SOA Exam ASTAM Spring 2024, Question 6

Study guide: Exam ASTAM

An actuary is investigating the severity of losses from a commercial line for reinsurance pricing, using a data set of 500 observations of loss severity, denoted $X$ . The actuary first proposes to estimate the loss severity using a frailty model with frailty random variable $\Lambda$ . You are given:

$\Lambda$ follows a Gamma distribution with mean $\alpha/\gamma$ and variance $\alpha/\gamma^2$ .
Given $\Lambda = \lambda$ , the conditional hazard rate of $X$ is $h_{X|\Lambda}(x|\lambda) = \lambda$ .

Part a (2 points)

Show that the unconditional survival function of $X$ is

$S_X(x) = \left(\frac{\gamma}{\gamma + x}\right)^{\alpha}, \qquad x \ge 0,$

which is the survival function for a Pareto distribution.

Part b (3 points)

You are given that the maximum likelihood estimates (MLEs) of $\alpha$ and $\gamma$ based on the 500 observations are $\hat\alpha = 4.19$ and $\hat\gamma = 6{,}726.8$ .

(i) Show that the 97.5% Value at Risk (VaR) of the losses using the fitted frailty model for $X$ is 9,500 to the nearest 100. You should calculate the value to the nearest 1.

(ii) Calculate the 97.5% Expected Shortfall (ES) of the losses using the fitted frailty model for $X$ .

Part c (2 points)

Using normalizing functions $c_n = \gamma n^{1/\alpha}$ and $d_n = -\gamma$ , show that the distribution function of $X$ is in the maximum domain of attraction (MDA) of the Fréchet distribution.

Part d (3 points)

The actuary then decides to estimate the risk measures, VaR and ES, using the Generalized Pareto Distribution (GPD) with a threshold of $d = 8{,}900$ for the excess loss distribution. The MLEs of the GPD parameters are $\hat\xi = 0.85$ and $\hat\beta = 1{,}595.6$ .

You are given the following table showing an excerpt from the largest 20 values from the sample of 500, in decreasing order.

Rank	500	499	...	484	483	482	481
Loss	38,439	22,491	...	8,916	8,402	8,342	8,110

(i) Show that the 97.5% VaR of the losses using the fitted GPD is 9,500 to the nearest 100. You should calculate the value to the nearest 1.

(ii) Calculate the 97.5% ES of the losses using the fitted GPD.

Part e (1 point)

Your colleague at your reinsurance company suggests that since the tail risk of the losses has been captured through fitting a fat-tailed frailty model, it is not necessary to perform excess loss analysis using an extreme value distribution. Critique this suggestion.

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