| | Preface |
| | |
| | Principal Notation |
| | |
| 1 | Risk Models |
| |
| 1.1 | Introduction |
| 1.2 | The Compound Binomial Model |
| 1.3 | The Compound Poisson Model |
| 1.4 | The Compound Mixed Poisson Model |
| 1.5 | The Compound Negative Binomial Model |
| 1.6 | A Note on the Individual Model |
| 1.7 | A Note on Reinsurance |
| |
| 1.7.1 | Proportional Reinsurance |
| 1.7.2 | Excess of Loss Reinsurance |
|
| 1.8 | Computation of the Distribution of S in the Discrete Case |
| 1.9 | Approximations to S |
| |
| 1.9.1 | The Normal Approximation |
| 1.9.2 | The Translated Gamma Approximation |
| 1.9.3 | The Edgeworth Approximation |
| 1.9.4 | The Normal Power Approximation |
|
| 1.10 | Premium Calculation Principles |
| |
| 1.10.1 | The Expected Value Principle |
| 1.10.2 | The Variance Principle |
| 1.10.3 | The Standard Deviation Principle |
| 1.10.4 | The Modified Variance Principle |
| 1.10.5 | The Principle of Zero Utility |
| 1.10.6 | The Mean Value Principle |
| 1.10.7 | The Exponential Principle |
| 1.10.8 | The Esscher Principle |
| 1.10.9 | The Distortion Principle |
| 1.10.10 | The Percentage Principle |
| 1.10.11 | Desirable Properties |
|
| 1.11 | Risk Measures |
| |
| 1.11.1 | Introduction |
| 1.11.2 | Representation of Convex and Coherent Risk Measures |
|
|
| 2 | Utility Theory |
| |
| 2.1 | The Expected Utility Hypothesis |
| 2.2 | The Zero Utility Premium |
| 2.3 | Optimal Insurance |
| 2.4 | The Position of the Insurer |
| 2.5 | Pareto-Optimal Risk Exchanges |
|
| 3 | Credibility Theory |
| |
| 3.1 | Introduction |
| 3.2 | Bayesian Credibility |
| |
| 3.2.1 | The Poisson-Gamma Model |
| 3.2.2 | The Normal-Normal Model |
| 3.2.3 | Is the Credibility Premium Formula Always Linear? |
|
| 3.3 | Empirical Bayes Credibility |
| |
| 3.3.1 | The Bühlmann Model |
| 3.3.2 | The Bühlmann-Straub Model |
| 3.3.3 | The Bühlmann-Straub Model with Missing Data |
|
| 3.4 | General Bayes Methods |
| 3.5 | Hilbert Space Methods |
| 3.6 | Bonus-Malus Systems |
|
| 4 | Claims Reserving |
| |
| 4.1 | Introduction |
| 4.2 | Classical Claims Reserving Methods |
| |
| 4.2.1 | The Chain-Ladder Method |
| 4.2.2 | The Loss-Development Method |
| 4.2.3 | The Additive Method |
| 4.2.4 | The Cape Cod Method |
| 4.2.5 | The Bornhuetter-Ferguson Method |
| 4.2.6 | The Cross-Classified Model |
|
| 4.3 | The Dirichlet Model |
|
| 5 | The Cramér-Lundberg Model |
| |
| 5.1 | Definition of the Cramér-Lundberg Process |
| 5.2 | A Note on the Model and Reality |
| 5.3 | A Differential Equation for the Ruin Probability |
| 5.4 | The Adjustment Coefficient |
| 5.5 | Lundberg's Inequality |
| 5.6 | The Cramér-Lundberg Approximation |
| 5.7 | Reinsurance and Ruin |
| |
| 5.7.1 | Proportional Reinsurance |
| 5.7.2 | Excess of Loss Reinsurance |
|
| 5.8 | The Severity of Ruin, the Capital Prior to Ruin and
the Distribution of inf {Ct : t ≥0} |
| 5.9 | The Laplace Transform of ψ |
| 5.10 | Approximations to ψ |
| |
| 5.10.1 | Diffusion Approximations |
| 5.10.2 | The deVylder Approximation |
| 5.10.3 | The Beekman-Bowers Approximation |
|
| 5.11 | Subexponential Claim Size Distributions |
| 5.12 | The Time to Ruin |
| 5.13 | Seal's Formulae |
| 5.14 | Finite Time Lundberg Inequalities |
| 5.15 | Capital Injections |
|
| 6 | The Renewal Risk Model |
| |
| 6.1 | Definition of the Renewal Risk Model |
| 6.2 | The Adjustment Coefficient |
| 6.3 | Lundberg's Inequality |
| |
| 6.3.1 | The Ordinary Case |
| 6.3.2 | The General Case |
|
| 6.4 | The Cramér-Lundberg Approximation |
| |
| 6.4.1 | The Ordinary Case |
| 6.4.2 | The General Case |
|
| 6.5 | Diffusion Approximations |
| 6.6 | Subexponential Claim Size Distributions |
| 6.7 | Finite Time Lundberg Inequalities |
|
| 7 | The Ammeter Risk Model |
| |
| 7.1 | Mixed Poisson Risk Processes |
| 7.2 | Definition of the Ammeter Risk Model |
| 7.3 | Lundberg's Inequality and the Cramér-Lundberg Approximation |
| |
| 7.3.1 | The Ordinary Case |
| 7.3.2 | The General Case |
|
| 7.4 | The Subexponential Case |
| 7.5 | Finite Time Lundberg Inequalities |
|
| 8 | Change of Measure Techniques |
| |
| 8.1 | The Method |
| 8.2 | The Cramér-Lundberg Case |
| 8.3 | The Renewal Case |
| |
| 8.3.1 | Markovisation via the Time Since the Last Claim |
| 8.3.2 | Markovisation via the Time till the Next Claim |
|
| 8.4 | The Ammeter Risk Model |
|
| 9 | The Markov Modulated Risk Model |
| |
| 9.1 | Definition of the Markov Modulated Risk Model |
| 9.2 | The Lundberg Exponent and Lundberg's Inequality |
| 9.3 | The Cramér-Lundberg Approximation |
| 9.4 | Subexponential Claim Sizes |
| 9.5 | Finite Time Lundberg Inequalities |
|
| | |
| A | Stochastic Processes |
| B | Martingales |
| C | Renewal Processes |
| D | Brownian Motion |
| E | Random Walds and the Wiener-Hopf Factorisation |
| F | Subexponential Distributions |
| G | Concave and Convex Functions |
| | |
| | |
| | Table of Distribution Functions |
| | |
| | References |
| | |
| | Index |
