Measuring Loss Reserve Uncertainty William H. Panning EVP, Willis Re Casualty Actuarial Society

1. Measuring Loss Reserve Uncertainty William H. Panning EVP, Willis Re Casualty Actuarial Society Annual Meeting, November 2006 2006 Hachemeister Award Presentation

2. Agenda • What is Loss Reserve Uncertainty (LRU) and why is it important? • How does this new method for measuring LRU differ from existing methods? • How does this new method work and how do we know it is accurate? • What are the practical advantages and limitations of this method?

3. What is Loss Reserve Uncertainty (LRU) and why is it important?

4. 1.1 What is LRU and why is it important? • LRU is a measure of potential loss reserve development --the degree to which actual future loss payments may ultimately deviate – favorably or unfavorably – from the currently forecast amounts that constitute loss reserves • LRU differs from pricing uncertainty -- the potential deviation between forecast loss (when a policy is written) and paid losses plus estimated reserve (at the end of an accident year).

5. 1.2 Why does measuring LRU matter? • ERM: LRU is a key component of Enterprise Risk Management, which requires measuring and managing all of a firm’s significant risks • Capital needs: knowing LRU assists a firm in determining the appropriate amount of capital to hold and whether some risk reduction action may be appropriate • Interpreting calendar year deviations: LRU can tell us whether their magnitude is significant and worthy of attention • Comparisons with other firms: are we better or worse?

6. Issues Estimating surplus adequacy Capital allocation and pricing Managerial feedback: is this deviation significant? Reinsurance Audiences Management Rating agencies and regulators Analysts and investors Management: actuary, CFO, CEO Management Rating agencies Management 1.3 To whom should LRU matter?

7. 2. How does this new method for measuring LRU differ from existing methods?

8. Existing Methods Most are ad hoc (algorithmic democracy), lack criteria for fit Others require costly, opaque software, specialized expertise None have been validated Chain ladder focus creates bias Susceptible to statistical pitfalls New Method Based on a standard, minimum squared error linear regression Simple, transparent, implemented in Excel spreadsheet Validated using simulated data Use of regression minimizes bias Avoids these pitfalls 2.1 How does this new method differ from existing ones?

9. 2.2 Additional objectives & features of the new method • It should be based on widely-available public data • Schedule P Part 3 Paid Loss triangles • The results should enable comparisons • Across different lines of business within a firm • For the same line of business across different firms • Between forecast and actual calendar year payments • The results should be scalable • Unaffected by irrelevant differences in the size of reserves • Applicable to reserves estimated by other methods

10. 3. How does this new method work and how do we know it is accurate?

11. 3.1 The starting point: Paid Loss Triangles Note that development years start with DY0 The cumulative numbers in each row converge to ultimate values. Reserve = sum of ultimates minus boxed diagonal values

12. 3.2 Essential steps in estimating LRU Step 1: Use the numbers already available to find a common underlying pattern for estimating future loss payments • The chain ladder method does this by calculating link ratios: the average ratio of numbers in a DY to the corresponding numbers in the preceding DY Step 2: Measure the variability of the available numbers around this underlying pattern Step 3: Use this measure to estimate the variability and correlation of forecast future payments and total reserve NOTE: Steps 2 and 3 depend crucially on doing Step 1 correctly

13. 3.3 Three statistical pitfalls in Step 1, and their solutions • The chain ladder has no objective criterion for measuring and maximizing goodness of fit to existing data • Solution: use linear regression, to minimize total squared error • The use of cumulative data creates serial correlation • Solution: use incremental data Development year 0 1 2 . . . Acc Yr: (A+ea) (A+ea)+(B+eb) (A+ea)+(B+eb)+(C+ec) . . . 3. Heteroskedasticity (non-constant SD): σ(ea) ≠ σ(eb) ≠ σ(ec) • Solution: analyze each development year separately

14. 3.4 Estimating reserves from incremental paid loss data Use these numbers (X) To fit these numbers (Y) Then use these numbers To forecast these numbers • Use linear regression to fit paid losses in future DY’s (up to DY7), with DY0 as the independent variable in all cases • Use estimated regression coefficients to forecast future loss payments

15. 3.5 Steps in calculating the standard deviation of reserves • Use the Salkever (textbook) method to calculate the standard deviation of each forecast future paid loss • Use the linear regression results to calculate the variance-covariance matrix of forecast errors for each future DY • Finally, aggregate these results to obtain LRU by Development Year, Calendar Year, and Total Reserve

16. Calculating the SD of Forecast Paid Losses for DY2

17. 3.6 How do we know that the method is accurate? • We created 10,000 simulated paid loss triangles where we knew the true underlying values (ultimate paid losses as well as the actual paid losses at any given point in the process) • We used the new method to estimate ultimate paid losses • We compared the known true values to the estimates obtained from the simulated triangles • These estimates were, on average, identical to the true values underlying the simulation • For estimated reserves • For loss reserve uncertainty, which includes parameter risk • NOTE: No other method for estimating LRU has been validated (to the best of my knowledge)

18. 3.7 Validation statistics

19. 4. What are the practical advantages and limitations of this method?

20. 4.1 Avantage: The results of this method are scalable • This method measures LRU in dollars. A better measure is the coefficient of variation, or CV, which is LRU as a % of the estimated reserve • CV is unaffected by reserve size, and so can be compared • across different business lines for the same firm • or across the same line for different firms • I believe that the CV can be legitimately applied to reserve estimates obtained in other ways (e.g., using claims data)

21. 4.2 Advantage: Comparisons across lines of business Σ Coefs: Ratio of remaining payments to dollars paid in initial development year (measures length of payout) E(Res): Estimated Reserve SD/E(Res): Coefficient of Variation, or standard deviation of reserve divided by estimated reserve SD/E(CY): Coefficient of Variation of forecast payments in the next Calendar Year NOTE: All results are based on 2003 Schedule P Part 3

22. 4.3a Advantage: Comparisons across firms: PPA Σ Coefs: Ratio of remaining payments to dollars paid in initial development year (measures length of payout) E(Res): Estimated Reserve SD/E(Res): Coefficient of Variation, or standard deviation of reserve divided by estimated reserve SD/E(CY): Coefficient of Variation of forecast payments in the next Calendar Year NOTE: All results are based on 2003 Schedule P Part 3

23. 4.3c Advantage: Comparisons across firms: WC Σ Coefs: Ratio of remaining payments to dollars paid in initial development year (measures length of payout) E(Res): Estimated Reserve SD/E(Res): Coefficient of Variation, or standard deviation of reserve divided by estimated reserve SD/E(CY): Coefficient of Variation of forecast payments in the next Calendar Year NOTE: All results are based on 2003 Schedule P Part 3

24. 4.4a Comparisons of forecast versus actual paids: WC

25. 4.4b Comparisons of forecast versus actual paids: CMP

26. 4.5a Limitations of this method: data peculiarities

27. 4.5b Limitations of this method: data peculiarities

28. 4.5c Limitations of this method: data peculiarities

29. 4.6 What are the limitations of this method? • Some versions of this method permit a gradual speedup or slowdown in payment patterns over time, but otherwise it assumes a stable past and future environment with regard to • Underwriting criteria • Exposure types • Reinsurance parameters • Legal and Regulatory environments • Estimating the tail is difficult with this method as with others • It necessarily relies on public data, and so does not reflect claims-level data available only to the firm

30. 5. Conclusions

31. 5.1 Conclusions • This method produces validated results • It is the only method that has been validated • It is reasonably simple • It avoids serious statistical pitfalls • It is based on standard textbook methods • It can be implemented in a spreadsheet • It can be explained to colleagues and superiors • It enables intra-firm comparisons of different lines of business • It enables comparisons of a line of business across firms • It enables detection of emerging problems that need attention

32. 5.2 How can I learn more about this method? Send me an email at Bill.Panning@Willis.com