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Figure 6.7: QQ Plot of Daily S&P 500 Tail Shocks against the EVT Distribution

Figure 6.7: QQ Plot of Daily S&P 500 Tail Shocks against the EVT Distribution

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Calculating VaR and ES from EVT

• We are ultimately interested not in QQ plots but rather in

portfolio risk measures such as VaR.

• Using the loss quantile F-11-p defined above by



• The VaR from the EVT combined with the variance

model is now easily calculated as



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



68



Calculating VaR and ES from EVT



69



• We usually calculate the VaR taking Φ-1p to be the pth

quantile from the standardized return so that

• But we now take F-11-p to be the (1-p)th quantile of the

standardized loss so that

• The expected shortfall can be computed using

• Where



when ξ<1



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Calculating VaR and ES from EVT



70



• In general, the ratio of ES to VaR for fat-tailed

distribution will be higher than that of the normal.

• When using the Hill approximation of the EVT tail the

previous formulas for VaR and ES show that we have a

particularly simple relationship, namely



• so that for fat-tailed distributions where ξ > 0, the fatter

the tail, the larger the ratio of ES to VaR:

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Calculating VaR and ES from EVT



71



• The preceding formula shows that when ξ= 0.5 then the

ES to VaR ratio is 2

• Thus even though the 1% VaR is the same in the two

distributions by construction, the ES measure reveals the

differences in the risk profiles of the two distributions,

which arises from one being fat-tailed

• The VaR does not reveal this difference unless the VaR is

reported for several extreme coverage probabilities, p.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Figure 6.8: Tail Shapes of the Normal Distribution

(blue) and EVT (red)



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



72



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Figure 6.7: QQ Plot of Daily S&P 500 Tail Shocks against the EVT Distribution

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