| |
| |
| Preface | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Modeling | |
| |
| |
| |
| The model-based approach | |
| |
| |
| |
| Organization of this book | |
| |
| |
| |
| Random variables | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Key functions and four models | |
| |
| |
| |
| Basic distributional quantities | |
| |
| |
| |
| Moments | |
| |
| |
| |
| Quantiles | |
| |
| |
| |
| Generating functions and sums of random variables | |
| |
| |
| |
| Tails of distributions | |
| |
| |
| |
| Measures of Risk | |
| |
| |
| |
| Actuarial Models | |
| |
| |
| |
| Characteristics of actuarial models | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| The role of parameters | |
| |
| |
| |
| Continuous models | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Creating new distributions | |
| |
| |
| |
| Selected distributions and their relationships | |
| |
| |
| |
| The linear exponential family | |
| |
| |
| |
| TVaR for continuous distributions | |
| |
| |
| |
| Extreme value distributions | |
| |
| |
| |
| Discrete distributions and processes | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| The Poisson distribution | |
| |
| |
| |
| The negative binomial distribution | |
| |
| |
| |
| The binomial distribution | |
| |
| |
| |
| The (a, b, 0) class | |
| |
| |
| |
| Counting processes | |
| |
| |
| |
| Truncation and modification at zero | |
| |
| |
| |
| Compound frequency models | |
| |
| |
| |
| Further properties of the compound Poisson class | |
| |
| |
| |
| Mixed Poisson distributions | |
| |
| |
| |
| Mixed Poisson processes | |
| |
| |
| |
| Effect of exposure on frequency | |
| |
| |
| |
| An inventory of discrete distributions | |
| |
| |
| |
| TVaR for discrete distributions | |
| |
| |
| |
| Multivariate models | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Sklar�s theorem and copulas | |
| |
| |
| |
| Measures of dependency | |
| |
| |
| |
| Tail dependence | |
| |
| |
| |
| Archimedean copulas | |
| |
| |
| |
| Elliptical copulas | |
| |
| |
| |
| Extreme value copulas | |
| |
| |
| |
| Archimax copulas | |
| |
| |
| |
| Frequency and severity with coverage modifications | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Deductibles | |
| |
| |
| |
| The loss elimination ratio and the effect of inflation for ordinary deductibles | |
| |
| |
| |
| Policy limits | |
| |
| |
| |
| Coinsurance, deductibles, and limits | |
| |
| |
| |
| The impact of deductibles on claim frequency | |
| |
| |
| |
| Aggregate loss models | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Model choices | |
| |
| |
| |
| The compound model for aggregate claims | |
| |
| |
| |
| Analytic results | |
| |
| |
| |
| Computing the aggregate claims distribution | |
| |
| |
| |
| The recursive method | |
| |
| |
| |
| The impact of individual policy modifications on aggregate payments | |
| |
| |
| |
| Inversion methods | |
| |
| |
| |
| Calculations with approximate distributions | |
| |
| |
| |
| Comparison of methods | |
| |
| |
| |
| The individual risk model | |
| |
| |
| |
| TVaR for aggregate losses | |
| |
| |
| |
| Discrete-time ruin models | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Process models for insurance | |
| |
| |
| |
| Discrete, finite-time ruin probabilities | |
| |
| |
| |
| Continuous-time ruin models | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| The adjustment coefficient and Lundberg�s inequality | |
| |
| |
| |
| An integrodifferential equation | |
| |
| |
| |
| The maximum aggregate loss | |
| |
| |
| |
| Cramer�s asymptotic ruin formula and Tijms' approximation | |
| |
| |
| |
| The Brownian motion risk process | |
| |
| |
| |
| Brownian motion and the probability of ruin | |
| |
| |
| |
| Construction Of Empirical Models | |
| |
| |
| |
| Review of mathematical statistics | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Point estimation | |
| |
| |
| |
| Interval estimation | |
| |
| |
| |
| Tests of hypotheses | |
| |
| |
| |
| Estimation for complete data | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| The empirical distribution for complete, individual data | |
| |
| |
| |
| Empirical distributions for grouped data | |
| |
| |
| |
| Estimation for modified data | |
| |
| |
| |
| Point estimation | |
| |
| |
| |
| Means, variances, and interval estimation | |
| |
| |
| |
| Kernel density models | |
| |
| |
| |
| Approximations for large data sets | |
| |
| |
| |
| Parametric Statistical Methods | |
| |
| |
| |
| Parameter estimation | |
| |
| |
| |
| Method of moments and percentile matching | |
| |
| |
| |
| Maximum likelihood estimation | |
| |
| |
| |
| Variance and interval estimation | |
| |
| |
| |
| Non-normal confidence intervals | |
| |
| |
| |
| Bayesian estimation | |
| |
| |
| |
| Estimation for discrete distributions | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Model selection | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Representations of the data and model | |
| |
| |
| |
| Graphical comparison of the density and distribution functions | |
| |
| |
| |
| Hypothesis tests | |
| |
| |
| |
| Selecting a model | |
| |
| |
| |
| Estimation and model selection for more complex models | |
| |
| |
| |
| Extreme value models | |
| |
| |
| |
| Copula models | |
| |
| |
| |
| Models with covariates | |
| |
| |
| |
| Five examples | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Time to death | |
| |
| |
| |
| Time from incidence to report | |
| |
| |
| |
| Payment amount | |
| |
| |
| |
| An aggregate loss example | |
| |
| |
| |
| Another aggregate loss example | |
| |
| |
| |
| Comprehensive exercises | |
| |
| |
| |
| Adjusted Estimates | |
| |
| |
| |
| Interpolation and smoothing | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Polynomial interpolation and smoothing | |
| |
| |
| |
| Cubic spline interpolation | |
| |
| |
| |
| Approximating functions with splines | |
| |
| |
| |
| Extrapolating with splines | |
| |
| |
| |
| Smoothing splines | |
| |
| |
| |
| Credibility | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Limited fluctuation credibility theory | |
| |
| |
| |
| Greatest accuracy credibility theory | |
| |
| |
| |
| Empirical Bayes parameter estimation | |
| |
| |
| |
| Simulation | |
| |
| |
| |
| Simulation | |
| |
| |
| |
| Basics of simulation | |
| |
| |
| |
| Examples of simulation in actuarial modeling | |
| |
| |
| |
| Examples of simulation in finance | |
| |
| |
| |
| An inventory of continuous distributions | |
| |
| |
| |
| An inventory of discrete distributions | |
| |
| |
| |
| Frequency and severity relationships | |
| |
| |
| |
| The recursive formula | |
| |
| |
| |
| Discretization of the severity distribution | |
| |
| |
| |
| Numerical optimization and solution of systems of equations | |
| |
| |
| References | |
| |
| |
| Index | |