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

7 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

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