Best Estimates for Reserves Glen Barnett and Ben Zehnwirth or find us on

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

Ratio techniques and extensions j-1 j x{x{ }y}y (Mack 1993) Chain Ladder Ratio E( y | x ) = bx andVar( y | x ) =  2 x - weighted least squares with w = 1 / x - weighted average with w = x b = = = ^  x y · 1 / x  x 2 · 1 / x  y / x · x  x x  y y  x x

Ratio techniques are regression estimators y x trend y/x Equivalently, y = bx +  where E(  ) = 0 and Var (  ) =  2 x

· Chain ladder is weighted regression through origin · Derives standard errors of link ratios, forecasts for chain ladder · Introduces weighting (  : Var(  )  2 x  ) parameter:  Ave. Dev. Factor  Chain Ladder  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 is underfitted 1982: low incurred development 1984: high incurred development 1984 is overfitted

The best line has an intercept - typical with real, exposure adjusted data Why is E(y|x) = bx not satisfied by the data? y x

(Murphy, 1995) - Including an intercept will give a better fit. - Unbiased “Development Factors”. y = a + bx +  where Var (  ) =  2 x  Works wholly within a regression framework Advocates use of intercept Derives standard errors of forecasts for 

Ratios with Intercepts ELRF Parameters Delta (  ) = 1 AIC = 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

(Venter, 1996) Incremental at dev. period j Cumulative at dev. period j-1 y – x = a + (b – 1) x +   j-1 j x{x{ }y}y Case (ii) b = 1, a  0 a = Ave (incrementals) Use link-ratios for projection Abandon Ratios - No predictive power Case (i) b > 1, a = 0 ^

Plot of Development Year 1 vs Development Year 0 Cumulative (1) vs. Cumulative (0) Corr= P-value=0.764 Incremental (1) vs. Cumulative (0)

Weighted Standardized Residuals of Cape Cod model y – x = a +  a = Ave (y–x) ^

Forecasts Cape Cod ModelChain Ladder Ratios Model

Trend Parameter For Incrementals y – x = a 0 + a 1 w + (b – 1) x +  Intercept Acc Yr trend “Ratio” j-1 j x{x{ }y}y 1 2 n  w An increasing trend down the accident years will ‘induce’ a correlation between (y-x) and x. where Var(  )  2 x  Extended Link Ratio Family of Models

Wtd. Std. Residualsvs. Payment Year Wtd. Std. Residualsvs. Fitted Values Commonly have changing payment year trends Often have non-normality Even this extended family of models is generally inadequate: (ABC) (Pan6)

Part II d t = w+d Development year Payment year Accident year w 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

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 d  y(d)y(d) 11 22 Development trend for single accident year, data on log scale : (Data = Trends + Random Fluctuations) log(p(d)) = y(d) =  +   i +  d y(0) =  +  0 y(1) =  +  1 +  1 y(2) =  +  1 +  2 +  2  d i=1

d p(d)p(d) On the original (dollar) scale, each payment has a lognormal distribution, related by the trends.

All years - trends in 3 directions log(p(w,d)) = y(w,d) =  w +   i +   j +  w,d d i=1 w+d j=1 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.

Wtd. Std. Residualsvs. Payment Year Fitting a single trend to changing trends - estimates as an average trend - changes show up in the residuals

if the modeling framework “works”, it should be hard to differentiate between real data and data simulated from an identified model Checking the modeling framework 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 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.

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). Noisy data is not necessarily hard to predict Wtd. Std. Residuals vs. Pay. Year Wtd. Std. Residuals vs. Acc. Year Wtd. Std. Residuals vs. Dev. Year Wtd. Std. Residuals vs. Fitted Values Residual plots after removing a single development year and payment year trend.

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): Noisy data is not necessarily hard to predict

Part III 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. Prediction intervals and uncertainty 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.

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. Risk Based Capital

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.