Two Approaches to Calculating
Correlated Reserve Indications
Across Multiple Lines of Business
Gerald Kirschner
Deloitte Consulting
CAS Spring Meeting
June 2008
Presentation Structure


Correlation – a general discussion
Case studies




Description
Results using correlation matrix approach
Results using bootstrap approach
Conclusion
What this presentation is not


Presentation is not going to discuss how to
simulate ranges of reserves for a single line of
business
The discussion here is of methodologies that
might be appropriate for reserve analysis, but
are probably not appropriate for Enterprise
Risk Management or Capital Adequacy
analyses
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.
Ideally would want to directly model effects of
causality, but not always able to do so
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:
0.00
0.25
Values at 75th
percentile
223.8
226.7
Ratio of Values at
75th percentile
0.0%
1.3%
0.50
0.75
1.00
229.2
231.5
233.7
2.4%
3.4%
4.4%
Correlation
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
at the 75th percentile
Correlation
0.25
0.50
0.75
1.00
223.8
247.7
295.4
390.8
Ratio of values at 75th percentile
1.3%
2.3%
3.8%
5.8%
2.4%
4.3%
7.3%
11.0%
3.4%
6.2%
10.4%
15.8%
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
Index
1
2
3
4
5
A
155
138
164
122
107
B
154
125
100
198
128
Perfect Inverse Correlation
Rank to Use
A
B
5
4
4
1
2
5
1
2
3
3
A
107
122
138
155
164
Resulting Joint Dist.
B
A+B
198
305
154
276
128
266
125
280
100
264
Range of Joint Dist.
264
Low
305
High
No Correlation
Rank to Use
A
B
1
1
2
2
3
3
4
4
5
5
Perfectly Correlated
Rank to Use
A
B
5
3
4
2
2
5
1
1
3
4
Resulting Joint Dist.
A
B
A+B
155
154
309
138
125
263
164
100
264
122
198
320
107
128
235
Resulting Joint Dist.
A
B
A+B
107
100
207
122
125
247
138
128
266
155
154
309
164
198
362
Range of Joint Dist. Range of Joint Dist.
235
207
Low
Low
320
362
High
High
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
Uncorrelated reserves
Sorted from low to high
Iteration Class A Class B
1
100
1000
2
200
2000
3
300
3000
4
400
4000
5
500
5000
Ranking of draws
from Correlated Normal
Iteration Class A Class B
1
3
5
2
5
2
3
1
1
4
4
4
5
2
3
3
Correlated reserves
Iteration Class A Class B
1
300
5000
2
500
2000
3
100
1000
4
400
4000
5
200
3000
2nd value from Class A, 3rd value from Class B
Bootstrap Correlation
Methodology
Variability Parameters Calculated from Original Data
2 (2A,1A) (2A,2A) (2A,3A)
Triangle B
Development Year
1
2
3
4
1 (1B,1B) (1B,2B) (1B,3B) (1B,4B)
AY
AY
Triangle A
Development Year
1
2
3
4
1 (1A,1A) (1A,2A) (1A,3A) (1A,4A)
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
2 (2A,2A) (1A,2A) (2A,3A)
Development Year
1
2
3
4
1 (2B,1B) (3B,2B) (1B,3B) (3B,1B)
AY
AY
Development Year
1
2
3
4
1 (2A,1A) (3A,2A) (1A,3A) (3A,1A)
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
 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
Case Study 1




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
Case Study 1:
Rank correlation results
11,000,000
10,000,000
Dollars (000 omitted)
75th Percentile
0% corr
9,000,000
25% corr
50% corr
8,000,000
75% corr
100% corr
7,000,000
Mean Value
6,000,000
5,000,000
4,000,000
3,000,000
2,000,000
1%
ile
5%
ile
10% 20% 30% 40% 50% 60% 70% 80% 90% 95% 99%
ile
ile
ile
ile
ile
ile
ile
ile
ile
ile
ile
Estimated 75th
Percentiles
0% corr.
4,640,039
25% corr.
4,697,602
50% corr.
4,739,459
75% corr.
4,794,767
100% corr.
4,836,166
Percent
Change
from Zero
Percentile
n/a
1.2%
2.1%
3.3%
4.2%
Case Study 1:
Add Bootstrap results
11,000,000
0% corr
25% corr
50% corr
75% corr
100% corr
Bootstrap
10,000,000
Dollars (000 omitted)
9,000,000
8,000,000
75th Percentile
7,000,000
Mean Value
6,000,000
5,000,000
4,000,000
Estimated 75th
Percentiles
0% corr.
4,640,039
25% corr.
4,697,602
50% corr.
4,739,459
75% corr.
4,794,767
100% corr.
4,836,166
Bootstrap
3,000,000
2,000,000
1%
ile
5%
ile
10% 20% 30% 40% 50% 60% 70% 80% 90% 95% 99%
ile
ile
ile
ile
ile
ile
ile
ile
ile
ile
ile
4,755,952
Percent
Change
from Zero
Percentile
n/a
1.2%
2.1%
3.3%
4.2%
2.5%
Case Study 1 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
Case Study 2

Three lines of business – data from AM Best





Workers Compensation
Commercial Auto
Other Liability Occurrence
Run a 2,000 iteration simulation exercise for
each line
Examine the results for the aggregated
reserve indication at different percentiles of
the aggregate distribution
Case Study 2: Range of outcomes
at different levels of correlation
Aggregated Reserve Indications for WC + CAL + OL (Occurrence)
Aggregated Reserve Indication ($ Millions)
$585
75th Percentile
0% correl
25% Correl
50% correl
75% correl
95% correl
100% correl
Boot Uncorr
Boot Corr
$580
$575
$570
Correlation
%age
Mean Value
$565
$560
$555
$550
5%
10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 99%
Percentile
Estimated
75th %ile
Value
% Change
from
0% Corr.
0%
25%
50%
75%
100%
567,604,508
568,099,864
568,520,544
568,716,734
569,782,406
n/a
0.1%
0.2%
0.2%
0.4%
Uncorrel.
Bootstrap
567,604,508
n/a
Correlated
Bootstrap
567,897,784
0.1%
Case Study 2 Conclusions


Correlation assumptions do not change
the overall reserve distribution
significantly
Bootstrap indicates ~ 25% correlation
exists in the 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 might be important for ones with
volatile reserves, especially as one moves
further towards a tail of the aggregate
distribution