SOA Exam ASTAM Sample Question 5

Study guide: Exam ASTAM

You are given that $X_1, X_2, \ldots, X_n$ are independent and identically distributed copies of a continuous loss random variable $X$ , where $X$ represents a firm's aggregate daily losses from operational risks. Let $M_n = \max\left(X_1, X_2, \ldots, X_n\right)$ .

The n-day Expected Maximum (nDEM) risk measure is defined as $\rho(X) = \mathrm{E}\left[M_n\right]$ .

Currently, the firm uses the 95% VaR risk measure, denoted $Q_{0.95}(X)$ . A colleague suggests changing to the nDEM risk measure, using $n = 20$ for consistency with the current 1-in-20-day standard.

Part a

Show that $\Pr\left[M_{20} > Q_{0.95}(X)\right] = 0.64$ to the nearest 0.01. You should calculate the probability to the nearest 0.001.

Part b

Now suppose that $X$ is exponentially distributed with mean 1. Let $\alpha(n) = \dfrac{n-1}{n}$ .

Show that the $\alpha(n)$ -VaR of $X$ is $Q_{\alpha(n)}(X) = \log n$ .

Part c

Show that the $\alpha(n)$ -Expected Shortfall of $X$ is $ES_\alpha(X) = 1 + \log n$ (as printed). You may use without proof the memoryless property of the exponential distribution.

Part d

You are given that

The exponential distribution is in the Maximum Domain of Attraction (MDA) of the Gumbel distribution.
The mean of the standard Gumbel distribution is 0.5772.
The normalizing functions for the exp(1) distribution, under the Fisher-Tippett-Gnedenko theorem, are $c_n = 1$ and $d_n = \log n$ .
(i) Explain what it means to say that the exponential distribution is in the MDA of the Gumbel distribution.
(ii) Assume that $n$ is sufficiently large that the normalized $n$ -day maximum loss can be approximated by the standard Gumbel distribution. Show that

$Q_{\alpha(n)}(X) < E[M_n] < ES_{\alpha(n)}(X)$

Part e

Explain why the ordering in (d) is likely to hold more generally.

Answer this question and see the worked solution