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Loss Models: From Data to Decisions

Stuart A. Klugman, Harry H. Panjer, and Gordon E. Willmot
John Wiley
Publication Date: 
Number of Pages: 
Wiley Series in Probability and Statistics
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Samuel Broverman
, on

An understanding of loss processes is a crucial componenent in the determination of appropriate insurance coverage and insurance costs and is fundamental to actuarial work. This book provides comprehensive coverage of the mathematical, statistical and probabilistic foundations of loss models in general. The book has many examples and a large number of exercises.

In this 3rd edition of the book, the authors have incorporated several topics that have become of increasing importance to actuaries in recent years. In particular, a discussion of Tail Value at Risk (TVaR) has been added as well as coverage of copula models and their estimation. Previous editions included a discussion of the exponential family of distributions and their application to Bayesian credibility, but the exposition has been improved in this edition, and the concepts will be more accessible to students studying this topic. The final chapter of the book covers simulation, but one gets the feeling that the coverage of the topic is a little bit rushed. I would recommend that this topic get a more thorough treatment in a future edition.

Portions of earlier editions of the book have been used as the official reference material for some topics tested on the professional examinations of the Society of Actuaries and Casualty Actuarial Society. Beginning with the Fall 2009 session of the SOA/CAS examinations, this edition will be the sole official reference for the examination on Construction and Evaluation of Actuarial Models (Exam C).

This book has developed into a valuable reference for actuaries, covering conprehensively the topics related to modelling and estimation of models. The book reviews topics from the elementary concepts of random variables and basic distributional quantities to risk theory and estimation of non-parametric and parametric distributions. As an official reference for the SOA/CAS exams, this book will be necessary for all academic programs in actuarial science. It will also serve as an important reference for practisiing actuaries.

Samuel Broverman is Professor of Actuarial Science at the University of Toronto.




1. Modeling. 

1.1 The model-based approach.

1.2 Organization of this book.

2. Random variables. 

2.1 Introduction.

2.2 Key functions and four models.

3. Basic distributional quantities. 

3.1 Moments.

3.2 Quantiles.

3.3 Generating functions and sums of random variables.

3.4 Tails of distributions.

3.5 Measures of Risk.


4. Characteristics of Actuarial Models. 

4.1 Introduction.

4.2 The role of parameters.

5. Continuousmodels. 

5.1 Introduction.

5.2 Creating new distributions.

5.3 Selected distributions and their relationships.

5.4 The linear exponential family.

5.5 TVaR for continuous distributions.

5.6 Extreme value distributions.

6. Discrete distributions and processes. 

6.1 Introduction.

6.2 The Poisson distribution.

6.3 The negative binomial distribution.

6.4 The binomial distribution.

6.5 The (a, b, 0) class.

6.6 Counting processes .

6.7 Truncation and modification at zero.

6.8 Compound frequency models.

6.9 Further properties of the compound Poisson class.

6.10 Mixed Poisson distributions.

6.11 Mixed Poisson processes.

6.12 Effect of exposure on frequency.

6.13 An inventory of discrete distributions.

6.14 TVaR for discrete distributions.

7. Multivariatemodels. 

7.1 Introduction.

7.2 Sklar’s theorem and copulas.

7.3 Measures of dependency.

7.4 Tail dependence.

7.5 Archimedean copulas.

7.6 Elliptical copulas.

7.7 Extreme value copulas.

7.8 Archimax copulas.

8. Frequency and severity with coverage modifications. 

8.1 Introduction.

8.2 Deductibles.

8.3 The loss elimination ratio and the effect of inflation for ordinary deductibles.

8.4 Policy limits.

8.5 Coinsurance, deductibles, and limits.

8.6 The impact of deductibles on claim frequency.

9. Aggregate loss models. 

9.1 Introduction.

9.2 Model choices.

9.3 The compound model for aggregate claims.

9.4 Analytic results.

9.5 Computing the aggregate claims distribution.

9.6 The recursive method.

9.7 The impact of individual policy modifications on aggregate payments.

9.8 Inversion methods.

9.9 Calculations with approximate distributions.

9.10 Comparison of methods.

9.11 The individual risk model.

9.12 TVaR for aggregate losses.

10. Discrete-time ruin models. 

10.1 Introduction.

10.2 Process models for insurance.

10.3 Discrete, finite-time ruin probabilities.

11. Continuous-time ruin models. 

11.1 Introduction.

11.2 The adjustment coefficient and Lundberg’s inequality.

11.3 An integrodifferential equation.

11.4 The maximum aggregate loss.

11.5 Cramer’s asymptotic ruin formula and Tijms? approximation.

11.6 The Brownian motion risk process.

11.7 Brownian motion and the probability of ruin.


12. Review of mathematical statistics. 

12.1 Introduction.

12.2 Point estimation.

12.3 Interval estimation.

12.4 Tests of hypotheses.

13. Estimation for complete data. 

13.1 Introduction.

13.2 The empirical distribution for complete, individual data.

13.3 Empirical distributions for grouped data.

14. Estimation for modified data. 

14.1 Point estimation.

14.2 Means, variances, and interval estimation.

14.3 Kernel density models.

14.4 Approximations for large data sets.


15. Parameter estimation. 

15.1 Method of moments and percentile matching.

15.2 Maximum likelihood estimation.

15.3 Variance and interval estimation.

15.4 Non-normal confidence intervals.

15.5 Bayesian estimation.

15.6 Estimation for discrete distributions.

15.6.7 Exercises.

16. Model selection. 

16.1 Introduction.

16.2 Representations of the data and model.

16.3 Graphical comparison of the density and distribution functions.

16.4 Hypothesis tests.

16.5 Selecting a model.

17. Estimation and model selection for more complex models. 

17.1 Extreme value models.

17.2 Copula models.

17.3 Models with covariates.

18. Five examples. 

18.1 Introduction.

18.2 Time to death.

18.3 Time from incidence to report.

18.4 Payment amount.

18.5 An aggregate loss example.

18.6 Another aggregate loss example.

18.7 Comprehensive exercises.


19. Interpolation and smoothing. 

19.1 Introduction.

19.2 Polynomial interpolation and smoothing.

19.3 Cubic spline interpolation.

19.4 Approximating functions with splines.

19.5 Extrapolating with splines.

19.6 Smoothing splines.

20. Credibility. 

20.1 Introduction.

20.2 Limited fluctuation credibility theory.

20.3 Greatest accuracy credibility theory.

20.4 Empirical Bayes parameter estimation.


21. Simulation. 

21.1 Basics of simulation.

21.2 Examples of simulation in actuarial modeling.

21.3 Examples of simulation in finance.

Appendix A: An inventory of continuous distributions.

Appendix B: An inventory of discrete distributions.

Appendix C: Frequency and severity relationships.

Appendix D: The recursive formula.

Appendix E: Discretization of the severity distribution.

Appendix F: Numerical optimization and solution of systems of equations.