Introduction xi
CHAPTER 1
Investment Guarantees 1
CHAPTER 2
Modeling Long-Term Stock Returns 15
CHAPTER 3
Maximum Likelihood Estimation for Stock Return Models 47
vii
Contents
Introduction 1
Major Benefit Types 4
Contract Types 5
Equity-Linked Insurance and Options 7
Provision for Equity-Linked Liabilities 11
Pricing and Capital Requirements 14
Introduction 15
Deterministic or Stochastic? 15
Economical Theory or Statistical Method? 17
The Data 18
The Lognormal Model 24
Autoregressive Models 27
ARCH(1) 28
Regime-Switching Lognormal Model (RSLN) 30
The Empirical Model 36
The Stable Distribution Family 37
General Stochastic Volatility Models 38
The Wilkie Model 39
Vector Autoregression 45
Introduction 47
Properties of Maximum Likelihood Estimators 49
Some Limitations of Maximum Likelihood Estimation 52

Using MLE for TSE and S&P Data 53
Likelihood-Based Model Selection 60
Moment Matching 63
Introduction 65
Quantile Matching 66
The Canadian Calibration Table 67
Quantiles for Accumulation Factors: The Empirical Evidence 68
The Lognormal Model 70
Analytic Calibration of Other Models 72
Calibration by Simulation 75
Bayesian Statistics 77
Markov Chain Monte Carlo—An Introduction 79
The Metropolis-Hastings Algorithm (MHA) 81
MCMC for the RSLN Model 85
Simulating the Predictive Distribution 90
Introduction 95
The Stochastic Processes 96
Simulating the Stock Return Process 97
Notation 98
Guaranteed Minimum Maturity Benefit 100
Guaranteed Minimum Death Benefit 101
Example 101
Guaranteed Minimum Accumulation Benefit 102
GMAB Example 104
Stochastic Simulation of Liability Cash Flows 108
The Voluntary Reset 112
Introduction 115
The Guarantee Liability as a Derivative Security 116

Replication and No-Arbitrage Pricing 116
The Black-Scholes-Merton Assumptions 123
The Black-Scholes-Merton Results 124
The European Put Option 126
The European Call Option 128
Put-Call Parity 128
Dividends 129
Exotic Options 130
Introduction 133
Black-Scholes Formulae for Segregated Fund Guarantees 134
Pricing by Deduction from the Separate Account 142
The Unhedged Liability 143
Examples 151
Introduction 157
The Quantile Risk Measure 159
The Conditional Tail Expectation Risk Measure 163
Quantile and CTE Measures Compared 167
Risk Measures for GMAB Liability 169
Risk Measures for VA Death Benefits 173
Decisions 177
Capital Requirements: Actuarial Risk Management 180
Capital Requirements: Dynamic-Hedging Risk Management 184
Emerging Costs with Solvency Capital 188
Example: Emerging Costs for 20-Year GMAB 189
Sources of Uncertainty 195
Random Sampling Error 196
Variance Reduction 201
Parameter Uncertainty 213
Model Uncertainty 219
CHAPTER 12
Guaranteed Annuity Options 221
CHAPTER 13
Equity-Indexed Annuities 237
APPENDIX A
Mortality and Survival Probabilities 265
APPENDIX B
The GMAB Option Price 271
APPENDIX C
Actuarial Notation 273
REFERENCES 275
INDEX 281