Loss Models: From Data to Decisions, 3rd Edition
Written by three renowned authorities in the actuarial field, Loss Models, Third Edition upholds the reputation for excellence that has made this book required reading for the Society of Actuaries (SOA) and Casualty Actuarial Society (CAS) qualification examinations. This update serves as a complete presentation of statistical methods for measuring risk and building models to measure loss in real-world events.
This book maintains an approach to modeling and forecasting that utilizes tools related to risk theory, loss distributions, and survival models. Random variables, basic distributional quantities, the recursive method, and techniques for classifying and creating distributions are also discussed. Both parametric and non-parametric estimation methods are thoroughly covered along with advice for choosing an appropriate model. Features of the Third Edition include:
- Extended discussion of risk management and risk measures, including Tail-Value-at-Risk (TVaR)
- New sections on extreme value distributions and their estimation
- Inclusion of homogeneous, nonhomogeneous, and mixed Poisson processes
- Expanded coverage of copula models and their estimation
- Additional treatment of methods for constructing confidence regions when there is more than one parameter
The book continues to distinguish itself by providing over 400 exercises that have appeared on previous SOA and CAS examinations. Intriguing examples from the fields of insurance and business are discussed throughout, and all data sets are available on the book's FTP site, along with programs that assist with conducting loss model analysis.
Loss Models, Third Edition is an essential resource for students and aspiring actuaries who are preparing to take the SOA and CAS preliminary examinations. It is also a must-have reference for professional actuaries, graduate students in the actuarial field, and anyone who works with loss and risk models in their everyday work.
To explore our additional offerings in actuarial exam preparation visit www.wiley.com/go/actuarialexamprep.
PART I: INTRODUCTION.
1.1 The model-based approach.
1.2 Organization of this book.
2. Random variables.
2.2 Key functions and four models.
3. Basic distributional quantities.
3.3 Generating functions and sums of random variables.
3.4 Tails of distributions.
3.5 Measures of Risk.
PART II: ACTUARIAL MODELS.
4. Characteristics of actuarial models.
4.2 The role of parameters.
5. Continuous models.
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.
6.2 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. Multivariate models.
7.2 Sklara??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.
8.3 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.
9.2 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.
10.2 Process models for insurance.
10.3 Discrete, finite-time ruin probabilities.
11. Continuous-time ruin models.
11.2 The adjustment coefficient and Lundberga??s inequality.
11.3 An integrodifferential equation.
11.4 The maximum aggregate loss.
11.5 Cramera??s asymptotic ruin formula and Tijms' approximation.
11.6 The Brownian motion risk process.
11.7 Brownian motion and the probability of ruin.
PART III: CONSTRUCTION OF EMPIRICAL MODELS.
12. Review of mathematical statistics.
12.2 Point estimation.
12.3 Interval estimation.
12.4 Tests of hypotheses.
13. Estimation for complete data.
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.
PART IV: PARAMETRIC STATISTICAL METHODS.
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.
16. Model selection.
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.
18.2 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.
PART V: ADJUSTED ESTIMATES.
19. Interpolation and smoothing.
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.
20.2 Limited fluctuation credibility theory.
20.3 Greatest accuracy credibility theory.
20.4 Empirical Bayes parameter estimation.
PART VI: 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.
HARRY H. PANJER, PhD, is Professor Emeritus in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. Past president of both the Canadian Institute of Actuaries and the Society of Actuaries, Dr. Panjer has published numerous articles on risk modeling in the fields of finance and actuarial science.
GORDON E. WILLMOT, PhD, is Munich Re Chair in Insurance and Professor in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. Dr. Willmot has authored or coauthored over sixty published articles in the areas of risk theory, queueing theory, distribution theory, and stochastic modeling in insurance.
Thoroughly revised and updated to include new material related to Exam C (old Exam 4) of the Society of Actuaries' accreditation program
New sections have been added, with coverage on extreme value distributions and their estimation
Additional methods are now discussed, including homogeneous, nonhomogeneous, and mixed Poisson processes
- The authors have expanded the treatment of copula models and their estimation
- Additional treatment of methods for constructing confidence regions when there is more than one parameter, is now included
- Intriguing examples from the fields of insurance and business relate the book's content to real-world events
- A related FTP site features all data sets along with programs that assist with conducting loss model analysis
- Companion software, datasets, and over 400 sample test exercises, with worked-out solutions, are available in various online products
“This book will be necessary for all academic programs in actuarial science. It will also serve as an important reference for practicing actuaries.” (Mathematical Assoc. of America, June 2009)