Best Estimates for Reserves Glen Barnett and Ben Zehnwirth email: GlenBarnett@insureware.com, BenZehnwirth@insureware.com, or find us on http://www.insureware.com Summary I. Ratio techniques and extensions • Ratios are regressions • Regressions have assumptions assume when using ratios) • Assumptions need to be checked (know what you (When do ratios work?) • Assumptions often don't hold • What does this suggest? Summary II. Statistical modeling framework (Probabilistic Trend Family of models) • Model the logarithms of the incrementals • Parameters to pick up trends in the three directions • Probability Distribution to every cell • Assumptions generally met • Assessing stability of trends (confidence about the future) III. Reserve Figure Ratio techniques and extensions j-1 j x{ }y E(y|x) = bx and Var(y|x) = 2 x (Mack 1993) - weighted least squares with w = 1/x - weighted average with w = x ^ b= xy ·1/x x2 ·1/x = y/x · x x = y x Chain Ladder Ratio Equivalently, y = bx + where E( ) = 0 and Var ( ) = 2x y y/x x trend Ratio techniques are regression estimators · Chain ladder is weighted regression through origin · Derives standard errors of link ratios, forecasts for chain ladder · Introduces weighting (d: Var() = 2xd) parameter: d = 2: Ave. Dev. Factor d = 1: Chain Ladder d = 0: Ordinary regression through origin Regression methodology advantages · Standard Errors of ratios and forecasts · Testing of assumptions Incurred losses analysed by Mack(1993) Residuals of chain ladder ratios Wtd. Std. Residuals vs. Fitted Values 2 1 0 -1 0 5000 10000 15000 20000 Is E(y|x) = bx satisfied by the data? 25000 30000 1982: low incurred development 1984: high incurred development Unadjusted Data vs. Dev. Year Unadjusted Data vs. Dev. Year 2.5e4 2.5e4 2e4 2e4 1.5e4 1.5e4 1e4 1e4 5e3 5e3 0 0 0 1 2 3 4 5 6 7 8 9 1982 is underfitted 0 1 2 1 1 0 0 -1 -1 83 84 85 86 87 4 5 6 7 8 9 Wtd. Std. Residuals vs. Acc. Year 2 82 3 1984 is overfitted Wtd. Std. Residuals vs. Acc. Year 81 2 88 89 81 82 83 84 85 86 87 88 89 Why is E(y|x) = bx not satisfied by the data? y x The best line has an intercept - typical with real, exposure adjusted data y = a + bx + Var ( ) = 2xd - Including an intercept will give a better fit. (Murphy, 1995) where - Unbiased “Development Factors”. • Works wholly within a regression framework • Advocates use of intercept • Derives standard errors of forecasts for d=012 Ratios with Intercepts ELRF Parameters Devel. Period 00-01 01-02 02-03 03-04 04-05 05-06 06-07 07-08 08-09 Est. 4,329.21 4,159.69 4,235.92 2,188.79 3,562.27 743.46 792.81 **** **** Intercept S.E. 516.31 2,531.37 2,814.52 1,133.11 2,031.41 2,387.33 153.13 **** **** P-value 0.00007 0.15144 0.19266 0.12557 0.17778 0.78494 0.12147 **** **** Ratio(Slope) Est. Ratio-1 1.21445 0.21445 1.06962 0.06962 0.91968 -0.08032 1.03341 0.03341 0.92675 -0.07325 1.00608 0.00608 0.99109 -0.00891 1.01689 0.01689 1.00922 0.00922 S.E. 0.42131 0.35842 0.24743 0.07443 0.11023 0.12204 0.00824 0.01495 **** P-value 0.6264 0.8524 0.7586 0.67679 0.55389 0.96477 0.47483 0.46142 **** Delta (d) = 1 AIC = 760.5 In order for the test to be conducted at an overall 5% level, a parameter is regarded as insignificant if the corresponding P-Value is greater than 0.00320. y – x = a + (b – 1) x + (Venter, 1996) Cumulative at dev. period j-1 Incremental at dev. period j j-1 j x{ }y Case (i) b > 1, a = 0 Use link-ratios for projection Case (ii) b = 1, a 0 a^ = Ave (incrementals) Abandon Ratios - No predictive power Plot of Development Year 1 vs Development Year 0 Cumulative (1) vs. Incremental (1) vs. Cumulative (0) Cumulative (0) 12000 9000 7000 8000 5000 3000 4000 1000 0 0 0 2000 5000 2000 Corr=-0.117 P-value=0.764 5000 Weighted Standardized Residuals of Cape Cod model y–x=a+ ^a = Ave (y–x) Wtd. Std. Residuals vs. Dev. Year Wtd. Std. Residuals vs. Acc. Year 2 2 1 1 0 0 -1 -1 -2 -2 1 2 3 4 5 6 7 8 9 81 Wtd. Std. Residuals vs. Pay. Year 82 83 84 85 86 87 88 89 Wtd. Std. Residuals vs. Fitted Values 2 2 1 1 0 0 -1 -1 -2 -2 82 83 84 85 86 87 88 89 90 5000 10000 15000 20000 25000 Forecasts Chain Ladder Ratios Model Accident Mean Standard Coeff. Of Year Forecast Error Variation 1981 0 0 *** 1982 155 148 0.96 1983 616 586 0.95 1984 1633 702 0.43 1985 2779 1404 0.51 1986 3671 1976 0.54 1987 5455 2190 0.4 1988 10934 5351 0.49 1989 10668 6335 0.59 1990 16360 24606 1.5 Total 52272 26883 0.51 Cape Cod Model Accident Mean Standard Coeff. Of Year Forecast Error Variation 1981 0 0 *** 1982 172 41 0.24 1983 483 464 0.96 1984 1112 498 0.45 1985 1971 1167 0.59 1986 4230 1526 0.36 1987 6908 1651 0.24 1988 10283 3231 0.31 1989 14905 3644 0.24 1990 19367 4521 0.23 Total 59430 8425 0.14 An increasing trend down the accident years will ‘induce’ a correlation between (y-x) and x. Trend Parameter For Incrementals 1 2 w n j-1 j x{ }y y – x = a0 + a1w + (b – 1) x + Intercept Acc Yr trend “Ratio” where Var() = 2xd Extended Link Ratio Family of Models Even this extended family of models is generally inadequate: Wtd. Std. Residuals vs. Payment Year 1 Commonly have changing payment year trends 0.5 0 (ABC) -0.5 -1 -1.5 78 79 80 81 82 83 84 85 86 87 Wtd. Std. Residuals vs. Fitted Values 2 (Pan6) Often have non-normality 1 0 -1 0 2500000 5000000 7500000 10000000 Part II Trends occur in three directions: 0 1 Development year d 1 2 Payment year t = w+d w Accident year Trend properties of loss development arrays • Trends in payment year direction project onto the other two directions and vice versa • Changing trends can be hard to pick up in the presence of noise, unless main trends are removed first (regression as a form of adjustment) • Modeling a changing trend as a single trend will result in pattern in the residual plots Underlying Trends in the Data Projection of trends onto other directions Changing trends hard to pick without removing main development and payment year trend. Distribution of data about those trends (Data = Trends + Random Fluctuations) Development trend for single accident year, data on log scale: y(0) = + 0 y(1) = + 1 + 1 y(2) = + 1 + 2 + 2 2 y(d) 1 0 1 2 3 4 d log(p(d)) = y(d) = + i + d i=1 5 d On the original (dollar) scale, each payment has a lognormal distribution, related by the trends. p(d) 0 1 2 3 4 5 d All years - trends in 3 directions d w+d i=1 j=1 log(p(w,d)) = y(w,d) = w+ i + j + w,d Different levels for accident years Payment year trends You would never use all these parameters at the same time parsimony is as important as flexibility. A model with too many parameters will give poor forecasts. Fitting a single trend to changing trends Wtd. Std. Residuals vs. Payment Year 2 1 0 -1 -2 78 79 80 81 82 83 84 85 86 87 88 89 90 91 - estimates as an average trend - changes show up in the residuals Checking the modeling framework • if the modeling framework “works”, it should be hard to differentiate between real data and data simulated from an identified model • if you create (simulate) data, you should be able to identify the (known) changing trends in the data; mean forecasts should usually be within about 2 standard errors of the true mean Smooth data can conceal changing trends Individual Link Ratios by Delay 2.50 2.25 2.00 1.75 1.50 1.25 1.00 0-1 Very smooth data 1-2 2-3 3-4 4-5 5-6 6-7 7-8 Very smooth ratios (real array - ABC - values in paper) 8-9 9-10 Smooth data can conceal changing trends Wtd. Std. Residuals vs. Dev. Year Wtd. Std. Residuals vs. Acc. Year 2 2 1 1 0 0 -1 -1 0 1 2 3 4 5 6 7 8 9 10 77 78 Wtd. Std. Residuals vs. Pay. Year 2 1 1 0 0 -1 -1 78 79 80 81 82 83 84 85 80 81 82 83 84 85 86 87 Wtd. Std. Residuals vs. Fitted Values 2 77 79 86 87 9.00 10.00 11.00 12.00 Residuals after removing all accident year and development year trends. Noisy data is not necessarily hard to predict • Volatile (noisy) data can be predictable (within model uncertainty) if trends are stable The data in the following example is very volatile (noisy) see the paper for this real data array (Pan6). Wtd. Std. Residuals vs. Dev. Year Wtd. Std. Residuals vs. Acc. Year 1 1 0 0 -1 -1 -2 Residual plots after removing a single development year and payment year trend. -2 0 1 2 3 4 5 86 Wtd. Std. Residuals vs. Pay. Year 87 88 89 90 91 92 93 94 95 96 Wtd. Std. Residuals vs. Fitted Values 1 1 0 0 -1 -1 -2 -2 86 87 88 89 90 91 92 93 94 95 96 10.00 11.00 12.00 13.00 14.00 15.00 Noisy data is not necessarily hard to predict • change in development year trend (the trend between 0-1 is different from the later years) • no obvious trend changes in other directions • wider spread for first two development years • single superimposed inflation parameter is not significantly different from 0 one accident year level, two development year trends, no payment year trend, weighted regression Noisy data is not necessarily hard to predict • residual plots and other diagnostics for that model are good • forecast of this model yields an outstanding mean forecast of $20.6 million and a standard deviation of $9.3 million, so the standard deviation is high (volatile data). • It is important to see how the forecasts compare as we remove the most recent years (validation): Years in Estimation 86-96 86-95 86-94 86-93 86-92 N 44 41 35 29 25 Trend standard (dev period 1+) error 0.6250 0.1432 0.6102 0.1479 0.6149 0.1681 0.5024 0.1977 0.5631 0.2143 Mean Standard Fcst Error 20,352,011 9,136,870 21,410,781 9,839,127 21,037,520 10,654,173 19,755,944 11,647,274 18,567,664 11,529,359 Part III Prediction intervals and uncertainty • any single figure will be wrong, but we can find the probability of lying in a range. • include both process risk and parameter risk. Ignoring parameter risk leads to underreserving. • forecast distributions are accurate if assumptions about the future remain true. • Distribution of sum of payment year totals important for dynamic financial analysis. Distributions for future underwriting years important for pricing. • For a fixed security level on all the lines combined, the risk margin per line decreases as the number of lines increases. Risk Based Capital • future uncertainty in loss reserves should be based on a probabilistic model, which might not be related to reserves carried by the in the past. • uncertainty for each line should be based on a probabilistic model that describes the particular line • experience may be unrelated to the industry as a whole. • Security margins should be selected formally. Implicit risk margins may be much less or much more than required. Booking the Reserve • extract information, in terms of trends, stability of trends and distributions about trends, for the loss development array. Validation analysis. • formulate assumptions about future. If recent trends unstable, try to identify the cause, and use any relevant business knowledge. • select percentile (use distribution of reserves, combined security margin, and available risk capital). Increased uncertainty about future trends may require a higher security margin. Other Benefits of the Statistical Paradigm • Credibility - if a trend parameter estimate for an individual company is not credible, it can be formally shrunk towards an industry estimate. • Segmentation and layers - often the statistical model (parameter structure) identified for a combined array applies to some of its segments. These ideas can also be applied to territories etc. and to layers.