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Weigh the generalized extreme value (GEV) method against POT when it comes to quantifying extreme risk, noting what separates them.

FRM® Part II · Market Risk Measurement and Management · Parametric Approaches (II): Extreme Value · FRM2.MR.3.d

The idea

The generalized extreme value (GEV) approach and peaks over threshold (POT) both rest on EVT but use the data differently. GEV divides the sample into blocks, for example months or years, and fits the distribution of each block's maximum with location, scale and shape parameters. POT fits a generalized Pareto distribution, with scale and shape, to every observation above a high threshold. The shape parameter ξ describes the same tail in both. GEV keeps only one observation per block, so it discards other large losses in the same block and needs a long history. Block size is a bias-variance choice. POT uses all the large observations, so it is more data-efficient and is the usual choice in financial risk. But it needs a threshold, and its exceedances cluster. Exam point: GEV is block-based, POT is threshold-based.

Formulas

Probability that a block maximum exceeds a level

P(block maximum > x) = 1 − F(x)ⁿ

F(x)
the probability that a single iid observation is at or below x
n
the number of observations in the block

Independent, identically distributed observations. The probability is not n × (1 − F(x)), which double counts overlapping events.

A practice question

From the question bank, with the reason each wrong option is wrong.

What distinguishes the GEV approach from POT?

  1. GEV models the excesses over a high threshold, whereas POT models block maxima.Why not: Reverses the two methods.
  2. GEV models the centre of the distribution, whereas POT models the tail.Why not: Both are tail methods; GEV applies only to maxima of blocks.
  3. GEV is non-parametric, whereas POT assumes normally distributed excesses.Why not: GEV is a parametric family, and POT excesses follow the generalized Pareto, not the normal.
  4. GEV models block maxima, whereas POT models excesses over a high threshold. - correct

GEV fits one extreme per block (location, scale, shape); POT fits all observations above a threshold with the GP (scale, shape). That is the essential difference.

A harder one

A GEV distribution fitted to annual maximum losses has shape parameter xi = 0.30. A GP distribution is then fitted to the exceedances of the same loss series over a high threshold. What shape parameter should the GP fit be expected to give, apart from sampling error?

  1. About 3.3Why not: Uses the tail index 1/xi, which is not the shape parameter.
  2. About -0.30Why not: Reverses the sign; the same tail has the same sign of shape in both.
  3. About 0.30 - correct
  4. About 0.00Why not: Assumes the excess is exponential, which would apply only to a thin-tailed (Gumbel-type) tail.

The GEV and GP describe the same tail, and the shape parameter xi is shared. Differences between the two estimates come from sampling error and from different data usage, not from the parameter definition.

The hardest tier

Daily losses are iid. A risk manager considers the maximum loss over a 21-day month, the object modelled by GEV with 21-day blocks. What is the probability that the monthly maximum exceeds the daily 99 percent VaR?

  1. 21.0 percentWhy not: Adds the 1 percent daily probability 21 times, ignoring overlap between days.
  2. 1.0 percentWhy not: Treats the maximum as exceeding VaR with the same probability as a single day.
  3. 81.0 percentWhy not: Reports the probability that the maximum stays below VaR, the complement of what was asked.
  4. 19.0 percent - correct

The maximum is below VaR only if all 21 days are: 0.99^21 = 0.8097. So the probability of exceeding is 1 - 0.8097 = 0.1903. Maxima have distribution F^n, the starting point of block-maxima theory.

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