For the workers’ compensation claims of a construction company you are given:

1. The annual number of claims follows the Poisson distribution with mean 20.
2. Claim sizes X follow the lognormal distribution with [math]\mu = 4.2 [/math] and [math]\sigma = 1.1[/math] .
3. The company retains the first 500 of each claim.
4. Annual aggregate retained claims approximately follow the normal distribution.
5. [math]\operatorname{E}[(X \wedge 500)^2] = 26189[/math]

Determine the 90th percentile of the aggregate distribution of retained claims.

• Less than 2900
• At least 2900, but less than 3100
• At least 3100, but less than 3300
• At least 3300, but less than 3500
• At least 3500

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: C

[[math]] \operatorname{E}[ X \wedge 500 ] = \exp(4.2 + \frac{1.1^2}{2}) \Phi(0.73) + 500(1-\Phi(1.83)) = 110.5 [[/math]]

Let S denote the aggregate distribution of retained claims.

[[math]] \begin{aligned} &\operatorname{E}[ S ] = \operatorname{E}[ N ] \operatorname{E}[ X \wedge 500] = 20(110.5) = 2210 \\ &\operatorname{E}[ S ] = \operatorname{E}[ N ]\operatorname{E}[ X \wedge 500] + \operatorname{E}[ ( X \wedge 500)^ 2 )\operatorname{E}[ N ] = 20(26,189) = 523, 780 \end{aligned} [[/math]]

The 90th percentile of [math]S[/math] is [math]2210 + 1.282 \sqrt{523,780} = 3137.82.[/math]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.