# P2.T5.22.6. Blocks (BM) approach to extreme value theory (EVT)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning objectives: Compare and contrast the generalized extreme value and POT approaches to estimating extreme risks. Evaluate the tradeoffs involved in setting the threshold level when applying the generalized Pareto (GP) distribution. Explain the multivariate EVT for risk management

Questions:

22.6.1. In applying extreme value theory (EVT) to their investment firm's financial returns (i.e., market risks), Patricia and Peter are trying to decide between the classic blocks (aka, blocks maxima) approach and the modern peaks over threshold (POT) approach. Patricia is more familiar with the blocks approach and she makes the following points, but one statement is inaccurate. In regard to the blocks approach, each of the following is true EXCEPT which statement is false?

a. If the parent distribution is a Levy distribution, the extremes fall in the attraction domain of the Frechet distribution
b. For a given historical window, shorter blocks (and therefore more blocks) achieve lower variance but with more bias
c. Blocks is superior to POT because it uses the data more efficiently and its Gumbel distribution instance successfully avoids the trade-off between bias and variance
d. They can address time dependency in their loss data with longer (and therefore fewer) blocks and less bias (aka, better accuracy) but at the cost of higher variance

22.6.2. Patricia has been asked by the Chief Risk Officer (CRO) to estimate value at risk (VaR) at high confidence levels. Initially, she fits a generalized extreme value (GEV) distribution to the loss data. However, some members of the ultimate audience (the board) do not understand the math. She observes that Dowd(†) explains, "[t]here are also short-cut ways to estimate VaR (or ES) using EV theory. These are based on the idea that if ξ > 0, the tail of an extreme loss distribution follows a power-law times a slowly varying function: F(x) = k(x)*x^(-1/ξ) where k(x) varies slowly with x. For example, if we assume for convenience that k(x) is approximately constant, then [this equation] becomes: F(x) ≈ k*x^(-1/ξ)". Notice the assumption that k(x) is approximately constant.

Consequently, she decides to estimate extremely confident VaRs with a power-law shortcut. Specifically, when Patricia sets xi, ξ = 4.0, she observes that the 99.50% VaR is $6.0 million. If the loss tail follows the power law, which is nearest to the implied 99.90% VaR? a.$5.0
b. $9.0 c.$13.0
d. \$21.0

22.6.3. Peter showed his boss an extremely confident value at risk (VaR) for one of the firm's portfolios using a peaks over threshold (POT) approach. His boss observed that a disadvantage of the POT approach to extreme values (EVT) is that their choice of a threshold parameter was somewhat arbitrary. He asked Peter to show the VaR under a blocks (aka, block maxima) approach with an extremely high confidence level of 99.90%. He shows Peter an example given the following parameters: n = 120 observations, μ = 3.00%; scale, σ = 1.33%; and tail, ξ = 0.80. In this case, the 99.90% GEV (block maxima) is given by μ - σ/ξ*[1-(-n* LN(α))^(-ξ)] = 3.00-1.33/0.80*[1-(-120* LN(99.90%))^(-0.80)] = 10.40%.

Specifically, the blocks assumed parameters are the following: location, n = 160 observations, μ = 2.00%; scale, σ = 1.50%; and tail, ξ = 0.90. Which of the following is nearest to this 99.90% confident VaR.

a. 99.90% VaR equals 6.60%
b. 99.90% VaR equals 7.70%
c. 99.90% VaR equals 9.00%
d. 99.90% VaR equals 13.50%