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P2.T7.705. Extreme value theory (EVT)

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705.1. Techstreet Hedge Fund has active positions in cryptocurrency futures contracts and employs the peaks-over-threshold (POT) approach variation to characterize extreme potential loss tails according to extreme value theory (EVT). The fund estimates both value at risk (VaR) and expected shortfall (ES) at the 99.0% confidence level. Their choice of threshold, denoted (u) and shown below, is set at 3.0% which 30 losses among 1,000 observations; e.g., this might be sub-daily observations within a high-frequency environment. The full set of parameters are shown below, and further included are the formulas for both VaR and ES implied by the POT approach:

Each of the following statements is true EXCEPT which is false?

a. The POT 99.0% value at risk (VaR) is about 5.0
b. The POT 99.0% expected shortfall (ES) is about 7.15
c. An increase in the value of the tail index, ξ, will increase both the VaR and the ES
d. An increase in the value of the threshold, u, will decrease both the VaR and the ES

705.2. Stephanie is a Risk Analyst who is trying to model the loss tail of an operation loss distribution that exhibits very low-probability but high-impact events. She decides to lean on extreme value theory (EVT) and consults with her colleagues. Among the following pieces of advice, each is basically true (ie., good advice) EXCEPT FOR which is likely untrue?

a. If the loss data exhibits clustering (aka, time dependency), an easy solution is to employ a generalized extreme value (GEV) distribution according to the "block maxima" approach
b. If she does employ a generalized extreme value (GEV) distribution but cannot identify the parent loss distribution, it is advisable to assume a Frechet distribution where the tail index (ξ) is greater than zero
c. For either EVT approach--i.e., the generalized extreme value (GEV) or peaks-over-threshold (POT)--it is acceptable if the losses are not i.i.d., but the parent distribution must be known in order to estimate the limiting distribution
d. If she prefers to rely on maximum likelihood estimation (MLE), regression or semi-parametric parameter estimation techniques throughout, and thusly avoid arbitrary or judgmental calibration of parameters, then then GEV is probably better suited than POT

705.3. Kevin Dowd outlines two basic approaches to extreme value tails. The first approach characterizes the set of each maximum loss within non-overlapping "blocks" of time (block maxima) with the generalized extreme value (GEV) distribution. The second approach characterizes the set of extreme losses above a threshold, regardless of their timing, with a generalized Pareto (GP) distribution. In regard to the extreme value theory (EVT) approach, which of the following statements is TRUE?

a. POT GP can produce an expected shortfall (ES) estimate, but GEV cannot return an ES
b. Unconditional EVT is useful when forecasting value at risk (VaR) or expected shortfall (ES) over a long horizon period, but for a short horizon conditional EVT is probably better
c. If the threshold selected in the GP distribution equals zero, then the peaks-over-threshold generalized Pareto (GP POT) collapses to the generalized extreme value (GEV) distribution
d. Due to the central limit theorem (CLT), elliptical copulas are justified is modeling the dependence structure of a multivariate extreme value distribution

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