Actuarial Notes

Nonlife Actuarial Models: Theory, Methods and Evaluation

By Yiu-Kuen Tse · Cambridge University Press · 2009

Study guides: Exam MAS-I, Exam MAS-II

A textbook for senior undergraduates on the theory, methods and empirical implementation of nonlife actuarial models. Its preface says it covers all the topics of Exam C of the Society of Actuaries (Exam 4 of the Casualty Actuarial Society) as per the 2007 Basic Education Catalog, and some beyond them, such as risk measures and ruin theory. It is in four parts — loss models, risk and ruin, credibility, and model construction and evaluation — with an appendix reviewing statistics, many worked examples, and Excel computation notes in several chapters.

On the syllabus

  • Exam MAS-I — objectives B1–B4 and B7–B9; Sections 1.1, 2.2, 2.3, 2.4, 2.5, 10.1, 10.2, 11.1, 12.1, 12.3 and 13.2.
  • Exam MAS-II — objectives A1–A2; Chapters 6.1–6.3, 7.1–7.4, 8.1–8.2 and 9.1–9.2.

Preface

Notation and convention

Part I Loss models

1 Claim-frequency distribution

  • 1.1 Claim frequency, claim severity, and aggregate claim
  • 1.2 Review of statistics
  • 1.3 Some discrete distributions for claim frequency
  • 1.4 The (a, b, 0) class of distributions
  • 1.5 Some methods for creating new distributions
    • 1.5.1 Compound distribution
    • 1.5.2 Mixture distribution
  • 1.6 Excel computation notes
  • 1.7 Summary and conclusions
  • Exercises

2 Claim-severity distribution

  • 2.1 Review of statistics
    • 2.1.1 Survival function and hazard function
    • 2.1.2 Mixed distribution
    • 2.1.3 Expected value of function of random variable
    • 2.1.4 Distribution of function of random variable
  • 2.2 Some continuous distributions for claim severity
  • 2.3 Some methods for creating new distributions
  • 2.4 Tail properties of claim severity
  • 2.5 Effects of coverage modifications
    • 2.5.1 Deductible
    • 2.5.2 Policy limit
    • 2.5.3 Coinsurance
    • 2.5.4 Effects of inflation
    • 2.5.5 Effects of deductible on claim frequency
  • 2.6 Excel computation notes
  • 2.7 Summary and conclusions
  • Exercises

3 Aggregate-loss models

  • 3.1 Individual risk and collective risk models
  • 3.2 Individual risk model
    • 3.2.1 Exact distribution using convolution
    • 3.2.2 Exact distribution using the De Pril recursion
    • 3.2.3 Approximations of the individual risk model
  • 3.3 Collective risk model
    • 3.3.1 Properties of compound distributions
    • 3.3.2 Panjer recursion
    • 3.3.3 Approximations of the collective risk model
    • 3.3.4 Compound Poisson distribution and individual risk model
  • 3.4 Coverage modifications and stop-loss reinsurance
  • 3.5 Summary and conclusions
  • Exercises

Part II Risk and ruin

4 Risk measures

  • 4.1 Uses of risk measures
  • 4.2 Some premium-based risk measures
  • 4.3 Axioms of coherent risk measures
  • 4.4 Some capital-based risk measures
    • 4.4.1 Value-at-Risk (VaR)
    • 4.4.2 Conditional tail expectation and related measures
  • 4.5 More premium-based risk measures
    • 4.5.1 Proportional hazard transform and risk-adjusted premium
    • 4.5.2 Esscher transform and risk-adjusted premium
  • 4.6 Distortion-function approach
  • 4.7 Wang transform
  • 4.8 Summary and conclusions
  • Exercises

5 Ruin theory

  • 5.1 Discrete-time surplus and events of ruin
  • 5.2 Discrete-time ruin theory
    • 5.2.1 Ultimate ruin in discrete time
    • 5.2.2 Finite-time ruin in discrete time
    • 5.2.3 Lundberg’s inequality in discrete time
  • 5.3 Continuous-time surplus function
  • 5.4 Continuous-time ruin theory
    • 5.4.1 Lundberg’s inequality in continuous time
    • 5.4.2 Distribution of deficit
  • 5.5 Summary and conclusions
  • Exercises

Part III Credibility

6 Classical credibility

  • 6.1 Framework and notations
  • 6.2 Full credibility
    • 6.2.1 Full credibility for claim frequency
    • 6.2.2 Full credibility for claim severity
    • 6.2.3 Full credibility for aggregate loss
    • 6.2.4 Full credibility for pure premium
  • 6.3 Partial credibility
  • 6.4 Variation of assumptions
  • 6.5 Summary and discussions
  • Exercises

7 Bühlmann credibility

8 Bayesian approach

  • 8.1 Bayesian inference and estimation
    • 8.1.1 Posterior distribution of parameter
    • 8.1.2 Loss function and Bayesian estimation
    • 8.1.3 Some examples of Bayesian credibility
  • 8.2 Conjugate distributions
    • 8.2.1 The gamma–Poisson conjugate distribution
    • 8.2.2 The beta–geometric conjugate distribution
    • 8.2.3 The gamma–exponential conjugate distribution
  • 8.3 Bayesian versus Bühlmann credibility
  • 8.4 Linear exponential family and exact credibility
  • 8.5 Summary and discussions
  • Exercises

9 Empirical implementation of credibility

  • 9.1 Empirical Bayes method
  • 9.2 Nonparametric estimation
  • 9.3 Semiparametric estimation
  • 9.4 Parametric estimation
  • 9.5 Summary and discussions
  • Exercises

Part IV Model construction and evaluation

10 Model estimation and types of data

  • 10.1 Estimation
    • 10.1.1 Parametric and nonparametric estimation
    • 10.1.2 Point and interval estimation
    • 10.1.3 Properties of estimators
  • 10.2 Types of data
    • 10.2.1 Duration data and loss data
    • 10.2.2 Complete individual data
    • 10.2.3 Incomplete individual data
    • 10.2.4 Grouped data
  • 10.3 Summary and discussions
  • Exercises

11 Nonparametric model estimation

  • 11.1 Estimation with complete individual data
    • 11.1.1 Empirical distribution
    • 11.1.2 Kernel estimation of probability density function
  • 11.2 Estimation with incomplete individual data
    • 11.2.1 Kaplan–Meier (product-limit) estimator
    • 11.2.2 Nelson–Aalen estimator
  • 11.3 Estimation with grouped data
  • 11.4 Excel computation notes
  • 11.5 Summary and discussions
  • Exercises

12 Parametric model estimation

  • 12.1 Methods of moments and percentile matching
  • 12.2 Bayesian estimation method
  • 12.3 Maximum likelihood estimation method
    • 12.3.1 Complete individual data
    • 12.3.2 Grouped and incomplete data
  • 12.4 Models with covariates
  • 12.5 Modeling joint distribution using copula
  • 12.6 Excel computation notes
  • 12.7 Summary and discussions
  • Exercises

13 Model evaluation and selection

  • 13.1 Graphical methods
  • 13.2 Misspecification tests and diagnostic checks
    • 13.2.1 Kolmogorov–Smirnov test
    • 13.2.2 Anderson–Darling test
    • 13.2.3 Chi-square goodness-of-fit test
    • 13.2.4 Likelihood ratio test
  • 13.3 Information criteria for model selection
  • 13.4 Summary and discussions
  • Exercises

14 Basic Monte Carlo methods

  • 14.1 Monte Carlo simulation
  • 14.2 Uniform random number generators
  • 14.3 General random number generators
    • 14.3.1 Inversion method
    • 14.3.2 Acceptance–rejection method
    • 14.3.3 Generation of correlated random variables
  • 14.4 Specific random number generators
    • 14.4.1 Some continuous distributions
    • 14.4.2 Some discrete distributions
  • 14.5 Accuracy and Monte Carlo sample size
  • 14.6 Variance reduction techniques
    • 14.6.1 Antithetic variable
    • 14.6.2 Control variable
    • 14.6.3 Importance sampling
  • 14.7 Excel computation notes
  • 14.8 Summary and discussions
  • Exercises

15 Applications of Monte Carlo methods

  • 15.1 Monte Carlo simulation for hypothesis test
    • 15.1.1 Kolmogorov–Smirnov test
    • 15.1.2 Chi-square goodness-of-fit test
  • 15.2 Bootstrap estimation of p-value
  • 15.3 Bootstrap estimation of bias and mean squared error
  • 15.4 A general framework of bootstrap
  • 15.5 Monte Carlo simulation of asset prices
    • 15.5.1 Wiener process and generalized Wiener process
    • 15.5.2 Diffusion process and lognormal distribution
    • 15.5.3 Jump–diffusion process
  • 15.6 Summary and discussions
  • Exercises

Appendix: Review of statistics

Answers to exercises

References

Index

Related readings

  • Introduction to Mathematical Statistics (Hogg et al. - 2018) — assigned with it for Domain B (Statistics) of the MAS-I outline

Sources