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Investment Guarantees Modeling and Risk Management for Equity-Linked Life Insurance MARY HARDY John Wiley & Sons, Inc Investment Guarantees Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding The Wiley Finance series contains books written specifically for finance and investment professionals as well as sophisticated individual investors and their financial advisors Book topics range from portfolio management to e-commerce, risk management, financial engineering, valuation and financial instrument analysis, as well as much more For a list of available titles, visit our Web site at wwwWileyFinance.com Investment Guarantees Modeling and Risk Management for Equity-Linked Life Insurance MARY HARDY John Wiley & Sons, Inc ϱ This book is printed on acid-free paper ᭺ Copyright ᮊ 2003 by Mary Hardy All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01928, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, e-mail: permcoordinator࠽wiley.com Limit of 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publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Hardy, Mary, 1958Investment guarantees : modeling and risk management for equity-linked life insurance / Mary Hardy p cm – (Wiley finance series) Includes bibliographical references and index ISBN 0-471-39290-1 (cloth : alk paper) Insurance, Life-mathematical models Risk management–Mathematical models title II Series HG8781.H313 2003 368.32’0068’1–dc21 Printed in the United States of America 10 2002034200 Acknowledgments his work has been supported by the National Science and Engineering Research Council of Canada, and by the Actuarial Education and Research Fund I would also like to thank the members of the Department of Statistics at the London School of Economics and Political Science for their hospitality while the book was being completed, especially Anthony Atkinson, Angelos Dassios, Martin Knott, and Ragnar Norberg I would like to thank Taylor and Francis, publishers of the Scandinavian Actuarial Journal, for permission to reproduce material from Bayesian Risk Management for Equity-Linked Insurance in Chapter I learned a great deal from my fellow members of the magnificent Canadian Institute of Actuaries Task Force on Segregated Funds In particular, I would like to thank Geoffrey Hancock, who has provided invaluable advice and assistance during the preparation of this book Also, thanks to Martin Le Roux, David Gilliland, and the two Chairs, Simon Curtis and Murray Taylor, who had a lot to put up with, not least from me I have been very lucky to work with some wonderful colleagues and students over the years, many of whom have contributed directly or indirectly to this book In particular, thanks to Andrew Cairns, Julia Wirch, David Wilkie, Judith Chan, Karen Chau, Geoff Thiessen, Yuan Tao, So-Yuen Kim, Anping Wang, Boyang Liu, Harry Panjer, and Sheauwen Yang Thanks also to Glen Harris, who introduced me to regime-switching models It is a special privilege to work with Ken Seng Tan at the University of Waterloo and with Howard Waters at Heriot-Watt University My brother, Peter Hardy, worked with me to prepare the RSLN software (Hardy and Hardy 2002), which is a useful complement to this work It was good fun working with him Mostly I would like to express my deepest gratitude to my husband, Phelim Boyle, for his unstinting encouragement, support, and patience; culinary contributions; and unwavering readiness to share with me his encyclopedic knowledge of finance T M H v Contents Introduction xi CHAPTER Investment Guarantees Introduction Major Benefit Types Contract Types Equity-Linked Insurance and Options Provision for Equity-Linked Liabilities Pricing and Capital Requirements 11 14 CHAPTER Modeling Long-Term Stock Returns 15 Introduction Deterministic or Stochastic? Economical Theory or Statistical Method? The Data The Lognormal Model Autoregressive Models ARCH(1) Regime-Switching Lognormal Model (RSLN) The Empirical Model The Stable Distribution Family General Stochastic Volatility Models The Wilkie Model Vector Autoregression CHAPTER Maximum Likelihood Estimation for Stock Return Models Introduction Properties of Maximum Likelihood Estimators Some Limitations of Maximum Likelihood Estimation 15 15 17 18 24 27 28 30 36 37 38 39 45 47 47 49 52 vii 272 APPENDIX B Clearly EQ [Ht1 eϪrt1 ] ‫ ס‬PS0 (t1 ) Also EQ [Ht2 eϪrt2 ] ‫ ס‬EQ [EQ [Ht2 eϪrt2 ͉Ft‫ם‬1 ]] ‫ ס‬EQ [Ft‫ם‬1 eϪrt1 P(t2 Ϫ t1 )] ‫ ס‬EQ [(Ft1Ϫ ‫ ם‬Ht1 )eϪrt1 ] P(t2 Ϫ t1 ) ‫( ס‬S0 eϪmt1 ‫ ם‬PS0 (t1 )) P(t2 Ϫ t1 ) And, similarly, EQ [Ht3 eϪrt3 ] ‫ ס‬EQ [EQ [Ht3 eϪrt3 ͉Ft‫ם‬2 ]] ‫ ס‬EQ [Ft‫ם‬2 eϪrt2 ] P(t3 Ϫ t2 ) ‫ ס‬EQ [(Ft2Ϫ ‫ ם‬Ht2 )eϪrt2 ] P(t3 Ϫ t2 ) ‫͕ ס‬EQ [Ft‫ם‬1 eϪm(t2 Ϫt1 )Ϫrt1 ] ‫ ם‬EQ [Ht2 eϪrt2 ]͖P(t3 Ϫ t2 ) ‫͕ ס‬EQ [(Ft1Ϫ ‫ ם‬Ht1 )eϪrt1 ]eϪm(t2 Ϫt1 ) ‫ ם‬EQ [Ht2 eϪrt2 ]͖P(t3 Ϫ t2 ) ‫͕ ס‬S0 eϪmt2 ‫ ם‬EQ [Ht1 eϪrt1 ]eϪm(t2 Ϫt1 ) ‫ ם‬EQ [Ht2 eϪrt2 ]͖ P(t3 Ϫ t2 ) ‫͕ ס‬S0 eϪmt2 ‫ ם‬eϪm(t2 Ϫt1 ) PS0 (t1 ) ‫( ם‬S0 eϪmt1 ‫ ם‬PS0 (t1 )) P(t2 Ϫ t1 ))͖ P(t3 Ϫ t2 ) This gives a total option price of PS0 (t1 ) ‫( ם‬S0 eϪmt1 ‫ ם‬PS0 (t1 ))(1 ‫ ם‬P(t3 Ϫ t2 ))P(t2 Ϫ t1 ) ‫ם‬P(t3 Ϫ t2 ) (S0 eϪmt2 ‫ ם‬eϪm(t2 Ϫt1 ) PS0 (t1 )) APPENDIX C Actuarial Notation e have generally used standard actuarial notation in this book, with the exception that we are generally measuring term and duration in months Standard actuarial notation uses the following conventions: W t px is the probability that a life currently aged x survives to age x ‫ ם‬t t qx is the probability that a life currently aged x dies before age x ‫ ם‬t ␮x,t is the force of mortality at age x ‫ ם‬t for a life currently age x The force of mortality is also known as the mortality transition intensity or hazard rate It is defined as Ϫ d t px t px dt a¨ x:n is the expected present value of an annuity of per time unit, paid at the start of each time unit until the life age x dies, or until n time units expire, whichever is sooner For an interest rate of r, continuously compounded, the equation for the annuity is nϪ1 a¨ x:n ‫ ס‬Α t px eϪr t t‫ס‬0 The force of interest is the continuously compounded interest rate v is the annual discount factor; for a force of interest r, v ‫ ס‬eϪ r Tx is the random future lifetime of a life currently aged x years 273 274 APPENDIX C In this book we have used these symbols adapted to allow for the two decrements, death and withdrawal The superscript ␶ indicates that both decrements are allowed for; d indicates decrement by death and w indicates decrements by withdrawal The specific notation used is it is assumed to take the value t ‫ ס‬1.0 ␶ t px,u is the probability that a policyholder currently aged x years and u months survives and does not withdraw for a further t months ␶ ␶ t qx,u is Ϫ t px,u w t qx is the probability that a policyholder currently aged x years withdraws before t months expire d t qx is the probability that a policyholder currently aged x years dies in force before t months expire d u ͉t qx is the probability that a policyholder aged x years is still in force after u months, but dies in force before the expiry of a further t months If the t is omitted, it is assumed to take the value t ‫ ס‬1.0 ␶ t qx is the probability that a policyholder aged x years dies or lapses the policy before t months expire (d ) ␮x,t is the force of mortality experienced by a life aged x years and t months a¨ ␶x:n iЈ is the value of an annuity of per month paid monthly in advance for n months, contingent on the survival, in force (the ␶ indicates the double decrement function), of a life age x The rate of interest is iЈ per month, which means that the discount factor for the payment due at t is (1 ‫ ם‬iЈ)Ϫt References Akaike, H (1974) A new look at statistical model identification IEEE Trans Aut Control, 19, 716–723 Annuity Guarantee Working Party (AGWP) (1997) Reserving for Annuity Guarantees Published by the Faculty of Actuaries and Institute of Actuaries Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D (1997, November) Thinking coherently RISK, 10, 68–71 Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D (1999) Coherent measures of risk Mathematical Finance, 9(3), 203–228 Bacinello, G., & Ortu, F (1993) Pricing equity-linked life insurance with endogenous minimum guarantees Insurance: Mathematics and Economics, 12, 245–257 Bakshi G., Cao, C., & Chen, Z (1999) Pricing and hedging long-term options Journal of Econometrics, 94, 277–183 Black, F., & Scholes, M., (1973) The pricing of options and corporate liabilities Journal of Political Economy, 81, 637–654 Bollen, N P B (1998) Valuing options in regime switching models Journal of Derivatives, 6, 38–49 Bollerslev, T (1986) Generalized autoregressive conditional heteroskedasticity Journal of Econometrics, 31, 307–327 Boyle, P P (1977) Options: A Monte Carlo approach Journal of Financial Economics, 4(4), 323–338 Boyle, P P., & Boyle, F P (2001) Derivatives: The tools that changed finance United Kingdom: Risk Books Boyle, P P., Broadie, M., & Glasserman, P (1997) Monte Carlo methods for security pricing Journal of Economic Dynamics and Control, 21, 1267–1321 Boyle, P P., Cox, S., Dufresne, D., Gerber, H., Mueller, H., Pedersen, H., Pliska, S., Sherris, M., Shiu, E., Tan, K S (1998) Financial economics Chicago: The Actuarial Foundation Boyle, P P & Emmanuel, D (1980) Discretely adjusted option hedges Journal of Financial Economics, 8, 259–282 275 276 REFERENCES Boyle, P P., & Hardy, M R (1996) Reserving for maturity guarantees (96-18) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Boyle, P P., & Hardy, M R (1998) Reserving for maturity guarantees: Two approaches Insurance: Mathematics and Economics, 21, 113–127 Boyle, P P., & Schwartz, E S (1977) Equilibrium prices of guarantees under equity-linked contracts Journal of Risk and Insurance, 44(4), 639–660 Boyle, P P., Siu, T K., & Yang, H (2002) A two level binomial tree for risk measurement Research Report 325 University of Hong Kong, Dept of Statistics and Actuarial Science Boyle, P P., & Tan, K S (2002) Valuation of ratchet options (02) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Boyle P P., & Tan, K S (2003) Quasi Monte Carlo methods with applications to actuarial science Monograph sponsored by Actuarial Education and Research Fund; 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(1996) Markov chain Monte Carlo in practice London: Chapman and Hall/CRC Hamilton, J D (1989) A new approach to the economic analysis of non-stationary time series Econometrica, 57, 357–384 Hamilton, J D., & Susmel, R (1994) Autoregressive conditional heteroskedasticity and changes in regime Journal of Econometrics, 64, 307–333 Hancock, G H (2001) Policy liabilities using stochastic valuation methods Canadian Institute of Actuaries Segregated Fund Symposium Hancock, G H (2002) Private communication Hardy, M R (1998) Maturity guarantees for segregated fund contracts; hedging and reserving (98-07) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Hardy, M R (1999) Stock return models for segregated fund investment guarantees (99-12) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Hardy, M R (2001) A regime switching model of long term stock returns North American Actuarial Journal, 5(2), 41–53 Hardy, M R (2002) Bayesian risk management for equity-linked insurance Scandinavian Actuarial Journal, 3, 185–211 Hardy, M R., & Hardy, P G (2002) Regime switching lognormal model (rsln.xls) Excel workbook available from www.soa.org Harris, G R (1999) Markov chain Monte Carlo estimation of regime switching vector autoregressions ASTIN Bulletin, 29, 47–80 Huber, P P (1997) A review of Wilkie’s stochastic asset model British Actuarial Journal, 3, 181–210 Hull, J C (1989) Options futures and other derivative securities New Jersey: Prentice Hall Klugman, S A., Panjer, H H., & Willmot, G E (1998) Loss models; From data to decisions New York: Wiley Kolkiewicz, W A., & Tan, K S (1999) Unit linked life insurance contracts with lapse rates depending on economic factors (99-09) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research 278 REFERENCES Lee, H (2002) Pricing equity-indexed annuities embedded with exotic options Contingencies, Jan/Feb 2002, 34–38 Leland, H (1995) Option pricing and replication with transactions costs Journal of Finance, 40, 1283–1301 Lin, X S., & Tan, K S (2002) Valuation of equity-indexed annuities under stochastic interest rates (02) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Manistre, B J., & Hancock, G H (2002) Variance of the CTE estimator Working paper, MMC Enterprise Risk Consulting, Toronto Maturity Guarantees Working Party (MGWP) (1980) Journal of the Institute of Actuaries, 107, 103–209 McCulloch, J H (1996) Financial applications of stable distributions Handbook of Statistics, 14, 393–425 Merton, R C (1973) Theory of rational option pricing Bell Journal of Economics and Management Science, 4, 141–183 Morgan, M S (1990) The history of econometric ideas Cambridge: Cambridge University Press Nolan, J P (1998) Parameterization and modes of stable distributions Statistics and Probability Letters, 38, 187–195 Nolan, J P (2000) Information on stable distributions [On-line] Available: http://www.cas.american.edu/jpnolan/stable.html Nonnemacher, D J F., & Russ, J (1997) Equity linked life insurance in Germany: Quantifying the risk of additional policy reserves Proceedings of the 7th AFIR conference, Cairns, 719–738 Pagan, A R., & Schwert, G W (1990) Alternative models for conditional stock volatility Journal of Econometrics, 45, 267–290 Panjer, H H., & Sharp, K P (1998) Report on Canadian Economic Statistics Canadian Institute of Actuaries Panjer, H H., & Tan, K S (1995) Graduation of Canadian individual insurance mortality experience: 1986–1992 Canadian Institute of Actuaries Panneton, C.-M.(1999) The impact of the distribution of stock market returns on the cost of the segregated fund long term guarantees Segregated Funds Symposium Proceedings Canadian Institute of Actuaries Pelsser, A (2002) Pricing and hedging guaranteed annuity options via static option replication Working Paper, Erasmus University at Rotterdam, Netherlands [On-line] Available: http://www.few.eur.nl/few/people/ pelsser Persson, S.-A., & Aase, K (1994) Valuation of the minimum guaranteed return embedded in life insurance products Journal of Risk and Insurance, 64(4), 599–617 Press, W H., Teukolsky, S A., Vetterling, W T., & Flannery, B P (1992) Numerical recipes in C Cambridge: Cambridge University Press References 279 Roberts, G O (1996) Markov chain concepts related to sampling algorithms In Gilks, W R., Richardson, S., & Spiegelhalter, D J (Eds.) Markov chain Monte carlo in practice (pp 45–57) London: Chapman and Hall/CRC Ross, S M (1996) Simulation San Francisco: Morgan Kaufmann Publishers Schwartz, G (1978) Estimating the dimension of a model Annals of Statistics, 6, 461–464 Segregated Funds Task Force (SFTF) (2002) Report of the Task Force on Segregated Fund Investment Guarantees Canadian Institute of Actuaries [On-line] Available: http://www.actuaries.ca/publications/2002/ 202012e.pdf Streiff, T F., & DiBiase, C A (1999) Equity indexed annuities Dearborn Financial Publishing, USA Tiong, S (2001) Valuing equity-indexed annuities North American Actuarial Journal, 4(4), 149–170 Wang, S X (1995) Insurance pricing and increased limits ratemaking by proportional hazard transforms Insurance: Mathematics and Economics, 17, 43–54 Webber, N., & James, J (2000) Interest rate modelling: Financial engineering London: Wiley Wilkie, A D (1986) A stochastic investment model for actuarial use Transactions of the Faculty of Actuaries, 39, 341–381 Wilkie, A D (1995) More on a stochastic asset model for actuarial use British Actuarial Journal, (V), 777–964 Windcliffe, H., Le Roux, M., Forsythe, P., Vetzal, K (2002) Understanding the behavior and hedging of segregated funds offering the reset feature North American Actuarial Journal, 6(2), 107–124 Windcliffe, H., Forsythe, P., Vetzal, K (2001) Valuation of segregated funds shout options with maturity extensions Insurance: Mathematics and Economics, 29, 1–21 Wirch, J L., & Hardy, M R (1999) A synthesis of risk measures for capital adequacy Insurance: Mathematics and Economics, 25, 337–347 Wright, I D (1997) A stochastic approach to pension scheme funding and asset allocation Ph.D thesis, Heriot-Watt University, Edinburgh, Scotland Yang, S (2001) Reserving, pricing and hedging for guaranteed annuity options Ph.D thesis, Heriot-Watt University, Edinburgh, Scotland Zhang, P G (1998) Exotic options: A guide to second generation options (2nd Ed.) River Edge, NJ: World Scientific Publishing Company Index a(55) mortality table, 225 Acceptance probability (for MCMC), 83, 85, 86 Acceptance-rejection method (for MCMC), 82 Accumulation factors, 68–76, 94, 125 Actuarial approach (to risk management), 3, 12 capital requirements for GMAB with actuarial risk management, 180 emerging cost analysis using actuarial risk management, 178 for guaranteed annuity options, 228 risk measure for GMAB using actuarial risk management, 170 risk measure for VA GMDB using actuarial risk management, 173 Actuarial notation, 273–274 Administration fees See Management expense ratio Akaike information criterion (AIC), 61 American options, Annual ratchet (equity-indexed annuities), 240 Antithetic variates, 204 Arbitrage See No-arbitrage assumption Asian options, 7, 131 Asymptotic MLE results, 50 At-the-money, Autocorrelation, 16, 26, 27 Autoregression, 17 Autoregressive (AR) model, 27, 55, 61, 73, 82 Autoregressive conditionally heteroscedastic (ARCH) models ARCH(1), 28, 29, 31, 47, 56, 61, 82 GARCH(1,1), 29, 31, 47, 57, 61, 76, 82, 97, 126 using ARCH and GARCH models, 29 Averaging, 253 Bayesian parameter estimation, 77–94 See also Markov chain Monte Carlo Bayes’ theorem, 77 Bias of an estimator, 50 Binomial model example for option pricing, 117–124 Black-Scholes hedge, 125 Black-Scholes-Merton assumptions, 124, 133 Black-Scholes-Merton theory, 10, 115–132 Black-Scholes option pricing formula, 10, 124, 145 European put option (BSP), 126 European call option (BSC), 128, 250 European call option with dividends, 130, 245 for the GMAB, 139 for the GMDB, 136 for the GMMB, 134 Bootstrap method for quantiles, 69 Brownian bridge, 39 Burn-in (for MCMC), 80 281 282 Calibration See Left-tail calibration, or parameter estimation Call option, Canadian calibration table, 67 Canadian Institute of Actuaries Task Force on Segregated Funds (SFTF), 17, 65–69, 169 Candidate distribution (for MCMC), 82 Cap rate (equity-indexed annuities), 241, 251 Cash-flow analysis, 193 Cauchy distribution, 38 Certificate of deposit, 237, 243 Coherence criteria for risk measures, 168 Compound annual ratchet (CAR), 240, 247 Conditional tail expectation (CTE), 158, 163–176, 181, 200, 208, 230, 262 Conditionally heteroscedastic models See Autoregressive conditionally heteroscedastic (ARCH) models Confidence interval for simulated quantile risk measure, 160 Consols (U.K government bonds), 224 Control variate, 161, 207–211, 253 Counterparty risk, 11, 236 Cox-Ingersoll-Ross model, 223 Cramer-Rao lower bound for the variance of an estimator, 51 Data mining, 45 Delta method (of maximum likelihood estimation), 51 Deterministic methods deterministic techniques, 2, 3, 15 deterministic valuation, Discrete hedging error See hedging error Dividends, effect on Black-Scholes option price, 129, 135 INDEX Dynamic hedging, 3, 11, 13, 120 capital requirements for GMAB with dynamic hedging, 184 emerging costs analysis with dynamic hedging, 179 for equity-indexed annuities, 260 for guaranteed annuity options, 230 risk measures with dynamic hedging, 170 for separate account guarantees, 133–156 for VA death benefits, 174 Efficient market hypothesis, 17, 45 Emerging cost analysis, 177–194 Empirical model, 36 Equitable Life (U.K.), 13 Equity-indexed annuities (EIA), 1, 6, 10, 130, 237–263 Equity participation, Esscher transforms, 246 European option, European call option (BSC), 128, 250 European call option with dividends, 130, 245 European put option (BSP), 126 for segregated fund guarantees, 134 Exotic options, 130 Expected information, 50 Expected shortfall, 158 Family-of-funds benefit, Floor rate, 240, 250 FTSE All Share index, 225 Fund-by-fund benefit, Generalized-ARCH (GARCH) model See Autoregressive conditionally heteroscedastic (ARCH) models Geometric Brownian motion (GBM), 16, 24, 125 Gibbs sampler, 81 Guaranteed annuity option (GAO), 5, 13, 221–236 283 Index Guaranteed annuity rate (GAR), 222 Guaranteed minimum accumulation benefit (GMAB), 4, 5, 6, 16, 21 Black-Scholes formula, 139, 271 control variate method, 208 dynamic hedging for GMAB, 133 emerging costs for GMAB, 189 with hedging error and transactions costs, 151 modeling the guarantee liability, 102, 104–108, 110 model uncertainty, 220 option price, 271–272 parameter uncertainty, 217 risk measures, 169, 171 sampling error, 197 solvency capital for GMAB example, 180 with voluntary reset, 112, 171 Guaranteed minimum death benefit (GMDB), 4, 5, hedge formula, 136 with hedging error and transactions costs, 151 modeling the guarantee liability, 99, 101–102, 151 parameter uncertainty, 215 quantile risk measure, 163 risk measures, 158, 173 Guaranteed minimum income benefit (GMIB), 5, 6, 221 See also Guaranteed annuity option Guaranteed minimum maturity benefit (GMMB), 4, 5, 6, 9, 16 Black-Scholes formula, 134 CTE risk measure, 167 hedge costs, 136 hedge error, 145 historical evidence, 23 modeling the guarantee liability, 99–102 unhedged liability, 151 Guaranteed minimum surrender benefit (GMSB), Hedging error, 144, 146–149, 152 High water mark (HWM) (equityindexed annuity), 242, 258 Hurdle rate, 190 Importance sampling, 211 Indexation benefit, 237 Information matrix, 50, 54, 56 Insurance risk, Interest rate modeling, 39, 42, 224 Interest rate risk, 223 Interest spread, 243 In-the-money, Invariant (stationary) distribution for Markov regime-switching process, 34, 58 Joint probability density function, 47, 49 KPTP (equity-indexed annuity, point-to-point strike price), 244 Law of one price See No-arbitrage Left-tail calibration, 65–76, 220 Levy process, 37–38 Life annuity, 6, 222 Life-contingent risks, 1, Life-of-contract guarantee, 248 Likelihood-based model selection, 60 Likelihood function, 47–49, 78, 83 Likelihood ratio test, 60 Lognormal model, 16, 24, 53, 61, 66, 70 Log-return random variable, 27, 67 Lookback option, 131, 259 Low discrepancy sequences, 212 Management expense ratio (MER), 5, 99, 134 Margin offset, 99, 100, 133, 143, 158 Markov chain Monte Carlo parameter estimation (MCMC), 77–94 burn-in, 80 candidate distribution, 82 Gibbs sampler, 80 284 Markov chain Monte Carlo parameter estimation (continued) Metropolis-Hastings Algorithm (MHA), 80–85 parameter uncertainty, 213 for the RSLN model, 85–89 Maturity Guarantees Working Party (MGWP) U.K., 12, 17, 39 Maximum likelihood estimation (MLE), 47–63, 65, 66, 72, 73, 78 AR(1) model, 55 ARCH and GARCH models, 56 asymptotic minimum variance, 50 asymptotic normal distribution, 51 asymptotic unbiasedness, 50 conditions for asymptotic properties, 49, 52 delta method, 51 lognormal model, 53 RSLN model, 57 Metropolis-Hastings Algorithm (MHA), 80–85 Minimum variance estimator, 50 Model selection, 60 Model uncertainty and model error, 150, 195, 219–220 Moment matching for parameter estimation, 63 variance reduction technique, 203 Monte Carlo method for option pricing, 131, 253, 258 Mortality and survival probabilities, 265 Mortality risk, 135 Move-based strategy for rebalancing hedge, 144 Multivariate models Wilkie, 39– 45 vector autoregression, 45 Mutual fund, Net present value of future loss (NPVFL), 190 Net present value of liability (NPV), 107, 108, 113 INDEX No-arbitrage, 8, 9, 116 Nondiversifiable risk, Nonoverlapping data, 68 Nonstationary models, 52 October 1987 stock market crash, 16, 26 Office of the Superintendent of Financial Institutions (OSFI) in Canada, 15, 16, 169 Options, 7–11 American, 7, 10 Asian 7, 10 Black-Scholes-Merton pricing theory, 115–129 in equity-linked insurance, in-the-money, 8, 120 out-of-the-money, Parameter estimation, 47–63, 77–94 Parameter uncertainty, 77, 195, 213–219 Participation rate, 6, 239, 246, 250, 251 Path-dependent benefit, 16 Periodicity of random number generators, 97 Physical measure See P-measure P-measure, 11, 115, 120, 147, 159, 223 Point-to-point indexation (PTP), 239 Policyholder behavior, 96, 113 Posterior distribution, 78, 80, 86, 88, 90 p-quantile, 66 Predictive distribution, 79, 90, 94, 214 Premium principles, 158 Pricing and capital requirements, 14 Pricing using B-S-M valuation, 142 Prior distribution, 78, 81 Profit testing See Emerging cost analysis Put-call parity, 9, 10, 128 Put option, Index Q-measure, 11, 115, 119, 125–126, 147, 150, 159, 223 Quantile, 66–76 Quantile matching, 66 Quantile risk measure, 158, 159–163, 167–173, 198 Random number generators, 97, 104 Random walk (stock price process), 17 Random-walk Metropolis algorithm, 85 Ratcheted premium, 241 Real-world measure See P-measure Rebalancing the replicating (hedge) portfolio, 115 Regime-switching lognormal (RSLN) model, 30–36, 47, 57, 77 comparison with other models, 61 hedging and the RSLN model, 152 invariant (stationary) distribution for regime process, 34, 58 left-tail calibration, 74 Markov chain Monte Carlo parameter estimation, 85–89 maximum likelihood estimation, 57 parameters for examples, 104 probability function for RSLN model, 34 simulation, 98 sojourn distribution probability function, 33, 74 stress testing for parameter uncertainty, 218 transition matrix, 32 Regime-switching autoregressive (RSAR) model, 59, 225 Reinsurance, 11 Replicating portfolio, 10, 11, 115, 116 Reset option for segregated fund policies, 112–114, 171 Risk management actuarial approach, 3, 12, 13, 158 ad hoc approach, 13 dynamic hedging, 3, 11, 158 reinsurance, 11 285 Risk measures, 12, 157–176 Risk-neutral measure (Q-measure), 11, 22, 115, 119, 125, 150, 159 Sample paths (for MCMC), 84, 89, 91 Sampling error, 195, 196–201 Sampling variability, 75, 76, 160 S&P 500 total return index, 18–25 AR(1) model, 55 ARCH and GARCH models, 56 likelihood-based model selection, 61 lognormal model, 53 maximum likelihood parameter estimation, 53–64 MCMC parameter estimation, 86–90 RSLN model, 35, 57 S&P/TSX-Composite index, 18 See also TSE 300 index Schwartz-Bayes criterion (SBC), 60 Segregated fund contracts, 1, 2, 5, 9, 11, 21, 65, 67, 133 See also GMAB, GMDB, and GMMB Self-financing hedge, 123, 150 Separate account insurance, 2, 65, 133 Simple annual ratchet (SAR), 257 Sojourn time (R), 32 Solvency capital, 158 Stable model, 37, 61 Standard error CTE estimate, 165, 183 expected value, 197 quantile estimate, 160 Static hedge, 123 Static replication for guaranteed annuity options, 235 Stationary distributions, 49 Stochastic simulation for left-tail calibration, 75 Stochastic simulation of liabilities, 16, 108 actuarial approach, 95–114 cash-flow analysis, 110, 154 CTE risk measure, 165 distribution function, 108–109 286 Stochastic simulation of liabilities (continued) density function, 108–110 quantile risk measure, 159 stock return process, 97 Stochastic volatility models, 38 Stock price index, Stress testing (for parameter uncertainty), 217 Strike price, 7, 121 Systematic risk, Systemic risk, Tail risk, Tail-VaR, 158 Task Force on Segregated Funds See Canadian Institute of Actuaries Task Force on Segregated Funds (SFTF) Term structure of interest rates, 223 Time-based strategy for rebalancing hedge, 144 Tracking error See hedging error Transactions costs, 149 Transactions costs and hedging error reserve (t V T &H ), 180 Transition matrix (for RSLN model), 32 Trinomial lattice approximation, 256 TSE 300 total return index, 18–25, 72 AR(1) model, 55 ARCH and GARCH models, 56 calibration table, 67 empirical evidence for quantiles, 68 likelihood-based model selection, 61 lognormal model, 53 INDEX maximum likelihood parameter estimation, 53–64, 218 MCMC parameter estimation, 86–90 RSLN model, 35, 57 U.K FTSE All-Share total return index, 59 Unbiased estimator, 50 Unhedged liability, 133, 143 Unit-linked insurance, 1, 6, 133, 221 Universal life, Value-at-risk (VaR), 12, 158 Variable annuity (VA), 1, 2, 6, 10, 133, 138, 143, 158 Variable-annuity death benefits, 173– 176, 215 See also Guaranteed minimum death benefit (GMDB) Variable-annuity guaranteed living benefits (VAGLB), See also Guaranteed minimum maturity benefit (GMMB) Variance reduction, 131, 201–213 Vector autoregressive model, 45 Volatility, 18, 22, 28, 30, 38 general stochastic volatility models, 38 market (implied) volatility, 22 stochastic volatility, 26, 28, 30, 38, 150 volatility bunching, 26, 27, 37 White noise process, 27 Wilkie model, 17, 39– 45 Withdrawals, 96, 100 [...]... chapters, different risk management strategies require different levels of capital (for the same level of risk) , and therefore the implied price for the guarantee would vary The approach of this book is that all of these issues are really facets of the same issue The first requirement for pricing or for determination of capital requirements is a credible estimate of the distribution of the liabilities,... comprehensive and self-contained introduction to modeling and risk management for equity-linked life insurance A feature of the book is the combination of econometric analysis of investment models with their application in pricing and risk management The focus is on the stochastic modeling of embedded guarantees that depend on equity performance In the major part of the book the contracts that are used to... progressing through modeling, and finally moving on to risk management In more detail, the structure of the book is as follows The first chapter introduces the contracts and some of the basic ideas from financial economics that will be utilized in later chapters The next four chapters cover some of the econometrics of modeling equity processes In Chapter 2, we introduce a number of families of models that... The management of the segregated funds is often independent of the insurer 6 INVESTMENT GUARANTEES A policyholder may withdraw some or all of his or her segregated fund account at any time, though there may be a penalty on early withdrawals The insurer usually offers a range of funds, including fixed interest, balanced (a mixture of fixed interest and equity), broad-based equity, and perhaps a higher -risk. .. Put Options Although the risks associated with equity-linked insurance are new to insurers, at least, relative to life-contingent risks, they are very familiar to practitioners and academics in the field of derivative securities The payoffs under equity-linked insurance contracts can be expressed in terms of options There are many books on the theory of option pricing and risk management In this book... hedge the option payoff A feature of the replicating portfolio is that it changes over time, so the theory also requires the balance of stocks and bonds to be rearranged at frequent intervals over the term of the contract The stock price, St , is the random variable in the payoff equations for the options (we assume that the risk- free rate of interest is fixed) The Provision for Equity-Linked Liabilities... Examples 133 134 142 143 151 CHAPTER 9 Risk Measures Introduction The Quantile Risk Measure The Conditional Tail Expectation Risk Measure Quantile and CTE Measures Compared Risk Measures for GMAB Liability Risk Measures for VA Death Benefits CHAPTER 10 Emerging Cost Analysis Decisions Capital Requirements: Actuarial Risk Management Capital Requirements: Dynamic-Hedging Risk Management Emerging Costs with Solvency... that are useful in the risk management of equity-linked insurance into a single volume, and to focus specifically on the parts of the theory that are most relevant This also enables us to develop the theory into practical methods for insurance companies, and to illustrate these with specific reference to equity linked contracts There are two common approaches to risk management of equity-linked insurance,... value-at -risk (VaR) concept of finance, though generally applied over longer time periods by the insurance companies than by the banks The underlying principle of this method of calculating the capital requirements is that the capital is assumed to be invested in risk- free bonds The use of the quantile of the distribution as a risk measure is not actually fundamental to this approach, and other risk measures... and high severity in that, if the guarantee does bite, the potential liability is very large The assessment and management of financial risk is a very different proposition to the management of insurance risk The management of insurance risk relies heavily on diversification With many thousands of policies in force on lives that are largely independent, it is clear from the central limit theorem that there