SOA Exam ASTAM Fall 2024, Question 3

An insurance company sells one-year insurance policy on a risk with the following loss frequency distribution

Number of losses	Probability
0	0.5
1	0.3
2	0.1
3	0.1

You are also given the following information.

The severity of each loss $X_j$ follows a lognormal distribution with parameters $\mu = 7$ and $\sigma = 0.5$ .
The number of losses and the amount of each loss are assumed to be mutually independent.
The policy has an ordinary deductible of $d = 500$ per loss.
$N^L$ denotes the number of loss events in a single year.
$N^P$ denotes the number of payments in a single year.

Part a (3 points)

(i) Calculate the expected value of the aggregate ground-up loss.

(ii) Calculate the standard deviation of the aggregate ground-up loss.

(iii) Explain briefly why the Normal Approximation would not be a good method of approximating the distribution of the aggregate ground-up loss for this policy.

Part b (2 points)

Calculate the loss elimination ratio.

Part c (2 points)

Let

$I_j = \begin{cases} 0 & X_j \le d \\ 1 & X_j > d \end{cases} \qquad \text{and let } E[I_j] = v$

so that $N^P = \sum_{j=1}^{N^L} I_j$ (where we define $N^P = 0$ when $N^L = 0$ ).

Prove that the PGF for $N^P$ is

$P_{N^P}(z) = P_{N^L}(1 - v(1 - z)), \text{ where } v = 0.9419.$

Part d (2 points)

(i) Calculate the probability that the aggregate payment from the coverage is 0.

(ii) Calculate the probability that there is exactly one payment made in a year.

Part e (3 points)

The insurer purchases an excess-of-loss reinsurance policy such that the insurer's own maximum payment on an individual claim is 2,000. The reinsurance premium is set at 120% of the expected annual reinsurance claim payments.

(i) Calculate the annual reinsurance premium for this policy.

(ii) The amount of each loss in the next year is expected to increase uniformly by 10% due to inflation, and all contract terms remain the same. As a result, the reinsurance premium for next year is expected to be increased by $h$ .

Without further calculation, state with reasons whether $h$ is less than, equal to, or greater than 10%.

Answer this question and see the worked solution