# P2.T5.22.5. Peaks-over-threshold (POT) approach to extreme value theory (EVT)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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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:

22.5.1. The board has asked Claire to develop risk estimates at very high confidence levels; e.g., 99.5%, 99.9%. Given the realities of her dataset (e.g., high time dependency), she may need to rely on extreme value theory (EVT) and she recognizes the two branches of univariate EVT are block maxima and peaks-over-threshold (POT), She is comparing the univariate EVT approaches to non-parametric approaches such as historical simulation.

As she evaluates EVT, with respect to these two univariate EVT approaches (i.e., block maxima and POT), each of the following statements is true, and is true about both approaches, EXCEPT which statement is false?

a. Each of the approaches automatically and easily fits distributions to a stochastic process with autocorrelated clusters
b. Each of the fitted EV parametric distributions has a parameter that can be increased to model increasingly heavier tails
c. She can fit a parametric distribution of extreme values to actual datasets without knowledge of the underlying distribution
d. Unlike basic historical simulation, the EVT VaR or ES can return value(s) that exceeds the greatest loss in the actual historical dataset

22.5.2. Ralph employed the peaks-over-threshold (POT) approach under extreme value theory (EVT) to the estimation of value at risk (VaR) for his firm's portfolio. He has four years of data, n = 1,000 days. He would prefer to apply a maximum likelihood estimation method (aka, MLE) to parameter estimation, but he thinks he needs to choose the threshold first. When he selects a threshold of u = 4.0%, he observes N(u) = 28 exceedances such that N(u)/n = 2.80%. Fitting the data generates these parameters: scale, β = 0.80%,; and tail index, ξ = 0.310. Based on these assumptions, the 99.90% confident VaR is 8.670%.

He subsequently decides to increases the threshold from 4.0% to 5.0% while retaining the tail index at 0.310 and the scale parameter at 0.80%. The number of exceptions declines to N(u) = 17 such that N(u)/n = 17/1,000 = 1.70%. His revised 99.90% VaR is similar at 8.630%. In regard to choosing a threshold, which of the following statements is TRUE?

a. The higher threshold offers the benefit of lower bias but the drawback of higher variance
b. The higher threshold offers the benefit of lower variance but the drawback of higher bias
c. Because the desired confidence is 99.90% and 0.10% = 1/1,000, the (optimal) threshold should be set equal to the dataset's second-worst return
d. He can avoid a subjective choice of the threshold by applying an analytical maximum log-likelihood estimation (MLE) function to all three parameters simultaneously

22.5.3. Peter wants to estimate value at risk (VaR) under the peaks-over-threshold (POT) approach to extreme values. To estimate the parameters, he selects a threshold of 3.0% and observes N(u) = 45 exceedances out of n = 750 days, so that N(u)/n = 45/750 = 6.0%. The scale parameter, β = 0.90%, and the tail index, ξ = 0.250. Which is nearest to the 99.50% VaR?

a. 3.90%
b. 6.10%
c. 8.40%
d. 9.70%