SOA Exam ASTAM Fall 2024, Question 4

Study guide: Exam ASTAM

An insurance company has a portfolio of insurance policies for which the loss random variable, $X > 0$ , has the following survival function:

$S_X(x) = \left(\frac{\lambda}{\lambda + \theta x}\right)^{\frac{1}{\theta}} \qquad \lambda > 0, \theta > 0$

Part a (3 points)

(i) Show that the hazard rate function for $X$ is $h(x) = \dfrac{1}{\lambda + \theta x}$ .

(ii) Briefly categorize the tail behavior of $X$ , based on $h(x)$ .

Part b (1 point)

The insurance company is deciding whether to use the Value at Risk or Expected Shortfall risk measure to determine its solvency capital needed to cover adverse loss experience.

Describe one advantage and one disadvantage of using the Expected Shortfall risk measure, as compared with the Value at Risk measure.

Part c (3 points)

Now let $\lambda = 2000$ and $\theta = 0.4$ .

(i) Show that for the distribution of $X$ the 99% Value at Risk (VaR), denoted $Q_{99\%}$ , is 26,550 to the nearest 10. You should calculate the value to the nearest 1.

(ii) You are given that the mean excess loss function for $X$ is

$e(d) = \frac{\lambda + \theta d}{1 - \theta} \quad \text{for } 0 < \theta < 1.$

Calculate the confidence level $\alpha$ such that the $\alpha$ -Expected Shortfall, denoted $ES_\alpha$ , is the same as the 99% VaR calculated in (c)(i).

(iii) In general, suppose that $Q_\beta = ES_\alpha$ for some $\alpha, \beta$ . Explain why it is necessary that $\alpha \le \beta$ .

Part d (3 points)

(i) Identify the functions $c_n$ and $d_n$ (not involving $x$ ) such that for any $x$ , $\theta > 0$ ,

$nS(c_nx + d_n) = \left(1 + \theta x\right)^{-1/\theta}$

(ii) Explain the significance of this result in terms of the relationship between $\theta$ and the tail behavior of $X$ .

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SOA Exam ASTAM Fall 2024, Question 4

Part a (3 points)

Part b (1 point)

Part c (3 points)

Part d (3 points)

Concepts tested

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