For a collective risk model the number of losses, N, has a Poisson distribution with [math]\lambda = 20[/math]. The common distribution of the individual losses has the following characteristics:

1. [math]\operatorname{E}[ X ] = 70 [/math]
2. [math]\operatorname{E}[ X \wedge 30] = 25 [/math]
3. [math]\operatorname{Pr}( X \gt 30) = 0.75 [/math]
4. [math]\operatorname{E}[ X^2 | X \gt 30] = 9000 [/math]

An insurance covers aggregate losses subject to an ordinary deductible of 30 per loss.

Calculate the variance of the aggregate payments of the insurance.

• 54,000
• 67,500
• 81,000
• 94,500
• 108,000

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

Key: B

Losses in excess of the deductible occur at a Poisson rate of [math]\lambda^* = [1 − F (30)]\lambda = 0.75(20) = 15.[/math] The expected payment squared per payment is

[[math]] \begin{aligned} &\operatorname{E}[( X − 30)^2 | X \gt 30) = \operatorname{E}[ X^2 − 60 X + 900 | X \gt 30) \\ &\operatorname{E}[ X^2 − 60( X − 30) − 900 | X \gt 30] \\ &=9000-60 \frac{70-25}{0.75} - 900 = 4500. \end{aligned} [[/math]]

The variance of S is the expected number of payments times the second moment, 15(4500) =67,500.

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