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Show how EVT extends to several variables at once and why that helps risk management.

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

The idea

Multivariate EVT extends the tail modelling to several risk factors at once. The standard approach models each variable's tail with univariate EVT, using GEV or GP. It then models the dependence between extremes separately, for example with an extreme value copula or a tail dependence coefficient. This matters because correlation estimated from typical days says little about how assets behave in a crisis. The Gaussian copula has no asymptotic tail dependence for any correlation below 1, so it understates the chance of joint crashes and overstates diversification. Multivariate EVT supports portfolio tail VaR, contagion analysis and stress scenarios. The limitation is data. Joint extremes are far rarer than single-asset extremes, and the number of observations in the joint tail shrinks quickly as the dimension grows. Exam point: marginals alone are not enough.

Formulas

Joint exceedance probability from a conditional probability

P(both) = P(A) × P(B | A)

P(A)
the probability that the first asset exceeds its VaR
P(B | A)
the probability that the second does, given that the first has

Multiplying the two marginal probabilities assumes independence and understates joint extremes when tails are dependent.

Expected joint exceedances of independent variables

expected joint days = n × p^(d)

n
the number of observations
p
each variable's exceedance probability
d
the number of variables

Independence only. The count collapses as d grows, which is why joint-tail data are so scarce in multivariate EVT.

A practice question

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

Which approach is typical in multivariate EVT?

  1. Model each tail with univariate EVT and join them with a copula-type dependence structure. - correct
  2. Fit a multivariate normal distribution to the exceedances of all variables.Why not: A multivariate normal has thin tails and no tail dependence, which defeats the purpose.
  3. Fit a GEV to the sum of the returns of all variables.Why not: This collapses the problem to one dimension and loses the dependence structure.
  4. Apply EVT only to the largest risk factor and treat the others as constants.Why not: Ignores the tails and the co-movements of the other risk factors.

Separating the marginal tails from the dependence structure (copula or dependence function) is the standard multivariate EVT framework, and lets each part be modelled and checked on its own.

A harder one

Two equity indices have a correlation of 0.6, and a risk manager models them with a bivariate normal distribution. The history shows the two indices often fall sharply on the same day. Which explanation is correct?

  1. The correlation is too low; raising it to 0.95 would produce the same tail dependence as observed.Why not: Under a Gaussian copula, extremes are asymptotically independent for any correlation below 1, however high.
  2. A bivariate normal has no asymptotic tail dependence for any correlation below 1, so it understates joint extremes. - correct
  3. The normal model overstates joint crashes because its tails are too fat to describe ordinary days.Why not: The normal tails are too thin, which understates joint crashes.
  4. Joint crashes only reflect the marginal tails; if each index has a correct GP tail, no dependence modelling is needed.Why not: Joint tail probability needs the dependence structure as well as the marginals.

A Gaussian dependence structure makes extremes in one variable asymptotically unrelated to extremes in the other, whatever the correlation. Empirical joint crashes call for a model with tail dependence, such as an extreme value or Student t copula.

The hardest tier

Two assets each have a 1 percent daily probability of exceeding their own 99 percent VaR. Given that one exceeds its VaR, the probability that the other also does is 30 percent. Using 250 trading days a year, what are the joint daily exceedance probability and the expected number of joint-exceedance days per year?

  1. 0.01 percent per day, about 0.025 days per yearWhy not: Applies independence (1 percent x 1 percent) and ignores the 30 percent conditional probability.
  2. 30 percent per day, about 75 days per yearWhy not: Takes the conditional probability as the joint probability, ignoring the 1 percent chance of the first exceedance.
  3. 0.60 percent per day, about 1.5 days per yearWhy not: Double counts the joint event by adding the two conditional routes, 1% x 30% twice.
  4. 0.30 percent per day, about 0.75 days per year - correct

Joint probability = P(first exceeds) x P(second exceeds | first) = 0.01 x 0.30 = 0.003, i.e. 0.30 percent; times 250 gives 0.75 days. Independence would give 0.0001, so joint crashes are 30 times more likely than independence implies.

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