September 11, 2007

1. VR 7: Testing Assumptions Underlying Estimates of Loss Reserves Presented by: Chris Gross chris.gross@cgconsult.com (651)293-8008 September 11, 2007

2. Loss Reserving: Methods or Models • Debate as to the use of models for reserve development vs traditional “methods” • Whether you think of them as such or not, traditionalmethods are models. • Like any (necessarily imperfect) model of reality, traditional reserving methods carry with them certain assumptions. • Understanding which assumptions might be violated and the potential ramifications for the estimates is important.

3. Why Statistical Analysis of Assumptions? • We are easily fooled by unaided observation

6. Why Statistical Analysis of Assumptions? • We are easily fooled by unaided observation. • Finding patterns where there are none • Statistical tools provide an objective balance to this basic human trait. • They are easy to apply and embedded in tools we already use.

7. Excel Regression Functions • SLOPE • INTERCEPT • STEYX • TREND • FORECAST • RSQ • LINEST

8. Chain Ladder Method Assumptions • Linear Relationship of Incremental Losses (y=mx) • Linear relationship between incremental loss amounts and previous cumulative amount? Intercept=zero? • What does it mean if the assumption does not hold? • “Last three” selection implies a change • Does there appear to be a change? • What does this mean? • Loss development factors are uncorrelated • Do they appear correlated? • What are the implications for estimated reserves?

9. Data

10. Y=mX ?

11. Y≠mX !

12. Age to Age Factors Changing?

13. Age to Age Factors Changing!

14. Age to Age Factors Independent?

15. Inconclusive

16. Bornhuetter-Ferguson Assumptions • Loss development is directly related to premium • Non-correlation of development from age to age

17. Losses related to premium? All amount in $millions

18. Losses not related to premium!

19. Observations of Residuals • “Predict” past paid and incurred data using your modeled relationship. • Compare these predictions to actual data. • Ask whether you expect future values to be biased or unbiased, given what you see in the residuals.

20. Residuals

21. Multiple Algorithms and Reserving • Why is reserve data organized into aggregated loss triangles? • What information is lost? • What are the advantages of using multiple algorithms? • What are the disadvantages of using multiple algorithms? • How much weight do you give to each?

22. Residuals Revisited

23. Residuals Revisited

24. Comparison of Methods

25. Correlation & Covariance

26. Minimum Variance Weighted Estimate • Choose weight vector w to minimize combined variance: w´ S w • Basic answer provided by calculating the matrix inverse of S, summing by row or column, and then normalizing to sum to 1. • Negative weights probably should be avoided, so minimization is a bit more involved • For the example here, the optimal (non-negative) weighting is:

27. Summary • You make implicit assumptions every time you perform reserve analysis. • An understanding of what these assumptions are is beneficial to determining where the methods are strong and where they are weak. • Some basic statistical techniques in your toolkit can help you test these assumptions, as well as point you in the direction of development of other methods that may be more appropriate.