1. Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business Gerald Kirschner Classic Solutions Casualty Loss Reserve Seminar September 2002

2. Presentation Structure Background – simulating reserves stochastically Correlation – a general discussion Case study Description Results using correlation matrix approach Results using bootstrap approach Conclusion

3. Simulating reserves stochastically using a chain-ladder method Begin with a traditional loss triangle Calculate link ratios Calculate mean and standard deviation of the link ratios

4. Simulating reserves stochastically using a chain-ladder method Think of the observed link ratios for each development period as coming from an underlying distribution with mean and standard deviation as calculated on the previous slide Make an assumption about the shape of the underlying distribution – easiest assumptions are Lognormal or Normal

5. Simulating reserves stochastically using a chain-ladder method For each link ratio that is needed to square the original triangle, pull a value at random from the distribution described by 1. Shape assumption (i.e. Lognormal or Normal) 2. Mean 3. Standard deviation

6. Simulating reserves stochastically using a chain-ladder method Simulated values are shown in red Random draw

7. Simulating reserves stochastically using a chain-ladder method Square the triangle using the simulated link ratios to project one possible set of ultimate accident year values. Sum the accident year results to get a total reserve indication. Repeat 1,000 or 5,000 or 10,000 times. Result is a range of outcomes.

8. Simulating reserves stochastically using other methods Numerous options for enhancing this basic approach Logarithmic transformation of link ratios before fitting Inclusion of a parameter risk adjustment Use of incremental data instead of cumulative Extend payout via tail factor extrapolation

9. Simulating reserves stochastically via bootstrapping Bootstrapping is a different way of arriving at the same place Bootstrapping does not care about the underlying distribution – instead bootstrapping assumes that the historical observations contain sufficient variability in their own right to help us predict the future

10. Simulating reserves stochastically via bootstrapping 1. Keep current diagonal intact 2. Apply average link ratios to “back- cast” a series of fitted historical payments Ex: 1,988 = 2,300 

11. Simulating reserves stochastically via bootstrapping 3. Convert both actual and fitted triangles to incrementals 4. Look at difference between fitted and actual payments to develop a set of Residuals

12. Simulating reserves stochastically via bootstrapping 5. Create a “false history” by making random draws, with replacement, from the triangle of residuals. Combine the random draws with the recast historical data to come up with the “false history”.

13. Simulating reserves stochastically via bootstrapping 6. Calculate link ratios from the data in the cumulated false history triangle 7. Use the link ratios to square the false history data triangle 8. Several additional steps are described in the paper… 9. Repeat process N times to get N different reserve indications.

14. Pros / Cons of each method Chain-ladder Pros More flexible - not limited by observed data Chain-ladder Cons More assumptions Potential problems with negative values Bootstrap Pros Do not need to make assumptions about underlying distribution Bootstrap Cons Variability limited to that which is in the historical data

15. Correlation Correlation vs. causality Correlation is a a way of measuring the “strength of relationship” between two sets of numbers. Causality is the relation between a cause (something that brings about a result) and its effect. Can have correlation between two things without causality – both could be influenced by an unknown third item.

16. Effects of Correlation Suppose we have two lines, A & B, whose reserve indications exhibit correlation Strength of the correlation is irrelevant if we only care about the mean reserve indication for A + B: mean (A + B) = mean (A) + mean (B) Strength of correlation matters when we look towards the ends of the distribution of (A+B).

17. Effects of Correlation: Example 1 2 lines of business, N (100,25) 75 th percentile of A+B at different levels of correlation between A and B: CorrelationValues at 75 th percentile Ratio of Values at 75 th percentile 0.00223.80.0% 0.25226.71.3% 0.50229.22.4% 0.75231.53.4% 1.00233.74.4%

18. Effects of Correlation: Example 2 Same idea, but increase variability of distributions for lines A and B:

19. Correlation methodologies Method 1: relies on the user to specify a correlation matrix that describes the relative strength of relationship between the lines of business by analyzed. Will use rank correlation technique to develop a correlated reserve indication. Method 2: uses the bootstrap process to maintain any correlations that might be implicit in the historical data. No other information is needed to develop the correlated reserve indication.

20. Rank correlation example

21. Method 1 approach Model user must determine (through other means) the relative relationships between the lines of business being modeled Information is entered into a correlation matrix Uncorrelated reserve indications generated for each line and sorted from low to high

22. Method 1 approach continued Model creates a Normal distribution for each line with mean = average reserve indication for each class, standard deviation = standard deviation of reserve indications for each class Normal distributions are correlated using the user-entered correlation matrix Pull values from correlated normal distribution – drawing N correlated values, where N = # simulated reserve indications

23. Method 1 approach continued Use relative positioning of the correlated Normal draws as the basis for pulling values from the sorted table of uncorrelated reserve indications to create correlated reserve indications across the lines of business

24. Example of Method 1 approach 2 nd value from Class A, 3 rd value from Class B 12 3

25. Bootstrap Correlation Methodology

26. Pros / Cons of each approach Correlation Matrix Pros More flexible - not limited by observed data Correlation Matrix Cons Requires modeler to do additional work to quantify the correlations between lines Bootstrap Correlation Pros Do not need to make assumptions about underlying correlations Bootstrap Cons Results reflect only those correlations that were in the historical data

27. Case Study Three lines of business All produce approximately the same mean reserve indication, but with different levels of volatility around the mean Run a 5,000 iteration simulation exercise for each line Examine the results for the aggregated reserve indication at different percentiles of the aggregate distribution

28. Rank correlation results

29. Add Bootstrap results

30. Case Study Conclusions Mean aggregated reserve = 4.33B Reserves at the 75 th percentile range from 4.64B to 4.84B Bootstrap tells us that there does appear to be some level of correlation in underlying data

31. General Conclusions To calculate an aggregate reserve distribution, must understand and be able to quantify the dependencies between underlying lines of business Correlation is probably not an important issue for lines of business with non-volatile reserve ranges, but can be important for ones with volatile reserves, especially as one moves further towards a tail of the aggregate distribution