Stochastic Claims Reserving Methods in Insurance
by Wü; thrich, Mario V.; Merz, MichaelEdition: 1st
Format: Hardcover
Pub. Date: 2008-06-09
Publisher(s): Wiley
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Summary
This book covers all the theory and practical advice that actuaries need in order to determine the claims reserves for non-life insurance. The book describes all the mathematical methods used to estimate loss reserves and shares the authors' practical experience, which is essential in showing which of the methods should be applied in any given situation. They focus on the mathematical description of relevant stochastic models, showing the reader how to estimate total claims reserves whilst quantifying the total uncertainty in the reserves (prediction errors in total ultimate claims).
Author Biography
Mario V. Wüthrich holds a Ph.D. in mathematics from ETH Zurich (The Swiss Federal Institute of Technology Zurich). He completed his postdoctoral work on statistical physics in 2000 at the University of Nijmegen in The Netherlands. From 2000 to 2005, he held an actuarial position at Winterthur Insurance (Switzerland) where he was responsible for claims reserving in non-life insurance, as well as developing and implementing the Swiss Solvency Test. Since 2005, he has served as senior researcher and lecturer at ETH Zurich with teaching duties in actuarial and financial mathematics. He serves on the board of the Swiss Association of Actuaries (SAA) and is joint editor of the Bulletin SAA.
Michael Merz has been Assistant Professor for Statistics, Risk and Insurance at the University of Tübingen since October 2006. He was awarded the internationally renowned SCOR Actuarial Prize 2004 for his doctoral thesis in risk theory. After completing his doctorate, he worked in the actuarial department of the Baloise insurance company in Basel/Switzerland and gained valuable practical working experience in actuarial science and quantitative risk management. His main research interests are actuarial science and quantitative risk management, with special emphasis on claims reserving and risk theory. He is a referee for many academic journals and has published extensively in leading academic journals, including the ASTIN Bulletin and the Scandinanvian Actuarial Journal.
Table of Contents
| Preface | p. xi |
| Acknowledgement | p. xiii |
| Introduction and Notation | p. 1 |
| Claims process | p. 1 |
| Accounting principles and accident years | p. 2 |
| Inflation | p. 3 |
| Structural framework to the claims-reserving problem | p. 5 |
| Fundamental properties of the claims reserving process | p. 7 |
| Known and unknown claims | p. 9 |
| Outstanding loss liabilities, classical notation | p. 10 |
| General remarks | p. 12 |
| Basic Methods | p. 15 |
| Chain-ladder method (distribution-free) | p. 15 |
| Bornhuetter-Ferguson method | p. 21 |
| Number of IBNyR claims, Poisson model | p. 25 |
| Poisson derivation of the CL algorithm | p. 27 |
| Chain-Ladder Models | p. 33 |
| Mean square error of prediction | p. 33 |
| Chain-ladder method | p. 36 |
| Mack model (distribution-free CL model) | p. 37 |
| Conditional process variance | p. 41 |
| Estimation error for single accident years | p. 44 |
| Conditional MSEP, aggregated accident years | p. 55 |
| Bounds in the unconditional approach | p. 58 |
| Results and interpretation | p. 58 |
| Aggregation of accident years | p. 63 |
| Proof of Theorems 3.17, 3.18 and 3.20 | p. 64 |
| Analysis of error terms in the CL method | p. 70 |
| Classical CL model | p. 70 |
| Enhanced CL model | p. 71 |
| Interpretation | p. 72 |
| CL estimator in the enhanced model | p. 73 |
| Conditional process and parameter prediction errors | p. 74 |
| CL factors and parameter estimation error | p. 75 |
| Parameter estimation | p. 81 |
| Bayesian Models | p. 91 |
| Benktander-Hovinen method and Cape-Cod model | p. 91 |
| Benktander-Hovinen method | p. 92 |
| Cape-Cod model | p. 95 |
| Credible claims reserving methods | p. 98 |
| Minimizing quadratic loss functions | p. 98 |
| Distributional examples to credible claims reserving | p. 101 |
| Log-normal/Log-normal model | p. 105 |
| Exact Bayesian models | p. 113 |
| Overdispersed Poisson model with gamma prior distribution | p. 114 |
| Exponential dispersion family with its associated conjugates | p. 122 |
| Markov chain Monte Carlo methods | p. 131 |
| Buhlmann-Straub credibility model | p. 145 |
| Multidimensional credibility models | p. 154 |
| Hachemeister regression model | p. 155 |
| Other credibility models | p. 159 |
| Kalman filter | p. 160 |
| Distributional Models | p. 167 |
| Log-normal model for cumulative claims | p. 167 |
| Known variances [sigma superscript 2 subscript j] | p. 170 |
| Unknown variances | p. 177 |
| Incremental claims | p. 182 |
| (Overdispersed) Poisson model | p. 182 |
| Negative-Binomial model | p. 183 |
| Log-normal model for incremental claims | p. 185 |
| Gamma model | p. 186 |
| Tweedie's compound Poisson model | p. 188 |
| Wright's model | p. 199 |
| Generalized Linear Models | p. 201 |
| Maximum likelihood estimators | p. 201 |
| Generalized linear models framework | p. 203 |
| Exponential dispersion family | p. 205 |
| Parameter estimation in the EDF | p. 208 |
| MLE for the EDF | p. 208 |
| Fisher's scoring method | p. 210 |
| Mean square error of prediction | p. 214 |
| Other GLM models | p. 223 |
| Bornhuetter-Ferguson method, revisited | p. 223 |
| MSEP in the BF method, single accident year | p. 226 |
| MSEP in the BF method, aggregated accident years | p. 230 |
| Bootstrap Methods | p. 233 |
| Introduction | p. 233 |
| Efron's non-parametric bootstrap | p. 234 |
| Parametric bootstrap | p. 236 |
| Log-normal model for cumulative sizes | p. 237 |
| Generalized linear models | p. 242 |
| Chain-ladder method | p. 244 |
| Approach 1: Unconditional estimation error | p. 246 |
| Approach 3: Conditional estimation error | p. 247 |
| Mathematical thoughts about bootstrapping methods | p. 248 |
| Synchronous bootstrapping of seemingly unrelated regressions | p. 253 |
| Multivariate Reserving Methods | p. 257 |
| General multivariate framework | p. 257 |
| Multivariate chain-ladder method | p. 259 |
| Multivariate CL model | p. 259 |
| Conditional process variance | p. 264 |
| Conditional estimation error for single accident years | p. 265 |
| Conditional MSEP, aggregated accident years | p. 272 |
| Parameter estimation | p. 274 |
| Multivariate additive loss reserving method | p. 288 |
| Multivariate additive loss reserving model | p. 288 |
| Conditional process variance | p. 295 |
| Conditional estimation error for single accident years | p. 295 |
| Conditional MSEP, aggregated accident years | p. 297 |
| Parameter estimation | p. 299 |
| Combined Multivariate CL and ALR method | p. 308 |
| Combined CL and ALR method: the model | p. 308 |
| Conditional cross process variance | p. 313 |
| Conditional cross estimation error for single accident years | p. 315 |
| Conditional MSEP, aggregated accident years | p. 319 |
| Parameter estimation | p. 321 |
| Selected Topics I: Chain-Ladder Methods | p. 331 |
| Munich chain-ladder | p. 331 |
| The Munich chain-ladder model | p. 333 |
| Credibility approach to the MCL method | p. 335 |
| MCL Parameter estimation | p. 340 |
| CL Reserving: A Bayesian inference model | p. 346 |
| Prediction of the ultimate claim | p. 351 |
| Likelihood function and posterior distribution | p. 351 |
| Mean square error of prediction | p. 354 |
| Credibility chain-ladder | p. 359 |
| Examples | p. 361 |
| Markov chain Monte Carlo methods | p. 364 |
| Selected Topics II: Individual Claims Development Processes | p. 369 |
| Modelling claims development processes for individual claims | p. 369 |
| Modelling framework | p. 370 |
| Claims reserving categories | p. 376 |
| Separating IBNeR and IBNyR claims | p. 379 |
| Statistical Diagnostics | p. 391 |
| Testing age-to-age factors | p. 391 |
| Model choice | p. 394 |
| Age-to-age factors | p. 396 |
| Homogeneity in time and distributional assumptions | p. 398 |
| Correlations | p. 399 |
| Diagonal effects | p. 401 |
| Non-parametric smoothing | p. 401 |
| Distributions | p. 405 |
| Discrete distributions | p. 405 |
| Binomial distribution | p. 405 |
| Poisson distribution | p. 405 |
| Negative-Binomial distribution | p. 405 |
| Continuous distributions | p. 406 |
| Uniform distribution | p. 406 |
| Normal distribution | p. 406 |
| Log-normal distribution | p. 407 |
| Gamma distribution | p. 407 |
| Beta distribution | p. 408 |
| Bibliography | p. 409 |
| Index | p. 417 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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