Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business...

Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business

Gerald KirschnerClassic 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 traditional loss triangle

Calculate link ratios

Calculate mean and standard deviation of the link ratios

Acc.Year 12 24 36 48

1 1,000 1,500 1,750 2,0002 1,200 2,000 2,3003 1,800 2,5004 2,100

Development Age

Acc.Year 12 - 24 24 - 36 36 - 48

1 1.500 1.167 1.1432 1.667 1.1503 1.389

Mean 1.500 1.157 1.143Std. Deviation 0.1179 0.0082 0

Link Ratios

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


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. Mean3. Standard deviation

Acc.Year 12 - 24 24 - 36 36 - 48

1 1.500 1.167 1.1432 1.667 1.1503 1.389

Mean 1.500 1.157 1.143Std. Deviation 0.1179 0.0082 0

Acc.Year 12 - 24 24 - 36 36 - 48

1 1.500 1.167 1.143

2 1.667 1.150 1.143

3 1.389 1.163 1.1434 1.419 1.145 1.143

Link Ratios

Link Ratios


Simulated values are shown in red

Lognormal Distribution, mean 1.5, standard deviation 0.1179

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

1.168 1.249 1.330 1.411 1.492 1.573 1.654 1.735 1.816 1.897

% o

f T

ota

l O

bs

erv

ati

on

s

Random draw

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

Actual Cumulative Historical Data

Acc.Year 12 24 36 48

1 1,000 1,500 1,750 2,0002 1,200 2,000 2,3003 1,800 2,5004 2,100

Ave Link Ratio 1.500 1.157 1.143

Recast Cumulative Historical Data

Acc.Year 12 24 36 48

1 1,008 1,512 1,750 2,0002 1,325 1,988 2,3003 1,667 2,5004 2,100

Development Age

Development Age


1. Keep current diagonal intact

2. Apply average link ratios to “back-cast” a series of fitted historical payments

Ex: 1,988 =

2,300


Actual Incremental Historical Data

Acc.Year 12 24 36 48

1 1,000 500 250 2502 1,200 800 3003 1,800 7004 2,100

Recast Cumulative Historical Data

Acc.Year 12 24 36 48

1 1,008 504 238 2502 1,325 663 3123 1,667 8334 2,100

Development Age

Development Age

3. Convert both actual and fitted triangles to incrementals

4. Look at difference between fitted and actual payments to develop a set of ResidualsResiduals

Acc.Year 12 24 36 48

1 (0.259) (0.183) 0.801 0.0002 (3.437) 5.340 (0.699)3 3.266 (4.619)4 0.000

Development Age


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”.

Random Draw from Residuals

Acc.Year 12 24 36 48

1 0.801 (0.183) 3.266 0.8012 5.340 (4.619) (0.259)3 (0.699) (3.437)4 5.340

False History

Acc.Year 12 24 36 48

1 1,034 500 288 2632 1,519 544 3083 1,638 7344 2,345

Development Age

Development Age


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.

Cumulated False History

Acc.Year 12 24 36 48

1 1,034 1,534 1,822 2,0842 1,519 2,063 2,3713 1,638 2,3724 2,345

Ave Link Ratio 1.424 1.166 1.144

Squaring of the Cumulated False History

Acc.Year 12 24 36 48

1 1,034 1,534 1,822 2,0842 1,519 2,063 2,371 2,7133 1,638 1,638 1,909 2,1854 2,345 3,339 3,892 4,454

Development Age

Development Age

Pros / Cons of each methodChain-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

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 percentile

Ratio of Values at 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

at the 75th percentile 223.8 247.7 295.4 390.8

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 examplePerfectly Correlated

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 41 155 1542 138 1253 164 100 A B A+B A B A+B A B A+B4 122 198 107 198 305 155 154 309 107 100 2075 107 128 122 154 276 138 125 263 122 125 247

138 128 266 164 100 264 138 128 266155 125 280 122 198 320 155 154 309164 100 264 107 128 235 164 198 362

Low 264 Low 235 Low 207High 305 High 320 High 362

Rank to Use

Resulting Joint Dist.

Range of Joint Dist.

No CorrelationRank to Use



Rank to Use



Perfect Inverse Correlation

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

2nd value from Class A, 3rd value from Class B

Iteration Class A Class B1 100 10002 200 20003 300 30004 400 40005 500 5000

Uncorrelated reservesSorted from low to high

1


Ranking of drawsfrom Correlated Normal

2 3

Iteration Class A Class B12345

Correlated reserves


Correlated reserves

Iteration Class A Class B12345 200 3000

Correlated reserves

Bootstrap Correlation Methodology

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)

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)

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)

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)

Development Year Development Year

Correlated Bootstrapping - Reshuffling of variability parameters in Triangle B

Variability Parameters Calculated from Original Data

Triangle A Triangle BDevelopment Year Development Year

AY

AY

AY

AY

Calculated Variability Parameters

Randomly Selected Variability Parameters to be used in the creation of one possible pseudo-history

Pros / Cons of each approachCorrelation 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 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

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

10,000,000

11,000,000

1%ile

5%ile

10%ile

20%ile

30%ile

40%ile

50%ile

60%ile

70%ile

80%ile

90%ile

95%ile

99%ile

Do

llars

(00

0 o

mit

ted

)

0% corr

25% corr

50% corr

75% corr

100% corr

Mean Value

75th PercentilePercentChange

from Zero Percentile

0% corr. 4,640,039 n/a25% corr. 4,697,602 1.2%50% corr. 4,739,459 2.1%75% corr. 4,794,767 3.3%100% corr. 4,836,166 4.2%

Estimated 75th Percentiles

Add Bootstrap results

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

10,000,000

11,000,000

1%ile

5%ile

10%ile

20%ile

30%ile

40%ile

50%ile

60%ile

70%ile

80%ile

90%ile

95%ile

99%ile

Do

llars

(00

0 o

mit

ted

)

0% corr

25% corr

50% corr

75% corr

100% corr

Bootstrap

Mean Value

75th PercentilePercentChange

from Zero Percentile

0% corr. 4,640,039 n/a25% corr. 4,697,602 1.2%50% corr. 4,739,459 2.1%75% corr. 4,794,767 3.3%100% corr. 4,836,166 4.2%

Bootstrap 4,755,952 2.5%

Estimated 75th Percentiles

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

Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business...

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