Approximation of Aggregate Losses

1
Approximation of Aggregate Losses

• Dmitry Papush
• Commercial Risk Reinsurance Company
• CAS Seminar on Reinsurance
• June 7, 1999
• Baltimore, MD

2
Approximation of an Aggregate Loss Distribution

• Usual Frequency - Severity Approach
• Analyze Number of Claims Distribution
• and Claim Size Distribution separately, then
  convolute.

3
The Problem

• How to approximate an Aggregate Loss Distribution
  if there is no individual claim data available?
• - What type of distribution to use?

4
Method Used

• Choose severity and frequency distributions
• Simulate number of claims and individual claims
  amounts and the corresponding aggregate loss
• Repeat many times (5,000) to obtain a sample of
  Aggregate Losses
• For different Distributions calculate Method of
  Moments parameter estimators using the simulated
  sample of Aggregate Losses
• Test the Goodness of fit

5
Assumptions for Frequency and Severity Used

• Frequency Negative Binomial
• Severity Five parameter Pareto, Lognormal
• Layers 0 - 250K (Low Retention) 0 - 1000K
  (High Retention)
• 750K xs 250K (Working Excess) 4M xs 1M
  (High Excess)

6
Distributions Used to Approximate Aggregate Losses

• Lognormal
• Normal
• Gamma

7
Gamma Distribution

• b-a
  xa-1exp(-x/b)
• f(x)
• G(a)
• Mean a b
• Variance a b2

8
Goodness of Fit Tests

• Percentile Matching
• Limited Expected Loss Costs

9
Example 1.

• Small Book of Business, Low Retention
• Expected Number of Claims 50,
• Layer 0 - 250K,
• Severity - 5 Parameter Pareto

10
Example 1.
11
Example 2.

• Large Book of Business, Low Retention
• Expected Number of Claims 500,
• Layer 0 - 250K,
• Severity - 5 Parameter Pareto

12
Example 2.
13
Example 3.

• Small Book of Business, High Retention
• Expected Number of Claims 50,
• Layer 0 - 1,000K,
• Severity - 5 Parameter Pareto

14
Example 3.
15
Example 4.

• Large Book of Business, High Retention
• Expected Number of Claims 500,
• Layer 0 - 1,000K,
• Severity - 5 Parameter Pareto

16
Example 4.
17
Example 5.

• Working Excess Layer
• Layer 750K xs 250K,
• Expected Number of Claims 20,
• Severity - 5 Parameter Pareto

18
Example 5.
19
Example 6.

• High Excess Layer
• Layer 4M xs 1M,
• Expected Number of Claims 10,
• Severity - 5 Parameter Pareto

20
Example 6.
21
Example 7.

• High Excess Layer
• Layer 4M xs 1M,
• Expected Number of Claims 10,
• Severity - Lognormal

22
Example 7.
23
Results

• Normal works only for very large number of claims
• Lognormal is too severe on the tail
• Gamma is the best approximation out of the three