Figure 2.6: Cumulative P/L from Traders with HS and RM VaRs
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Evidence from the 2008-2009 Crisis
Performance difference between HS and RM VaRs
•The RM trader will lose less in the fall of 2008 and
earn much more in 2009.
•The HS trader takes more losses in the fall of 2008 and
is not allowed to invest sufficiently in the market in 2009
•The HS VaR reacts too slowly to increases in volatility
as well as to decreases in volatility.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Probability of Breaching the HS VaR
• Assume that the S&P 500 market returns are
generated by a time series process with dynamic
volatility and normal innovations
• Assume that innovation to S&P 500 returns each
day is drawn from the normal distribution with
mean zero and variance equal to
• We can write:
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Probability of Breaching the HS VaR
• Simulate 1,250 return observations from above equation
• Starting on day 251, compute each day the 1-day, 1%
VaR using Historical Simulation
• Compute the true probability that we will observe a loss
larger than the HS VaR we have computed
• This is the probability of a VaR breach
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 2.7: Actual Probability of Loosing More than
the 1% HS VaR When Returns Have Dynamics
Variance
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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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
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