Dedication | p. v |
Preface | p. vii |
Introduction | p. ix |
Claims Reserving and Pricing with Run-Off Triangles | p. 1 |
The evolving nature of claims and reserves | p. 1 |
Chain ladder methods | p. 4 |
Basic chain ladder method | p. 5 |
Inflation-adjusted chain ladder method | p. 8 |
The average cost per claim method | p. 11 |
The Bornhuetter-Ferguson or loss ratio method | p. 14 |
An example in pricing products | p. 19 |
Statistical modeling and the separation technique | p. 26 |
Problems | p. 27 |
Loss Distributions | p. 35 |
Introduction to loss distributions | p. 35 |
Classical loss distributions | p. 36 |
Exponential distribution | p. 36 |
Pareto distribution | p. 39 |
Gamma distribution | p. 43 |
Weibull distribution | p. 45 |
Lognormal distribution | p. 47 |
Fitting loss distributions | p. 51 |
Kolmogorov-Smirnoff test | p. 52 |
Chi-square goodness-of-fit tests | p. 54 |
Akaike information criteria | p. 58 |
Mixture distributions | p. 58 |
Loss distributions and reinsurance | p. 61 |
Proportional reinsurance | p. 62 |
Excess of loss reinsurance | p. 62 |
Problems | p. 68 |
Risk Theory | p. 77 |
Risk models for aggregate claims | p. 77 |
Collective risk models | p. 78 |
Basic properties of compound distributions | p. 79 |
Compound Poisson, binomial and negative binomial distributions | p. 79 |
Sums of compound Poisson distributions | p. 85 |
Exact expressions for the distribution of S | p. 87 |
Approximations for the distribution of S | p. 92 |
Individual risk models for S | p. 94 |
Basic properties of the individual risk model | p. 95 |
Compound binomial distributions and individual risk models | p. 97 |
Compound Poisson approximations for individual risk models | p. 98 |
Premiums and reserves for aggregate claims | p. 99 |
Determining premiums for aggregate claims | p. 99 |
Setting aside reserves for aggregate claims | p. 103 |
Reinsurance for aggregate claims | p. 107 |
Proportional reinsurance | p. 109 |
Excess of loss reinsurance | p. 111 |
Stop-loss reinsurance | p. 116 |
Problems | p. 120 |
Ruin Theory | p. 129 |
The probability of ruin in a surplus process | p. 129 |
Surplus and aggregate claims processes | p. 129 |
Probability of ruin in discrete time | p. 132 |
Poisson surplus processes | p. 132 |
Probability of ruin and the adjustment coefficient | p. 134 |
The adjustment equation | p. 135 |
Lundberg's bound on the probability of ruin [psi] (U) | p. 138 |
The probability of ruin when claims are exponentially distributed | p. 140 |
Reinsurance and the probability of ruin | p. 146 |
Adjustment coefficients and proportional reinsurance | p. 147 |
Adjustment coefficients and excess of loss reinsurance | p. 149 |
Problems | p. 152 |
Credibility Theory | p. 159 |
Introduction to credibility estimates | p. 159 |
Classical credibility theory | p. 161 |
Full credibility | p. 161 |
Partial credibility | p. 163 |
The Bayesian approach to credibility theory | p. 164 |
Bayesian credibility | p. 164 |
Greatest accuracy credibility theory | p. 170 |
Bayes and linear estimates of the posterior mean | p. 172 |
Predictive distribution for X[subscript n+] 1 | p. 175 |
Empirical Bayes approach to credibility theory | p. 176 |
Empirical Bayes credibility - Model 1 | p. 177 |
Empirical Bayes credibility - Model 2 | p. 180 |
Problems | p. 183 |
No Claim Discounting in Motor Insurance | p. 191 |
Introduction to No Claim Discount schemes | p. 191 |
Transition in a No Claim Discount system | p. 193 |
Discount classes and movement in NCD schemes | p. 193 |
One-step transition probabilities in NCD schemes | p. 195 |
Limiting distributions and stability in NCD models | p. 198 |
Propensity to make a claim in NCD schemes | p. 204 |
Thresholds for claims when an accident occurs | p. 205 |
The claims rate process in an NCD system | p. 208 |
Reducing heterogeneity with NCD schemes | p. 212 |
Problems | p. 214 |
Generalized Linear Models | p. 221 |
Introduction to linear and generalized linear models | p. 221 |
Multiple linear regression and the normal model | p. 225 |
The structure of generalized linear models | p. 230 |
Exponential families | p. 232 |
Link functions and linear predictors | p. 236 |
Factors and covariates | p. 238 |
Interactions | p. 238 |
Minimally sufficient statistics | p. 244 |
Model selection and deviance | p. 245 |
Deviance and the saturated model | p. 245 |
Comparing models with deviance | p. 248 |
Residual analysis for generalized linear models | p. 252 |
Problems | p. 258 |
Decision and Game Theory | p. 265 |
Introduction | p. 265 |
Game theory | p. 267 |
Zero-sum two-person games | p. 268 |
Minimax and saddle point strategies | p. 270 |
Randomized strategies | p. 273 |
The Prisoner's Dilemma and Nash equilibrium in variable-sum games | p. 278 |
Decision making and risk | p. 280 |
The minimax criterion | p. 283 |
The Bayes criterion | p. 283 |
Utility and expected monetary gain | p. 288 |
Rewards, prospects and utility | p. 290 |
Utility and insurance | p. 292 |
Problems | p. 295 |
References | p. 304 |
Basic Probability Distributions | p. 309 |
Some Basic Tools in Probability and Statistics | p. 313 |
Moment generating functions | p. 313 |
Convolutions of random variables | p. 316 |
Conditional probability and distributions | p. 317 |
The double expectation theorem and E(X) | p. 319 |
The random variable V(X/Y) | p. 322 |
Maximum likelihood estimation | p. 324 |
An Introduction to Bayesian Statistics | p. 327 |
Bayesian statistics | p. 327 |
Conjugate families | p. 328 |
Loss functions and Bayesian inference | p. 329 |
Answers to Selected Problems | p. 335 |
Claims reserving and pricing with run-off triangles | p. 335 |
Loss distributions | p. 335 |
Risk theory | p. 337 |
Ruin theory | p. 338 |
Credibility theory | p. 338 |
No claim discounting in motor insurance | p. 340 |
Generalized linear models | p. 340 |
Decision and game theory | p. 341 |
Index | p. 345 |
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