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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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