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Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business Gerald Kirschner Classic Solutions Casualty Loss Reserve Seminar September 2002 Presentation Structure Background – simulating reserves stochastically Correlation – a general discussion Case study Description Results using correlation matrix approach Results using bootstrap approach Conclusion Simulating reserves stochastically using a chain-ladder method Begin with a Acc. Year 12 Development Age 24 36 48 traditional loss 1 1,000 1,500 1,750 2,000 triangle 2 3 1,200 1,800 2,000 2,500 2,300 Calculate link 4 2,100 ratios Acc. Link Ratios Calculate mean Year 12 - 24 24 - 36 36 - 48 and standard 1 1.500 1.167 1.143 2 1.667 1.150 deviation of the 3 1.389 link ratios Mean 1.500 1.157 1.143 Std. Deviation 0.1179 0.0082 0 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 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 Simulating reserves stochastically using a chain-ladder method Acc. Link Ratios Year 12 - 24 24 - 36 36 - 48 1 1.500 1.167 1.143 2 1.667 1.150 3 1.389 Lognormal Distribution, mean 1.5, standard deviation 0.1179 Mean 1.500 1.157 1.143 30.0% Std. Deviation 0.1179 0.0082 0 % of Total Observations 25.0% 20.0% 15.0% 10.0% Acc. Link Ratios 5.0% Year 12 - 24 24 - 36 36 - 48 0.0% 1 1.500 1.167 1.143 1.168 1.249 1.330 1.411 1.492 1.573 1.654 1.735 1.816 1.897 2 1.667 1.150 1.143 3 1.389 1.163 1.143 4 1.419 1.145 1.143 Random draw Simulated values are shown in red 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. 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 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 Simulating reserves stochastically via bootstrapping Actual Cumulative Historical Data 1. Keep current Acc. Development Age diagonal intact Year 12 24 36 48 1 1,000 1,500 1,750 2,000 2. Apply average link 2 1,200 2,000 2,300 ratios to “back- 3 1,800 2,500 4 2,100 cast” a series of Ave Link Ratio 1.500 1.157 1.143 fitted historical payments Recast Cumulative Historical Data Acc. Development Age Ex: 1,988 = Year 12 24 36 48 2,300 1 1,008 1,512 1,750 2,000 2 1,325 1,988 2,300 3 1,667 2,500 4 2,100 Simulating reserves stochastically via bootstrapping 3. Convert both actual and Actual Incremental Historical Data fitted triangles to Acc. Year 12 Development Age 24 36 48 incrementals 1 2 1,000 1,200 500 800 250 300 250 4. Look at difference 3 1,800 700 between fitted and actual 4 2,100 payments to develop a set of Residuals Recast Cumulative Historical Data Residuals Acc. Development Age Acc. Development Age Year 12 24 36 48 Year 12 24 36 48 1 1,008 504 238 250 1 (0.259) (0.183) 0.801 0.000 2 1,325 663 312 2 (3.437) 5.340 (0.699) 3 1,667 833 3 3.266 (4.619) 4 2,100 4 0.000 Simulating reserves stochastically via bootstrapping Random Draw from Residuals 5. Create a “false Acc. Development Age history” by making Year 1 12 24 36 0.801 (0.183) 3.266 48 0.801 random draws, with 2 5.340 (4.619) (0.259) replacement, from 3 4 (0.699) (3.437) 5.340 the triangle of residuals. Combine False History the random draws with the recast Acc. Development Age Year 12 24 36 48 historical data to come 1 1,034 500 288 263 up with the “false 2 1,519 544 308 3 1,638 734 history”. 4 2,345 Simulating reserves stochastically via bootstrapping Cumulated False History 6. Calculate link ratios from Acc. Development Age the data in the cumulated Year 12 24 36 48 false history triangle 1 2 1,034 1,519 1,534 2,063 1,822 2,371 2,084 7. Use the link ratios to 3 1,638 2,372 square the false history 4 2,345 data triangle Ave Link Ratio 1.424 1.166 1.144 8. Several additional steps are described in the paper… Squaring of the Cumulated False History 9. Repeat process N times to Acc. Development Age get N different reserve Year 12 24 36 48 indications. 1 1,034 1,534 1,822 2,084 2 1,519 2,063 2,371 2,713 3 1,638 1,638 1,909 2,185 4 2,345 3,339 3,892 4,454 Pros / Cons of each method Chain-ladder Pros Bootstrap Pros More flexible - not Do not need to limited by observed make assumptions data about underlying Chain-ladder Cons distribution More assumptions Bootstrap Cons Potential problems Variability limited to with negative values that which is in the historical data 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. 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). Effects of Correlation: Example 1 2 lines of business, N (100,25) 75th percentile of A+B at different levels of correlation between A and B: Correlation Values at 75th Ratio of Values at percentile 75th percentile 0.00 223.8 0.0% 0.25 226.7 1.3% 0.50 229.2 2.4% 0.75 231.5 3.4% 1.00 233.7 4.4% Effects of Correlation: Example 2 Same idea, but increase variability of distributions for lines A and B: Standard Deviation Value 25 50 100 200 Value for 0.00 correlation 223.8 247.7 295.4 390.8 at the 75th percentile Correlation Ratio of values at 75th percentile 0.25 1.3% 2.3% 3.8% 5.8% 0.50 2.4% 4.3% 7.3% 11.0% 0.75 3.4% 6.2% 10.4% 15.8% 1.00 4.4% 8.0% 13.4% 20.2% 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. Rank correlation example Perfect Inverse Correlation No Correlation Perfectly Correlated Rank to Use Rank to Use Rank to Use A B A B A B 5 4 1 1 5 3 4 1 2 2 4 2 2 5 3 3 2 5 1 2 4 4 1 1 Index A B 3 3 5 5 3 4 1 155 154 2 138 125 Resulting Joint Dist. Resulting Joint Dist. Resulting Joint Dist. 3 164 100 A B A+B A B A+B A B A+B 4 122 198 107 198 305 155 154 309 107 100 207 5 107 128 122 154 276 138 125 263 122 125 247 138 128 266 164 100 264 138 128 266 155 125 280 122 198 320 155 154 309 164 100 264 107 128 235 164 198 362 Range of Joint Dist. Range of Joint Dist. Range of Joint Dist. Low 264 Low 235 Low 207 High 305 High 320 High 362 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 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 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 Example of Method 1 approach 1 2 3 Uncorrelated reserves Ranking of draws Correlated reserves Sorted from low to high from Correlated Normal Iteration Class A Class B Iteration Class A Class B Iteration Class A Class B 1 100 1000 1 3 5 1 300 5000 2 200 2000 2 5 2 2 500 2000 3 300 3000 3 1 1 3 100 1000 4 400 4000 4 4 4 4 400 4000 5 500 5000 5 2 3 5 200 3000 2nd value from Class A, 3rd value from Class B Bootstrap Correlation Methodology Variability Parameters Calculated from Original Data Triangle A Triangle B Development Year Development Year 1 2 3 4 1 2 3 4 1 (1A,1A) (1A,2A) (1A,3A) (1A,4A) 1 (1B,1B) (1B,2B) (1B,3B) (1B,4B) AY AY 2 (2A,1A) (2A,2A) (2A,3A) 2 (2B,1B) (2B,2B) (2B,3B) 3 (3A,1A) (3A,2A) 3 (3B,1B) (3B,2B) 4 (4A,1A) 4 (4B,1B) Calculated Variability Parameters Correlated Bootstrapping - Reshuffling of variability parameters in Triangle B Development Year Development Year 1 2 3 4 1 2 3 4 1 (2A,1A) (3A,2A) (1A,3A) (3A,1A) 1 (2B,1B) (3B,2B) (1B,3B) (3B,1B) AY AY 2 (2A,2A) (1A,2A) (2A,3A) 2 (2B,2B) (1B,2B) (2B,3B) 3 (3A,1A) (1A,1A) 3 (3B,1B) (1B,1B) 4 (1A,1A) 4 (1B,1B) Randomly Selected Variability Parameters to be used in the creation of one possible pseudo-history Pros / Cons of each approach Correlation Matrix Pros Bootstrap Correlation More flexible - not Pros limited by observed Do not need to make data assumptions about Correlation Matrix Cons underlying correlations Requires modeler to do Bootstrap Cons additional work to Results reflect only quantify the correlations those correlations that between lines were in the historical data 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 Rank correlation results 11,000,000 10,000,000 0% corr 75th Percentile 9,000,000 25% corr Percent Dollars (000 omitted) 50% corr Change 8,000,000 75% corr Estimated 75th from Zero 100% corr Percentiles Percentile 7,000,000 0% corr. 4,640,039 n/a 25% corr. 4,697,602 1.2% Mean Value 50% corr. 4,739,459 2.1% 6,000,000 75% corr. 4,794,767 3.3% 5,000,000 100% corr. 4,836,166 4.2% 4,000,000 3,000,000 2,000,000 1% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95% 99% ile ile ile ile ile ile ile ile ile ile ile ile ile Add Bootstrap results 11,000,000 0% corr 10,000,000 25% corr 50% corr 75th Percentile 9,000,000 75% corr Percent Dollars (000 omitted) 100% corr Change 8,000,000 Bootstrap Estimated 75th from Zero Percentiles Percentile 7,000,000 0% corr. 4,640,039 n/a Mean Value 25% corr. 4,697,602 1.2% 6,000,000 50% corr. 4,739,459 2.1% 75% corr. 4,794,767 3.3% 5,000,000 100% corr. 4,836,166 4.2% 4,000,000 Bootstrap 4,755,952 2.5% 3,000,000 2,000,000 1% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95% 99% ile ile ile ile ile ile ile ile ile ile ile ile ile Case Study Conclusions Mean aggregated reserve = 4.33B Reserves at the 75th percentile range from 4.64B to 4.84B Bootstrap tells us that there does appear to be some level of correlation in underlying data 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

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