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
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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
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• 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
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• 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
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