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Basic Life Insurance Mathematics - Norberg.pdf
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Basic Life Insurance Mathematics
Ragnar Norberg
Version: September 2002Contents
1 Introduction 5
1.1 Banking versus insurance . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Banking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 With-pro t contracts: Surplus and bonus . . . . . . . . . . . . . 14
1.6 Unit-linked insurance . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Issues for further study . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Payment streams and interest 19
2.1 Basic de nitions and relationships . . . . . . . . . . . . . . . . . 19
2.2 Application to loans . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Mortality 28
3.1 Aggregate mortality . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Some standard mortality laws . . . . . . . . . . . . . . . . . . . . 33
3.3 Actuarial notation . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Select mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Insurance of a single life 39
4.1 Some standard forms of insurance . . . . . . . . . . . . . . . . . . 39
4.2 The principle of equivalence . . . . . . . . . . . . . . . . . . . . . 43
4.3 Prospective reserves . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Thiele's di
erential equation . . . . . . . . . . . . . . . . . . . . . 52
4.5 Probability distributions . . . . . . . . . . . . . . . . . . . . . . . 56
4.6 The stochastic process point of view . . . . . . . . . . . . . . . . 57
5 Expenses 59
5.1 A single life insurance policy . . . . . . . . . . . . . . . . . . . . 59
5.2 The general multi-state policy . . . . . . . . . . . . . . . . . . . . 62
6 Multi-life insurances 63
6.1 Insurances depending on the number of survivors . . . . . . . . . 63
1
CONTENTS
27 Markov chains in life insurance 67
7.1 The insurance policy as a stochastic process . . . . . . . . . . . . 67
7.2 The time-continuous Markov chain . . . . . . . . . . . . . . . . . 68
7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.4 Selection phenomena . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.5 The standard multi-state contract . . . . . . . . . . . . . . . . . 79
7.6 Select mortality revisited . . . . . . . . . . . . . . . . . . . . . . 86
7.7 Higher order moments of present values . . . . . . . . . . . . . . 89
7.8 A Markov chain interest model . . . . . . . . . . . . . . . . . . . 94
7.8.1 The Markov model . . . . . . . . . . . . . . . . . . . . . . 94
7.8.2 Di
erential equations for moments of present values . . . 95
7.8.3 Complement on Markov chains . . . . . . . . . . . . . . . 98
7.9 Dependent lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.9.2 Notions of positive dependence . . . . . . . . . . . . . . . 101
7.9.3 Dependencies between present values . . . . . . . . . . . . 103
7.9.4 A Markov chain model for two lives . . . . . . . . . . . . 103
7.10 Conditional Markov chains . . . . . . . . . . . . . . . . . . . . . 106
7.10.1 Retrospective fertility analysis . . . . . . . . . . . . . . . 106
8 Probability distributions of present values 109
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Calculation of probability distributions of present values by elementary
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3 The general Markov multistate policy . . . . . . . . . . . . . . . 111
8.4 Di
erential equations for statewise distributions . . . . . . . . . . 112
8.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9 Reserves 119
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.2 General de nitions of reserves and statement of some relationships
between them . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.3 Description of payment streams appearing in life and pension
insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.4 The Markov chain model . . . . . . . . . . . . . . . . . . . . . . . 126
9.5 Reserves in the Markov chain model . . . . . . . . . . . . . . . . 131
9.6 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10 Safety loadings and bonus 145
10.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . 145
10.2 First and second order bases . . . . . . . . . . . . . . . . . . . . . 146
10.3 The technical surplus and how it emerges . . . . . . . . . . . . . 147
10.4 Dividends and bonus . . . . . . . . . . . . . . . . . . . . . . . . . 149
10.5 Bonus prognoses . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.7 Including expenses . . . . . . . . . . . . . . . . . . . . . . . . . . 161
CONTENTS
310.8 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
11 Statistical inference in the Markov chain model 167
11.1 Estimating a mortality law from fully observed life lengths . . . . 167
11.2 Parametric inference in the Markov model . . . . . . . . . . . . . 172
11.3 Con dence regions . . . . . . . . . . . . . . . . . . . . . . . . . . 176
11.4 More on simultaneous con dence intervals . . . . . . . . . . . . . 177
11.5 Piecewise constant intensities . . . . . . . . . . . . . . . . . . . . 179
11.6 Impact of the censoring scheme . . . . . . . . . . . . . . . . . . . 183
12 Heterogeneity models 185
12.1 The notion of heterogeneity { a two-stage model . . . . . . . . . 185
12.2 The proportional hazard model . . . . . . . . . . . . . . . . . . . 187
13 Group life insurance 190
13.1 Basic characteristics of group insurance . . . . . . . . . . . . . . 190
13.2 A proportional hazard model for complete individual policy and
claim records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
13.3 Experience rated net premiums . . . . . . . . . . . . . . . . . . . 194
13.4 The uctuation reserve . . . . . . . . . . . . . . . . . . . . . . . . 195
13.5 Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . 197
14 Hattendor
and Thiele 198
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
14.2 The general Hattendor
theorem . . . . . . . . . . . . . . . . . . 199
14.3 Application to life insurance . . . . . . . . . . . . . . . . . . . . . 201
14.4 Excerpts from martingale theory . . . . . . . . . . . . . . . . . . 205
15 Financial mathematics in insurance 212
15.1 Finance in insurance . . . . . . . . . . . . . . . . . . . . . . . . . 212
15.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3 A Markov chain nancial market - Introduction . . . . . . . . . . 218
15.4 The Markov chain market . . . . . . . . . . . . . . . . . . . . . . 219
15.5 Arbitrage-pricing of derivatives in a complete market . . . . . . . 226
15.6 Numerical procedures . . . . . . . . . . . . . . . . . . . . . . . . 229
15.7 Risk minimization in incomplete markets . . . . . . . . . . . . . 229
15.8 Trading with bonds: How much can be hedged? . . . . . . . . . . 232
15.9 The Vandermonde matrix in nance . . . . . . . . . . . . . . . . 235
15.10Two properties of the Vandermonde matrix . . . . . . . . . . . . 236
15.11Applications to nance . . . . . . . . . . . . . . . . . . . . . . . . 237
15.12Martingale methods . . . . . . . . . . . . . . . . . . . . . . . . . 240
A Calculus 4
B Indicator functions 9
C Distribution of the number of occurring events 12
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