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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
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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 (know what you assume when using ratios) • Assumptions need to be checked (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
j j-1 }y x{ y/x · x xy ·1/x y ^ b = = = x x2 ·1/x x Ratio techniques and extensions E(y|x) = bx and Var(y|x) = 2x - weighted least squares with w = 1/x - weighted average with w = x (Mack 1993) Chain Ladder Ratio
y y/x x trend Equivalently, y = bx + Ratio techniques are regression estimators where E() = 0 and Var () = 2x
· Chain ladder is weighted regression through origin · Derives standard errors of link ratios, forecasts for chain ladder · Introduces weighting (d: Var(e) = s2xd) 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 Is E(y|x) = bx satisfied by the data?
1982: low incurred development 1984: high incurred development 1982 is underfitted 1984 is overfitted
y x Why is E(y|x) = bx not satisfied by the data? The best line has an intercept - typical with real, exposure adjusted data
y = a + bx + where Var ( ) = 2x - Including an intercept will give a better fit. - Unbiased “Development Factors”. (Murphy, 1995) • Works wholly within a regression framework • Advocates use of intercept • Derives standard errors of forecasts for
Ratios with Intercepts ELRF Parameters Delta () = 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.
j j-1 }y x{ y – x = a + (b – 1) x + (Venter, 1996) Cumulative at dev. period j-1 Incremental at dev. period j 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
Incremental (1) vs. Cumulative (1) vs. Cumulative (0) Cumulative (0) 12000 9000 7000 8000 5000 3000 4000 1000 0 0 2000 5000 0 2000 5000 Corr=-0.117 P-value=0.764 Plot of Development Year 1vs Development Year 0
^ a = Ave (y–x) Weighted Standardized Residuals of Cape Cod model y – x = a +
Forecasts Chain Ladder Ratios Model Cape Cod Model
j j-1 }y x{ An increasing trend down the accident years will ‘induce’ a correlation between (y-x) and x. Trend Parameter For Incrementals 1 y – x = a0 + a1w + (b – 1) x + 2 w n Acc Yr trend Intercept “Ratio” where Var(e) = s2xd Extended Link Ratio Family of Models
Wtd. Std. Residuals vs. Payment Year 1 0.5 0 -0.5 -1 -1.5 78 79 80 81 82 83 84 85 86 87 Wtd. Std. Residuals vs. Fitted Values 2 1 0 -1 0 2500000 5000000 7500000 10000000 Even this extended family of models is generally inadequate: Commonly have changing payment year trends (ABC) Often have non-normality (Pan6)
Development year 0 1 d 1 2 Payment year t = w+d w Accident year Part II Trends occur in three directions:
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
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 y(d) 2 1 d 3 0 1 2 4 5 d i=1 log(p(d)) = y(d) = + i + d
d 3 0 1 2 4 5 On the original (dollar) scale, each payment has a lognormal distribution, related by the trends. p(d)
All years - trends in 3 directions d i=1 w+d j=1 log(p(w,d)) = y(w,d) = w+ i + j + w,d Payment year trends Different levels for accident years 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
Individual Link Ratios by Delay 2.50 2.25 2.00 1.75 1.50 1.25 1.00 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 Smooth data can conceal changing trends Very smooth data (real array - ABC - values in paper) Very smooth ratios
Smooth data can conceal changing trends 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 Wtd. Std. Residuals vs. Dev. Year vs. Acc. Year 1 1 Residual plots after removing a single development year and payment year trend. 0 0 -1 -1 -2 -2 0 1 2 3 4 5 86 87 88 89 90 91 92 93 94 95 96 Wtd. Std. Residuals Wtd. Std. Residuals vs. Pay. Year 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):
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.