Common mistake 1 with Extreme value theory (tail risk)

Choosing the threshold too low, which mixes the body of the distribution into the tail fit.

Common mistake 2 with Extreme value theory (tail risk)

Choosing it too high, which leaves too few points and a noisy ξ.

Common mistake 3 with Extreme value theory (tail risk)

Running it on prices or monthly data with a few hundred points. It needs a long series of returns.

Common mistake 4 with Extreme value theory (tail risk)

Reading VaR at 99.9% from a few years of data as if it were precise. The confidence interval is wide.

EconStatsCraft › Extreme value theory (tail risk)

Extreme value theory (tail risk)

Fits a generalised Pareto distribution to the losses beyond a high threshold and uses it to estimate VaR and Expected Shortfall far out in the tail, compared with historical and Normal estimates, with a mean excess plot to judge the threshold.

What it needs

Return series (daily works best)

The intuition

Ordinary statistics are built from the middle of the data, where most observations are. Risk lives in the tail, where there are very few. Extreme value theory says that whatever the full distribution looks like, the part above a high threshold has a known shape, the generalised Pareto. Fitting only that part uses the few extreme losses efficiently and lets you estimate losses worse than any you have seen.

When to use it

Measuring risk at very high confidence (99.5% and beyond), stress testing, operational and insurance losses, and any series with fat tails where the Normal is plainly wrong. It is core FRM Part II market risk material.

Where the estimator comes from

The Pickands–Balkema–de Haan theorem states that for a wide class of distributions the excess Y=L-u given L>u converges to a generalised Pareto as u rises: G(y)=1-(1+\xi y/\beta)^{-1/\xi}. The tail probability is then P(L>x)\approx\frac{N_u}{n}\left(1+\xi\frac{x-u}{\beta}\right)^{-1/\xi}. Setting this equal to 1-q and solving for x gives VaR; integrating the tail beyond VaR gives ES =\frac{\text{VaR}+\beta-\xi u}{1-\xi} for \xi<1.

The model

P(L-u>y\mid L>u)\approx\left(1+\frac{\xi y}{\beta}\right)^{-1/\xi},\qquad \text{VaR}_q = u+\frac{\beta}{\xi}\left[\left(\frac{n}{N_u}(1-q)\right)^{-\xi}-1\right],\qquad \text{ES}_q=\frac{\text{VaR}_q+\beta-\xi u}{1-\xi}

What it assumes

Losses are independent and identically distributed. Volatility clustering breaks this; a common fix is to fit EVT to GARCH-filtered residuals.
The threshold is high enough for the GPD approximation to hold, but low enough to leave enough exceedances to fit.
The tail shape is stable over the sample period.

Statistical properties

ξ > 0 corresponds to Pareto-type tails (Student t, financial returns); ξ = 0 to thin tails (Normal, lognormal); ξ < 0 to bounded tails.
With ξ ≥ 0.5 the variance is infinite; with ξ ≥ 1 even the mean is.
Estimates beyond the data become less precise the further out they go; VaR at 99.9% from a few thousand points has a wide interval.
The method is applied to i.i.d. losses; GARCH filtering first is common practice for returns.

How to read the output

The shape parameter ξ is the headline. ξ > 0 means a heavy, power-law tail; ξ = 0 an exponential tail; ξ < 0 a tail with a finite end point.
Compare the three VaR rows. When EVT sits well above the Normal, the Normal is understating extreme risk.
Historical VaR cannot exceed the worst loss in the sample; EVT can, which is the point of using it.
A mean excess plot that rises roughly linearly above the threshold supports a GPD tail with ξ > 0.

A worked reading

The returns sample has 1,500 daily returns. Taking the worst 5% as the tail sets the threshold at a 1.38% loss and leaves 75 exceedances. The fitted shape is ξ = −0.085, close to zero, so this tail is roughly exponential rather than power-law, and the scale is 0.84%. At 99%, EVT gives a VaR of 2.64% and an ES of 3.32%; historical simulation gives 2.76% and 3.38%; the Normal gives only 2.23% and 2.56%. Even with a thin-ish tail, the Normal understates the 1-in-100 day loss by about 0.4 points.

Symptom → diagnosis → what to do

If you see this	It means	Do this
ξ jumps around as the threshold moves	Too few exceedances	Lower the threshold or use more data
EVT VaR below historical VaR	Thin tail, or threshold too high	Check the mean excess plot
Negative ξ on returns	A short or calm sample	Usual for some samples; read it as no fat tail here
Huge negative VaR	Prices used instead of returns	Compute returns first

Common mistakes

Choosing the threshold too low, which mixes the body of the distribution into the tail fit.
Choosing it too high, which leaves too few points and a noisy ξ.
Running it on prices or monthly data with a few hundred points. It needs a long series of returns.
Reading VaR at 99.9% from a few years of data as if it were precise. The confidence interval is wide.

Where to read this properly

McNeil, Frey and Embrechts, Quantitative Risk Management (2015), ch. 5.
Embrechts, Klüppelberg and Mikosch, Modelling Extremal Events (1997).
GARP, FRM Part II, Market Risk Measurement and Management: Extreme Value Theory.

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Extreme value theory (tail risk)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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