Modern Actuarial Risk Theory : Using R

Modern Actuarial Risk Theory Contains What Every Actuary Needs To Know About Non-life Insurance Mathematics. It Starts With The Standard Material Like Utility Theory, Individual And Collective Model And Basic Ruin Theory. Other Topics Are Risk Measures And Premium Principles, Bonus-malus Systems, Ordering Of Risks And Credibility Theory. It Also Contains Some Chapters About Generalized Linear Models, Applied To Rating And Ibnr Problems. As To The Level Of The Mathematics, The Book Would Fit In A Bachelors Or Masters Program In Quantitative Economics Or Mathematical Statistics. This Second And Much Expanded Edition Emphasizes The Implementation Of These Techniques Through The Use Of R. This Free But Incredibly Powerful Software Is Rapidly Developing Into The De Facto Standard For Statistical Computation, Not Just In Academic Circles But Also In Practice. With R, One Can Do Simulations, Find Maximum Likelihood Estimators, Compute Distributions By Inverting Transforms, And Much More.--publisher. 1. Utility Theory And Insurance -- 1.1. Introduction -- 1.2. The Expected Utility Model -- 1.3. Classes Of Utility Functions -- 1.4. Stop-loss Reinsurance -- 1.5. Exercises -- 2. The Individual Risk Model -- 2.1. Introduction -- 2.2. Mixed Distributions And Risks -- 2.3. Convolution -- 2.4. Transforms -- 2.5. Approximations -- 2.5.1. Normal Approximation -- 2.5.2. Translated Gamma Approximation -- 2.5.3. Np Approximation -- 2.6. Application: Optimal Reinsurance -- 2.7. Exercises -- 3. Collective Risk Models -- 3.1. Introduction -- 3.2. Compound Distributions -- 3.2.1. Convolution Formula For A Compound Cdf -- 3.3. Distributions For The Number Of Claims -- 3.4. Properties Of Compound Poisson Distributions -- 3.5. Panjer's Recursion -- 3.6. Compound Distributions And The Fast Fourier Transform -- 3.7. Approximations For Compound Distributions -- 3.8. Individual And Collective Risk Model -- 3.9. Loss Distributions: Properties, Estimation, Sampling -- 3.9.1. Techniques To Generate Pseudo-random Samples -- 3.9.2. Techniques To Compute Ml-estimates -- 3.9.3. Poisson Claim Number Distribution -- 3.9.4. Negative Binomial Claim Number Distribution -- 3.9.5. Gamma Claim Severity Distributions -- 3.9.6. Inverse Gaussian Claim Severity Distributions -- 3.9.7. Mixtures/combinations Of Exponential Distributions -- 3.9.8. Lognormal Claim Severities -- 3.9.9. Pareto Claim Severities -- 3.10. Stop-loss Insurance And Approximations -- 3.10.1. Comparing Stop-loss Premiums In Case Of Unequal Variances 76 -- 3.11. Exercises -- 4. Ruin Theory -- 4.1. Introduction -- 4.2. The Classical Ruin Process -- 4.3. Some Simple Results On Ruin Probabilities -- 4.4. Ruin Probability And Capital At Ruin -- 4.5. Discrete Time Model -- 4.6. Reinsurance And Ruin Probabilities -- 4.7. Beekman's Convolution Formula -- 4.8. Explicit Expressions For Ruin Probabilities -- 4.9. Approximation Of Ruin Probabilities -- 4.10. Exercises -- 5. Premium Principles And Risk Measures -- 5.1. Introduction -- 5.2. Premium Calculation From Top-down -- 5.3. Various Premium Principles And Their Properties -- 5.3.1. Properties Of Premium Principles -- 5.4. Characterizations Of Premium Principles -- 5.5. Premium Reduction By Coinsurance -- 5.6. Value-at-risk And Related Risk Measures -- 5.7. Exercises -- 6. Bonus-malus Systems -- 6.1. Introduction -- 6.2. A Generic Bonus-malus System -- 6.3. Markov Analysis -- 6.3.1. Loimaranta Efficiency -- 6.4. Finding Steady State Premiums And Loimaranta Efficiency -- 6.5. Exercises -- 7. Ordering Of Risks -- 7.1. Introduction -- 7.2. Larger Risks -- 7.3. More Dangerous Risks -- 7.3.1. Thicker-tailed Risks -- 7.3.2. Stop-loss Order -- 7.3.3. Exponential Order -- 7.3.4. Properties Of Stop-loss Order -- 7.4. Applications -- 7.4.1. Individual Versus Collective Model -- 7.4.2. Ruin Probabilities And Adjustment Coefficients -- 7.4.3. Order In Two-parameter Families Of Distributions -- 7.4.4. Optimal Reinsurance -- 7.4.5. Premiums Principles Respecting Order -- 7.4.6. Mixtures Of Poisson Distributions -- 7.4.7. Spreading Of Risks -- 7.4.8. Transforming Several Identical Risks -- 7.5. Incomplete Information -- 7.6. Comonotonic Random Variables -- 7.7. Stochastic Bounds On Sums Of Dependent Risks -- 7.7.1. Sharper Upper And Lower Bounds Derived From A Surrogate -- 7.7.2. Simulating Stochastic Bounds For Sums Of Lognormal Risks -- 7.8. More Related Joint Distributions -- 7.8.1. More Related Distributions -- 7.8.2. Copulas -- 7.9. Exercises -- 8. Credibility Theory -- 8.1. Introduction -- 8.2. The Balanced Buhlmann Model -- 8.3. More General Credibility Models -- 8.4. The Buhlmann-straub Model -- 8.4.1. Parameter Estimation In The Buhlmann-straub Model -- 8.5. Negative Binomial Model For The Number Of Car Insurance Claims -- 8.6. Exercises -- 9. Generalized Linear Models -- 9.1. Introduction -- 9.2. Generalized Linear Models -- 9.3. Some Traditional Estimation Procedures And Glms -- 9.4. Deviance And Scaled Deviance -- 9.5. Case Study I: Analyzing A Simple Automobile Portfolio -- 9.6. Case Study Ii: Analyzing A Bonus-malus System Using Glm -- 9.6.1. Glm Analysis For The Total Claims Per Policy -- 9.7. Exercises -- 10. Ibnr Techniques -- 10.1. Introduction -- 10.2. Two Time-honored Ibnr Methods -- 10.2.1. Chain Ladder -- 10.2.2. Bornhuetter-ferguson -- 10.3. A Glm That Encompasses Various Ibnr Methods -- 10.3.1. Chain Ladder Method As A Glm -- 10.3.2. Arithmetic And Geometric Separation Methods -- 10.3.3. De Vijlder's Least Squares Method -- 10.4. Illustration Of Some Ibnr Methods -- 10.4.1. Modeling The Claim Numbers In Table 10.1 -- 10.4.2. Modeling Claim Sizes -- 10.5. Solving Ibnr Problems By R -- 10.6. Variability Of The Ibnr Estimate -- 10.6.1. Bootstrapping -- 10.6.2. Analytical Estimate Of The Prediction Error -- 10.7. An Ibnr-problem With Known Exposures -- 10.8. Exercises -- 11. More On Glms -- 11.1. Introduction -- 11.2. Linear Models And Generalized Linear Models -- 11.3. The Exponential Dispersion Family -- 11.4. Fitting Criteria -- 11.4.1. Residuals -- 11.4.2. Quasi-likelihood And Quasi-deviance -- 11.4.3. Extended Quasi-likelihood -- 11.5. The Canonical Link -- 11.6. The Irls Algorithm Of Nelder And Wedderburn -- 11.6.1. Theoretical Description -- 11.6.2. Step-by-step Implementation -- 11.7. Tweedie's Compound Poisson -- Gamma Distributions -- 11.7.1. Application To An Ibnr Problem -- 11.8. Exercises -- The 'r' In Modern Art -- Appendix 1. A Short Introduction To R -- Appendix 2. Analyzing A Stock Portfolio Using R -- Appendix 3. Generating A Pseudo-random Insurance Portfolio. By Rob Kaas ... [et Al.]. Expanded Edition Of: Modern Actuarial Risk Theory. 2003. Includes Bibliographical References And Index.