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Old 05-13-2018, 01:41 PM
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gauchodelpaso gauchodelpaso is offline
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Default STAM - Textbook Example 5.7 Reinsurance Exposure Rating

On example 5.7 in page 183 of the 4th edition of the textbook by Brown and Lennox the authors imply a relationship between Increased Limit Factors ILFs and the cumulative distribution of limited loss. It seems to make sense, but I wanted to know if there was a more solid base for the relationship.

I see that ILFs are similar to ratios of E[X^u] with varying u, one being the basic in the denominator. I can also see that E[X^5,000,000] = 14,000,000, as the exercise mentions it's the highest limit that the insureds can buy, and from there all the others result. The basic would be

E[X^500,000) = E[X^5,000,000) / 1.6 = 8,750,000
E[X^1,000,000) = 1.15 * 8,750,000 = 10,062,500
E[X^2,000,000) = 1.35 * 8,750,000 = 11,812,500

and then the losses for the reinsurance layer would be

14,000,000 - 10,062,500 = 3,937,500.

There is a shorter answer.

Losses in insurance layer =

70% * 20,000,000 (1 - 1.15/1.6) = 3,937,500

The reference to the cumulative function is OK, but it was not clear to me directly (even after correcting as per the errata).

The cumulative distribution using the losses at different limits would be (using M as millions)

F(0.5M) = 8.75/14 = 0.625
F(1M) = 10.0625/14 = 0.71875
F(2M) = 11.8125/14 = 0.84375
F(5M) = 14/14 = 1.000
and are the same as in the book as the ratios of ILFs to the ILF at the high limit of the reinsurance.
But there is a shorter way to get to the answer as shown above.
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