May 02

Extreme Value Theory (EVT), Highlights for the FRM exam: Part 3

by David Harper, CFA, FRM, CIPM

FRM | Risk |

I earlier gave some background on EVT (see Part 1 and Part 2). The GARP-assigned EVT reading is, in my opinion, sort of a slog. If you are rusty on the math, it's doesn't try to help. That's okay because you won't need to perform EVT calculations (note: we don't have AIMs yet, so I cannot be 100% certain, but I am pretty sure). Also, there is a small typo in the Kalyvas reading that might confuse a careful reader. It concerns the tail index (see details at bottom), so don't get distracted by the tail index. The tail index is the reciprocal of the shape parameter, but what matters is the shape parameter.

Here is my advice: after you read the assigned reading, take at look at the unassigned, brief discussion of EVT in Chapter 5 of Jorion's Value at Risk. Once again, Jorion rescues with clear prose.

In regard to EVT, we care about three parameters: location, scale and shape. Location and scale "normalize" the loss observations into normal variables; e.g., if the location (mean) is -$200 and the scale (standard deviation) is $30, then a loss of -$260 is a 2.0 standard deviation exceedence ("in excess" of the threshold). As Jorion writes, "the all-important shape parameter determines the speed at which the tail disappears."

Briefly, here then are highlights about EVT:

  • Traditional risk measurement (i.e., imposing a normal distribution on loss events) suffers because the normal distribution tends to underestimate extreme (low frequency, high severity) loss events. EVT solves this with a laser-like focus on the tail of the distribution, where losses happen.
  • The traditional EVT approach employs a block maxima model. Time is sliced into equal intervals. The largest loss within each interval is plucked. Collect each of those "largest loss within the time block" into a distribution, that is described by a generalized extreme value (GEV) distribution.
  • The modern approach is peaks over threshold (POT). Under this "point process" method, time is ignored. If an event-loss (an exceedence) is worse than the threshold, it contributes to the distribution. This distribution is often described by a generalized Pareto distribution (GPD).
  • The GEV is given by the following cumulative distribution function (CDF)

 

  • The y is the normalized variable. Greek xi is the shape parameter. Financial returns tend to show heavy- or fat-tails (i.e., they are leptokurtotic. That's when kurtosis > 3). Therefore, of the three sub-classes of GEV we really care about the Frechet distribution, where the shape parameter is greater than zero (>0). For stocks, Jorion says the shape param tends to lie between 0.2 and 0.4. Again, larger shape param = heavy tail.
  • In regard to POT approach, we can employ semi-parametric or parametric. Parametric further divides into conditional or unconditional.
  • Expected shortfall (ES) is an important and helpful risk metric. VaR does not tell us anything about losses in excess of itself (a devastating deficiency, if you think about it). ES is elegant, it is the expected loss if (conditional on) the loss exceeds VaR. Plus, according to Dowd, ES is coherent and satisfies subadditivity.
  • There are two big model "design" issues: tail size and time dependency. If we increase the tail size (index) too much, the threshold moves toward a the middle and we defeat the purpose of EVT (and the shape index, as the reciprocal, moves toward zero, and we have converge to a Gumbel distribution). Time dependency is thematic: since volatility clusters (e.g., high volatility today tends to be followed by high volatility tomorrow), the assumption of independence is technically violated and renders the definition of time intervals important but fragile.

Regarding the typo in the Kalyvas reading: Xi is called both the shape parameter and the tail index, and then the statement, "the lower tail index, the fatter the tail." But I think they meant to say that alpha is the tail index, where the tail index is the reciprocal [tail = 1/shape] of the shape parameter. What you care about for the the exam is that "a larger shape parameter corresponds to fatter tails."

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