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Modeling and Risk Management for Equity-Linked Life Insurance... Modeling and Risk Management for Equity-Linked Life Insurance... Hardy, Mary, 1958-Investment guarantees : modeling and

Guarantees

MARY HARDY

John Wiley & Sons, Inc.

Modeling and Risk Management for Equity-Linked Life Insurance

Guarantees

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Guarantees

MARY HARDY

John Wiley & Sons, Inc.

Modeling and Risk Management for Equity-Linked Life Insurance

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Copyright 2003 by Mary Hardy All rights reserved.

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Hardy, Mary,

1958-Investment 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)

1 Insurance, Life-mathematical models 2 Risk management–Mathematical models.

1 title II Series.

HG8781.H313 2003

368.32’0068’1–dc21 2002034200 Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

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Acknowledgments

T

Bayesian Risk Management for Equity-Linked Insurance

his work has been supported by the National Science and EngineeringResearch Council of Canada, and by the Actuarial Education andResearch Fund I would also like to thank the members of the Department

of Statistics at the London School of Economics and Political Science fortheir hospitality while the book was being completed, especially AnthonyAtkinson, Angelos Dassios, Martin Knott, and Ragnar Norberg

I would like to thank Taylor and Francis, publishers of the ScandinavianActuarial Journal, for permission to reproduce material from

in Chapter 5

I learned a great deal from my fellow members of the magnificentCanadian Institute of Actuaries Task Force on Segregated Funds In partic-ular, I would like to thank Geoffrey Hancock, who has provided invaluableadvice and assistance during the preparation of this book Also, thanks toMartin Le Roux, David Gilliland, and the two Chairs, Simon Curtis andMurray 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 dents over the years, many of whom have contributed directly or indirectly

stu-to this book In particular, thanks stu-to Andrew Cairns, Julia Wirch, DavidWilkie, 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 aspecial privilege to work with Ken Seng Tan at the University of Waterlooand 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 wasgood 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 hisencyclopedic knowledge of finance

M H

Deterministic or Stochastic? 15Economical Theory or Statistical Method? 17

Autoregressive Models 27

Regime-Switching Lognormal Model (RSLN) 30

The Stable Distribution Family 37General Stochastic Volatility Models 38

Vector Autoregression 45

Properties of Maximum Likelihood Estimators 49Some Limitations of Maximum Likelihood Estimation 52

Analytic Calibration of Other Models 72Calibration by Simulation 75

Markov Chain Monte Carlo—An Introduction 79The Metropolis-Hastings Algorithm (MHA) 81MCMC for the RSLN Model 85Simulating the Predictive Distribution 90

Stochastic Simulation of Liability Cash Flows 108

The Guarantee Liability as a Derivative Security 116

CONTENTS

The Quantile Risk Measure 159The Conditional Tail Expectation Risk Measure 163Quantile and CTE Measures Compared 167Risk Measures for GMAB Liability 169Risk Measures for VA Death Benefits 173

Capital Requirements: Actuarial Risk Management 180Capital Requirements: Dynamic-Hedging Risk Management 184Emerging Costs with Solvency Capital 188Example: Emerging Costs for 20-Year GMAB 189

Sources of Uncertainty 195Random Sampling Error 196

Parameter Uncertainty 213

CONTENTS

Introduction

This book is designed for all practitioners working in equity-linkedinsurance, whether in product design, marketing, pricing and valuation,

or risk management It is written with actuaries in mind, but it should also

be interesting to other investment professionals The material in this bookforms the basis of a one-semester graduate course for students of actuarialscience, insurance, and finance The aim is to provide a comprehensiveand self-contained introduction to modeling and risk management forequity-linked life insurance A feature of the book is the combination ofeconometric analysis of investment models with their application in pricingand risk management

The focus is on the stochastic modeling of embedded guarantees thatdepend on equity performance In the major part of the book the contractsthat are used to illustrate the methods are single premium, separate accountproducts This class includes variable annuities in the United States, seg-regated fund contracts in Canada, and unit-linked contracts in the UnitedKingdom The investment guarantees associated with this type of productare usually payable contingent on the policyholder’s death, and in somecases also apply to survival benefits For these contracts, the insurer’s lia-bility at the expiry of the contract is the excess, if any, of the guaranteedminimum payout and the amount of the policyholder’s separate account.Generally, the probability of the guarantee actually resulting in a benefit issmall In the language of finance, we say that the guarantees are usually deepout-of-the-money In the past this has led to a certain complacency, but it

is now recognized that the risk management of these contracts represents

a major challenge to insurers, particularly where the investment guaranteeapplies to maturity benefits, and where separate account products haveproved popular with policyholders

This book took shape as a result of my membership in the CanadianInstitute of Actuaries Task Force on Segregated Fund contracts Afterthat Task Force completed its report, there was a clear demand for someeducational material to help actuaries understand the methods that wererecommended in the report, and that were subsequently mandated by theregulators Also, many actuaries and regulators in the United States took agreat interest in the report, and the demand for relevant educational materialbegan to come also from across the United States Meanwhile, in the United

There are two common approaches to risk management of equity-linkedinsurance, particularly separate account products such as variable annuities

or segregated funds The “actuarial” approach uses the distribution ofthe guarantee liabilities discounted at the risk-free rate of interest Thedynamic-hedging approach uses financial engineering, and assumes that aportfolio of bonds and stocks is used to replicate the guarantee payoff.The replicating portfolio must be rebalanced at frequent intervals, as theunderlying stock price changes The actuarial approach is commonly usedfor risk management of investment guarantees by insurance companies inNorth America and in the United Kingdom The dynamic-hedging approach

is used by financial engineers in banks and hedge funds, and occasionally

in insurance companies It has been the case since the earliest equity-linkedcontracts were issued that many practitioners who use one of these methodsharbor a deep distrust of the other method, often based on a lack ofunderstanding of the other side’s methodology

In this book both approaches are presented, discussed, and extensivelyillustrated with examples This should help practitioners on either side ofthe fence talk to each other, at the very least My own view is that bothmethods have their merits, and that the best approach is to use both, inappropriate combination

I have included in Chapter 7 an introduction to the concepts of arbitrage pricing, replication, and the risk-neutral measure I am aware thatmany people who read this book will be very familiar with this material,but I am also aware of a great deal of misunderstanding surrounding thesevery fundamental issues For example, there are many actuaries workingwith investment guarantees who do not fully comprehend the role of the -measure By focusing solely on the important concepts, I hope to facilitate

no-a better understno-anding of the finno-ancino-al economics no-approno-ach In order tokeep the book to a manageable project, I have not generally included thecomplication of stochastic interest rates, except in Chapter 12, where it isnecessary to explain the annuitization liability under the guaranteed annuity

INTRODUCTION

Introduction

option (GAO) contract This is often dealt with in the more technicalliterature on equity-linked insurance, such as Persson and Aase (1994) andLin and Tan (2001)

The book is presented in a progressive, linear structure, starting withmodels, progressing through modeling, and finally moving on to risk man-agement In more detail, the structure of the book is as follows

The first chapter introduces the contracts and some of the basic ideasfrom financial economics that will be utilized in later chapters The nextfour chapters cover some of the econometrics of modeling equity processes

In Chapter 2, we introduce a number of families of models that havebeen proposed for equity returns

In Chapter 3, we discuss parameter estimation for some of the models,using maximum likelihood estimation (MLE) We also discuss ways of usingthe likelihood to rank the appropriateness of the models for the data.Because MLE tends to fit the center of the distribution, and may not fitthe tails particularly well for some processes, in Chapter 4 we discuss how

to adjust the maximum likelihood parameters to improve the fit in otherparts of the distribution This may be important where the far tail of theequity return distribution is critical in the distribution of the investmentguarantee payout This chapter, incidentally, explains how to satisfy thecalibration requirements of the Canadian Institute of Actuaries task forcereport on segregated funds (SFTF 2000)

Chapter 5 describes how to use the Markov chain Monte Carlo(MCMC) method for parameter estimation This is a Bayesian methodfor parameter estimation that provides a powerful method for assessingparameter uncertainty

Having decided on a model for equity returns, and estimated appropriateparameters, we can start to model the investment guarantees In Chapter 6,

we explain how to use stochastic simulation to model the distribution of theliability outgo for an equity-linked contract This is the basis of the actuarialapproach to risk management

We then move on to the dynamic-hedging approach This needssome elementary results from financial economics, which are presented inChapter 7

Then, in Chapter 8, we apply the methods to investment guarantees.This chapter goes beyond the pure pricing information provided by theBlack-Scholes-Merton framework We also assess the liability that is notcovered by the Black-Scholes hedge The three sources of this unhedgedliability are

Transactions costs from rebalancing the hedge

Hedging errors arising from discrete hedging intervals

Additional hedging costs arising from the use of realistic equity models,under which the Black-Scholes hedge is no longer self-financing

1.

2.

3.

In Chapter 9, we discuss how to use risk measures to quantify the tailrisk from a distribution; risk measures can also be used for pricing The mostcommon risk measure in finance is value at risk (VaR) This is a quantilerisk measure More recent theory favors the conditional tail expectation riskmeasure, also known as Tail-VaR Both are described in Chapter 9, withexamples of application to benefits such as variable annuities and segregatedfunds

Chapter 10 describes stochastic emerging cost modeling This allows

us to bring together the actuarial and dynamic-hedging approaches andcompare them in a systematic way Emerging cost modeling is a powerfultool for making decisions about policy design, pricing, and risk management.Because stochastic simulation is the fundamental tool for analyzing theliabilities for equity-linked insurance, it is useful to discuss the error anduncertainty associated with the method and to consider ways to reducethe variability of results In Chapter 11, we examine three sources offorecast uncertainty The first is random sampling variation It is possible

to reduce the effect of this using variance reduction techniques, and theseare described with examples where they are useful in modeling embeddedinvestment guarantees The second is uncertainty in parameter estimation;this is where the Bayesian approach of Chapter 5 is particularly useful Wediscuss how to apply Bayesian methods to quantify the effect of parameteruncertainty Finally, we discuss model uncertainty—that is, how to assessthe risk from the possibility that stock returns in the future follow a differentmodel than that used in forecasts

The final two chapters expand the application of the methods to twodifferent types of equity-linked contracts The first is the U.K unit-linkedcontract with guaranteed annuity option (GAO) This has similarities withthe guaranteed minimum income benefit associated with some variableannuity contracts Issued in the early 1980s, at a time of very high long-term interest rates, the problems of stochastic interest rates and lack ofdiversification of risk associated with investment guarantees are, unfortu-nately, exemplified in the serious problems experienced by a number ofU.K insurers arising from maturing GAO contracts Chapter 12 discussesthe actuarial and the dynamic-hedging approaches to risk management ofGAOs In Chapter 13, we discuss equity-indexed annuities (EIA) Theseoffer a combination of minimum return guarantee plus participation instock appreciation for some equity index The benefits appear quite sim-ilar to the variable annuity with maturity guarantee However, as weshall demonstrate, the structure of the product is quite different Theactuarial approach is not appropriate for EIA contracts, and a com-mon approach to risk management is a static strategy, effectively usingoptions purchased from a third party to reinsure the investment guaranteeliability

INTRODUCTION

in Chapter 3, where this model is shown to provide a superior fit tomonthly stock return data Also, the model is easy to understand and ismathematically tractable However, although I am partial to the RSLNmodel myself, nothing in the later chapters depends on it, so feel free to useyour own favorite model, subject to some quantitative assessment (along thelines of Chapters 3 through 5) of how well it models the stock return process.For those interested in exploring the RSLN model further, the Society ofActuaries intends to make available a Microsoft Excel workbook for fittingthe two-regime model to stock return data The workbook calculates thelikelihood for given parameters and data; calculates the maximum likelihoodfor given data; calculates the distribution function; tests the left tail against

a left-tail calibration table (see Chapter 4); and generates random paths forthe stock index for a given set of parameters (see Hardy and Hardy 2002).After I had written the major part of the book, one of the extensivelyused stock return indices changed its name and composition The TSE 300index has been repackaged as the S&P/TSX Composite index It is still thebroad-based Canadian total return index, but is no longer restricted to 300companies

Although many people have helped with this work at various stages, allremaining errors are my responsibility I am receptive to hearing of any; feelfree to e-mail me at mrhardy uwaterloo.ca