Preface.

**PART I: INTRODUCTION.**

**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.

**PART II: ACTUARIAL MODELS.**

**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.

**PART III: CONSTRUCTION OF EMPIRICAL MODELS.**

**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.

**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.

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.

**PART V: ADJUSTED ESTIMATES.**

**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.

**PART VI: SIMULATION.**

**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.

References.