SOA Exam ASTAM Sample Question 7

Study guide: Exam ASTAM

Aggregate claims for a portfolio are assumed to follow a compound Poisson distribution, with Poisson parameter $\lambda = 50$ . The individual claims distribution is lognormal, with parameters $\mu = 6.0, \quad \sigma^2 = 1.5$ .

Part a

Show that, if $f(x)$ is the density function for the lognormal $(\mu, \sigma)$ distribution, then

$\int_d^\infty x^k f(x)\,dx = e^{k\mu + k^2\sigma^2/2}\left\{1 - \Phi\left(\frac{\log d - \mu - k\sigma^2}{\sigma}\right)\right\}$

Part b

The insurer shares the risk with a reinsurer under an excess of loss agreement with retention 3000.

(i) Calculate the expected number of claims involving the reinsurer.
(ii) Show that the expected value of an individual reinsurance payment, given that the loss involves the reinsurer, is 2722.
(iii) Show that the reinsurer's expected aggregate claim cost is 6,900 to the nearest 10.
(iv) Calculate the direct insurer's expected aggregate claims cost after reinsurance.

Part c

Calculate the standard deviation of the reinsurer's aggregate claims under the excess of loss contract.

Part d

The insurer is considering changing to a proportional reinsurance arrangement, under which the reinsurer pays $(1 - \alpha)$ of every claim. You are given that the expected cost of claims under the new contract is the same as under the excess of loss contract. Show that the proportion retained by the insurer is $\alpha = 0.84$ to the nearest 0.01.

Part e

Calculate the standard deviation of the reinsurer's claims under the proportional reinsurance contract.

Part f

Comment briefly on the implications of the different standard deviations on the pricing of the two reinsurance contracts.

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