- Preface xiii PART I INTRODUCTION
- 1 Modeling
- 3 1.1 The model-based approach
- 3 1.2 Organization of this book
- 5
- 2 Random variables
- 7
- 2.1 Introduction
- 7 2.2 Key functions and four models
- 9
- 3 Basic distributional quantities
- 19 3.1 Moments
- 19 3.2 Percentiles
- 27 3.3 Generating functions and sums of random variables
- 29 3.4 Tails of distributions
- 31 3.5 Measures of Risk
- 38 PART II ACTUARIAL MODELS
- 4 Characteristics of Actuarial Models
- 49
- 4.1 Introduction
- 49 4.2 The role of parameters
- 49
- 5 Continuous models
- 59 5.1 Introduction
- 59 5.2 Creating new distributions
- 59 5.3 Selected distributions and their relationships
- 72 5.4 The linear exponential family
- 75
- 6 Discrete distributions
- 79 6.1 Introduction
- 79 6.2 The Poisson distribution
- 80 6.3 The negative binomial distribution
- 83 6.4 The binomial distribution
- 85 6.5 The (a, b, 0) class
- 86 6.6 Truncation and modification at zero
- 89
- 7 Advanced discrete distributions
- 95 7.1 Compound frequency distributions
- 95 7.2 Further properties of the compound Poisson class
- 101 7.3 Mixed frequency distributions
- 107 7.4 Effect of exposure on frequency
- 114 7.5 An inventory of discrete distributions
- 114
- 8 Frequency and severity with coverage modifications
- 117 8.1 Introduction
- 117 8.2 Deductibles
- 117 8.3 The loss elimination ratio and the effect of inflation for ordinary deductibles
- 122 8.4 Policy limits
- 125 8.5 Coinsurance, deductibles, and limits
- 127 8.6 The impact of deductibles on claim frequency
- 131
- 9 Aggregate loss models
- 137 9.1 Introduction
- 137 9.2 Model choices
- 140 9.3 The compound model for aggregate claims
- 141 9.4 Analytic results
- 155 9.5 Computing the aggregate claims distribution
- 159 9.6 The recursive method
- 161 9.7 The impact of individual policy modifications on aggregate payments
- 173 9.8 The individual risk model
- 176 PART III CONSTRUCTION OF EMPIRICAL MODELS
- 10 Review of mathematical statistics
- 187 10.1 Introduction
- 187 10.2 Point estimation
- 188 10.3 Interval estimation
- 196 10.4 Tests of hypotheses
- 198
- 11 Estimation for complete data
- 203 11.1 Introduction
- 203 11.2 The empirical distribution for complete, individual data
- 207 11.3 Empirical distributions for grouped data
- 211
- 12 Estimation for modified data
- 217 12.1 Point estimation
- 217 12.2 Means, variances, and interval estimation
- 225 12.3 Kernel density models
- 236 12.4 Approximations for large data sets
- 240 PART IV PARAMETRIC STATISTICAL METHODS
- 13 Frequentist estimation
- 253 13.1 Method of moments and percentile matching
- 253 13.2 Maximum likelihood estimation
- 259 13.3 Variance and interval estimation
- 272 13.4 Non-normal confidence intervals
- 280 13.5 Maximum likelihood estimation of decrement probabilities
- 282
- 14 Frequentist Estimation for discrete distributions
- 285 14.1 Poisson
- 285 14.2 Negative binomial
- 289 14.3 Binomial
- 291 14.4 The (a, b, 1) class
- 293 14.5 Compound models
- 297 14.6 Effect of exposure on maximum likelihood estimation
- 299 14.7 Exercises
- 300
- 15 Bayesian estimation
- 305 15.1 Definitions and Bayes theorem
- 305 15.2 Inference and prediction
- 309 15.3 Conjugate prior distributions and the linear exponential family
- 320 15.4 Computational issues
- 322
- 16 Model selection
- 323 16.1 Introduction
- 323 16.2 Representations of the data and model
- 324 16.3 Graphical comparison of the density and distribution functions
- 325 16.4 Hypothesis tests
- 330 16.5 Selecting a model
- 342 PART V CREDIBILITY
- 17 Introduction and Limited Fluctuation Credibility
- 357 17.1 Introduction
- 357 17.2 Limited fluctuation credibility theory
- 359 17.3 Full credibility
- 360 17.4 Partial credibility
- 363 17.5 Problems with the approach
- 366 17.6 Notes and References
- 367 17.7 Exercises
- 367
- 18 Greatest accuracy credibility
- 371 18.1 Introduction
- 371 18.2 Conditional distributions and expectation
- 373 18.3 The Bayesian methodology
- 377 18.4 The credibility premium
- 385 18.5 The Buhlmann model
- 388 18.6 The Buhlmann-Straub model
- 392 18.7 Exact credibility
- 397 18.8 Notes and References
- 401 18.9 Exercises
- 402
- 19 Empirical Bayes parameter estimation
- 415 19.1 Introduction
- 415 19.2 Nonparametric estimation
- 418 19.3 Semiparametric estimation
- 428 19.4 Notes and References
- 430 19.5 Exercises
- 430 PART VI SIMULATION
- 20 Simulation
- 437 20.1 Basics of simulation
- 437 20.2 Simulation for specific distributions
- 442 20.3 Determining the sample size
- 448 20.4 Examples of simulation in actuarial modeling
- 450 Appendix A: An inventory of continuous distributions
- 459 A.1 Introduction
- 459 A.2 Transformed beta family
- 463 A.3 Transformed gamma family
- 467 A.4 Distributions for large losses
- 470 A.5 Other distributions
- 471 A.6 Distributions with finite support
- 473 Appendix B: An inventory of discrete distributions
- 475 B.1 Introduction
- 475 B.2 The (a, b, 0) class
- 476 B.3 The (a, b, 1) class
- 477 B.4 The compound class
- 480 B.5 A hierarchy of discrete distributions
- 482 Appendix C: Frequency and severity relationships
- 483 Appendix D: The recursive formula
- 485 Appendix E: Discretization of the severity distribution
- 487 E.1 The method of rounding
- 487 E.2 Mean preserving
- 488 E.3 Undiscretization of a discretized distribution
- 488 Appendix F: Numerical optimization and solution of systems of equations
- 491 F.1 Maximization using Solver
- 491 F.2 The simplex method
- 495 F.3 Using Excel(R) to solve equations
- 496 References 501.
- (source: Nielsen Book Data)
- 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. Continuous models. 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. Multivariate models. 7.1 Introduction. 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. 8.1 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 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. 12.1 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. Index.
- (source: Nielsen Book Data)
Praise for the Third Edition "This book provides in-depth coverage of modelling techniques used throughout many branches of actuarial science...The exceptional high standard of this book has made it a pleasure to read." Annals of Actuarial Science Newly organized to focus exclusively on material tested in the Society of Actuaries' Exam C and the Casualty Actuarial Society's Exam 4, Loss Models: From Data to Decisions, Fourth Edition continues to supply actuaries with a practical approach to the key concepts and techniques needed on the job. With updated material and extensive examples, the book successfully provides the essential methods for using available data to construct models for the frequency and severity of future adverse outcomes. The book continues to equip readers with the tools needed for the construction and analysis of mathematical models that describe the process by which funds flow into and out of an insurance system. Focusing on the loss process, the authors explore key quantitative techniques including random variables, basic distributional quantities, and the recursive method, and discuss techniques for classifying and creating distributions. Parametric, non-parametric, and Bayesian estimation methods are thoroughly covered along with advice for choosing an appropriate model. New features of this Fourth Edition include: * Expanded discussion of working with large data sets, now including more practical elements of constructing decrement tables * Added coverage of methods for simulating several special situations * An updated presentation of Bayesian estimation, outlining conjugate prior distributions and the linear exponential family as well as related computational issues Throughout the book, numerous examples showcase the real-world applications of the presented concepts, with an emphasis on calculations and spreadsheet implementation. A wealth of new exercises taken from previous Exam C/4 exams allows readers to test their comprehension of the material, and a related FTP site features the book's data sets. Loss Models, Fourth Edition is an indispensable resource for students and aspiring actuaries who are preparing to take the SOA and CAS examinations. The book is also a valuable reference for professional actuaries, actuarial students, and anyone who works with loss and risk models. To explore our additional offerings in actuarial exam preparation visit
www.wiley.com/go/c4actuarial .
(source: Nielsen Book Data)
An update of one of the most trusted books on constructing and analyzing actuarial models 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 .
(source: Nielsen Book Data)