Risk Modelling in General Insurance From Principles to Practice 1st Edition by Roger J Gray, Susan M Pitts ISBN 0521863945 9780521863940

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ISBN 10: 0521863945 
ISBN 13: 9780521863940
Author: Roger J. Gray, Susan M. Pitts

Knowledge of risk models and the assessment of risk is a fundamental part of the training of actuaries and all who are involved in financial, pensions and insurance mathematics. This book provides students and others with a firm foundation in a wide range of statistical and probabilistic methods for the modelling of risk, including short-term risk modelling, model-based pricing, risk-sharing, ruin theory and credibility. It covers much of the international syllabuses for professional actuarial examinations in risk models, but goes into further depth, with worked examples, exercises and detailed case studies. The authors also use the statistical package R to demonstrate how simple code and functions can be used profitably in an actuarial context. The authors’ engaging and pragmatic approach, balancing rigour and intuition and developed over many years of teaching the subject, makes this book ideal for self-study or for students taking courses in risk modelling.

Risk Modelling in General Insurance From Principles to Practice 1st Table of contents:

1: Introduction
1.1 The aim of this book
1.2 Notation and prerequisites
1.2.1 Probability
1.2.2 Statistics
1.2.3 Simulation
1.2.4 The statistical software package R
2: Models for claim numbers and claim sizes
2.1 Distributions for claim numbers
2.1.1 Poisson distribution
2.1.2 Negative binomial distribution
2.1.3 Geometric distribution
2.1.4 Binomial distribution
2.1.5 A summary note on R
2.2 Distributions for claim sizes
2.2.1 A further summary note on R
2.2.2 Normal (Gaussian) distribution
2.2.3 Exponential distribution
2.2.4 Gamma distribution
2.2.5 Fat-tailed distributions
2.2.6 Lognormal distribution
2.2.7 Pareto distribution
2.2.8 Weibull distribution
2.2.9 Burr distribution
2.2.10 Loggamma distribution
2.3 Mixture distributions
2.4 Fitting models to claim-number and claim-size data
2.4.1 Fitting models to claim numbers
2.4.2 Fitting models to claim sizes
Exercises
3: Short term risk models
3.1 The mean and variance of a compound distribution
3.2 The distribution of a random sum
3.2.1 Convolution series formula for a compound distribution
3.2.2 Moment generating function of a compound distribution
3.3 Finite mixture distributions
3.4 Special compound distributions
3.4.1 Compound Poisson distributions
3.4.2 Compound mixed Poisson distributions
3.4.3 Compound negative binomial distributions
3.4.4 Compound binomial distributions
3.5 Numerical methods for compound distributions
3.5.1 Panjer recursion algorithm
3.5.2 The fast Fourier transform algorithm
3.6 Approximations for compound distributions
3.6.1 Approximations based on a few moments
3.6.2 Asymptotic approximations
3.7 Statistics for compound distributions
3.8 The individual risk model
3.8.1 The mean and variance for the individual risk model
3.8.2 The distribution function and moment generating function for the individual risk model
3.8.3 Approximations for the individual risk model
Exercises
4: Model based pricing – setting premiums
4.1 Premium calculation principles
4.1.1 The expected value principle (EVP)
4.1.2 The standard deviation principle (SDP)
4.1.3 The variance principle (VP)
4.1.4 The quantile principle (QP)
4.1.5 The zero utility principle (ZUP)
4.1.6 The exponential premium principle (EPP)
4.1.7 Some desirable properties of premium calculation principles
4.1.8 Other premium calculation principles
4.2 Maximum and minimum premiums
4.3 Introduction to credibility theory
4.4 Bayesian estimation
4.4.1 The posterior distribution
4.4.2 The wider context of decision theory
4.4.3 The binomial/beta model
4.4.4 The Poisson/gamma model
4.4.5 The normal/normal model
4.5 Bayesian credibility theory
4.5.1 Bayesian credibility estimates under the Poisson/gamma model
4.5.2 Bayesian credibility premiums under the normal/normal model
4.6 Empirical Bayesian credibility theory: Model 1 – the Bühlmann model
4.7 Empirical Bayesian credibility theory: Model 2 – the Bühlmann–Straub model
Exercises
5: Risk sharing – reinsurance and deductibles
5.1 Excess of loss reinsurance
5.1.1 Reinsurance claims
5.1.2 Simulation results
5.1.3 Aggregate claims model with excess of loss reinsurance
5.2 Proportional reinsurance
5.3 Deductibles (policy excesses)
5.4 Retention levels and reinsurance costs
5.5 Optimising the reinsurance contract
5.6 Optimising reinsurance contracts based on maximising expected utility
5.6.1 Excess of loss reinsurance
5.6.2 Proportional reinsurance
5.7 Optimising reinsurance contracts based on minimising the variance of aggregate claims
5.7.1 Minimising Var[S I] subject to fixed E[S I]5.7.2 Minimising Var[S R] subject to fixed Var[S I]5.7.3 Comparing stop loss and equivalent proportional reinsurance arrangements
5.7.4 Minimising Var[S I] + Var[S R]5.7.5 Minimising the sum of variances when two independent risks are shared between two insurers
5.8 Optimising reinsurance contracts for a group of independent risks based on minimising the varian
5.8.1 Optimal relative retentions in the case of excess of loss reinsurance
5.8.2 Optimal relative retentions in the case of proportional reinsurance
Exercises
6: Ruin theory for the classical risk model
6.1 The classical risk model
6.1.1 The relative safety loading
6.1.2 Ruin probabilities
6.2 Lundberg’s inequality and the adjustment coefficient
6.2.1 Properties of the adjustment coefficient
6.2.2 Proof of Lundberg’s inequality
6.2.3 When does the adjustment coefficient exist?
6.3 Equations for ψ(u) and φ(u): the ruin probability and the survival probability
6.4 Compound geometric representations for ψ(u) and φ(u): the ruin probability and the survival pr
6.5 Asymptotics for the probability of ruin
6.6 Numerical methods for ruin quantities
6.6.1 Numerical calculation of the adjustment coefficient
6.6.2 Numerical calculation of the probability of ruin
Exercises
7: Case studies
7.1 Case study 1: comparing premium setting principles
7.1.1 Case 1 – in the presence of an assumed model
7.1.2 Case 2 – without model assumptions, using bootstrap resampling
7.2 Case study 2: shared liabilities – who pays what?
7.2.1 Case 1 – exponential losses
7.2.2 Case 2 – Pareto losses
7.2.3 Case 3 – lognormal losses
7.3 Case study 3: reinsurance and ruin
7.3.1 Introduction
7.3.2 Proportional reinsurance
7.3.3 Proportional reinsurance with exponential claim sizes
7.3.4 Excess of loss reinsurance in a layer
7.3.5 Excess of loss reinsurance in a layer with exponential claim sizes

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