SOA Exam ASTAM Fall 2023, Question 2

Study guide: Exam ASTAM

An insurer is analyzing its auto insurance claims with a focus on the right tails of the distributions.

Part a (1 point)

Losses are categorized into two categories: bodily injury and vehicle repairs.

State with reasons which of these categories is more likely to have a heavy-tailed loss distribution.

Part b (1 point)

(i) State whether an increasing mean excess loss function indicates a heavy-tailed or light-tailed distribution.
(ii) State whether an increasing hazard rate function indicates a heavy-tailed or light-tailed distribution.

Part c (3 points)

You are given that $X \sim \text{Pareto}(\alpha, \theta_X)$ .

(i) Prove that $e_X(d) = \dfrac{\theta_X + d}{\alpha - 1}$ for $\alpha > 1$ .
(ii) Determine an expression for the hazard rate function for $X$ , $h_X(x)$ , in terms of $\alpha$ and $\theta_X$ .

Part d (2 points)

You are given that $Y \sim \text{Weibull}(\tau, \theta_Y)$ .

(i) Derive an expression for the hazard rate function for $Y$ , $h_Y(y)$ , in terms of $\tau$ and $\theta_Y$ .
(ii) Determine the range of values of $\tau$ for which $Y$ is heavy-tailed, based on the hazard rate function.

Part e (2 points)

You are given that $\alpha = 2$ , $\tau = \tfrac{1}{2}$ , $\theta_X = 20$ , and $\theta_Y = 10$ . Use the limiting ratio of the survival function of $X$ to the survival function of $Y$ to compare their relative tail weights.

Part f (1 point)

You are given that $Y$ is in the MDA of the Gumbel distribution. Briefly describe the implication of this on the moments of $Y$ .

Answer this question and see the worked solution