SOA Exam ASTAM Sample Question 14

Study guide: Exam ASTAM

Two severity models, denoted by $X_1$ and $X_2$ , are created to estimate a ground-up loss. You are given the following information.

(i) $X_1$ has a density function $f_1(x) = 0.001\beta e^{-0.001x} + 0.002(1-\beta)e^{-0.002x}, \qquad x > 0,$ where $\beta$ is a constant between 0 and 1.
(ii) The density of $X_2$ is proportional to the density of an exponential distribution with mean 500 in $(0,\ 500)$ and proportional to the density of a two-parameter Pareto distribution with $\alpha = 3$ and $\theta = 2000$ in $(500,\ \infty)$ .
(iii) The probability that the loss is no more than 500 is the same under both models.

You are given that the mean of the ground-up loss is 800, and $\beta$ is determined by matching the first moment.

Part a

Show that $\beta = 0.6$ .

If $X_1$ is used to model the ground-up loss, calculate the mean excess loss over a deductible of 200.

If $X_1$ is used to model the ground-up loss, calculate the Value at Risk of the loss at the 90% security level.

Determine the density function of $X_2$

If $X_2$ is used to model the ground-up loss, calculate the Value at Risk of the loss at the 90% security level.

Compare the tails of the two severity models. Justify your answer.