Figure 6.2: QQ Plot of Daily S&P 500 GARCH Shocks
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Filtered Historical Simulation Approach
• We have seen the pros and cons of both databased and model-based approaches.
• The Filtered Historical Simulation (FHS) attempts
to combine the best of the model-based with the
best of the model-free approaches in a very
intuitive fashion.
• FHS combines model-based methods of variance
with model-free method of distribution in the
following fashion.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Filtered Historical Simulation Approach
• Assume we have estimated a GARCH-type model
of our portfolio variance.
• Although we are comfortable with our variance
model, we are not comfortable making a specific
distributional assumption about the standardized
returns, such as a Normal or a ~t ( d ) distribution.
• Instead we would like the past returns data to tell
us about the distribution directly without making
further assumptions.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Filtered Historical Simulation Approach
• To fix ideas, consider again the simple example of a
GARCH(1,1) model
• where
• Given a sequence of past returns,
we can estimate the GARCH model.
• Next we calculate past standardized returns from
the observed returns and from the estimated
standard deviations as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Filtered Historical Simulation Approach
• We will refer to the set of standardized returns as
• To calculate the 1-day VaR using the percentile of the
database of standardized residuals
• Expected shortfall (ES) for the 1-day horizon is
• The ES is calculated from the historical shocks via
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Filtered Historical Simulation Approach
• where the indicator function 1(*) returns a 1 if the
argument is true and zero if not
• FHS can generate large losses in the forecast
period even without having observed a large loss
in the recorded past returns
• FHS deserves serious consideration by any risk
management team
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
The Cornish-Fisher Approximation to
VaR
We consider a simple alternative way of
calculating Value at Risk, which has certain
advantages:
• First, it allows for skewness and excess kurtosis.
• Second, it is easily calculated from the empirical
skewness and excess kurtosis estimates from the
standardized returns.
• Third, it can be viewed as an approximation to the
VaR from a wide range of conditionally
nonnormal distributions.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Cornish-Fisher Approximation
to VaR
• Standardized portfolio returns is defined by
• where D(0,1) denotes a distribution with a mean
equal to 0 and a variance equal to 1
• i.i.d. denotes independently and identically
distributed
• The Cornish-Fisher VaR with coverage rate, p, can
be calculated as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Cornish-Fisher Approximation
to VaR
• Where
• Where
is the skewness and is the excess kurtosis
of the standardized returns
• If we have neither skewness nor excess kurtosis so that
.
, then we get the quantile of the
normal distribution
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Cornish-Fisher Approximation
to VaR
• Consider now for example the one percent VaR,
where
• Allowing for skewness and kurtosis we can
calculate the Cornish-Fisher 1% quantile as
• and the portfolio VaR can be calculated as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Cornish-Fisher Approximation to
VaR
• Thus, for example, if skewness equals –1 and excess
kurtosis equals 4, then we get
• which is much higher than the VaR number from a
normal distribution, which equals 2.33σPF,t+1
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Cornish-Fisher Approximation
to VaR
• The expected shortfall can be derived as
Where
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
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