# Bootstrap Cons

Document Sample

```					Two Approaches to Calculating
Correlated Reserve Indications

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
   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 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

Mean         1.500    1.157    1.143
Std. Deviation 0.1179 0.0082           0
Simulating reserves stochastically
   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
    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
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%

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
   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
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
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
 More flexible - not     Do not need to
limited by observed      make assumptions
 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
   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
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

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
   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
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
   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

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 2 posted: 9/26/2012 language: Unknown pages: 31