Cambridge University Press Risk Modelling in General Insurance: From Principles to Practice Roger J. Gray and Susan M.

Transcription

1 adjustment coefficient, 272 and Cramér Lundberg approximation, 302 existence, 279 and Lundberg s inequality, 272 numerical methods for, 303 properties, 272 and reinsurance (case study), 348 statistical estimation of, 310 aggregate claims, 91 aggregate risk model, see collective risk model asymptotic approximation, 126 Bühlmann model, 157, 176 Bühlmann Straub model, 157, 185 Bayes rule, 157 Bayes Theorem, 159 Bayesian credibility estimate, 170 Bayesian credibility theory, 170 empirical, 176 Bayesian estimation, 157 Bayes loss, 160 Bayesian criterion, 160 binomial/beta model, 161 Poisson/gamma model, 163 Bernoulli trials, independent, 16 Bessel function modified, 105 beta-binomial distribution, 58 between risk variance, 173 binomial distribution, 20 compound, 114 bootstrap, 323 parametric, 133 bootstrap confidence interval, 133 Burr distribution, 48 central moment, 4 claim numbers, 11 distributions for, 12 fitting models to data, 58 mixture distributions for, 56 claim sizes, 11 distributions for, 23 fitting models to data, 58 mixture distributions for, 54 classical risk model, 267 coefficient of variation (c.v.), 237 collateral data, 169 collective risk model, 90, 135, 139 compound distribution, 91 approximations for, 124 asymptotic approximation for, 127 compound binomial, 114 compound geometric, 110 compound mixed Poisson, 108 compound negative binomial, 110 compound Poisson, 103 counting random variable, 91 distribution function, 97 fast Fourier transform algorithm, 120, 123 mean, 93 moment generating function, 98 normal approximation, 125 Panjer recursion algorithm, 116 statistical estimation, 128 step random variable, 91 translated gamma approximation for, 125 variance,

2 390 conditional expectation formula, 5 conditional tail property, 42 conditional variance formula, 5 conjugate prior, 159 convolution power, 95 product, 137 correlation, 4 counting distributions, 11 counting random variable, 91 covariance, 4 Cramér Lundberg approximation, 302 Cramér von Mises distance function, 60, 72 credibility theory, 156 Bühlmann model, 157, 176 Bühlmann Straub model, 157, 185 Bayesian credibility, 170 Bayesian credibility theory, 169 between risk variance, 173 credibility data, 169 credibility estimate, 170 credibility factor, 156, 169 credibility premium, 156, 169 EBCT model 1, 176 EBCT model 2, 185 empirical Bayesian credibility theory (EBCT), 176 normal/normal model, 172 Poisson/gamma model, 163 structural parameters, 180, 189 within risk variance, 173 cumulant, 33, 141 cumulant generating function, 7, 33 random sum, 99 cumulative distribution function, 2 De Pril recursion, 140 decision rule, 160 decision theory, 159 deductible, 205, 223 delta method, 131, 309 direct insurer, 205 discrete Fourier transform, 119 discretisation, 118 distribution χ 2 n, 30 American Pareto, 40 Bernoulli, 22 beta, 57 binomial, 13 Burr, 48 compound, 91 compound geometric, 110 compound mixed Poisson, 108 compound negative binomial, 110 compound Poisson, 103 Erlang, 30 exponential, 25 fat-tailed, 32 finite mixture, 100 gamma, 28 Gaussian, 24 generalised (three-parameter) Pareto, 56 geometric, 18 heavy-tailed, 16 limited, 82 loggamma, 51 lognormal, 33 mixture, 43, 101 negative binomial, 13 normal, 24 Pareto, 40 Poisson, 13 shifted geometric, 110 shifted negative binomial, 110 thin-tailed, 23, 32 transformed Pareto, 49 translated Pareto, 94 Weibull, 45 distribution function, 2 distributions, threshold, 82 empirical Bayesian credibility theory (EBCT), 176 model 1, 176, 182, 184 model 2, 185 empirical distribution function, 9, 67 equal mixture, 102 excess, 205, 223 excess of loss, 206 excess of loss reinsurance, 82, 206 expectation, 3 expected frequency, 63 expected utility criterion, 372 expected value principle (EVP), 148 experience rating, 156 exponential premium principle (EPP), 150 exponentially bounded tail, 31 failure rate, 86 fast Fourier transform (FFT) algorithm, 119 for compound distributions, 120, 123

3 391 vs. Panjer algorithm, for probability of ruin, 306 fat tail, 11, 23, 31, 216 finite mixture distribution, 100 fitted frequency, see expected frequency fitting models to claim numbers, 58, 60 fitting models to claim sizes, 58, 65 force of mortality, 86 Fourier frequencies, 119 fourth central moment, 4, 29, 32 gamma function, 16 incomplete, 28 Gaussian distribution, see normal distribution geometric distribution, 27 compound, 110 shifted, 110 goodness-of-fit criterion, 63 hazard rate, 86 heavy tail, see fat tail, 16 heterogeneous individual risk model, 135 homogeneous individual risk model, 135 iid, 5, 216 independent random variables, 5 individual risk model, 90, 134, 138 compound Poisson approximation, 139 distribution function, 137 heterogeneous, 135 homogeneous, 135 mean, 136 moment generating function, 137 normal approximation, 139 skewness, 146 variance, 136, 146 insurance loss, 23 jth cumulant, 33 Jensen s inequality, 378 Kolomogorov Smirnov (K S) test statistic, 67 kurtosis coefficient of, 4, 32, 33 in terms of cumulants, 33 excess, 33 exponential distribution, 33 gamma distribution, 29 lack of memory property, 27 Lebesgue Stieltjes integral, 7 leptokurtic distribution, 32 likelihood function, 59, 158 loggamma distribution, 51 lognormal distribution, 33, 35 meanlog, 35 sdlog, 35 loss distributions, 23 loss function, 159, 160 Lundberg exponent, see adjustment coefficient Lundberg s inequality, 272, 302, 348 marginal distribution, 55 maximum likelihood estimator (MLE), 9, 59, 129, 309 meanlog, 35 method of moments, 58 method of moments estimator (MME), 58 method of percentiles, 60 minimax criterion, 159 minimum distance estimation, 60, 71 mixed Poisson distribution, compound, 108 mixture distribution, 43, 54 finite, 100 mixing distribution, 55 mixing proportions, 101 moment, 4 moment generating function, 6, 7, 28 compound distribution, 98 individual risk model, 137 random sum, 98 motor insurance, 13 negative binomial distribution, 13, 16, 110 compound, 110 shifted, 110 nested model, 64 net profit condition, 269 normal approximation for compound distribution, 125 individual risk model, 139 normal distribution, 24, 32 standardisation to N(0, 1), 25 normal/normal model, 165, 172 P-value, 63 Panjer recursion algorithm, 116 vs. FFT, parametric bootstrap, 133 Pareto distribution generalised (three-parameter), 56 transformed, 49

4 392 translated, 94 Pareto loss, 338 Pearson criterion, 63 plug-in estimator, 132, 315 Poisson distribution, 13, 20 compound, 103 compound mixed, 108 Poisson process, 12, 15, 22 in classical risk model, 267 inter-event times, 27 Poisson/gamma model, 163, 170 policy excess, 205, 223 posterior distribution, 158 power law decay, 40 premium, 147 credibility, 173 pure, 147 premium calculation principle, 148 desirable properties of, 152 expected value principle (EVP), 148 exponential premium principle (EPP), 150 quantile principle (QP), 149 standard deviation principle (SDP), 149 variance principle (VP), 149 zero utility principle (ZUP), 150 premium setting principles case study, 316 principal insurer, 205 prior distribution, 54, 158 probability density function, 2 probability generating function, 6 probability mass function, 3 probability of ruin, see ruin probability proportional reinsurance, 221, 351 quantile principle (QP), 149 quota share reinsurance, 222 R, 9, 22, 24 random sum, 91, 92 distribution function of, 97 mean, 93 moment generating function, 98 variance, 93 ratelog, 51 reference distribution, 23 reinsurance, 205, 218 case study, 332, 348 excess of loss, 206 function, 235, 349 layer, 356 optimising, 228 proportional, 206, 221, 235, 351 quota share, 222 stop loss, 235 reinsurance claim, 210 reinsurer, 205 relative safety loading, 270 relative security loading, 148, 149, 270 renewal theory, 296 renewal-type equation, 296 resampling, 323 retention level, 206 risk aversion, 151, 230, 233, 373, 375 coefficient of, 376 risk function, 160 Bayes, 160 risk loading, 148 risk model classical, 267 collective, 90 individual, 90, 134 short term, 90 risk parameter, 54, 172 risk retention, 227 risk sharing, 205, 332 ruin probability, 270 asymptotics, 296 case study, 348 compound geometric tail representation, 291, 295 finite-time, 270 integral equation for, 289 integro-differential equation for, 284 Lundberg s inequality for, 272 numerical methods for, 305 reinsurance and ruin, 348 statistical estimation, 309 sdlog, 35 shapelog, 51 shared liabilities case study, 332 shifted geometric distribution, 110 shifted negative binomial distribution, 110 short term risk models, 90 simulation, 9 inverse transform method, 26 skewness, 4 coefficient of, 4 in terms of cumulants, 33 standard deviation, 4

5 393 standard deviation principle (SDP), 149 standardised random variable, 143 states of nature, 159 statistical estimation, 58, 128, 308 step random variable, 91 stop loss reinsurance, 235 structural parameters, 180, 189 Student s t distribution, 32 survival probability, 271 compound geometric representation, 291, 295 integral equation for, 286 integro-differential equation for, 284 tail, 2 fat, 11, 23, 31 heavy, 11, 23, 31 thin, 23, 32 total claim amount, 91 transformed beta family, 56 Burr, 56 generalised (three-parameter) Pareto, 56 Pareto, 40, 42 translated gamma approximation, 125 uncertainty reduction, 234 unit mass at zero, 102 utility, 147, 368 utility function, 369, 370 variance, 4 variance principle (VP), 149 Weibull distribution, 45 alternative parameterisation, 46 within risk variance, 173 zero utility principle (ZUP), 150