Stochastic Claims Reserving Methods in Insurance

Acknowledgement.

1 Introduction and Notation.

1.1 Claims Process.

1.2 Structural Framework to the Claims-Reserving Problem.

1.3 Outstanding Loss Liabilities, Classical Notation.

1.4 General Remarks.

2 Basic Methods.

2.1 Chain-Ladder Method (Distribution-Free).

2.2 Bornhuetter–Ferguson Method.

2.3 Number of IBNyR Claims, Poisson Model.

2.4 Poisson Derivation of the CL Algorithm.

3 Chain-Ladder Models.

3.1 Mean Square Error of Prediction.

3.2 Chain-Ladder Method.

3.3 Bounds in the Unconditional Approach.

3.4 Analysis of Error Terms in the CL Method.

4 Bayesian Models.

4.1 Benktander–Hovinen Method and Cape–Cod Model.

4.2 Credible Claims Reserving Methods.

4.3 Exact Bayesian Models.

4.4 Markov Chain Monte Carlo Methods.

4.5 Bühlmann–Straub Credibility Model.

4.6 Multidimensional Credibility Models.

4.7 Kalman Filter.

5 Distributional Models.

5.1 Log-Normal Model for Cumulative Claims.

5.2 Incremental Claims.

6 Generalized Linear Models.

6.1 Maximum Likelihood Estimators.

6.2 Generalized Linear Models Framework.

6.3 Exponential Dispersion Family.

6.4 Parameter Estimation in the EDF.

6.5 Other GLM Models.

6.6 Bornhuetter–Ferguson Method, Revisited.

7 Bootstrap Methods.

7.1 Introduction.

7.2 Log-Normal Model for Cumulative Sizes.

7.3 Generalized Linear Models.

7.4 Chain-Ladder Method.  

7.5 Mathematical Thoughts about Bootstrapping Methods.

7.6 Synchronous Bootstrapping of Seemingly Unrelated Regressions.

8 Multivariate Reserving Methods.

8.1 General Multivariate Framework.

8.2 Multivariate Chain-Ladder Method.

8.3 Multivariate Additive Loss Reserving Method.

8.4 Combined Multivariate CL and ALR Method.

9 Selected Topics I: Chain-Ladder Methods.

9.1 Munich Chain-Ladder.

9.2 CL Reserving: A Bayesian Inference Model.

10 Selected Topics II: Individual Claims Development Processes.

10.1 Modelling Claims Development Processes for Individual Claims.

10.2 Separating IBNeR and IBNyR Claims.

11 Statistical Diagnostics.

11.1 Testing Age-to-Age Factors.

11.2 Non-Parametric Smoothing.

Appendix A: Distributions.

A.1 Discrete Distributions.

A.2 Continuous Distributions.

Bibliography.

Index.