More information will be available at the St. James Room 4 th floor 7 – 11 pm.

1. More information will be available at the St. James Room 4th floor 7 – 11 pm. ICRFS-ELRF will be available for FREE! Much of current discussion included in the software

2. ELRF MPTF PTF LRT SEE NEXTSLIDE BF/ELR Unique Benefits afforded by Paradigm Shift “Best Estimates for Reserves” is now included in 2005 CAS Syllabus of Examinations. Reviewhttp://casact.org/pubs/actrev/may01/latest.htm Models – Coin Versus Roulette Wheel MPTF discussed in Session 3 (3:15 Arlington) and 6 (10:30am White Hill)

3. Summary- Examples • Many myths grounded in a flawed paradigm Ranges? Confidence Intervals? Myths Loss Reserve Upgrades. Myths • Link Ratios cannot capture trends and volatility • Link Ratios can give very false indications • Must model Paids and CREs separately Cannot determine volatility in paids from incurreds! Some real life examples taken from “Best Estimates..”

4. Unique Benefits afforded by Paradigm Shift Modeling MPTF PTF PAD MULTIPLE LINES/ SEGMENTS/LAYERS REINSURANCE V@R RELATIONSHIPS/CORRELATIONS ADVERSE DEVELOPMENTCOVER CAPITAL ALLOCATION CREDIBILITY MODELLING EXCESS OF LOSS The Pleasure of Finding Things Out !

5. x x x x x x x x x x x x x x x x x x x x x x x x x x ProbabilisticModelling e.g. trends in the development year direction If we graph the data for an accident year against development year, we can see two trends. 0123456789101112

6. x x x x x x x x x x x x x x x x x x x x x x x x x x Probabilistic Modelling Could put a line through the points, using a ruler. Or could do something formally, using regression. 0123456789101112 Variance =

7. Introduction to Probabilistic Modelling Models Include More Than The Trends x x x (y – ŷ) x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 10 11 12 • The model is not just the trends in the mean, but the distribution about the mean (Data = Trends + Random Fluctuations)

8. o x o o o x x x x o x o o x x o o x o o x o x x x o Introduction to Probabilistic Modelling Simulating the Same “Features” in the Data 0 1 2 3 4 5 6 7 8 9 10 11 12 • Simulate “new” observations based in the trends and standard errors • Simulated data should be indistinguishable from the real data

9. - Real Sample: x1,…,xn - Random Sample from fitted distribution: y1,…,yn What does it mean to say a model gives a good fit? e.g. lognormal fit to claim size distribution Does not mean we think the model generated the data fitted lognormal - Fitted Distribution y’s look like x’s: — Model has probabilistic mechanisms that can reproduce the data

10. PROBABILISTIC MODEL S3 Real Data S2 S1 Based on Ratios Simulated triangles cannot be distinguished from real data – similar trends, trend changes in same periods, same amount of random variation about trends Models project past volatility into the future

11. j j-1 y X = Cum. @ j-1 Y = Cum. @ j y y x x x ELRF(Extended Link Ratio Family)x is cumu. at dev. j-1 and y is cum. at dev. j • Link Ratios are a comparison of columns • We can graph the ratios of Y to X y/x y/x

12. Mack (1993) Chain Ladder Ratio( Volume Weighted Average) Arithmetic Average

13. Intercept (Murphy (1994)) Since y already includes x: y = x + p Incremental Cumulative at j at j -1 Is b -1 significant ? Venter (1996)

14. j-1 j Cumulative Incremental p j-1 j } p x x x x x x Use link-ratios for projection x Abandon Ratios - No predictive power

15. p x Is assumption E(p|x) = a + (b-1) xtenable? Note: If corr(x, p) = 0, then corr((b-1)x, p) = 0 If x, p uncorrelated, no ratio has predictive power Ratio selection by actuarial judgement can’t overcome zero correlation.

16. Cumulative Incremental j-1 j } p j-1 j 90 91 92 w Condition 1: p x x x x x x x x w p Condition 2:

17. Now Introduce Trend Parameter For Incrementals p

18. Condition 3: Incremental Review 3 conditions: Condition 1: Zero trend Condition 2: Constant trend, positive or negative Condition 3: Non-constant trend

19. FORECASTING AND STATISTICAL MODELS FUTURE PAST (i) RECOGNIZE POTENTIAL ERRORS (ii) ON STRAIGHT STRETCHES NAVIGATE QUITE WELL

20. Past Future 1986 1987 1998 Probabilistic Modelling Trends occur in three directions: 0 1 Development year d Calendar year t = w+d w Accident year

21. - 0.2d d M3IR5 Data 0 1 2 3 4 5 6 7 8 9 10 11 12 13 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072 7427 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 100000 81873 67032 54881 44933 36788 30119 24660 20190 100000 81873 67032 54881 44933 36788 30119 24660 100000 81873 67032 54881 44933 36788 30119 100000 81873 67032 54881 44933 36788 100000 81873 67032 54881 44933 100000 81873 67032 54881 100000 81873 67032 100000 81873 100000 alpha = 11.513 -0.2 PAID LOSS = EXP(alpha - 0.2d)

22. 0.15 0.3 0.1 Probabilistic Modelling Axiomatic Properties of Trends

23. Sales Figures

24. Resultant development year trends

26. WHEN CAN ACCIDENT YEARS BE REGARDED AS DEVELOPMENT YEARS?GLEN BARNETT, BEN ZEHNWIRTH AND EUGENE DUBOSSARSKYAbstractThe chain ladder (volume-weighted average development factor) is perhaps the most widely used of the link ratio (age-to-age development factor) techniques, being popular among actuaries in many countries. The chain ladder technique has a number of interesting properties. We present one such property, which indicates that the chain ladder doesn’t distinguish between accident years and development years. While we have not seen a proof of this property in English language journals, it appears in Dannenberg, Kaas and Usman [1]. The result is also discussed in Kaas et al [2]. We give a simple proof that the chain ladder possesses this property and discuss its other implications for the chain ladder technique. It becomes clear that the chain ladder does not capture the structure of real triangles.

27. The Chain Ladder( Volume Weighted Average ) Transpose Invariance property Use Volume Weighted Average to project incremental data: Take incremental array, cumulate across, find ratios, project, and difference back to incremental data. Now: transpose incremental*, do Volume Weighted Average , transpose back  same forecasts! (equivalently, perform chain ladder ‘down’ not ‘across’: cumulate down, take ratios down, project down, difference back)

28. The Chain Ladder( Volume Weighted Average )

29. The Chain ladder (Volume weighted aveage) - Transpose Invariance property Chain ladder does not distinguish between accident and development directions. But they are notalike: raw data adjusted for trend in other direction

30. The Chain Ladder( Volume Weighted Average ) Additionally, chain ladder (and ratio methods in general) ignore abundant information in nearby data. * If you left out a point, how would you guess what it was? - observations at same delay very informative.

31. The Chain Ladder( Volume Weighted Average ) Additionally, chain ladder (and ratio methods in general) ignore information in nearby data. * If you left out a point, how would you guess what it was? - observations at same delay very informative. - nearby delays also informative (smooth trends) (could leave out whole development) Chain ladder ignores both

32. Unstable Trends, Low Process VariabilityData (ABC) Major Calendar Year shifts satisfying Condition 3

33. The plots indicate a shift from calendar periods 84-85-86. However, we cannot adjust for accident period trends to diagnostically view what is left over along the calendar periodsas we can with PTF models.This example is in “Best Estimates”

34. Do U assign zero weight to all years save last two or three?

35. The link-ratio type models cannot capture changes along the calendar periods (diagonals).Determine the optimal model and note that several of the ratios are set to 1. The residualsof the optimal model are displayed below.

36. Model Display

37. Note that as you move down the accident years the 16%+_ trend kicks in at earlierdevelopment periods. If variancewas not so small, we would not be able to see this on thegraphs of the data themselves – the trend change would be ‘obscured’ by randomfluctuation. 1977 Run-off 1978 Run-off 1979 Run-off

38. Link Ratios can give answersthat are much too high (LR high) - Case Study 5. Overview This case study illustrates how the residuals in ELRF can be very powerful indemonstrating that methods based on link ratios can sometimes give answers that aremuch too high. The ELRF module also allows us to assess the predictive power of link ratio methodscompared to trends in the incremental data. For the data studied, trends in the incremental data have much more predictive power. Moreover, link ratio methods do not capture many ofthe features of the data.

39. Bring up the Weighted Residual Plot using the button.

40. Residuals represent the data minus what has been fitted to the data. Observe that theresiduals vs calendar years (Wtd Std Res vs Cal Year) trend downwards (negative trend).

41. This means that the trends fitted to the data are much higher than the actual trends in thedata.Accordingly any forecast produced by this method will assume trends that are muchhigher than the trends in the data. Therefore, the forecast will be much too high.

42. Below are the forecasts based on volume weighted average ratio and the arithmeticaverage ratio. Table 5.1 - Summary of Forecasts

43. Do link ratio method have any predictive power for this data?

44. The Best Model in ELRF rarely uses link ratios and treats development periods as separate problems- Show them!

45. A good model for this data has the following trends and volatility about trends

46. With the calendar yeartrend of 8.71% ± 0.97%, we obtain a distribution with a mean $593,506,000,and a st. deviation of $42,191,000. Scenario 1If we revertto the trend of 18.3% ± 2.6% experienced from 1981-1984, we obtain a reservedistribution with the mean of$751,912,000 with a standard deviation of $79,509,000. Scenario 2.Returning to the calendar year trend changes, it is important to try and identify whatcaused those changes. We cannot just assume that the most recent trend (8.71% ±0.97%) will continue for the next 17 years. Modelling other data types such as CaseReserve Estimates (CRE) and Number of Claims Closed (NCC) can assist in formulatingassumptions about future calendar year trends. See below.This would make it easier to decide on a future trend scenario along the calendar years.

47. The estimated trend between 1974-1978 is 47.5% ± 3.2%. If we assume that trend onlyfor the next calendar year (1991-1992) and revert to a trend of 18.3% ± 2.6% thereafter,we obtain a reserve distribution with a mean of $1,007,496,000 and a standard deviationof $112,234,000. Scenario 3

48. Conclusions There is some evidence that even the mean of $593,506,000 based on scenario 1 is too high. Accordingly, scenarios 2 and 3 appear tobe even more unlikely given the features in these three triangles in the past.

49. Comparing PL vs CRE Overview This case study illustrates how comparing a model for the Paid Losses with a model forthe Case Reserve Estimates (CRE) gives additional critical information that cannot beextracted from the Incurred Losses triangle. For most portfolios, we find that CREs lagPaid Losses in respect of calendar year trends.