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
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VaR with Extreme Coverage Rates
• The tail of the portfolio return distribution conveys
information about the future losses.
• 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
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Figure 2.8: Relative Difference between NonNormal (Excess Kurtosis=3) and Normal VaR
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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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
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