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Figure 6.2: QQ Plot of Daily S&P 500 GARCH Shocks

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



14



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



17



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



18



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



19



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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Figure 6.2: QQ Plot of Daily S&P 500 GARCH Shocks

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