A Survival Model Approach to Non-life Run-off Triangle Estimation

1. A Survival Model Approach to Non-life Run-off Triangle Estimation Casualty Loss Reserve Seminar Washington, D.C. 18 September 2008 Brian Fannin

2. Agenda • Motivation • Brief Review of Survival Models • The Method • What Next?

3. Motivation

4. The Current Situation • Significant progress has been made in refining the way in which we analyze aggregate loss triangles • Ad hoc methods have been replaced by stochastic models • Variance of the estimate of loss reserves now receives a great deal of attention • Techniques which combine triangles- particularly paid and incurred- have been developed • Correlation between triangles or lines of business are becoming more sophisticated. A Survival Model Approach to Non-life Run-off Triangle Estimation

5. However … • We're still using the same aggregate data, presented in the same format in which it's been presented for over half a century. • The target estimator in most reserving exercises (and virtually every individual size of loss distribution) is the sum of nominal loss payments. Other quantities are of secondary interest • This amount is of limited use in determining the market cost to transfer the liabilities • When a discounted value is needed, typically the nominal estimate is shoe-horned into a payment pattern which has been derived elsewhere. • Further, although they do quite a bit to help us make better estimates, they tell us little about the underlying processes which affect the ultimate cost of claims. What causes the variance in the level of reserves? Put another way, can you answer the following questions? A Survival Model Approach to Non-life Run-off Triangle Estimation

6. Can you answer these questions? • What is the impact to our loss reserves of a 1% rise in the rate of medical inflation five years from now? • What effect does a delay in claim reporting have on case reserves? • What is the impact of changes in claims department staffing levels? • What is the probability that claims will remain open 5 years more or less than expected? • How long will our current liability book remain open? A Survival Model Approach to Non-life Run-off Triangle Estimation

7. The first step towards a different approach • Rather than looking at aggregate behavior, why not look at the life cycle of an individual claim? • A claim occurs (is "born") • The claim is reported (enters a population under observation) • Payment(s) are made • The claim is closed ("dies") • The language in parentheses is deliberate. A pool of claims can be regarded as being analagous to a population of lives. We'll look at casualty claims from a survival model perspective. A Survival Model Approach to Non-life Run-off Triangle Estimation

8. Brief Review of Survival Models

9. Survival Model Mathematics • Incredibly easy. • The function of interest S(x) measures the probability that a random variable will be greater than or equal to some fixed quantity, x. This is nothing more than the complement of the cumulative probabilty distribution, or S(x) = 1 – F(x). • When used to describe age, the function describes the probability that a life will survive to an age greater than x. A Survival Model Approach to Non-life Run-off Triangle Estimation

10. Survival Model Mathematics Continued aqx = 1 - apx A Survival Model Approach to Non-life Run-off Triangle Estimation

11. What is the Future Lifetime? • We use the random variable K(x) to describe the future lifetime for a life aged x. It's expectation and variance are derived using integration by parts, which results in the following expressions. A Survival Model Approach to Non-life Run-off Triangle Estimation

12. The method

13. The Method • Assemble data which tabulates claim closure rates by age of claim • Assume a binomial for the probability that a claim will close. Use method of moments to estimate the parameters by age • You're done! A Survival Model Approach to Non-life Run-off Triangle Estimation

14. Assembling the Data • I began by looking at a database of payment transactions. Age is defined as payment year minus accident year plus one. • I'll spare you the details, but using some SQL, I was able to arrange data so that it showed- for each age- the number of claims open and the number of claims which would be closed in the subsequent year. • It turns out that this can be represented via a triangle. On a personal level this was a disappointing result, but may be of some comfort to those who still like triangles. A Survival Model Approach to Non-life Run-off Triangle Estimation

15. Estimation of the Probabilities Closure probabilities by age are simply given by If we assume the probability of claim closure is binomial, then the sample estimate is an unbiased estimator of qx A Survival Model Approach to Non-life Run-off Triangle Estimation

16. Results Note that the probability of survival drops for the first six years, but then raises to a relatively constant value A Survival Model Approach to Non-life Run-off Triangle Estimation

17. Smoothing • Just as one can alter link ratios or tail factors via either judgement or some sort of regression or curve-fitting, one can smooth mortality rates. • The smoothing method employed in this case was the Whitaker-Henderson technique. This method has been on earth longer than you have and is somewhat subjective. However, I see at least one advantage: All of the survival probabilities are adjusted at the same time. Contrast this with the typical approach of adjusting each link-ratio individually. • The goal of the method is to create a set of factors which strikes a balance between smoothness and reproduction of the sample estimates. The following expression is minimized: • The parameter ε controls the relative weight one places on either smoothness or the sample values. A Survival Model Approach to Non-life Run-off Triangle Estimation

18. Smoothed Results A Survival Model Approach to Non-life Run-off Triangle Estimation

19. Comparison to Other Methods • Adler & Kline and Berquist & Sherman discuss a claim closure ratio defined as the ratio of claims closed in a particular interval to the total number of claims. Fisher & Lange use the same sort of ratio, but compare to number of claims reported. Unless you're working with report year data, the total number of claims is an estimate. As the estimate of ultimate count changes, so does the closure ratio. • Teng estimates a closure ratio equal to the number of closures at any age relative to the total reported to date. This is 1 – S(x). • In all cases, the authors do not suggest an underlying stochastic model for claim closure rates. Rather, the presumption is that the most recent experience will persist in the future. A Survival Model Approach to Non-life Run-off Triangle Estimation

20. What's Next?

21. What's Next? • Loads! • The two biggest missing items are • Incorporation of a payment model • Estimation of claim emergence • Model to forecast changing claim closure rates, i.e. change in mortality probabilities • More sophisticated graduation techniques A Survival Model Approach to Non-life Run-off Triangle Estimation

22. Conclusion • If you ignore the (significant!) issue of newly reported claims, I have answered at least two of the questions that I posed earlier. • What is the probability that claims will remain open 5 years more or less than expected? • How long will our current liability book remain open? That may not be a lot, but it's a start! A Survival Model Approach to Non-life Run-off Triangle Estimation

23. Thank you very much for your attention. Brian Fannin If you have any questions, please feel free to e-mail me at BFannin@MunichRe.com.