Contents Utility theory and insurance The individual risk model Collective risk models

Transcription

1 Contents There are stars in the galaxy. That used to be a huge number. But it s only a hundred billion. It s less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers Richard Feynman ( ) 1 Utility theory and insurance Introduction The expected utility model Classes of utility functions Stop-loss reinsurance Exercises The individual risk model Introduction Mixed distributions and risks Convolution Transforms Approximations Normal approximation Translated gamma approximation NP approximation Application: optimal reinsurance Exercises Collective risk models Introduction Compound distributions Convolution formula for a compound cdf Distributions for the number of claims Properties of compound Poisson distributions Panjer s recursion Compound distributions and the Fast Fourier Transform Approximations for compound distributions Individual and collective risk model Loss distributions: properties, estimation, sampling Techniques to generate pseudo-random samples Techniques to compute ML-estimates xv

2 xvi Contents Poisson claim number distribution Negative binomial claim number distribution Gamma claim severity distributions Inverse Gaussian claim severity distributions Mixtures/combinations of exponential distributions Lognormal claim severities Pareto claim severities Stop-loss insurance and approximations Comparing stop-loss premiums in case of unequal variances Exercises Ruin theory Introduction The classical ruin process Some simple results on ruin probabilities Ruin probability and capital at ruin Discrete time model Reinsurance and ruin probabilities Beekman s convolution formula Explicit expressions for ruin probabilities Approximation of ruin probabilities Exercises Premium principles and Risk measures Introduction Premium calculation from top-down Various premium principles and their properties Properties of premium principles Characterizations of premium principles Premium reduction by coinsurance Value-at-Risk and related risk measures Exercises Bonus-malus systems Introduction A generic bonus-malus system Markov analysis Loimaranta efficiency Finding steady state premiums and Loimaranta efficiency Exercises Ordering of risks Introduction Larger risks More dangerous risks Thicker-tailed risks

3 Contents xvii Stop-loss order Exponential order Properties of stop-loss order Applications Individual versus collective model Ruin probabilities and adjustment coefficients Order in two-parameter families of distributions Optimal reinsurance Premiums principles respecting order Mixtures of Poisson distributions Spreading of risks Transforming several identical risks Incomplete information Comonotonic random variables Stochastic bounds on sums of dependent risks Sharper upper and lower bounds derived from a surrogate Simulating stochastic bounds for sums of lognormal risks More related joint distributions; copulas More related distributions; association measures Copulas Exercises Credibility theory Introduction The balanced Bühlmann model More general credibility models The Bühlmann-Straub model Parameter estimation in the Bühlmann-Straub model Negative binomial model for the number of car insurance claims Exercises Generalized linear models Introduction Generalized Linear Models Some traditional estimation procedures and GLMs Deviance and scaled deviance Case study I: Analyzing a simple automobile portfolio Case study II: Analyzing a bonus-malus system using GLM GLM analysis for the total claims per policy Exercises IBNR techniques Introduction Two time-honored IBNR methods Chain ladder

4 xviii Contents Bornhuetter-Ferguson A GLM that encompasses various IBNR methods Chain ladder method as a GLM Arithmetic and geometric separation methods De Vijlder s least squares method Illustration of some IBNR methods Modeling the claim numbers in Table Modeling claim sizes Solving IBNR problems by R Variability of the IBNR estimate Bootstrapping Analytical estimate of the prediction error An IBNR-problem with known exposures Exercises More on GLMs Introduction Linear Models and Generalized Linear Models The Exponential Dispersion Family Fitting criteria Residuals Quasi-likelihood and quasi-deviance Extended quasi-likelihood The canonical link The IRLS algorithm of Nelder and Wedderburn Theoretical description Step-by-step implementation Tweedie s Compound Poisson gamma distributions Application to an IBNR problem Exercises The R in Modern ART A.1 A short introduction to R A.2 Analyzing a stock portfolio using R A.3 Generating a pseudo-random insurance portfolio Hints for the exercises Notes and references Tables Index

5