AIM: Describe the peaks-over-threshold (POT) approach.
Compute VaR and expected shortfall using the POT approach, given various parameter values.
Questions:
87.1 If X is a random i.i.d loss with distribution function F(x), and (u) is a threshold value of X, what function defines the peaks-over-threshold (POT) approach?
a. F(x) = P{ X <= x | X > u}
b. F(x) = P{ X <= x | X = u}
c. F(x) = P{ X - u <= x | X > u}
d. F(x) = P{ X - u <= x | X = u}
87.2 Assume the following GP parameters under POT approach to extreme values: scale (beta) = 0.9, shape/tail index (xi) = 0.15, threshold (u) = 4.0%, and the percentage of observations above the threshold (Nu/n) = 10.0%. What are, respectively, the 99.5% and 99.9% value at risk (VaR)? (note: variation on Dowd's Example 7.5)
a. 3.95% (99.5%) and 5.24% (99.9%)
b. 4.15% and 6.24%
c. 7.40% and 9.97%
d. 9.03% and 11.31%
87.3 Using the same assumptions and same POT approach (generalized Pareto distribution), what are, respectively, the 99.5% and 99.9% expected shortfall (ES)?
a. 7.40% (99.5%) and 9.97% (99.9%)
b. 9.06% and 12.08%
c. 10.22% and 14.65%
d. 12.62% and 16.68%
Answers:
87.1 C. F(x) = P{ X - u <= x | X > u}
Conditional on X exceeding the threshold (X>u), what is the probability that the loss in excess of the threshold (X-u) is less than or equal to x (i.e., CDF).
… note that F(x) is the parent distribution.
87.2 C. (7.404 @ 99.5% and 9.972 @ 99.9%)
See spreadsheet
87.3 B. (9.06 @ 99.5% and 12.08 @ 99.9%)
See spreadsheet
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