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Account for why extreme values matter in risk management and what makes dealing with them difficult.

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

The idea

Extreme values matter because solvency, capital and survival are decided by rare, very large losses rather than by ordinary days. VaR at 99 percent or more, expected shortfall and stress tests all live in the far tail. Dealing with them is hard for three linked reasons. First, data are scarce. With 1,000 daily observations a 99.9 percent quantile has about one observation beyond it, and a historical simulation can never produce a loss larger than the worst one in the sample. Second, standard models are fitted to the centre of the distribution, and a normal assumption badly understates the fat tails of real returns. Third, tail estimates are therefore noisy and must rely on extrapolation beyond the data. Exam point: scarce tail data plus fat tails make high-confidence VaR hard, which is the motivation for extreme value methods.

A practice question

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

Why do extreme values deserve separate treatment in market risk management?

  1. They are data errors, so they should be removed before estimating volatility and VaR.Why not: Treats genuine tail events as outliers to be cleaned, when they are exactly the information the risk manager needs.
  2. Rare, huge losses drive solvency, and the tail behaves unlike the centre. - correct
  3. They occur so often that they dominate the sample mean and the 95 percent VaR.Why not: Overstates their frequency: extremes are rare by definition and a 95 percent VaR is not an extreme quantile.
  4. The central limit theorem makes the normal VaR exact in the far tail.Why not: Misapplies the CLT, which concerns sums in the centre; fat tails make a normal VaR too low far out.

Solvency is decided by rare large losses, which sit in the tail, and the tail is not well described by models fitted to the bulk of the data. That is why extreme value methods exist.

A harder one

A risk manager runs historical simulation on 500 daily P&L observations to estimate the 99.9 percent VaR. How many sample observations are expected to lie beyond the true 99.9 percent VaR, and what follows?

  1. 5 observations, so the tail is adequately populatedWhy not: Uses 1 percent (99 percent confidence) instead of 0.1 percent: 500 x 0.01 = 5.
  2. 50 observations, so the tail is adequately populatedWhy not: Uses 10 percent instead of 0.1 percent: 500 x 0.10 = 50.
  3. 0.05 observations, so the tail holds no information at allWhy not: Uses 0.01 percent (99.99 percent confidence): 500 x 0.0001 = 0.05; the correct tail probability is 0.001.
  4. 0.5 observations, so the worst sample loss caps the estimate - correct

Expected tail count = n x (1 - c) = 500 x 0.001 = 0.5. With less than one expected observation beyond the quantile, historical simulation cannot go beyond the worst observed loss, so the 99.9 percent VaR is unreliable.

The hardest tier

A bank has 2,520 daily iid loss observations (about 10 years). Let x be the true 99.9 percent one-day VaR. What is the probability that the largest loss in the sample is below x, so that even the worst observation understates the true VaR?

  1. 92.0 percentWhy not: Gives the probability that at least one observation exceeds x, the complement of what was asked.
  2. 0.1 percentWhy not: Reports the single-day exceedance probability, ignoring that 2,520 independent days are in the sample.
  3. 8.0 percent - correct
  4. 2.5 percentWhy not: Reads the expected number of exceedances (2,520 x 0.001 = 2.52) as a percentage.

All 2,520 days must stay below x, each with probability 0.999: 0.999^2520 = 0.0804, about 8.0 percent. In roughly 1 sample in 12 historical simulation misses the true 99.9 percent VaR entirely, since it cannot exceed the largest observation.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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