1. More on Stochastic Reserving in General Insurance GIRO Convention, Killarney, October Peter England and Richard Verrall

2. Developments in Stochastic Reserving
Overview
The story so far
Bootstrapping recursive models (including Mack's model)
Working with incurred data
Bayesian recursive models
A comparison between bootstrapping and Bayesian methods
Bootstrapping and the Bornhuetter-Ferguson Technique
Including curve fitting for estimating tail factors

3. Developments in Stochastic Reserving
Background
England, P and Verrall, R (1999), Analytic and bootstrap estimates of prediction errors in claims reserving, Insurance: Mathematics and Economics 25, pp
England, P (2002), Addendum to “Analytic and bootstrap estimates of prediction errors in claims reserving”, Insurance: Mathematics and Economics 31, pp
England, PD and Verrall, RJ (2002), Stochastic Claims Reserving in General Insurance, British Actuarial Journal 8, III, pp
+ many other papers

4. Developments in Stochastic Reserving
Conceptual Framework

5. Developments in Stochastic Reserving
Example
Developments in Stochastic Reserving

6. Prediction Errors – “Chain Ladder” Structure
Developments in Stochastic Reserving

8. Over-Dispersed Poisson
Developments in Stochastic Reserving
Over-Dispersed Poisson

9. Example Predictor Structures
Developments in Stochastic Reserving
Example Predictor Structures
Chain Ladder
Hoerl Curve
Smoother

10. Variability in Claims Reserves
Developments in Stochastic Reserving
Variability in Claims Reserves
Variability of a forecast
Includes estimation variance and process variance
Problem reduces to estimating the two components

11. Developments in Stochastic Reserving
Prediction Variance
Individual cell
Row/Overall total

12. EMBLEM Demo ODP Chain Ladder with constant scale parameter
Developments in Stochastic Reserving

13. Developments in Stochastic Reserving
Mack’s Model
Mack, T (1993), Distribution-free calculation of the standard error of chain-ladder reserve estimates. ASTIN Bulletin, 22,
Specifies first two moments only:

14. Developments in Stochastic Reserving
Mack’s Model

15. Developments in Stochastic Reserving
Mack’s Model

16. ResQ Demo – Mack’s Model
Developments in Stochastic Reserving

17. Parameter Uncertainty - Bootstrapping
Developments in Stochastic Reserving
Bootstrapping is a simple but effective way of obtaining a distribution of parameters
The method involves creating many new data sets from which the parameters are estimated
The new data sets are created by sampling with replacement from the observed data (or residuals)
The model is re-fitted to each new data set
Results in a (“simulated”) distribution of the parameters

18. Reserving and Bootstrapping
Developments in Stochastic Reserving
Reserving and Bootstrapping
Any model that can be clearly defined can be bootstrapped
(see the England and Verrall papers for bootstrapping the ODP)

19. Bootstrapping Mack’s Model
Developments in Stochastic Reserving
Bootstrapping Mack’s Model

20. Bootstrapping Mack’s Model
Developments in Stochastic Reserving
Bootstrapping Mack’s Model

21. Recursive Models: Forecasting
Developments in Stochastic Reserving
Recursive Models: Forecasting
With recursive models, forecasting proceeds one-step at a time:
Move one-step ahead by multiplying the previous cumulative claims by the appropriate bootstrapped development factor
Include the process error by sampling a single observation from the underlying process distribution, conditional on the mean given by the previous step
Move to the next step
Note that the process error is included at each step before proceeding

22. Igloo Demo 1 – Bootstrapping Chain Ladder Model Only
Developments in Stochastic Reserving
Igloo Demo 1 – Bootstrapping Chain Ladder Model Only
ODP – with constant scale parameter
ODP – with non-constant scale parameters
Bootstrapping Mack’s Model

23. Negative Binomial Recursive Model
Developments in Stochastic Reserving
Negative Binomial Recursive Model
This is a recursive equivalent to the ODP model

24. Developments in Stochastic Reserving
Igloo Demo 2 – Bootstrapping Negative Binomial – Chain Ladder Model only
Negative Binomial – with constant scale parameter
Negative Binomial – with non-constant scale parameters
Compare results with ODP and Mack shown earlier:
ODP and Negative Binomial are very close
Results with non-constant scale parameters are close to Mack’s method

25. Bootstrapping Recursive Models: Advantages
Developments in Stochastic Reserving
Bootstrapping Recursive Models: Advantages
Consistent with traditional deterministic actuarial techniques
Individual points can be weighted out for n-year volume weighted averages, exclude high/low etc
Curve fitting can be incorporated
Bootstrap version of Mack’s model can be used where negative incrementals are encountered
For example: Incurred claims
Bootstrapping incurred claims:
Gives distribution of Ultimates and IBNR
Can be combined with Paid to Date to give distribution of Outstanding claims
Must be combined with (simulated) Paid to Incurred ratios to give distributions of payment cash flows

26. Reserving and Bayesian Methods
Developments in Stochastic Reserving
Reserving and Bayesian Methods
Any model that can be clearly defined can be fitted as a Bayesian model

27. Excel Demo – Gibbs Sampling ODP - Chain Ladder Model Only
Developments in Stochastic Reserving
Excel Demo – Gibbs Sampling ODP - Chain Ladder Model Only

28. Developments in Stochastic Reserving
Igloo Demo 3 – Bayesian Methods ODP, Negative Binomial and Mack’s model Comparison with bootstrapping
Developments in Stochastic Reserving

29. Bayesian Stochastic Reserving: Advantages
Developments in Stochastic Reserving
Bayesian Stochastic Reserving: Advantages
Overcomes some practical difficulties with bootstrapping
Sets of pseudo-data are not required, therefore far less RAM hungry when simulating
Arguably more statistically rigorous and theoretically appealing
Flexible approach
Informative priors
Bayesian Bornhuetter-Ferguson Method (see latest NAAJ)
Model uncertainty
Individual claims and additional covariates

30. The Bornhuetter-Ferguson Method and Bootstrapping
Developments in Stochastic Reserving
The Bornhuetter-Ferguson Method and Bootstrapping
Pseudo-development factors give simulated proportion of ultimate to emerge in each development year
BF prior loss ratio gives prior Ultimate
Adjust pseudo-data to take account of BF prior Ultimate and simulated proportions, before forecasting
Forecast based on adjusted pseudo-data
BUT, simulate the BF prior ultimate making assumptions about precision of prior
Add simulated forecasts to historic Paid-to-Date to give distribution of Ultimate.

31. A Bayesian Bornhuetter-Ferguson Method
Developments in Stochastic Reserving
A Bayesian Bornhuetter-Ferguson Method
A Bayesian framework is a natural candidate for a stochastic BF method
Bayesian recursive models offer the best way forward
Work in progress!
Verrall, RJ (2004), A Bayesian Generalised Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving, NAAJ, July 2004

32. Bootstrapping and Curve Fitting
Developments in Stochastic Reserving
Bootstrapping and Curve Fitting

33. Igloo Demo 4 – Bootstrapping Curve fitting and Tail Factors
Developments in Stochastic Reserving
Igloo Demo 4 – Bootstrapping Curve fitting and Tail Factors

34. Summary of Developments
Developments in Stochastic Reserving
Summary of Developments
ODP with non-constant scale parameters
Bootstrap version of Mack’s model
Recursive version of ODP: Negative Binomial model
Recursive models allow weighting out of points (exclude Max/Min, n-year volume weighted averages etc)
Bootstrap version of the Negative Binomial model
Curve fitting, tail factors and bootstrapping
Bayesian stochastic reserving
Any clearly defined model (ODP, Mack, NB, curve fitting etc)
A stochastic Bornhuetter-Ferguson method