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Figure 2.6: Cumulative P/L from Traders with HS and RM VaRs

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



27



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



28



29



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



30



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



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Figure 2.6: Cumulative P/L from Traders with HS and RM VaRs

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