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 xiAcknowledgement xiii1 Introduction and Notation 11.1 Claims process 11.1.1 Accounting principles and accident years 21.1.2 Inflation 31.2 Structural framework to the claims-reserving problem 51.2.1 Fundamental properties of the claims reserving process 71.2.2 Known and unknown claims 91.3 Outstanding loss liabilities, classical notation 101.4 General remarks 122 Basic Methods 152.1 Chain-ladder method (distribution-free) 152.2 Bornhuetter–Ferguson method 212.3 Number of IBNyR claims, Poisson model 252.4 Poisson derivation of the CL algorithm 273 Chain-Ladder Models 333.1 Mean square error of prediction 333.2 Chain-ladder method 363.2.1 Mack model (distribution-free CL model) 373.2.2 Conditional process variance 413.2.3 Estimation error for single accident years 443.2.4 Conditional MSEP, aggregated accident years 553.3 Bounds in the unconditional approach 583.3.1 Results and interpretation 583.3.2 Aggregation of accident years 633.3.3 Proof of Theorems 3.17, 3.18 and 3.20 643.4 Analysis of error terms in the CL method 703.4.1 Classical CL model 703.4.2 Enhanced CL model 713.4.3 Interpretation 723.4.4 CL estimator in the enhanced model 733.4.5 Conditional process and parameter prediction errors 743.4.6 CL factors and parameter estimation error 753.4.7 Parameter estimation 814 Bayesian Models 914.1 Benktander–Hovinen method and Cape–Cod model 914.1.1 Benktander–Hovinen method 924.1.2 Cape–Cod model 954.2 Credible claims reserving methods 984.2.1 Minimizing quadratic loss functions 984.2.2 Distributional examples to credible claims reserving 1014.2.3 Log-normal/Log-normal model 1054.3 Exact Bayesian models 1134.3.1 Overdispersed Poisson model with gamma prior distribution 1144.3.2 Exponential dispersion family with its associated conjugates 1224.4 Markov chain Monte Carlo methods 1314.5 Bühlmann–Straub credibility model 1454.6 Multidimensional credibility models 1544.6.1 Hachemeister regression model 1554.6.2 Other credibility models 1594.7 Kalman filter 1605 Distributional Models 1675.1 Log-normal model for cumulative claims 1675.1.1 Known variances σj 2 1705.1.2 Unknown variances 1775.2 Incremental claims 1825.2.1 (Overdispersed) Poisson model 1825.2.2 Negative-Binomial model 1835.2.3 Log-normal model for incremental claims 1855.2.4 Gamma model 1865.2.5 Tweedie’s compound Poisson model 1885.2.6 Wright’s model 1996 Generalized Linear Models 2016.1 Maximum likelihood estimators 2016.2 Generalized linear models framework 2036.3 Exponential dispersion family 2056.4 Parameter estimation in the EDF 2086.4.1 MLE for the EDF 2086.4.2 Fisher’s scoring method 2106.4.3 Mean square error of prediction 2146.5 Other GLM models 2236.6 Bornhuetter–Ferguson method, revisited 2236.6.1 MSEP in the BF method, single accident year 2266.6.2 MSEP in the BF method, aggregated accident years 2307 Bootstrap Methods 2337.1 Introduction 2337.1.1 Efron’s non-parametric bootstrap 2347.1.2 Parametric bootstrap 2367.2 Log-normal model for cumulative sizes 2377.3 Generalized linear models 2427.4 Chain-ladder method 2447.4.1 Approach 1: Unconditional estimation error 2467.4.2 Approach 3: Conditional estimation error 2477.5 Mathematical thoughts about bootstrapping methods 2487.6 Synchronous bootstrapping of seemingly unrelated regressions 2538 Multivariate Reserving Methods 2578.1 General multivariate framework 2578.2 Multivariate chain-ladder method 2598.2.1 Multivariate CL model 2598.2.2 Conditional process variance 2648.2.3 Conditional estimation error for single accident years 2658.2.4 Conditional MSEP, aggregated accident years 2728.2.5 Parameter estimation 2748.3 Multivariate additive loss reserving method 2888.3.1 Multivariate additive loss reserving model 2888.3.2 Conditional process variance 2958.3.3 Conditional estimation error for single accident years 2958.3.4 Conditional MSEP, aggregated accident years 2978.3.5 Parameter estimation 2998.4 Combined Multivariate CL and ALR method 3088.4.1 Combined CL and ALR method: the model 3088.4.2 Conditional cross process variance 3138.4.3 Conditional cross estimation error for single accident years 3158.4.4 Conditional MSEP, aggregated accident years 3198.4.5 Parameter estimation 3219 Selected Topics I: Chain-Ladder Methods 3319.1 Munich chain-ladder 3319.1.1 The Munich chain-ladder model 3339.1.2 Credibility approach to the MCL method 3359.1.3 MCL Parameter estimation 3409.2 CL Reserving: A Bayesian inference model 3469.2.1 Prediction of the ultimate claim 3519.2.2 Likelihood function and posterior distribution 3519.2.3 Mean square error of prediction 3549.2.4 Credibility chain-ladder 3599.2.5 Examples 3619.2.6 Markov chain Monte Carlo methods 36410 Selected Topics II: Individual Claims Development Processes 36910.1 Modelling claims development processes for individual claims 36910.1.1 Modelling framework 37010.1.2 Claims reserving categories 37610.2 Separating IBNeR and IBNyR claims 37911 Statistical Diagnostics 39111.1 Testing age-to-age factors 39111.1.1 Model choice 39411.1.2 Age-to-age factors 39611.1.3 Homogeneity in time and distributional assumptions 39811.1.4 Correlations 39911.1.5 Diagonal effects 40111.2 Non-parametric smoothing 401Appendix A: Distributions 405A.1 Discrete distributions 405A.1.1 Binomial distribution 405A.1.2 Poisson distribution 405A.1.3 Negative-Binomial distribution 405A.2 Continuous distributions 406A.2.1 Uniform distribution 406A.2.2 Normal distribution 406A.2.3 Log-normal distribution 407A.2.4 Gamma distribution 407A.2.5 Beta distribution 408Bibliography 409Index 417

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 xiAcknowledgement xiii1 Introduction and Notation 11.1 Claims process 11.1.1 Accounting principles and accident years 21.1.2 Inflation 31.2 Structural framework to the claims-reserving problem 51.2.1 Fundamental properties of the claims reserving process 71.2.2 Known and unknown claims 91.3 Outstanding loss liabilities, classical notation 101.4 General remarks 122 Basic Methods 152.1 Chain-ladder method (distribution-free) 152.2 Bornhuetter–Ferguson method 212.3 Number of IBNyR claims, Poisson model 252.4 Poisson derivation of the CL algorithm 273 Chain-Ladder Models 333.1 Mean square error of prediction 333.2 Chain-ladder method 363.2.1 Mack model (distribution-free CL model) 373.2.2 Conditional process variance 413.2.3 Estimation error for single accident years 443.2.4 Conditional MSEP, aggregated accident years 553.3 Bounds in the unconditional approach 583.3.1 Results and interpretation 583.3.2 Aggregation of accident years 633.3.3 Proof of Theorems 3.17, 3.18 and 3.20 643.4 Analysis of error terms in the CL method 703.4.1 Classical CL model 703.4.2 Enhanced CL model 713.4.3 Interpretation 723.4.4 CL estimator in the enhanced model 733.4.5 Conditional process and parameter prediction errors 743.4.6 CL factors and parameter estimation error 753.4.7 Parameter estimation 814 Bayesian Models 914.1 Benktander–Hovinen method and Cape–Cod model 914.1.1 Benktander–Hovinen method 924.1.2 Cape–Cod model 954.2 Credible claims reserving methods 984.2.1 Minimizing quadratic loss functions 984.2.2 Distributional examples to credible claims reserving 1014.2.3 Log-normal/Log-normal model 1054.3 Exact Bayesian models 1134.3.1 Overdispersed Poisson model with gamma prior distribution 1144.3.2 Exponential dispersion family with its associated conjugates 1224.4 Markov chain Monte Carlo methods 1314.5 Bühlmann–Straub credibility model 1454.6 Multidimensional credibility models 1544.6.1 Hachemeister regression model 1554.6.2 Other credibility models 1594.7 Kalman filter 1605 Distributional Models 1675.1 Log-normal model for cumulative claims 1675.1.1 Known variances σj 2 1705.1.2 Unknown variances 1775.2 Incremental claims 1825.2.1 (Overdispersed) Poisson model 1825.2.2 Negative-Binomial model 1835.2.3 Log-normal model for incremental claims 1855.2.4 Gamma model 1865.2.5 Tweedie’s compound Poisson model 1885.2.6 Wright’s model 1996 Generalized Linear Models 2016.1 Maximum likelihood estimators 2016.2 Generalized linear models framework 2036.3 Exponential dispersion family 2056.4 Parameter estimation in the EDF 2086.4.1 MLE for the EDF 2086.4.2 Fisher’s scoring method 2106.4.3 Mean square error of prediction 2146.5 Other GLM models 2236.6 Bornhuetter–Ferguson method, revisited 2236.6.1 MSEP in the BF method, single accident year 2266.6.2 MSEP in the BF method, aggregated accident years 2307 Bootstrap Methods 2337.1 Introduction 2337.1.1 Efron’s non-parametric bootstrap 2347.1.2 Parametric bootstrap 2367.2 Log-normal model for cumulative sizes 2377.3 Generalized linear models 2427.4 Chain-ladder method 2447.4.1 Approach 1: Unconditional estimation error 2467.4.2 Approach 3: Conditional estimation error 2477.5 Mathematical thoughts about bootstrapping methods 2487.6 Synchronous bootstrapping of seemingly unrelated regressions 2538 Multivariate Reserving Methods 2578.1 General multivariate framework 2578.2 Multivariate chain-ladder method 2598.2.1 Multivariate CL model 2598.2.2 Conditional process variance 2648.2.3 Conditional estimation error for single accident years 2658.2.4 Conditional MSEP, aggregated accident years 2728.2.5 Parameter estimation 2748.3 Multivariate additive loss reserving method 2888.3.1 Multivariate additive loss reserving model 2888.3.2 Conditional process variance 2958.3.3 Conditional estimation error for single accident years 2958.3.4 Conditional MSEP, aggregated accident years 2978.3.5 Parameter estimation 2998.4 Combined Multivariate CL and ALR method 3088.4.1 Combined CL and ALR method: the model 3088.4.2 Conditional cross process variance 3138.4.3 Conditional cross estimation error for single accident years 3158.4.4 Conditional MSEP, aggregated accident years 3198.4.5 Parameter estimation 3219 Selected Topics I: Chain-Ladder Methods 3319.1 Munich chain-ladder 3319.1.1 The Munich chain-ladder model 3339.1.2 Credibility approach to the MCL method 3359.1.3 MCL Parameter estimation 3409.2 CL Reserving: A Bayesian inference model 3469.2.1 Prediction of the ultimate claim 3519.2.2 Likelihood function and posterior distribution 3519.2.3 Mean square error of prediction 3549.2.4 Credibility chain-ladder 3599.2.5 Examples 3619.2.6 Markov chain Monte Carlo methods 36410 Selected Topics II: Individual Claims Development Processes 36910.1 Modelling claims development processes for individual claims 36910.1.1 Modelling framework 37010.1.2 Claims reserving categories 37610.2 Separating IBNeR and IBNyR claims 37911 Statistical Diagnostics 39111.1 Testing age-to-age factors 39111.1.1 Model choice 39411.1.2 Age-to-age factors 39611.1.3 Homogeneity in time and distributional assumptions 39811.1.4 Correlations 39911.1.5 Diagonal effects 40111.2 Non-parametric smoothing 401Appendix A: Distributions 405A.1 Discrete distributions 405A.1.1 Binomial distribution 405A.1.2 Poisson distribution 405A.1.3 Negative-Binomial distribution 405A.2 Continuous distributions 406A.2.1 Uniform distribution 406A.2.2 Normal distribution 406A.2.3 Log-normal distribution 407A.2.4 Gamma distribution 407A.2.5 Beta distribution 408Bibliography 409Index 417

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 xiAcknowledgement xiii1 Introduction and Notation 11.1 Claims process 11.1.1 Accounting principles and accident years 21.1.2 Inflation 31.2 Structural framework to the claims-reserving problem 51.2.1 Fundamental properties of the claims reserving process 71.2.2 Known and unknown claims 91.3 Outstanding loss liabilities, classical notation 101.4 General remarks 122 Basic Methods 152.1 Chain-ladder method (distribution-free) 152.2 Bornhuetter–Ferguson method 212.3 Number of IBNyR claims, Poisson model 252.4 Poisson derivation of the CL algorithm 273 Chain-Ladder Models 333.1 Mean square error of prediction 333.2 Chain-ladder method 363.2.1 Mack model (distribution-free CL model) 373.2.2 Conditional process variance 413.2.3 Estimation error for single accident years 443.2.4 Conditional MSEP, aggregated accident years 553.3 Bounds in the unconditional approach 583.3.1 Results and interpretation 583.3.2 Aggregation of accident years 633.3.3 Proof of Theorems 3.17, 3.18 and 3.20 643.4 Analysis of error terms in the CL method 703.4.1 Classical CL model 703.4.2 Enhanced CL model 713.4.3 Interpretation 723.4.4 CL estimator in the enhanced model 733.4.5 Conditional process and parameter prediction errors 743.4.6 CL factors and parameter estimation error 753.4.7 Parameter estimation 814 Bayesian Models 914.1 Benktander–Hovinen method and Cape–Cod model 914.1.1 Benktander–Hovinen method 924.1.2 Cape–Cod model 954.2 Credible claims reserving methods 984.2.1 Minimizing quadratic loss functions 984.2.2 Distributional examples to credible claims reserving 1014.2.3 Log-normal/Log-normal model 1054.3 Exact Bayesian models 1134.3.1 Overdispersed Poisson model with gamma prior distribution 1144.3.2 Exponential dispersion family with its associated conjugates 1224.4 Markov chain Monte Carlo methods 1314.5 Bühlmann–Straub credibility model 1454.6 Multidimensional credibility models 1544.6.1 Hachemeister regression model 1554.6.2 Other credibility models 1594.7 Kalman filter 1605 Distributional Models 1675.1 Log-normal model for cumulative claims 1675.1.1 Known variances σj 2 1705.1.2 Unknown variances 1775.2 Incremental claims 1825.2.1 (Overdispersed) Poisson model 1825.2.2 Negative-Binomial model 1835.2.3 Log-normal model for incremental claims 1855.2.4 Gamma model 1865.2.5 Tweedie’s compound Poisson model 1885.2.6 Wright’s model 1996 Generalized Linear Models 2016.1 Maximum likelihood estimators 2016.2 Generalized linear models framework 2036.3 Exponential dispersion family 2056.4 Parameter estimation in the EDF 2086.4.1 MLE for the EDF 2086.4.2 Fisher’s scoring method 2106.4.3 Mean square error of prediction 2146.5 Other GLM models 2236.6 Bornhuetter–Ferguson method, revisited 2236.6.1 MSEP in the BF method, single accident year 2266.6.2 MSEP in the BF method, aggregated accident years 2307 Bootstrap Methods 2337.1 Introduction 2337.1.1 Efron’s non-parametric bootstrap 2347.1.2 Parametric bootstrap 2367.2 Log-normal model for cumulative sizes 2377.3 Generalized linear models 2427.4 Chain-ladder method 2447.4.1 Approach 1: Unconditional estimation error 2467.4.2 Approach 3: Conditional estimation error 2477.5 Mathematical thoughts about bootstrapping methods 2487.6 Synchronous bootstrapping of seemingly unrelated regressions 2538 Multivariate Reserving Methods 2578.1 General multivariate framework 2578.2 Multivariate chain-ladder method 2598.2.1 Multivariate CL model 2598.2.2 Conditional process variance 2648.2.3 Conditional estimation error for single accident years 2658.2.4 Conditional MSEP, aggregated accident years 2728.2.5 Parameter estimation 2748.3 Multivariate additive loss reserving method 2888.3.1 Multivariate additive loss reserving model 2888.3.2 Conditional process variance 2958.3.3 Conditional estimation error for single accident years 2958.3.4 Conditional MSEP, aggregated accident years 2978.3.5 Parameter estimation 2998.4 Combined Multivariate CL and ALR method 3088.4.1 Combined CL and ALR method: the model 3088.4.2 Conditional cross process variance 3138.4.3 Conditional cross estimation error for single accident years 3158.4.4 Conditional MSEP, aggregated accident years 3198.4.5 Parameter estimation 3219 Selected Topics I: Chain-Ladder Methods 3319.1 Munich chain-ladder 3319.1.1 The Munich chain-ladder model 3339.1.2 Credibility approach to the MCL method 3359.1.3 MCL Parameter estimation 3409.2 CL Reserving: A Bayesian inference model 3469.2.1 Prediction of the ultimate claim 3519.2.2 Likelihood function and posterior distribution 3519.2.3 Mean square error of prediction 3549.2.4 Credibility chain-ladder 3599.2.5 Examples 3619.2.6 Markov chain Monte Carlo methods 36410 Selected Topics II: Individual Claims Development Processes 36910.1 Modelling claims development processes for individual claims 36910.1.1 Modelling framework 37010.1.2 Claims reserving categories 37610.2 Separating IBNeR and IBNyR claims 37911 Statistical Diagnostics 39111.1 Testing age-to-age factors 39111.1.1 Model choice 39411.1.2 Age-to-age factors 39611.1.3 Homogeneity in time and distributional assumptions 39811.1.4 Correlations 39911.1.5 Diagonal effects 40111.2 Non-parametric smoothing 401Appendix A: Distributions 405A.1 Discrete distributions 405A.1.1 Binomial distribution 405A.1.2 Poisson distribution 405A.1.3 Negative-Binomial distribution 405A.2 Continuous distributions 406A.2.1 Uniform distribution 406A.2.2 Normal distribution 406A.2.3 Log-normal distribution 407A.2.4 Gamma distribution 407A.2.5 Beta distribution 408Bibliography 409Index 417