Modelling Longevity Dynamics for Pensions and Annuity Business

Preface......Page 6
Contents......Page 14
1.1 Introduction......Page 22
1.2.1 Withdrawing from a fund......Page 23
1.2.2 Avoiding early fund exhaustion......Page 26
1.2.3 Risks in annuities-certain and in life annuities......Page 27
1.3.1 The life annuity as a financial transaction......Page 29
1.3.2 Actuarial values......Page 30
1.3.3 Technical bases......Page 33
1.4.1 Mutuality......Page 35
1.4.2 Solidarity......Page 37
1.4.3 ‘Tontine’ annuities......Page 39
1.5.1 The random present value of a life annuity......Page 41
1.5.2 Focussing on portfolio results......Page 42
1.5.3 A first insight into risk and solvency......Page 45
1.5.4 Allowing for uncertainty in mortality assumptions......Page 48
1.6.1 Immediate annuities versus deferred annuities......Page 52
1.6.2 The accumulation period......Page 54
1.6.3 The decumulation period......Page 57
1.6.4 The payment profile......Page 59
1.6.5 About annuity rates......Page 61
1.6.6 Variable annuities and GMxB features......Page 62
1.7 References and suggestions for further reading......Page 64
2.1 Introduction......Page 66
2.2.1 Cohort tables and period tables......Page 67
2.2.2 ‘Population’ tables versus ‘market’ tables......Page 68
2.2.3 The life table as a probabilistic model......Page 69
2.2.4 Select mortality......Page 70
2.3.1 The survival function......Page 72
2.3.2 Other related functions......Page 74
2.3.3 The force of mortality......Page 76
2.3.5 Assumptions for non-integer ages......Page 78
2.4 Summarizing the lifetime probability distribution......Page 79
2.4.1 The life expectancy......Page 80
2.4.2 Other markers......Page 81
2.4.3 Markers under a dynamic perspective......Page 83
2.5 Mortality laws......Page 84
2.5.1 Laws for the force of mortality......Page 85
2.5.2 Laws for the annual probability of death......Page 87
2.6.1 Some preliminary ideas......Page 88
2.6.2 The Whittaker–Henderson model......Page 89
2.6.3 Splines......Page 90
2.7 Some transforms of the survival function......Page 94
2.8.1 Some preliminary ideas......Page 95
2.8.2 Models for mortality at highest ages......Page 96
2.9.1 Observable heterogeneity factors......Page 98
2.9.2 Models for differential mortality......Page 99
2.9.3 Unobservable heterogeneity factors. The frailty......Page 101
2.9.4 Frailty models......Page 104
2.9.5 Combining mortality laws with frailty models......Page 106
2.10 References and suggestions for further reading......Page 108
3.1 Introduction......Page 110
3.2 Data sources......Page 111
3.2.2 Federal Planning Bureau......Page 112
3.2.4 Banking, Finance, and Insurance Commission......Page 113
3.3.1 Age-period life tables......Page 114
3.3.2 Exposure-to-risk......Page 116
3.3.3 Death rates......Page 117
3.3.5 Closure of life tables......Page 122
3.3.6 Rectangularization and expansion......Page 126
3.3.7 Life expectancies......Page 132
3.3.8 Variability......Page 134
3.3.9 Heterogeneity......Page 136
3.4.1 Observed death rates......Page 137
3.4.2 Smoothed death rates......Page 139
3.4.3 Life expectancies......Page 143
3.4.4 Relational models......Page 144
3.4.5 Age shifts......Page 148
3.5 Mortality trends throughout EU......Page 150
3.6 Conclusions......Page 156
4.1 Introduction......Page 158
4.2.1 Representing mortality dynamics: single-figures versus age-specific functions......Page 160
4.2.2 A discrete, age-specific setting......Page 161
4.3.1 Some preliminary ideas......Page 162
4.3.2 Reduction factors......Page 165
4.3.3 The exponential formula......Page 166
4.3.4 An alternative approach to the exponential extrapolation......Page 167
4.3.5 Generalizing the exponential formula......Page 168
4.3.6 Implementing the exponential formula......Page 169
4.3.8 Some exponential formulae used in actuarial practice......Page 170
4.3.9 Other projection formulae......Page 172
4.4.1 The cohort tables in a projected table......Page 173
4.4.2 From a double-entry to a single-entry projected table......Page 174
4.4.3 Age shifting......Page 176
4.5.1 Mortality laws and projections......Page 177
4.5.2 Expressing mortality trends via Weibull’s parameters......Page 181
4.5.3 Some remarks......Page 183
4.5.4 Mortality graduation over age and time......Page 184
4.6.1 Interpolation versus extrapolation: the limit table......Page 186
4.6.2 Model tables......Page 187
4.6.3 Projecting transforms of life table functions......Page 188
4.7.1 Some preliminary ideas......Page 190
4.7.2 The LC model......Page 192
4.7.3 From LC to the Poisson log-bilinear model......Page 193
4.8.1 Cohort approach versus period approach. APC models......Page 194
4.9.1 Landmarks in mortality projections......Page 196
4.9.2 Further references......Page 199
5.1 Introduction......Page 202
5.2.1 Specification......Page 207
5.2.2 Calibration......Page 209
5.2.3 Application to Belgian mortality statistics......Page 221
5.3.1 Specification......Page 224
5.3.2 Calibration......Page 227
5.3.3 Application to Belgian mortality statistics......Page 228
5.4.1 Motivation......Page 230
5.4.2 P-splines approach......Page 231
5.4.3 Smoothing in the Lee–Carter model......Page 233
5.4.4 Application to Belgian mortality statistics......Page 234
5.5.1 Motivation......Page 235
5.5.2 Selection procedure......Page 237
5.5.3 Application to Belgian mortality statistics......Page 238
5.6.1 Deviance and Pearson residuals......Page 239
5.6.2 Application to Belgian mortality statistics......Page 241
5.7.1 Time series modelling for the time indices......Page 242
5.7.2 Modelling of the Lee-Carter time index......Page 244
5.7.3 Modelling the Cairns-Blake-Dowd time indices......Page 249
5.8.1 Why bootstrapping?......Page 250
5.8.2 Bootstrap percentiles confidence intervals......Page 251
5.8.3 Application to Belgian mortality statistics......Page 253
5.9 Forecasting life expectancies......Page 255
5.9.2 Andreev–Vaupel projections......Page 256
5.9.3 Application to Belgian mortality statistics......Page 258
5.9.5 Back testing......Page 261
6.1 Introduction......Page 264
6.2.1 Model structure......Page 267
6.2.2 Error structure and model fitting......Page 269
6.2.4 Discussion......Page 274
6.3 Application to United Kingdom mortality data......Page 275
6.4 Cairns-Blake-Dowd mortality projection model: allowing for cohort effects......Page 284
6.5 P-splines model: allowing for cohort effects......Page 286
7.1 Introduction......Page 288
7.2.1 Mortality risks......Page 289
7.2.2 Representing longevity risk: stochastic modelling issues......Page 291
7.2.3 Representing longevity risk: some examples......Page 294
7.2.4 Measuring longevity risk in a static framework......Page 297
7.3.1 A risk management perspective......Page 314
7.3.2 Natural hedging......Page 320
7.3.3 Solvency issues......Page 324
7.3.4 Reinsurance arrangements......Page 339
7.4.1 Life insurance securitization......Page 351
7.4.2 Mortality-linked securities......Page 353
7.4.3 Hedging life annuity liabilities through longevity bonds......Page 358
7.5.1 The location of mortality risks in traditional life annuity products......Page 364
7.5.2 GAO and GAR......Page 367
7.5.3 Adding FIexibility to GAR products......Page 368
7.6 Allowing for longevity risk in pricing......Page 371
7.7.1 Comparing life annuity prices......Page 375
7.7.2 Life annuities versus income drawdown......Page 377
7.7.3 The ‘mortality drag’......Page 380
7.7.4 Flexibility in FInancing post-retirement income......Page 384
7.8 References and suggestions for further reading......Page 390
References......Page 394
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F......Page 411
L......Page 412
N......Page 414
R......Page 415
Y......Page 416