Market-Valuation Methods in Life and Pension Insurance

0521868777......Page 1
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.2 The life insurance market......Page 17
1.3 The policy holder's account......Page 20
1.4 Dividends and bonus......Page 22
1.5 Unit-linked insurance and beyond......Page 25
2.1 Introduction......Page 27
2.2.1 The technical reserve and the second order basis......Page 28
2.2.2 The technical reserve and the first order basis......Page 29
2.2.3 The undistributed reserve and the real basis......Page 31
2.3.2 Bonus payments and the bonus potential......Page 34
2.4 Liabilities and principles for valuation......Page 37
2.4.1 Absence of arbitrage......Page 39
2.4.2 Diversification......Page 40
2.4.3 The market-based liability revisited......Page 42
2.5 The liability and the payments......Page 43
2.5.1 The market-based liability revisited......Page 44
2.6.1 Intensity-based surrender option valuation......Page 46
2.6.2 Intervention-based surrender option valuation......Page 48
2.6.3 The market-based liability revisited......Page 52
2.7.1 A simple free policy option value......Page 53
2.7.2 Intervention-based free policy option valuation......Page 55
2.7.3 The market-based liability revisited......Page 59
3.1 Introduction......Page 61
3.2.2 A portfolio of insured......Page 62
3.2.3 Interest, accumulation and discount factors......Page 64
3.2.4 The insurer’s loss and the principle of equivalence......Page 65
3.2.5 Deterministic bond prices......Page 66
3.2.6 Hedging with zero coupon bonds......Page 68
Hedging a portfolio of pure endowments......Page 69
Hedging a portfolio of term insurances......Page 70
3.2.7 Market values and zero coupon bonds......Page 71
The first order basis......Page 72
Market value of guaranteed payments......Page 73
3.3 Zero coupon bonds and interest rate theory......Page 74
3.3.2 Forward rates......Page 75
3.3.4 Simple rates......Page 78
3.3.5 Market values and forward rates......Page 79
The individual bonus potential......Page 80
3.4 A numerical example......Page 81
3.5.1 A general bond......Page 87
3.5.2 Yield and duration......Page 89
3.6.1 Observing prices of zero coupon bonds......Page 90
3.6.2 The Nelson–Siegel parameterization......Page 91
3.7 Arbitrage-free pricing in discrete time......Page 94
3.7.1 Traded assets and information......Page 95
3.7.2 A two-period example......Page 96
3.7.3 Investment strategies and the value process......Page 98
3.7.4 The cost process......Page 99
3.7.5 Self-financing strategies and arbitrage......Page 100
Equivalent martingale measures......Page 101
What is a martingale?......Page 102
Martingale measures ensure absence of arbitrage......Page 103
Pricing with an equivalent martingale measure......Page 104
3.7.8 The two-period example revisited......Page 105
A very simple interest rate guarantee......Page 106
3.8 Models for spot rate in continuous time......Page 109
Estimation of parameters......Page 110
3.8.1 Affine models......Page 111
The Vasicek model......Page 112
3.9 Market values in insurance revisited......Page 114
4.1 Introduction......Page 117
4.2 Discrete-time insurance model......Page 119
4.2.1 Two-period model......Page 126
4.3.1 The one-period model......Page 131
4.3.2 The multi-period model......Page 137
4.4 The Black–Scholes model......Page 140
4.5 Continuous-time insurance model......Page 147
4.5.1 Terminal bonus......Page 151
Prudent investment......Page 153
4.5.2 Cash bonus......Page 154
Prudent investment......Page 155
4.5.3 Additional benefits......Page 156
Prudent investment......Page 158
Stochastic interest in the binomial model......Page 159
Stochastic interest in the Black–Scholes model......Page 160
Factor models......Page 161
5.1 Introduction......Page 162
5.2 Unit-linked insurance......Page 164
5.2.1 Unit-linked life insurance as a contingent claim......Page 165
5.3 The policy holder's account......Page 168
5.3.1 The financial market......Page 171
Explicit expression for the price of the guarantee......Page 172
Value of account payable upon death......Page 173
Account payable upon death under a pure unit-linked contract......Page 174
5.3.3 Terminal guarantees......Page 175
5.3.5 On the choice of model......Page 176
5.4 Hedging integrated risks under diversification......Page 177
5.5 Hedging integrated risk in a one-period model......Page 179
Variance minimization......Page 181
Risk minimization for a unit-linked contract......Page 182
Non-diversifiable mortality risk......Page 184
Modeling systematic mortality risk......Page 186
Trading with mortality risk......Page 187
5.6.1 Important concepts......Page 188
5.6.2 Unattainable claims......Page 190
5.6.3 The binomial model......Page 191
5.7 Hedging integrated risks......Page 193
Information......Page 194
Martingale measures......Page 195
5.7.2 Super-hedging......Page 197
5.7.3 Super-hedging for unit-linked contracts......Page 198
5.7.4 Risk minimization......Page 200
Related criteria......Page 202
5.7.5 Risk minimization for unit-linked contracts......Page 203
5.7.6 Mean-variance hedging......Page 206
5.7.7 Mean-variance hedging for unit-linked contracts......Page 209
5.7.8 Indifference pricing and hedging......Page 210
5.7.9 Indifference pricing for unit-linked contracts......Page 213
5.7.10 Quantile hedging......Page 214
5.8 Traditional life insurance......Page 215
6.1 Introduction......Page 216
6.2 The insurance contract......Page 219
6.3 Surplus......Page 226
6.4 Surplus-linked dividends......Page 231
6.5 Dividends linear in surplus......Page 233
6.6 Bonus......Page 238
6.7 Surplus- and bonus-linked dividends......Page 242
6.8 Dividends linear in surplus and bonus......Page 245
7.1 Introduction......Page 251
Bonds with continuously paid coupons......Page 252
Swaps and swap rates......Page 253
Relationships between swap rates and forward rates......Page 254
Swaptions......Page 255
7.3.1 Change of numeraire and forward measures......Page 256
The Vasicek model......Page 258
Bond options in the Vasicek model......Page 261
7.4 Swaps and beyond in discrete time......Page 262
Swaps and swap rates......Page 263
Relations between swap rates and LIBOR forward rates......Page 264
Caps and floors......Page 265
Black’s formula for caps and floors......Page 267
Black’s formula for swaptions......Page 268
Constant maturity swap (CMS) and CMS options......Page 269
7.5.1 A market model for pricing a floorlet......Page 270
7.7 A portfolio of contracts......Page 273
The swap account......Page 274
The total account......Page 275
Numerical example......Page 277
Random phenomena over time......Page 279
Change of measure for remaining lifetimes......Page 280
A.2 The risk-minimizing strategy......Page 282
A.3 Risk minimization for unit-linked contracts......Page 283
A.4 Mean-variance hedging for unit-linked contracts......Page 285
References......Page 288
Index......Page 294