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Figure 2.7: Actual Probability of Loosing More than the 1% HS VaR When Returns Have Dynamics Variance

# Figure 2.7: Actual Probability of Loosing More than the 1% HS VaR When Returns Have Dynamics Variance

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The Probability of Breaching the HS VaR

• where is the cumulative density function for a

standard normal random variable.

• If the HS VaR model had been accurate then this plot

should show a roughly flat line at 1%

• Here we see numbers as high as 16% and numbers

very close to 0%

• The HS VaR will overestimate risk when true market

volatility is low, which will generate a low

probability of a VaR breach

• HS will underestimate risk when true volatility is

high in which case the VaR breach volatility will be

high

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

31

VaR with Extreme Coverage Rates

• The tail of the portfolio return distribution conveys

• Reporting the entire tail of the return distribution

corresponds to reporting VaRs for many different

coverage rates

• Here p ranges from 0.01% to 2.5% in increments

• When using HS with a 250-day sample it is not

possible to compute the VaR when

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

32

Figure 2.8: Relative Difference between NonNormal (Excess Kurtosis=3) and Normal VaR

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

33

VaR with Extreme Coverage Rates

• Note that (from the above figure) as p gets close to

zero the nonnormal VaR gets much larger than the

normal VaR

• When p = 0.025 there is almost no difference

between the two VaRs even though the underlying

distributions are different

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

34

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Figure 2.7: Actual Probability of Loosing More than the 1% HS VaR When Returns Have Dynamics Variance

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