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# P2.T7.804. Extreme value theory (EVT) approaches (Dowd Ch. 7)

#### Nicole Seaman

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Learning objectives: Explain the importance and challenges of extreme values in risk management. Describe extreme value theory (EVT) and its use in risk management. Describe the peaks-over-threshold (POT) approach.

Questions:

804.1. (Please note: This question is obviously inspired by practice question #1 in GARP's 2017 Part 2 Practice Exam.) Freshzim Investment Bankers has an active position in commodity futures and is using the peaks-over-threshold (POT) approach according to extreme value theory (EVT) for estimating value at risk (VaR) and expected shortfall (ES). The firm's risk managers decided to set a threshold level of 6.00% to evaluate excess losses. This choice of the threshold is consistent with their finding that 3.0% of the observations are in excess of this threshold value. As displayed below, aside from the threshold choice (i.e., u = 3.05), empirical analysis suggests the two other distributional parameters: scale, β = 0.80; and shape (aka, tail index), ξ = 0.23. At the 99.0% confidence level, the position's VaR under the POT approach is 7.00%. Which is nearest to the corresponding 99.0% expected shortfall (ES)?

a. 8.34%
b. 9.99%
c. 10.50%
d. 12.47%

804.2. Sandra is trying to fit a distribution to the extreme loss tail of a historical financial return dataset. She has a good fit for the parent (aka, body and shoulders) distribution, but she has not settled on her extreme value theory (EVT) approach. Her situations includes the following five issues:

I. The distribution F(x) is actually unknown; i.e., it could be anything
II. The loss data somewhat cluster; that is, losses are not strictly i.i.d.
III. The parent (i.e., non extreme loss) distribution is well-characterized by a student's t distribution; that is, the parent is non-normal
IV. The end users might prefer the extreme loss tail distribution be characterized by a Gumbel so that F(x) has exponential tails
V. The end users do prefer that the extreme loss tail distribution itself exhibit right-skew; aka, positive skew

Her end users have expressed a preference for the generalized extreme-value distribution, frankly because they are more comfortable with the traditional block maxima approach. Which of these issues, in theory, effectively DISQUALIFIES the generalized extreme-value (GEV) distribution as a candidate for application?

a. Only I., because F(x) does need to be specified of course
b. Both I. and II., because F(x) needs to specified and the loss data must be i.i.d.
c. Both IV. and V., because GEV can be heavy-tailed but will have zero skew, and further should be the Weibull case
d. None of these issues disqualify the GEV distribution

804.3. To retrieve the value at risk (VaR) under the generalized extreme-value (GEV) distribution, Dowd shows the derivation that results in the following equation which can characterize a heavy-tailed Frechet distribution: Let's assume the following somewhat "realistic" parameters:
• location, µ = 3.0%,
• scale, σ = 0.70%,
• tail index, ξ = 0.60, and
If the sample size, n = 60, then which is nearest to the implied 99.90% VaR?

a. 6.50%
b. 8.14%
c. 11.90%
d. 15.75%