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Characterise extreme value theory (EVT) and how risk managers put it to use.

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

The idea

Extreme value theory (EVT) is the statistical theory of the limiting behaviour of the largest observations, so it models the tail directly instead of the whole distribution. Its core result, Fisher-Tippett, says that for independent, identically distributed data the suitably normalised maximum of a large block converges to the generalized extreme value (GEV) family, with a shape parameter ξ. When ξ = 0 the limit is Gumbel, with thin tails such as the normal. When ξ > 0 it is Fréchet, with fat power-law tails and a tail index of 1 ÷ ξ. When ξ < 0 it is Weibull, with a bounded tail. The Fréchet case is often used for financial losses, though ξ is uncertain in finite samples. Risk managers use EVT to estimate VaR and expected shortfall at very high confidence levels, to extrapolate beyond the sample and to design stress tests. The usual mistake is ignoring the iid assumption. With volatility clustering, apply EVT to standardised residuals.

Formulas

Power-law tail extrapolation

P(L > x2) = P(L > x1) × (x2 ÷ x1)^(−1 ÷ ξ)

P(L > x1)
the known probability that loss L exceeds x1
x2
the larger loss level
ξ
the shape parameter, so 1 ÷ ξ is the tail index

Fréchet-type tails with ξ > 0. Using ξ itself as the exponent instead of 1 ÷ ξ is the usual trap.

A practice question

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

Which description of extreme value theory (EVT) is correct?

  1. A method that fits the whole return distribution, weighted toward the centre, to estimate typical losses.Why not: Describes ordinary parametric fitting; EVT concentrates on the tail and ignores the centre.
  2. A simulation method that reprices the portfolio under scenarios drawn from history.Why not: Describes historical simulation, which has no tail model and cannot extrapolate.
  3. A volatility model that updates the variance with the latest squared return and the previous variance.Why not: Describes GARCH-type volatility modelling, not tail modelling.
  4. A body of theory on the limiting behaviour of the largest observations, used to model the tail directly. - correct

EVT studies the distribution of maxima or of exceedances above a high level and gives a parametric form for the tail, so it can be used for quantiles beyond the sample.

A harder one

Daily losses lie in the Frechet domain with shape parameter xi = 0.25, so the tail index is 1/xi = 4. Which statement is correct?

  1. All moments are finite because xi is below 1.Why not: Confuses the condition for a finite mean (xi < 1) with finiteness of all moments; here moments of order 4 and above are infinite.
  2. The variance is infinite because any xi above zero implies infinite variance.Why not: The variance is infinite only if xi is 0.5 or more (tail index 2 or less).
  3. Orders below 4 are finite: mean and variance exist but kurtosis does not. - correct
  4. Only moments of order below 0.25 are finite.Why not: Uses xi rather than the tail index 1/xi as the cutoff for moment order.

For a power-law tail with index alpha = 1/xi = 4, moments of order k exist only for k < 4. The variance (k = 2) exists, kurtosis (k = 4) does not, which is why sample kurtosis is unstable for such data.

The hardest tier

A risk manager models daily losses with a tail index of 4 (xi = 0.25). The probability that daily loss exceeds 5 percent is 1.00 percent. Using the power-law tail, P(L > x) proportional to x^(-4), what is the probability of a loss above 10 percent, closest to?

  1. 0.50 percentWhy not: Assumes the probability falls in proportion to the loss size (exponent 1), not x^(-4).
  2. 0.25 percentWhy not: Uses an exponent of 2 instead of the tail index of 4: 1.00 x 2^(-2).
  3. 0.84 percentWhy not: Uses xi = 0.25 as the exponent instead of 1/xi = 4: 1.00 x 2^(-0.25).
  4. 0.06 percent - correct

Doubling the loss from 5 to 10 percent multiplies the probability by 2^(-4) = 1/16, so 1.00 percent x 0.0625 = 0.0625 percent. Extrapolating the tail this way is a core practical use of EVT.

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