Stochastic Claims Reserving Methods in

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).

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.

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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Stochastic Claims Reserving Methods in Insurance

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