Fundamentals of Actuarial Mathematics

2.99 MB·English

by  S. David Promislow

Front Matter Page: v
Dedication Page: v
Preface Page: xix
CHANGES IN THE THIRD EDITION Page: xxi
Acknowledgements Page: xxiii
About the companion website Page: xxiv
Part 1 THE DETERMINISTIC LIFE CONTINGENCIES MODEL Page: 1
1 Introduction and motivation Page: 3
1.1 Risk and insurance Page: 3
1.2 Deterministic versus stochastic models Page: 4
1.3 Finance and investments Page: 5
1.4 Adequacy and equity Page: 5
1.5 Reassessment Page: 6
1.6 Conclusion Page: 6
2 The basic deterministic model Page: 7
2.1 Cash flows Page: 7
2.2 An analogy with currencies Page: 8
2.3 Discount functions Page: 9
Definition 2.1 Page: 10
2.4 Calculating the discount function Page: 11
2.5 Interest and discount rates Page: 12
Definition 2.2 Page: 12
Definition 2.3 Page: 12
Remark Page: 12
Remark Page: 12
2.6 Constant interest Page: 12
2.7 Values and actuarial equivalence Page: 13
Example 2.1 Page: 14
Solution Page: 14
Remark Page: 14
Definition 2.4 Page: 14
Notation Page: 15
Definition 2.5 Page: 15
Example 2.2 Page: 16
Solution Page: 16
Figure 2.1 Example 2.2 Page: 16
Figure 2.2 Example 2.3 Page: 17
Example 2.3 Page: 17
Solution Page: 17
2.8 Vector notation Page: 17
2.9 Regular pattern cash flows Page: 18
2.10 Balances and reserves Page: 20
2.10.1 Basic concepts Page: 20
Example 2.4 Page: 20
Figure 2.3 Example 2.4 Page: 20
Solution Page: 20
Notation Page: 21
Definition 2.6 Page: 21
Definition 2.7 Page: 21
2.10.2 Relation between balances and reserves Page: 22
Example 2.5 Page: 22
Figure 2.4 Example 2.5 Page: 23
Solution Page: 23
2.10.3 Prospective versus retrospective methods Page: 23
2.10.4 Recursion formulas Page: 24
2.11 Time shifting and the splitting identity Page: 26
Example 2.6 Page: 27
Solution Page: 27
*2.11 Change of discount function Page: 27
2.12 Internal rates of return Page: 28
Definition 2.8 Page: 29
Remark Page: 29
Theorem 2.1 Page: 29
Proof Page: 29
Theorem 2.2 Page: 30
*2.13 Forward prices and term structure Page: 30
Definition 2.9 Page: 32
Definition 2.10 Page: 32
Example 2.7 Page: 32
Solution Page: 32
2.14 Standard notation and terminology Page: 33
2.14.1 Standard notation for cash flows discounted with interest Page: 33
2.14.2 New notation Page: 34
2.15 Spreadsheet calculations Page: 34
Notes and references Page: 35
Exercises Page: 35
Type A exercises Page: 35
Type B exercises Page: 36
Spreadsheet exercises Page: 38
3 The life table Page: 39
3.1 Basic definitions Page: 39
3.2 Probabilities Page: 40
3.3 Constructing the life table from the values of qx Page: 41
3.4 Life expectancy Page: 42
3.5 Choice of life tables Page: 44
3.6 Standard notation and terminology Page: 44
3.7 A sample table Page: 45
Notes and references Page: 45
Exercises Page: 45
Type A exercises Page: 45
Type B exercises Page: 46
Spreadsheet exercise Page: 46
4 Life annuities Page: 47
4.1 Introduction Page: 47
4.2 Calculating annuity premiums Page: 48
Example 4.1 Page: 49
Solution Page: 50
Figure 4.1 Example 4.1 Page: 50
4.3 The interest and survivorship discount function Page: 50
4.3.1 The basic definition Page: 50
Notation Page: 51
4.3.2 Relations between yx for various values of x Page: 52
Example 4.2 Page: 53
Solution Page: 53
4.4 Guaranteed payments Page: 53
Example 4.3 Page: 54
Solution Page: 54
Example 4.4 Page: 54
Solution Page: 54
4.5 Deferred annuities with annual premiums Page: 55
Example 4.5 Page: 55
Figure 4.2 Example 4.6 Page: 56
Solution Page: 56
4.6 Some practical considerations Page: 56
4.6.1 Gross premiums Page: 56
4.6.2 Gender aspects Page: 56
4.7 Standard notation and terminology Page: 57
4.8 Spreadsheet calculations Page: 58
Exercises Page: 59
Type A exercises Page: 59
Type B exercises Page: 59
Spreadsheet exercises Page: 60
5 Life insurance Page: 61
5.1 Introduction Page: 61
5.2 Calculating life insurance premiums Page: 61
Notation Page: 61
Example 5.1 Page: 63
Figure 5.1 Example 5.1 Page: 63
Solution Page: 63
Example 5.2 Page: 64
Solution Page: 64
5.3 Types of life insurance Page: 64
5.4 Combined insurance–annuity benefits Page: 64
Example 5.3 Page: 65
Solution Page: 65
Example 5.4 Page: 65
Solution Page: 65
Example 5.5 Page: 65
Figure 5.2 Example 5.5 Page: 65
Solution Page: 66
Example 5.6 Page: 66
Figure 5.3 Example 5.6 Page: 66
Solution Page: 66
Example 5.7 Page: 67
Figure 5.4 Example 5.7 Page: 67
Solution Page: 67
Example 5.8 Page: 67
Solution Page: 67
Example 5.9 Page: 68
Remark Page: 68
Example 5.10 Page: 68
Solution Page: 68
Example 5.11 Page: 69
Solution Page: 69
5.5 Insurances viewed as annuities Page: 69
5.6 Summary of formulas Page: 70
5.7 A general insurance–annuity identity Page: 70
5.7.1 The general identity Page: 70
5.7.2 The endowment identity Page: 71
Remark Page: 72
5.8 Standard notation and terminology Page: 72
5.8.1 Single-premium notation Page: 72
5.8.2 Annual-premium notation Page: 73
5.8.3 Identities Page: 74
5.9 Spreadsheet applications Page: 74
Exercises Page: 74
Type A exercises Page: 74
Type B exercises Page: 75
Spreadsheet exercises Page: 77
6 Insurance and annuity reserves Page: 78
6.1 Introduction to reserves Page: 78
Example 6.1 Page: 79
Figure 6.1 Example 6.1 Page: 79
Solution Page: 79
Example 6.2 Page: 80
Figure 6.2 Example 6.2 Page: 80
Solution Page: 80
Remark Page: 81
6.2 The general pattern of reserves Page: 81
Example 6.3 Page: 81
Solution Page: 81
6.3 Recursion Page: 82
Remark Page: 83
Definition 6.1 Page: 83
6.4 Detailed analysis of an insurance or annuity contract Page: 83
6.4.1 Gains and losses Page: 83
Example 6.4 Page: 84
Solution Page: 84
6.4.2 The risk–savings decomposition Page: 85
Example 6.5 Page: 86
Solution Page: 86
Example 6.6 Page: 86
Solution Page: 86
6.5 Bases for reserves Page: 87
Example 6.7 Page: 87
Solution Page: 87
6.6 Nonforfeiture values Page: 88
6.7 Policies involving a return of the reserve Page: 88
Example 6.8 Page: 89
Solution Page: 89
6.8 Premium difference and paid-up formulas Page: 90
6.8.1 Premium difference formulas Page: 90
6.8.2 Paid-up formulas Page: 90
6.8.3 Level endowment reserves Page: 91
6.9 Standard notation and terminology Page: 91
6.10 Spreadsheet applications Page: 93
INPUT FORMULAS Page: 93
INPUT DATA FOR EACH PARTICULAR PROBLEM Page: 93
OUTPUT Page: 94
Exercises Page: 94
Type A exercises Page: 94
Type B exercises Page: 95
Spreadsheet exercise Page: 97
7 Fractional durations Page: 98
7.1 Introduction Page: 98
7.2 Cash flows discounted with interest only Page: 99
Example 7.1 Page: 100
Solution Page: 100
7.3 Life annuities paid mthly Page: 101
7.3.1 Uniform distribution of deaths Page: 101
Example 7.2 Page: 101
Solution Page: 101
7.3.2 Present value formulas Page: 102
Example 7.3 Page: 103
Solution Page: 103
Figure 7.1 Due and immediate quarterly annuity payments Page: 104
7.4 Immediate annuities Page: 104
Example 7.4 Page: 104
Solution Page: 104
7.5 Approximation and computation Page: 105
Example 7.5 Page: 106
Solution Page: 106
*7.6 Fractional period premiums and reserves Page: 106
7.7 Reserves at fractional durations Page: 107
Example 7.6 Page: 109
Solution Page: 109
7.8 Standard notation and terminology Page: 109
Exercises Page: 109
Type A exercises Page: 109
Type B exercises Page: 110
Spreadsheet exercise Page: 111
8 Continuous payments Page: 112
8.1 Introduction to continuous annuities Page: 112
8.2 The force of discount Page: 113
Definition 8.1 Page: 113
Proposition 8.1 Page: 114
Proof Page: 114
Remark Page: 114
8.3 The constant interest case Page: 114
8.4 Continuous life annuities Page: 115
8.4.1 Basic definition Page: 115
8.4.2 Evaluation Page: 116
8.4.3 Life expectancy revisited Page: 117
8.5 The force of mortality Page: 118
Definition 8.2 Page: 118
Example 8.1 Page: 119
Solution Page: 119
Definition 8.3 Page: 119
8.6 Insurances payable at the moment of death Page: 119
8.6.1 Basic definitions Page: 119
8.6.2 Evaluation Page: 120
Example 8.2 Page: 121
Solution Page: 121
Example 8.3 Page: 121
Solution Page: 122
8.7 Premiums and reserves Page: 122
Example 8.4 Page: 122
Solution Page: 123
8.8 The general insurance–annuity identity in the continuous case Page: 123
8.9 Differential equations for reserves Page: 124
8.10 Some examples of exact calculation Page: 125
8.10.1 Constant force of mortality Page: 126
8.10.2 Demoivre's law Page: 127
8.10.3 An example of the splitting identity Page: 128
Example 8.5 Page: 128
Solution Page: 128
8.11 Further approximations from the life table Page: 129
8.12 Standard actuarial notation and terminology Page: 131
Notes and references Page: 132
Exercises Page: 132
Type A exercises Page: 132
Type B exercises Page: 134
Spreadsheet exercise Page: 136
9 Select mortality Page: 137
9.1 Introduction Page: 137
9.2 Select and ultimate tables Page: 138
9.3 Changes in formulas Page: 139
Chapter 3 Page: 139
Chapter 4 Page: 139
Chapter 5 Page: 140
Chapter 6 Page: 140
Chapter 8 Page: 140
Spreadsheets Page: 140
9.4 Projections in annuity tables Page: 141
Example 9.1 Page: 142
Solution Page: 142
9.5 Further remarks Page: 142
Exercises Page: 142
Spreadsheet exercises Page: 143
10 Multiple-life contracts Page: 144
10.1 Introduction Page: 144
10.2 The joint-life status Page: 144
10.3 Joint-life annuities and insurances Page: 146
10.4 Last-survivor annuities and insurances Page: 147
10.4.1 Basic results Page: 147
10.4.2 Reserves on second-death insurances Page: 148
10.5 Moment of death insurances Page: 149
Definition 10.1 Page: 149
10.6 The general two-life annuity contract Page: 150
Example 10.1 Page: 151
Solution Page: 151
Example 10.2 Page: 151
Solution Page: 151
Example 10.3 Page: 151
Solution Page: 151
Example 10.4 Page: 152
Solution Page: 152
Remark Page: 152
10.7 The general two-life insurance contract Page: 152
10.8 Contingent insurances Page: 153
10.8.1 First-death contingent insurances Page: 153
10.8.2 Second-death contingent insurances Page: 154
10.8.3 Moment-of-death contingent insurances Page: 155
Example 10.5 Page: 155
Solution Page: 155
10.8.4 General contingent probabilities Page: 155
10.9 Duration problems Page: 156
Example 10.6 Page: 156
Solution Page: 156
Example 10.7 Page: 157
Solution Page: 157
Example 10.8 Page: 157
Solution Page: 157
Example 10.9 Page: 159
Solution Page: 159
*10.10 Applications to annuity credit risk Page: 159
10.11 Standard notation and terminology Page: 160
10.12 Spreadsheet applications Page: 161
Notes and references Page: 161
Exercises Page: 161
Type A exercises Page: 161
Type B exercises Page: 163
Spreadsheet exercise Page: 165
11 Multiple-decrement theory Page: 166
11.1 Introduction Page: 166
11.2 The basic model Page: 166
11.2.1 The multiple-decrement table Page: 167
11.2.2 Quantities calculated from the multiple-decrement table Page: 168
11.3 Insurances Page: 169
Definition 11.1 Page: 169
11.4 Determining the model from the forces of decrement Page: 170
Example 11.1 Page: 171
Solution Page: 171
11.5 The analogy with joint-life statuses Page: 171
11.6 A machine analogy Page: 171
11.6.1 Method 1 Page: 172
11.6.2 Method 2 Page: 173
Example 11.2 Page: 174
Solution Page: 174
Example 11.3 Page: 174
Solution Page: 174
11.7 Associated single-decrement tables Page: 175
11.7.1 The main methods Page: 175
11.7.2 Forces of decrement in the associated single-decrement tables Page: 176
11.7.3 Conditions justifying the two methods Page: 177
Example 11.4 Page: 178
Solution Page: 179
Example 11.5 Page: 179
Solution Page: 179
11.7.4 Other approaches Page: 180
Notes and references Page: 181
Exercises Page: 181
Type A exercises Page: 181
Type B exercises Page: 182
12 Expenses and profits Page: 184
12.1 Introduction Page: 184
Example 12.1 Page: 185
Solution Page: 185
12.2 Effect on reserves Page: 186
12.3 Realistic reserve and balance calculations Page: 187
Example 12.2 Page: 188
Solution Page: 188
Remark Page: 189
12.4 Profit measurement Page: 189
12.4.1 Advanced gain and loss analysis Page: 189
Example 12.3 Page: 190
Solution Page: 190
12.4.2 Gains by source Page: 191
Example 12.4 Page: 192
Solution Page: 193
12.4.3 Profit testing Page: 193
Example 12.5 Page: 193
Solution Page: 194
Definition 12.1 Page: 194
Example 12.6 Page: 194
Solution Page: 194
Definition 12.12 Page: 195
Example 12.7 Page: 195
Solution Page: 195
Notes and references Page: 196
Exercises Page: 196
Type A exercises Page: 196
Type B exercises Page: 197
*13 Specialized topics Page: 199
13.1 Universal life Page: 199
13.1.1 Description of the contract Page: 199
13.1.2 Calculating account values Page: 201
Example 13.1 Page: 202
Solution Page: 202
Example 13.2 Page: 203
Solution Page: 203
Example 13.3 Page: 203
Solution Page: 203
13.2 Variable annuities Page: 203
13.3 Pension plans Page: 204
13.3.1 DB plans Page: 204
Example 13.4 Page: 205
Solution Page: 205
13.3.2 DC plans Page: 206
Example 13.5 Page: 206
Solution Page: 206
Example 13.6 Page: 207
Solution Page: 207
Exercises Page: 207
Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL Page: 209
14 Survival distributions and failure times Page: 211
14.1 Introduction to survival distributions Page: 211
14.2 The discrete case Page: 212
Definition 14.1 Page: 212
Example 14.1 Page: 213
Solution Page: 213
14.3 The continuous case Page: 213
14.3.1 The basic functions Page: 214
Definition 14.2 Page: 214
14.3.2 Properties of μ Page: 214
14.3.3 Modes Page: 215
14.4 Examples Page: 215
14.5 Shifted distributions Page: 216
Example 14.2 Page: 217
Solution Page: 217
14.6 The standard approximation Page: 217
14.7 The stochastic life table Page: 219
14.8 Life expectancy in the stochastic model Page: 220
Example 14.3 Page: 221
Solution Page: 221
14.9 Stochastic interest rates Page: 221
Notes and references Page: 222
Exercises Page: 222
Type A exercises Page: 222
Type B exercises Page: 222
15 The stochastic approach to insurance and annuities Page: 224
15.1 Introduction Page: 224
15.2 The stochastic approach to insurance benefits Page: 225
15.2.1 The discrete case Page: 225
15.2.2 The continuous case Page: 226
Example 15.1 Page: 226
Solution Page: 226
15.2.3 Approximation Page: 226
15.2.4 Endowment insurances Page: 227
Example 15.2 Page: 228
Solution Page: 228
15.3 The stochastic approach to annuity benefits Page: 229
15.3.1 Discrete annuities Page: 229
Example 15.3 Page: 230
Solution Page: 230
Remark Page: 230
Remark Page: 230
15.3.2 Continuous annuities Page: 231
Example 15.4 Page: 232
First solution Page: 232
*15.4 Deferred contracts Page: 233
15.5 The stochastic approach to reserves Page: 233
Definition 15.1 Page: 234
Remark Page: 234
Definition 15.2 Page: 234
Example 15.5 Page: 235
Solution Page: 235
15.6 The stochastic approach to premiums Page: 235
15.6.1 The equivalence principle Page: 235
15.6.2 Percentile premiums Page: 236
Example 15.6 Page: 236
Solution Page: 236
Example 15.7 Page: 237
Solution Page: 237
15.6.3 Aggregate premiums Page: 237
Definition 15.3 Page: 238
Example 15.8 Page: 239
Solution Page: 240
Example 15.9 Page: 240
Solution Page: 240
15.6.4 General premium principles Page: 240
15.7 The variance of r L Page: 241
Example 15.10 Page: 243
Solution Page: 243
15.8 Standard notation and terminology Page: 243
Notes and references Page: 244
Exercises Page: 244
Type A exercises Page: 244
Type B exercises Page: 246
Spreadsheet exercise Page: 247
16 Simplifications under level benefit contracts Page: 248
16.1 Introduction Page: 248
16.2 Variance calculations in the continuous case Page: 248
16.2.1 Insurances Page: 249
16.2.2 Annuities Page: 249
16.2.3 Prospective losses Page: 249
16.2.4 Using equivalence principle premiums Page: 249
16.3 Variance calculations in the discrete case Page: 250
Example 16.1 Page: 251
Solution Page: 251
16.4 Exact distributions Page: 252
16.4.1 The distribution of Page: 252
16.4.2 The distribution of Page: 252
16.4.3 The distribution of L Page: 252
16.4.4 The case where T is exponentially distributed Page: 253
Example 16.2 Page: 253
Solution Page: 253
16.5 Some non-level benefit examples Page: 254
16.5.1 Term insurance Page: 254
Example 16.3 Page: 254
Solution Page: 254
16.5.2 Deferred insurance Page: 254
Figure 16.1 Graph of for various types of insurance Page: 255
16.5.3 An annual premium policy Page: 255
Figure 16.2 Graph of fL(u) for various types of insurance Page: 256
Exercises Page: 256
Type A exercises Page: 256
Type B exercises Page: 257
17 The minimum failure time Page: 259
17.1 Introduction Page: 259
17.2 Joint distributions Page: 259
17.3 The distribution of T Page: 261
17.3.1 The general case Page: 261
17.3.2 The independent case Page: 261
Example 17.1 Page: 261
Solution Page: 261
17.4 The joint distribution of (T, J) Page: 261
17.4.1 The distribution function for (T, J) Page: 261
Example 17.2 Page: 262
Solution Page: 262
Example 17.3 Page: 263
Solution Page: 263
17.4.2 Density and survival functions for (T, J) Page: 264
Figure 17.1 The graph of fT, J(t, j) Page: 265
17.4.3 The distribution of J Page: 265
Example 17.4 Page: 265
Solution Page: 265
17.4.4 Hazard functions for (T, J) Page: 266
Definition 17.1 Page: 266
17.4.5 The independent case Page: 266
Example 17.5 Page: 267
Solution Page: 267
Example 17.6 Page: 268
Solution Page: 268
17.4.6 Nonidentifiability Page: 268
Example 17.7 Page: 268
Solution Page: 268
Theorem 17.1 Page: 269
Proof Page: 269
17.4.7 Conditions for the independence of T and J Page: 269
Theorem 17.2 Page: 269
Proof Page: 270
17.5 Other problems Page: 270
Example 17.8 Page: 270
Solution Page: 270
17.6 The common shock model Page: 271
Example 17.9 Page: 271
Solution Page: 271
Example 17.10 Page: 271
Solution Page: 271
Example 17.11 Page: 271
Solution Page: 271
Example 17.12 Page: 272
Solution Page: 272
17.7 Copulas Page: 273
Definition 17.2 Page: 274
Example 17.13 Page: 275
Solution Page: 275
Notes and references Page: 276
Exercises Page: 276
Type A exercises Page: 276
Type B exercises Page: 277
Part III ADVANCED STOCHASTIC MODELS Page: 279
18 An introduction to stochastic processes Page: 281
18.1 Introduction Page: 281
Figure 18.1 Evolution of stock price Page: 282
18.2 Markov chains Page: 283
18.2.1 Definitions Page: 283
Definition 18.1 Page: 283
18.2.2 Examples Page: 284
Figure 18.2 Random walk Page: 286
Figure 18.3 Gambler's fortune Page: 286
18.3 Martingales Page: 286
Definition 18.2 Page: 286
18.4 Finite-state Markov chains Page: 287
18.4.1 The transition matrix Page: 287
Remark Page: 287
Example 18.1 Page: 287
Solution Page: 287
18.4.2 Multi-period transitions Page: 288
18.4.3 Distributions Page: 288
Example 18.2 Page: 288
Solution Page: 289
Remark Page: 289
*18.4.4 Limiting distributions Page: 289
*18.4.5 Recurrent and transient states Page: 290
Definition 18.3 Page: 290
Example 18.3 Page: 290
Solution Page: 290
Definition 18.4 Page: 291
Theorem 18.1 Page: 291
Proof Page: 291
Theorem 18.2 Page: 291
Proof Page: 291
Example 18.4 (Random walk with absorbing barriers) Page: 292
Solution Page: 292
18.5 Introduction to continuous time processes Page: 293
Notation Page: 293
Definition 18.5 Page: 293
Definition 18.6 Page: 293
18.6 Poisson processes Page: 293
Definition 18.7 Page: 294
Theorem 18.3 Page: 294
18.6.1 Waiting times Page: 295
18.6.2 Nonhomogeneous Poisson processes Page: 295
Definition 18.8 Page: 295
18.7 Brownian motion Page: 295
18.7.1 The main definition Page: 295
18.7.2 Connection with random walks Page: 296
*18.7.3 Hitting times Page: 297
*18.7.4 Conditional distributions Page: 298
18.7.5 Brownian motion with drift Page: 299
18.7.6 Geometric Brownian motion Page: 299
Notes and references Page: 299
Exercises Page: 300
19 Multi-state models Page: 304
19.1 Introduction Page: 304
Figure 19.1 A two life multi-state model Page: 305
19.2 The discrete-time model Page: 305
19.2.1 Non-stationary Markov Chains Page: 305
Figure 19.2 The healthy-unhealthy-deceased model Page: 306
Example 19.1 Page: 306
Solution Page: 306
19.2.2 Discrete-time multi-state insurances Page: 307
Notation Page: 307
Example 19.2 Page: 307
Solution Page: 307
Example 19.3 Page: 308
Solution Page: 308
Remark Page: 310
19.2.3 Multi-state annuities Page: 310
Example 19.4 Page: 310
Solution Page: 310
Example 19.5 Page: 311
Solution Page: 311
19.3 The continuous-time model Page: 311
Definition 19.1 Page: 311
19.3.1 Forces of transition Page: 311
Definition 19.2 Page: 312
Theorem 19.1 Page: 312
Proof Page: 312
Theorem 19.2 (Kolmogorov forward equations) Page: 312
Proof Page: 312
Theorem 19.3 Page: 314
Proof Page: 314
Theorem 19.4 Page: 315
Remark Page: 315
19.3.2 Path-by-path analysis Page: 316
Example 19.6 Page: 317
Solution Page: 317
19.3.3 Numerical approximation Page: 317
Example 19.7 Page: 318
Solution Page: 318
19.3.4 Stationary continuous time processes Page: 318
Example 19.8 Page: 319
Solution Page: 319
19.3.5 Some methods for non-stationary processes Page: 320
Example 19.9 Page: 320
Solution Page: 320
19.3.6 Extension of the common shock model Page: 321
Example 19.10 Page: 322
Solution Page: 322
Example 19.11 Page: 322
Solution Page: 322
19.3.7 Insurance and annuity applications in continuous time Page: 322
Example 19.12 Page: 323
Solution Page: 323
Example 19.13 Page: 324
Solution Page: 324
19.4 Recursion and differential equations for multi-state reserves Page: 324
Example 19.14 Page: 324
Solution Page: 325
Example 19.15 Page: 326
Solution Page: 326
19.5 Profit testing in multi-state models Page: 327
Example 19.16 Page: 327
Solution Page: 328
19.6 Semi-Markov models Page: 328
Notes and references Page: 328
Exercises Page: 329
20 Introduction to the Mathematics of Financial Markets Page: 333
20.1 Introduction Page: 333
20.2 Modelling prices in financial markets Page: 333
20.3 Arbitrage Page: 334
Definition 20.1 Page: 335
Definition 20.2 Page: 336
Remark Page: 336
Theorem 20.1 Page: 336
Proof Page: 336
Remark Page: 337
Theorem 20.2 Page: 337
Proof Page: 337
Definition 20.3 Page: 337
20.4 Option contracts Page: 337
20.5 Option prices in the one-period binomial model Page: 339
Example 20.1 Page: 342
Solution Page: 342
Theorem 20.3 (Put-call Parity) Page: 342
Proof Page: 342
20.6 The multi-period binomial model Page: 342
Definition 20.4 Page: 343
Example 20.2 Page: 344
Solution Page: 344
Example 20.3 Page: 345
Solution Page: 345
Figure 20.1 Example 20.3 Page: 345
20.7 American options Page: 346
Example 20.4 Page: 347
Solution Page: 347
20.8 A general financial market Page: 348
Figure 20.2 A finanical market with two risky assets Page: 349
Definition 20.5 Page: 350
Definition 20.6 Page: 350
Example 20.5 Page: 350
Solution Page: 350
20.9 Arbitrage-free condition Page: 351
Theorem 20.4 (The fundamental theorem of asset pricing) Page: 352
20.10 Existence and uniqueness of risk-neutral measures Page: 353
20.10.1 Linear algebra background Page: 353
20.10.2 The space of contingent claims Page: 353
Figure 20.3 A picture of the one period binomial market Page: 355
Figure 20.4 A market in which not all contingent claims are replicable Page: 355
Figure 20.5 A market that is not arbitage-free Page: 355
Figure 20.6 See Exercise 20.7 Page: 356
Example 20.6 Page: 356
Solution Page: 356
20.10.3 The Fundamental theorem of asset pricing completed Page: 357
Example 20.7 Page: 359
Solution Page: 359
20.11 Completeness of markets Page: 359
Definition 20.7 Page: 359
Theorem 20.5 Page: 359
Proof Page: 359
Example 20.8 Page: 361
Solution Page: 361
20.12 The Black–Scholes–Merton formula Page: 361
Remark Page: 364
20.13 Bond markets Page: 364
20.13.1 Introduction Page: 364
20.13.2 Extending the notion of conditional expectation Page: 366
Theorem 20.6 Page: 366
Proof Page: 366
20.13.3 The arbitrage-free condition in the bond market Page: 367
Theorem 20.7 Page: 367
Proof Page: 368
20.13.4 Short-rate modelling Page: 368
Example 20.9 Page: 369
Solution Page: 369
Figure 20.7 Example 20.9. Values of Sk(n), k = 0, 1, 2, 3 Page: 370
20.13.5 Forward prices and rates Page: 370
Figure 20.8 Example 20.9. Values of k = 1, 2, 3 Page: 370
20.13.6 Observations on the continuous time bond market Page: 371
Notes and references Page: 372
Exercises Page: 373
Part IV RISK THEORY Page: 375
21 Compound distributions Page: 377
21.1 Introduction Page: 377
21.2 The mean and variance of S Page: 379
21.3 Generating functions Page: 380
21.4 Exact distribution of S Page: 381
Example 21.1 Page: 381
Solution Page: 381
21.5 Choosing a frequency distribution Page: 381
21.6 Choosing a severity distribution Page: 383
21.7 Handling the point mass at 0 Page: 384
Example 21.2 Page: 384
Solution Page: 384
21.8 Counting claims of a particular type Page: 385
21.8.1 One special class Page: 385
Example 21.3 Page: 385
Solution Page: 385
21.8.2 Special classes in the Poisson case Page: 386
21.9 The sum of two compound Poisson distributions Page: 387
21.10 Deductibles and other modifications Page: 388
21.10.1 The nature of a deductible Page: 388
Example 21.4 Page: 389
Solution Page: 389
Example 21.5 Page: 389
Solution Page: 389
21.10.2 Some calculations in the discrete case Page: 389
Example 21.6 Page: 390
Solution Page: 390
21.10.3 Some calculations in the continuous case Page: 390
Example 21.7 Page: 390
Solution Page: 390
Example 21.8 Page: 391
Solution Page: 391
Example 21.9 Page: 391
Solution Page: 391
21.10.4 The effect on aggregate claims Page: 392
Example 21.10 Page: 392
Solution Page: 392
Example 21.11 Page: 393
Solution Page: 393
21.10.5 Other modifications Page: 393
Example 21.12 Page: 393
Solution Page: 393
21.11 A recursion formula for S Page: 393
21.11.1 The positive-valued case Page: 393
Proposition 21.1 Page: 394
Proof Page: 394
Theorem 21.1 (The recursion formula) Page: 394
Proof Page: 394
Theorem 21.2 Page: 396
Proof Page: 396
Example 21.13 Page: 397
Solution Page: 397
21.11.2 The case with claims of zero amount Page: 397
Example 21.14 Page: 398
Solution Page: 398
Notes and references Page: 398
Exercises Page: 398
22 Risk assessment Page: 403
22.1 Introduction Page: 403
22.2 Utility theory Page: 403
Example 22.1 Page: 404
Solution Page: 404
Example 22.2 Page: 405
Solution Page: 405
22.3 Convex and concave functions: Jensen's inequality Page: 406
22.3.1 Basic definitions Page: 406
Definition 22.1 Page: 406
Figure 22.1 Graphs of ud(x) and vd(x) Page: 407
Definition 22.2 Page: 407
22.3.2 Jensen's inequality Page: 407
Theorem 22.1 (Jensen's inequality) Page: 408
Proof Page: 408
22.4 A general comparison method Page: 408
Definition 22.3 Page: 409
Theorem 22.2 Page: 409
Proof Page: 410
Remark Page: 410
Theorem 22.3 (The cut condition) Page: 410
Proof Page: 410
Figure 22.2 The cut condition: X is less risky than Y Page: 411
Remark Page: 411
Theorem 22.4 Page: 412
Proof Page: 412
Remark Page: 412
22.5 Risk measures for capital adequacy Page: 412
22.5.1 The general notion of a risk measure Page: 412
22.5.1 Value-at-risk Page: 413
Definition 22.4 Page: 413
22.5.3 Tail value-at-risk Page: 413
Definition 22.5 Page: 414
Example 22.3 Page: 416
Solution Page: 416
Example 22.4 Page: 416
Solution Page: 416
22.5.4 Distortion risk measures Page: 417
Notes and references Page: 417
Exercises Page: 417
23 Ruin models Page: 420
23.1 Introduction Page: 420
Example 23.1 Page: 421
Solution Page: 421
Remark Page: 422
23.2 A functional equation approach Page: 422
Example 23.2 Page: 423
Solution Page: 423
23.3 The martingale approach to ruin theory Page: 424
23.3.1 Stopping times Page: 424
Example 23.3 Page: 425
Solution Page: 425
Figure 23.1 Tree for Example 23.3 Page: 425
23.3.2 The optional stopping theorem and its consequences Page: 426
Theorem 23.1 (Optional stopping theorem) Page: 426
Proof Page: 426
Corollary Page: 426
Proof Page: 427
Example 23.4 Page: 427
Solution Page: 427
Example 23.5 Page: 427
Solution Page: 427
Example 23.6 Page: 428
Solution Page: 428
Remark Page: 429
23.3.3 The adjustment coefficient Page: 429
Definition 23.1 Page: 429
Figure 23.2 Graph of the function ϕ Page: 430
Example 23.7 Page: 430
Solution Page: 430
Example 23.8 Page: 431
Solution Page: 431
23.3.4 The main conclusions Page: 431
Theorem 23.2 Page: 431
Proof Page: 431
Theorem 23.3 Page: 432
Theorem 23.4 Page: 432
23.4 Distribution of the deficit at ruin Page: 433
23.5 Recursion formulas Page: 434
23.5.1 Calculating ruin probabilities Page: 434
Remark Page: 436
23.5.2 The distribution of D(u) Page: 436
Example 23.9 Page: 437
Solution Page: 437
23.6 The compound Poisson surplus process Page: 438
23.6.1 Description of the process Page: 438
Figure 23.3 A realization of the continuous-time surplus process Page: 439
23.6.2 The probability of eventual ruin Page: 440
23.6.3 The value of ψ(0) Page: 440
23.6.4 The distribution of D(0) Page: 440
23.6.5 The case when X is exponentially distributed Page: 441
Theorem 23.5 Page: 441
23.7 The maximal aggregate loss Page: 441
Definition 23.2 Page: 441
Example 23.10 Page: 443
Solution Page: 443
Example 23.11 Page: 444
Solution Page: 444
Theorem 23.6 Page: 444
Notes and references Page: 445
Exercises Page: 445
24 Credibility theory Page: 449
24.1 Introductory material Page: 449
24.1.1 The nature of credibility theory Page: 449
24.1.2 Information assessment Page: 449
Example 24.1 Page: 449
Solution Page: 450
Example 24.2 Page: 451
Solution Page: 451
Example 24.3 Page: 452
Solution Page: 452
24.2 Conditional expectation and variance with respect to another random variable Page: 453
24.2.1 The random variable E(X|Y) Page: 453
Definition 24.1 Page: 453
Example 24.4 Page: 453
Solution Page: 453
Theorem 24.1 Page: 454
Proof Page: 454
Remark Page: 455
24.2.2 Conditional variance Page: 455
Definition 24.2 Page: 455
Theorem 24.2 Page: 455
Proof Page: 456
Remark Page: 456
Example 24.5 Page: 456
Solution Page: 456
24.3 General framework for Bayesian credibility Page: 457
Remark Page: 458
24.4 Classical examples Page: 459
Example 24.6 Page: 459
Solution Page: 459
Example 24.7 (Poisson–Gamma) Page: 461
Solution Page: 461
24.5 Approximations Page: 462
24.5.1 A general case Page: 462
24.5.2 The Bühlman model Page: 463
24.5.3 Bühlman–Straub Model Page: 464
Example 24.8 Page: 465
Solution Page: 465
24.6 Conditions for exactness Page: 465
Example 24.9 Page: 465
Solution Page: 465
Theorem 24.3 Page: 466
Proof Page: 466
Remark Page: 467
Remark Page: 467
Theorem 24.4 Page: 468
Proof Page: 468
24.7 Estimation Page: 469
24.7.1 Unbiased estimators Page: 469
24.7.2 Calculating in the credibility model Page: 470
24.7.3 Estimation of the Bülhman parameters Page: 470
Example 24.10 Page: 472
Solution Page: 472
24.7.4 Estimation in the Bülhman–Straub model Page: 472
Notes and references Page: 473
Exercises Page: 473
Back Matter Page: 477
Answers to exercises Page: 477
Chapter 2 Page: 477
Chapter 3 Page: 478
Chapter 4 Page: 478
Chapter 5 Page: 478
Chapter 6 Page: 479
Chapter 7 Page: 480
Chapter 8 Page: 480
Chapter 9 Page: 481
Chapter 10 Page: 481
Chapter 11 Page: 483
Chapter 12 Page: 483
Chapter 13 Page: 484
Chater 14 Page: 484
Chapter 15 Page: 484
Chapter 16 Page: 485
Chapter 17 Page: 486
Chapter 18 Page: 487
Chapter 19 Page: 488
Chapter 20 Page: 488
Chapter 21 Page: 489
Chapter 22 Page: 490
Chapter 23 Page: 490
Chapter 24 Page: 491
Appendix A review of probability theory Page: 493
A.1 Sample spaces and probability measures Page: 493
A.2 Conditioning and independence Page: 495
A.3 Random variables Page: 495
A.4 Distributions Page: 496
A.5 Expectations and moments Page: 497
A.6 Expectation in terms of the distribution function Page: 498
A.7 Joint distributions Page: 499
A.8 Conditioning and independence for random variables Page: 501
A.9 Moment generating functions Page: 502
Theorem A.1 Page: 503
Proof Page: 503
Theorem A.2 (The uniqueness theorem) Page: 503
A.10 Probability generating functions Page: 503
A.11 Some standard distributions Page: 505
A.11.1 The binomial distribution Page: 505
A.11.2 The Poisson distribution Page: 505
Example A.1 Page: 506
Solution Page: 506
A.11.3 The negative binomial and geometric distributions Page: 506
A.11.4 The continuous uniform distribution Page: 507
A.11.5 The normal distribution Page: 507
A.11.6 The gamma and exponential distributions Page: 509
A.11.7 The lognormal distribution Page: 510
A.11.8 The Pareto distribution Page: 511
A.12 Convolution Page: 511
A.12.1 The discrete case Page: 511
Example A.2 Page: 513
Solution Page: 513
A.12.2 The continuous case Page: 513
Example A.3 Page: 514
Solution Page: 514
Figure A.1 The region of integration in Example A.3 Page: 515
A.12.3 Notation and remarks Page: 515
A.13 Mixtures Page: 516
References Page: 517
Notation index Page: 519
Index Page: 523
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