10.5 Conservative strategy of risk management for pension plans In this section, we apply the results proven in the previous sections to construct an optimally immunized strategy of the interest rate risk management for pension plans. First, we shall investigate how to choose a basic TSIR, and next, how to arrange the term structure of assets in relationship to liabilities. 10.5.1 Conservative choice of TSIR Recall that ði1 ; …; in Þ are the interest rates predicted by the actuary (or risk manager) while ði01 ; …; i0n Þ are the interest rates which may occur in the future. So it is of great interest to investigate how the actuary’s choice of ði1 ; …; in Þ can influence the assumptions of Theorem 10.4.2. We have already noticed that decreasing the rates ði1 ; …; in Þ enlarges the set of interest rates ði01 ; …; i0n Þ such that i0k $ ik ; for all k; against which the portfolio satisfying (9.3.2) is totally immunized. But what happens e.g., on the set of all ði01 ; …; i0n Þ such that i0k , ik for every k ¼ 1; …; n? Since f ðk þ 1Þ 2 f ðkÞ ¼

k ikþ1 2 i0kþ1 Y 1 þ is ; 1 þ i0kþ1 s¼1 1 þ i0s

if we put a smaller but still nonnegative is 2 Dis instead of is ; for every s ¼ 1; …; n;

345

Stochastic approach to financial risk management

we arrive at a new difference, say fD ðk þ 1Þ 2 fD ðkÞ; which is of the following form fD ðk þ 1Þ 2 fD ðkÞ ¼

k ikþ1 2 Dikþ1 2 i0kþ1 Y 1 þ is 2 Dis , f ðk þ 1Þ 2 f ðkÞ; 1 þ i0kþ1 1 þ i0s s¼1

for all possible ði01 ; …; i0n Þ such that i0k , ikþ1 2 Dikþ1 for every k ¼ 1; …; n: Hence a more conservative choice of the rates ði1 ; …; in Þ results in a smaller (or the same, in the worst case) constant K bounding the increment of f for those ði01 ; …; i0n Þ: This improves, however, the lower bound (10.4.7). So a more conservative approach to the interest rates prediction not only increases the set of possible TSIRs against which the portfolio is fully immunized, but also improves the lower bound (10.4.7) for some other set of TSIRs as well. The above analysis and the results of Sections 9.3, 10.3 and 10.4 also give a strict mathematical justification for the following conservative strategy in asset-liability management.

In relation to your expectations of the interest rates in the future, use a conservative basic term structure of interest rates, i.e., interest rates lower than expected. Then arrange the portfolio of assets and liabilities so that the Axiom of Solvency is satisfied, given S $ 0: If the interest rates in the future are higher than assumed ones (which is quite likely then), the difference S0 2 S will be automatically bounded from below by a nonnegative quantity SðP0 ðnÞ=PðnÞ 2 1Þ:

ð10:5:1Þ

An interesting point is that the bound (10.5.1) is valid regardless of the choice of i1 : This enables to refine the conservative strategy by choosing i1 higher than the expected one which would cause that P0 ðnÞ=PðnÞ 2 1 may be quite often positive. Then also the lower bound SðP0 ðnÞ=PðnÞ 2 1Þ . 0; for any S . 0; implying the immunization postulate S0 2 S . 0: We provide a more precise analysis of that problem in Section 10.4. However, if the first interest rate i1 is not overestimated, it is very likely that the difference S0 2 S will be negative. Indeed, S0 2 S is bounded from below by SðP0 ðnÞ=PðnÞ 2 1Þ where the second factor is nonpositive, if all interest rates in the future are not smaller than the assumed ones. And, as a result, the bigger the S; the less satisfactory the immunization of the portfolio can be. So, if we predict the future interest rate relatively well, the probabilities that P0 ðnÞ=PðnÞ . 1 and that P0 ðnÞ=PðnÞ , 1 are approximately the same and we are faced with a multicriteria portfolio optimization problem. We shall discuss more deeply the question how to choose S in Section 10.5.2, showing that the choice S ¼ 0 is quite sensible, at least in the case of pension plans.

346

Chapter 10

10.5.2 Portfolio with a minimal immunization risk In Sections 10.3 and 10.4, we showed the following lower bound on expected change in the portfolio surplus: EðS0 2 SÞ $ EðSð f ðnÞ 2 1ÞÞ 2 K EðnS 2 DnomM ðAÞ þ DnomM ðLÞÞ;

ð10:5:2Þ

where DnomM ðAÞ and DnomM ðLÞ denote the nominal Macaulay durations of A and L; respectively, and K is an upper bound on the expected change in the slope of f (see Theorems 10.3.2 and 10.4.2 for more details). This lower bound is a sum of two expected products, each one consisting of two terms. In both the products one term is a function of the interest rate change only, while the other depends only on the portfolio structure. So the second term can be regarded as a measure of immunization risk and optimized. In this subsection we shall construct a funding method for pension plans which minimizes that measure. The resulting method has accrued assets defined by the smallest concave majorant of the accrued liabilities and is known to satisfy several other optimality criteria as well (see Section 4.3). The basic problem for managing defined benefit pension plans relies on funding a stream of liabilities {L1 ; …; Ln } by a stream of premiums {A1 ; …; An }. In order to prevent shifting of the funding process to an undetermined future, usually it is assumed that the premiums have nonincreasing present values, i.e., PVðA1 Þ $ PVðA2 Þ $ · · · $ PVðAn Þ;

ð10:5:3Þ

see Section 4.3. In this chapter we allow the assets and liabilities be random, so instead of (10.5.3) we shall assume that it holds only on the average, i.e., EðPVðA1 ÞÞ $ EðPVðA2 ÞÞ $ · · · $ EðPVðAn ÞÞ:

ð10:5:4Þ

A funding strategy {A1 ; …; An } will be called optimally immunized if, given S; it maximizes the right side of inequality (10.5.2). This is equivalent to minimizing the following measure of risk for an immunized portfolio Re ¼ EðnS 2 DnomM ðAÞ þ DnomM ðLÞÞ: This problem was solved in [28] Theorem 10.5.1 A funding strategy {Ap1 ; …; Apn } is optimally immunized under P p conditions (10.3.1) and (10.5.4) if and only if the sequence { ts¼1 EðPVðA Pn21 s ÞÞ}t¼1;…;n is Pn the smallest concave majorant of {EðPVðL1 ÞÞ; …; s¼1 EðPVðLs ÞÞ; s¼1 EðPVðLs ÞÞ; þEðSÞ}:

347

Stochastic approach to financial risk management

Proof To prove Theorem 10.5.1 let us observe that given S; we are looking for a solution to the following problem: 8 < min {n EðSÞ 2 EðDnomM ðAÞÞ þ EðDnomM ðLÞÞ} A1 ;…;An

:

subject to ð10:3:1Þ and ð10:5:4Þ:

Now observe that, by (10.3.8), EðnS 2 DnomM ðAÞ þ DnomM ðLÞÞ ¼

n21 X t X t¼1

EðPVðAs ÞÞ 2

t X

s¼1

! EðPVðLs ÞÞ ;

s¼1

where the of the equation does not depend on S: By assumpPt right-hand side P t tion, EðPVðA ÞÞ $ for every t ¼ 1; …; n 2 1; so s s¼1 s¼1 EðPVðLs ÞÞ EðnS 2 DnomM ðAÞ þ DnomM ðLÞÞ is minimized if and only if the surplus EðSt Þ ¼

t X s¼1

EðPVðAs ÞÞ 2

t X

EðPVðLs ÞÞ;

s¼1

is as small as possible for every t ¼ 1; …; n 2 1 over all decreasing sequences {EðPVðA1 ÞÞ; …; EðPVðA n ÞÞ}: This minimum ofthe surpluses is achieved if and only if Pt the sequence EðPVðA ¼ 1; …; n is the smallest concave majorant s ÞÞ; tP s¼1 t of the sequence consisting of all s¼1 EðPVðLs ÞÞ; for t ¼ 1; …; n 2 1; and with Pn EðPVðL ÞÞ þ EðS Þ being the nth element of the Indeed, condition s s¼1 Psequence. t (10.5.4) means that the sequence of cumulants EðPVðA s ÞÞ; t ¼ 1; …; n s¼1 must be concave, while the smallest concave majorant minimizes, by definition, all A surpluses EðS Þt ; t ¼ 1; …; n: The funding strategy defined by the smallest concave majorant of cumulated liabilities was proposed in 1998 by the first author to fund bridging pensions for miners in Poland. In Gajek and Ostaszewski [32] it was proved that this funding method is optimal in the sense that it maximally stabilizes the plan contributions. Finally, let us discuss the problem what value of S should be chosen in practice. Clearly, S , 0 means that the pension plan has a long-term deficit and cannot be accepted. On the other hand, if S . 0; the present value of accumulated premiums is larger than the net present value of accumulated liabilities, which could be justified to the pension plan contributor only by the safety reasons. However, the right-hand side of (10.5.2) depends on S only through Sð f ðnÞ 2 1Þ (in fact the second summand is independent of S—see (10.3.8) in the proof of Theorem 10.5.1). If ik # i0k for all k ¼ 1; …; n; the multiplier f ðnÞ 2 1 is negative and the larger S; the smaller (and negative) the lower bound (10.3.7) is. This shows that S ¼ 0 is a quite reasonable choice in the case of funding pension liabilities; some other arguments for this choice can be found in Hu¨rlimann [39] and Ostaszewski and Zwiesler [55]. Since S is a random variable, S should be zero at least on the average. In that case the risk of the

348

Chapter 10

portfolio reduces to Re ¼ EðDnomM ðAÞÞ 2 EðDnomM ðLÞÞ; Pt p which is minimized when the sequence is the s¼1 EðPVðA Pn s ÞÞ t¼1;…; n smallest concave majorant of the sequence EðPVðL1 ÞÞ; …; s¼1 EðPVðLs ÞÞ : It is also easy to notice that Re cannot be reduced to zero unless this latter sequence is concave.

Exercises Exercise 10.6.1 May 2003 SOA Course 6 Examination, Problem No. C-10. You are given the following with respect to an option-free bond portfolio. The value of the bond portfolio using the current yield curve is 800. The value of the bond portfolio using the current yield curve with a parallel shift upwards of 20 basis points is 788. The value of the bond portfolio using the current yield curve with a parallel shift downwards of 20 basis points is 813. (a) Estimate the value of the bond portfolio for a parallel shift upwards of 200 basis points in the yield curve. (b) Explain how the inclusion of convexity impacts your estimate. Solution Let us note that a basis point is 0.01%. (a) Based on the information given, we estimate duration of this bond as D<

Pði 2 DiÞ 2 Pði þ DiÞ 813 2 788 25 ¼ ¼ ¼ 7:8125: 2PðiÞðDiÞ 2 · 800 · 0:0020 3:2

The analogous convexity estimate is C¼

Pði 2 DiÞ 2 2PðiÞ þ Pði þ DiÞ 813 2 2 · 800 þ 788 1 ¼ 312:5: ¼ ¼ 0:0032 800 · 0:0022 PðiÞ · ðDiÞ2

Based on the duration value, approximate change in the value of the bond when rates rise by 200 basis points (bps) is DP ¼ 2Di · P · D ¼ 20:02 · 800 · 7:8 ¼ 2124:80: That is 15.60% of the value of the bond.

349

Stochastic approach to financial risk management

(b) Including convexity, changes the approximation to the one using the two terms Taylor series expansion: 1 DP < 2P · D · Di þ P · C · ðDiÞ2 ; 2 so that 1 1 DP < 2P · D · Di þ P · C · ðDiÞ2 ¼ 2800 · 7:8 · 0:02 þ · 800 · ð0:02Þ2 2 2 ¼ 2124:80 þ 0:16 ¼ 2124:64: The loss estimate is somewhat lowered by the inclusion of convexity, it is now 15.58% instead of 15.60%. A Exercise 10.6.2 You are managing a bond portfolio of 3 000 000 monetary units. You decide that the Macaulay duration of your portfolio should be exactly 10. You have only two securities to choose from for your investments: a zero-coupon bond of maturity 5 years, and a continuous perpetuity paying at the rate of 1 per year. Current force of interest is 5%. How much will you invest in each of these securities in order to have the desired Macaulay duration? Solution The Macaulay duration of the zero coupon bond will be exactly 5 years. The perpetuity is worth 1=d if the force of interest is d; and so its derivative with respect to d equals 21=d2 ; and its Macaulay duration is 1=d: At the given force of interest, this means its duration is 20. If you invest a portion of your portfolio w in the zero coupon bond and 1 2 w in the perpetuity, Macaulay duration of the entire portfolio will be w · 5 þ ð1 2 wÞ · 20 ¼ 10: Thus 5w 2 20w ¼ 10 2 20;

15w ¼ 10;

w¼

2 ; 3

12w¼

1 : 3

You should invest 2 000 000 in the 5-year zero coupon bond and 1 000 000 in the perpetuity. A Exercise 10.6.3 You manage a bond portfolio for a life insurance company. You have observed then when the 1-year spot interest rate rose from 3.00 to 3.20% and the 10-year spot rate rose from 3.50 to 3.75%, your portfolio lost 1.00% of its value. On the other hand, when the 1-year spot rate fell from 3.20 to 2.90% and the 10-year spot rate fell from 3.75 to 3.60%, your portfolio gained 0.75% of its value. Estimate the 1-year and 10-year key-rate durations (also know as partial durations) of your portfolio.

350

Chapter 10

Solution Denote the key-rate durations with respect to 1 and 10 years, respectively, by D1 ; D1 0 and the respective spot interest rates by i1 and i1 0: Then we have the following approximation: DP < 2D1 · Di1 2 D10 · Di10 : P Substituting the data given: 21:00% < 2D1 · ð0:20%Þ 2 D10 · ð0:25%Þ; 0:75% < D1 · ð0:30%Þ þ D10 · ð0:15%Þ; and this is a system of linear equations that solves to D1 ¼

5 ; 6

D10 ¼

10 : 3

A

Exercise 10.6.4 May 2001 SOA Course 6 Examination, Problem No. 24. You are given the following information with respect to a callable bond. The expected (but not certain) cash flows at 7% annual yield are: at time 1: 8.00, at time 2: 7.90, at time 3: 107.80. At 6% annual yield this bond’s price is 104.33, at 7% annual yield this bond’s price is 102.37, and at 8% annual yield the price is 99.76. The current yield is 7%. Calculate the ratio of the modified duration (duration with respect to the interest rate) based on expected cash flows to the effective duration estimated based on the data on price changes of the bond available. Solution 1 Macaulay duration ¼

8 7:90 107:80 þ2 þ3 2 1:07 1:07 1:073 ¼ 284:36488 ¼ 2:7777: 8 7:90 107:80 102:37371 þ þ 1:07 1:072 1:073

Modified duration ¼

2:7777 ¼ 2:5959943: 1:07

Effective duration estimate

D<

Aði 2 DiÞ 2 Aði þ DiÞ 104:33 2 99:76 ¼ ¼ 2:2320992: 2AðiÞ · Di 2 · 102:37 · 0:01

Stochastic approach to financial risk management

351

The ratio of the two is 2:5959943 ¼ 1:1630: 2:2320992

A

Exercise 10.6.5 May 2001 SOA Course 6 Examination, Problem No. 25. The current price of a bond is 100. The derivative of the price with respect to the yield to maturity is 2 700. The yield to maturity is 8%. Calculate the Macaulay duration of that bond. Solution Macaulay duration ¼ 2

dP 1 dP 1 ¼ 2ð1 þ iÞ : dd P di P

In this case, we obtain: 1:08 · 700 ·

1 ¼ 7:56: 100

A

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353

Chapter 11 Conclusion

Asset-liability management of pension plans is a relatively new area of financial economics and despite its rapid development there remain many unanswered questions created by it. Nevertheless, one can state several fundamental rules of financial risk management for pension plans, which, if used, will significantly reduce the risk of insolvency. These rules are generally derived from the ideas of modern financial economics, applied for both assets and liabilities. According to the Cardinal Rule of Valuation, as defined by Redington, we have, in this book, used consistent methodologies of valuation to both sides of the balance sheet. We treat both assets and liabilities of a pension plan as financial assets. Below we categorize the methodologies of asset-liability management based on the type of risk, as classified in Section 8.1. The C1 risk may be managed with the following methods: † moving the risk entirely to the plan participants (by adopting a defined contribution plan instead of defined benefit plan—see Section 1.2); † move only a fraction of risk to the plan participants, by having them choose an investment strategy from a variety presented to them by the plan sponsor (this idea can apply to both defined contribution and defined benefit plans—see Section 1.2); † optimization of the investment portfolio with consideration for risk reduction and diversification (see Sections 7.1 and 10.1). The C1 risk may also be reduced by using hedging, i.e., the purchase of assets (typically derivative securities) with negative correlation with assets held currently, so that the volatility of the combined portfolio is reduced (see Section 7.3). The C2 risk may be managed by using the following methods: † the choice of an appropriate valuation method for accrued liabilities and funding method for the calculation of normal cost (see Chapters 4 and 5);

354

Chapter 11

† optimal amortization of actuarial deficit or actuarial gains (see Section 4.3 and Chapter 5); † creation of an appropriate reserve for random fluctuation of liabilities (see Chapter 6). The C3 risk can be managed by doing the following: † the use of consistent methodologies of valuation of assets and liabilities (see Chapter 5, as well as Sections 7.2 and 7.4); † the choice of an investment portfolio, which matches the existing liabilities portfolio (see Sections 9.3, 10.1, 10.2, 10.3, 10.4, and 10.5). In particular, we have given a strict mathematical justification for conservative strategies in assetliability management proving their rationale in deterministic (Section 9.3) as well as in stochastic models (Section 10.5). Our approach is practically oriented, taking into account common regulatory requirements concerned with solvency. We also have these methods of managing risk of insufficient funding by the plan sponsor: † the choice of an appropriate method of normal cost calculation (see Section 4.3, Chapter 5 and Section 8.4); † optimal amortization of deficit (see Section 4.3 and Chapter 5). Management of pension plans involves a special duty to plan participants, which has been recognized by legal structures and governments around the world. Errors in plan valuation, investments, and asset-liability management can result in long-term consequences, which cannot be reversed, especially upon retirement of plan participants. This special duty has legal implications, expressed, for example, not only in the fiduciary duty assigned to key plan persons in the Employee Retirement Income Security Act (ERISA) in the United States, but also in the fact that in so many countries pension provision is delegated to the state. We believe that duties to plan participants have not only a legal, but also scientific aspects—one must keep abreast of all new developments in modeling and management of pension liabilities, assets, and their inter-relationship. The methodologies of financial risk management for pension plans are still under development. They are a part of the dynamic body of research in financial economics, which has affected many areas of business and finance. We can expect equally dynamic new developments in the financial research and practice in the future. One area that we believe to hold great promise is integrated optimization of funding and management practices, with clearly identified profit and surplus objectives, under strict solvency constraints. We believe our work to be an important contribution in this direction—by identifying optimal funding methodologies in both deterministic and random cases as well as discussing optimal long-term pension plan

Conclusion

355

management strategies. Practical implementations of those strategies are complicated by the random nature of the variables affecting pension plans. Our work has pointed out many approaches in handling these difficulties. Yet the gradual shift from deterministic methodologies used commonly half a century ago to the increasingly important stochastic models is not yet complete. We believe this shift to be the main trend of current, rather tectonic, shifts in the structure of financial intermediaries. We hope that our work can and will be helpful to all financial decision makers involved in providing retirements benefits, especially of defined structure.

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357

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361

Index A Accrued benefit, 163– 4, 180– 1, 215 Accrued liability, 163 – 165, 186 – 189, 208, 214, 217– 26, 231, 242 Accumulated benefit obligation (ABO), 166 Accumulated cost of insurance, 126 Accumulation factor, 14, 71, 187, 205 Accumulation function, 14– 5, 18 Actuarial deficit, 165, 182, 300 Actuarial gain, 162 – 3, 165 – 6, 172 – 3, 183, 190, 195 –6, 199, 203, 207– 8, 210, 213– 4, 217– 8, 223– 4, 232 Actuarial loss, 163, 183, 190, 196, 208, 218 Actuarial present value, 90, 101– 3, 111– 4, 116– 8, 125, 135, 143, 151, 162, 164– 5, 186– 7, 191– 3, 196– 8, 200, 203, 205, 212, 215, 220– 5, 228, 231, 274 Actuarial surplus, 165 Adapted stochastic process, 32 –3 Additional premium, 161, 208 Aggregate funding method, 98 –9, 116 Aggregate pension funding methods, 215 American option, 276 Amortization of the unfunded actuarial liability, 161 Amortized Value Method of asset valuation, 280, 283 Amount function, 13

Annuity, 4, 48 –55, 60, 63, 75, 101– 8, 110– 17, 119, 124, 133, 139, 143, 162, 172, 177– 8, 180, 186, 192, 203– 5, 215– 7, 219, 221– 3, 225– 6, 229– 32, 292 Annuity-due, 104, 106 –7, 111, 114– 7, 119, 124, 162, 204, 206, 231 Annuity immediate, 47, 49 – 51, 53 –4, 60 Arbitrage, 26, 67, 171, 263 –73, 275 –6, 279, 282, 319– 20, 323 Arbitrage free pricing theory, 26 Arrow-Debreu security, 264, 268 Asset-Liability Management, 332, 345, 354 Asset valuation methods, 280, 281 –2 Associated single decrement table, 145– 6, 153, 155 At-the-money option, 276 Attained Age Normal, 204 Authorized Control Level, 292 Average Market Value Method of asset valuation, 281, 283 Average Unit Value Method of asset valuation, 282 Axiom of Solvency, 168, 322, 324– 5, 327, 337– 43, 345 B Babcock equation, 313 – 4 Balducci assumption, 85 Benefit premium, 90, 92 –6, 101, 103– 7, 116, 119– 20, 122– 4,

362

126– 32, 134 –8, 142, 148– 9, 155– 6, 172 Benefit reserve, 125 –6, 129 –30, 132, 134– 5 Beta, 262, 272 Black-Scholes Formula, 277, 279 Blend of Cost and Market Values Method of asset valuation, 281, 283 Book Value Method of asset valuation, 280 C C1 risk, 298 C2 risk, 294 C3 risk, 294, 354 C4 risk, 289 Call option, 276, 278– 9, 321 Capital Assets Pricing Model (CAPM), 260 Capital assets, 5, 8, 255– 6, 260, 263– 5, 270, 273, 303, 315, 322, 336 Capital market, 4, 254, 256, 264, 266, 270, 273, 301, 318 Capitalized cost, 74– 5 Cardinal rule of valuation, 316 Cash balance plans, 9 Cash flow testing, 274, 294 Cash flow, 10, 19, 21 –2, 24, 45, 52, 63– 5, 167– 71, 257, 263– 4, 269– 70, 273 –5, 279 –80, 283 –4, 294, 303– 4, 306– 10, 316– 9, 321, 325, 327, 331, 350 Commutation functions, 83, 102, 107, 216, 226 Company Action Level, 291 –3 Competing risks, 145 Complete life expectancy, 78, 88 Compound discount method of depreciation, 70 Compound interest, 14, 70, 280 Compound interest method of depreciation, 70

Index

Confidence intervals for accrued liability, 242, 251 Constant force of mortality, 79, 88, 93, 99, 112– 3, 117 Constant force assumption, 85, 121, 123, 127, 134 Constant percentage method of depreciation, 70, 74 Continuous annuity, 50 –1, 143 Continuous compounding, 13, 16, 21– 2, 24, 30, 34, 50– 2, 66, 75 – 7, 83, 90, 92, 94– 5, 102– 3, 105– 6, 112, 119– 21, 123 –5 Contract premium, 121, 128, 137 Contract Value Method of asset valuation, 280 –4 Contribution pension plan, 23 Contributory plan, 213 Convexity, 307 –14, 318 –9, 321 – 2, 331– 2, 348 –9 Cost of float, 324 Cost Value Method of asset valuation, 280, 283 Currency swap, 277 Curtate life expectancy, 91 D David Ricardo Identity, 48 De Moivre’s Law, 79, 82, 84, 87– 9, 93, 107, 111, 128, 132 – 3, 141 –2, 144, 153 Declining balance method of depreciation, 70 – 1, 73 Declining index system, 26 Decreasing annuity, 51, 53 Deferred annuity, 48, 116, 124 Deferred life annuity, 27, 48, 96, 98 – 9, 111, 116, 124, 281 – 3, 295 Deferred Recognition Method of asset valuation, 281 –3 Deferred whole life insurance, 96, 98– 9

363

Index

Defined benefit plans, 6 – 10, 282, 298 Defined contribution plans, 6– 10, 298 Depreciation, 68 –74, 298 Derivative securities, 275 Discount factor, 14, 19, 21, 59, 71, 90– 91, 105, 163, 305 Discounted Cash Flow Method of asset valuation, 280, 283 Dollar duration, 303, 317 Dollar-weighted rate of return, 21, 24, 26 Duration, 77, 125, 134, 231, 303 –14, 317– 8, 321– 2, 324 –6, 330 –2, 340, 343, 346, 348 – 51 E Effective annual interest rate, 19, 21, 66, 184, 275, 309 Effective duration, 304, 321, 330, 350 Effective rate of return, 24 Efficient frontier, 260 –1 Efficient portfolio, 257, 260, 332 Employee Retirement Income Security Act (ERISA), 4, 160, 354 Endowment, 94– 5, 136, 138 Entry Age Normal, 164, 185 –6, 191, 203– 4, 208, 214– 6, 220– 2, 225– 6, 228 Equivalence Principle (EP), 119, 164 European option, 276 – 7 Expected unfunded actuarial liability, 166, 173, 190, 224 Exponential premium, 119 Exposure, 24, 127, 151, 277, 326, 340 F Fair Market Value Method of asset valuation, 280, 283 Filtration, 26, 31 Financial assets, 17, 33, 262 – 3, 273, 321

Financial security systems, 3, 159 –60 Fixed index system, 26 Force of interest, 15, 17 –9, 21 –2, 25, 31, 66, 92, 99, 105 –7, 112, 117, 167, 184, 233, 246, 271, 304 –7, 311, 313, 349 Force of mortality, 76– 7, 79 –81, 85– 6, 88, 93, 99, 107, 112 –3, 117 –18, 145, 171, 184, 206, 232, 238, 305 Forward, 15, 65– 8, 275, 326, 342 Fractional age assumptions, 84 Fractional life expectancy, 84– 5 Frozen Attained Age, 204, 209 Frozen Initial Liability (Attained Age Normal) (FIL(AAN)) funding method, 217, 219 – 20 Frozen Initial Liability (Entry Age Normal) (FIL(EAN)) funding method, 216, 219 – 20, 225 Frozen Initial Liability, 196 –7, 199, 203, 209, 216– 7, 219– 20, 225 Fundamental Principle of Pension Plan Actuarial Valuation, 164– 5, 167 Fundamental Theorem of Asset Pricing, 26, 171, 267, 269 – 71, 273, 321 Funded ratio return, 336 Funded ratio, 317, 336 Funding ratio, 317 Futures, 275 –6, 278 Futures-spot parity, 275 G Gauss Formula, 72 Generally Acctepted Accounting Principles (GAAP), 64, 218 Gompertz model of mortality, 80 H Hazard rate, 76, 100, 184, 232, 238 Hedging, 276

364

Historical model of Value-at-Risk, 281, 335 Human capital, 3 Hyperbolic assumption, 86, 88

I Immunization, 168, 315– 20, 323– 6, 337– 8, 340, 342– 3, 345– 6 In-the-money option, 276 Increasing annuity, 50 –2, 55 Increasing perpetuity, 50 –2 Indifference curve, 254– 5 Individual Aggregate funding method, 215– 6 Individual Level Premium (ILP), 165, 191, 196, 208, 214 – 5, 229 Individual Retirement Accounts (IRA), 5 Interest measurement in a fund, 22 Interest measurement in a group of funds, 26 Interest rate swap, 277 Internal rate of return (IIR), 21, 63 Intrinsic value of an option, 276 Investment Year Method, 26

Index

M Macaulay convexity, 307 – 9, 311 Macaulay duration, 304 – 6, 308– 9, 313, 317, 324– 6, 332, 340, 343, 346, 349, 351 Macaulay M-squared, 307 Makeham’s model of mortality, 63, 80 Mandatory Control Level, 292– 3 Market risk premium, 262, 272 Marking to market, 63 Martingale fairness axiom, 26 Martingale, 26, 30, 36 –8, 45, 271 Martingale-fair, 26, 30, 36– 8 Merger of funds, 42 Model calibration, 321 Modern Portfolio Theory, 256 Modified duration, 304, 350 Monetary duration, 303 –4, 307 Mortality table, 82– 3, 85, 87, 89, 102, 106, 186, 231, 238, 250 Mortgage amortization, 49, 57 –8, 62, 293– 4, 319, 321 M-squared, 307– 8 Multiple decrements, 144 Multiple lives models, 138– 9 Multivariate immunization, 320

L Life annuity, 4, 75, 102 – 7, 110 – 2, 115– 7, 119, 124, 162, 180, 192, 204– 5, 217, 221, 225, 231 – 2 Life insurance premium, 101, 118, 120 Life insurance reserve, 125, 127 Life insurance, 10, 69, 75, 82, 89– 99, 101, 103, 105, 113, 118 –25, 127– 30, 132 –5, 138 –9, 155, 160– 1, 164, 167, 172, 204, 233, 274, 280, 294– 5, 300, 331 – 2, 349 Loan amortization, 56 Loss-at-issue, 119, 123 – 4

N Net single premium, 90, 103, 133, 204 Nominal convexity, 310 Nominal Macaulay duration, 306, 324, 326, 340, 343, 346 Nominal rate of return, 13 Normal cost, 4, 6, 9 – 11, 90, 161 –6, 169, 172– 3, 181– 7, 191– 3, 195– 7, 199 –200, 202 –4, 208 –9, 212– 3, 215 –7, 219 –31, 300– 1, 354

365

Index

O

R

Old Age, Survivors, and Disability Insurance (OASDI), 5, 167 Optimal payoff of a liability, 167 Option, 61, 114, 212, 275– 9, 321 –2, 348 Option-adjusted convexity, 322 Option-adjusted duration, 322 Out-of-the-money option, 276

Random future lifetime, 75, 105, 184, 240 Rate of return, 4, 21– 2, 24 –6, 30– 1, 34, 36– 43, 63, 68, 173– 6, 183, 244, 255– 7, 261– 2, 265, 271– 2, 274, 277– 8, 281, 283, 295, 297 – 300, 313, 317– 8, 330, 332– 6 Real assets, 8, 62, 71 Regulatory Action Level, 291 –2 Replication of a security Retirement plans, 6 Risk premium, 66, 262, 272 Risk-Based Capital, 334 Risk-free rate of return, 261 –2, 265, 271, 332 Risk-neutral probability measure, 26, 270, 273– 4 Rule of Moments, 92, 94 –5

P Parametric model of Value-at-Risk, 335– 6 Participation Unit Method, 25 – 6 Pay-as-you-go, 5– 6 Pension plan deficit, 4, 8, 10, 23, 75, 77, 106, 159– 61, 163– 7, 169, 171– 2, 175– 6 Pension plan surplus, 320, 330 Percentile premium, 119 Perpetuity, 48, 50– 4, 75, 110, 272, 284, 349 Perpetuity-due, 51 Perpetuity-immediate, 48, 50, 53– 4 Planned premium, 161 Poohsticks formulas, 140 Portfolio Method, 26 Principle of Asset-Liability Management, 290 Principle of Consistency of Valuation of Assets and Liabilities, 282 Principle of no-arbitrage, 267 Projected benefit obligation (PBO), 166 Projected Unit Credit funding method, 228 Pure endowment, 94– 5 Put option, 276, 278 Put-call parity, 276 – 8

S Salary function, 216 Select and ultimate table, 83, 87 Select cohort, 83 Self-financing portfolio, 29, 260 Self-financing strategy, 26 Short sales, 257 Short-Straddle model, 332 Shortfall control, 332, 334 Simple interest, 14, 17, 22, 24, 26 Simulation model of Value-at-Risk, 321, 334– 5, 337 Single benefit premium, 90, 92– 6, 101, 103– 7, 116, 122, 124, 135, 148, 155– 6, 172 Sinking fund method of depreciation, 70 Sinking fund, 58– 9, 70, 321 Social insurance, 4 –7, 167, 177 –8 Standard and Poor’s (S&P) Index, 262

366

Standard Valuation Law, 274, 280 State price vector, 267, 269 –70, 279 Statutory accounting principles (SAP), 69 Straight-line method of depreciation, 70– 3 Submartingale, 26, 45 Sum of the year’s digits method of depreciation, 70 – 3 Supermartingale, 26, 45, 337– 8, 340– 4 Supplemental cost, 161, 216, 220 – 1, 226 Surplus immunization, 323 Survival function within the pension plan, 106 Survival function, 76, 78, 81, 106 Swap, 277

Index

Unfunded actuarial liability, 165 –6, 168– 9, 172 –3, 182, 190, 208, 214– 7, 220 –2, 224, 231 Uniform distribution of deaths (UDD) assumption, 83– 4, 146 –7, 151, 205 Uniform distribution of deaths (UDD) in the associated single decrement table, 83 – 4, 146 – 7, 151, 205 Uniform distribution of deaths (UDD), 83– 4, 146– 7, 151, 205 Unit credit funding method, 219, 227– 8 Unit Credit, 163, 180, 183, 185, 204, 208, 212– 3, 217– 9, 221– 4, 227– 8, 232, 242 Unit normal cost, 197, 200, 202 – 3, 212 V

T Target benefit plans, 9 Temporary annuity, 75, 162 Temporary life annuity, 102, 106, 115 Temporary life expectancy, 89 Term life insurance, 69, 93– 4, 132 Term structure of interest rates, 65, 256, 323– 4, 327, 338, 341, 345 Terminal benefit reserve, 125 Threshold rate of return, 332 Time value of the option, 276 Time-weighted rate of return, 24 Trend-Line Method of asset valuation, 281, 284

Valuation rate, 90, 102, 161, 172– 3, 184, 280 Value-at-Risk, 334 –6 Variation, coefficient of, 7, 71, 204, 332 Varying annuities, 50 W Weibull’s model of mortality, 80 Whole life insurance, 69, 90– 93, 96– 9, 105, 113, 118– 24, 128– 30, 132, 134– 5, 138, 155, 233 Write-Up Method of asset valuation, 64, 281– 3 Y

U Underlying, 275 –8 Underwriting loss, 324

Yield curve, 65 – 8, 256, 303, 315, 317, 320, 348 Yield rate, 21, 64, 70