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Figure 6.6: QQ Plot of S&P 500 GARCH Shocks against the Asymmetric t Distribution

Figure 6.6: QQ Plot of S&P 500 GARCH Shocks against the Asymmetric t Distribution

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Extreme Value Theory (EVT)



53



• Typically, the biggest risks to a portfolio is the sudden

occurrence of a single large negative return.

• Having an as precise as possible knowledge of the

probabilities of such extremes is therefore at the essence

of financial risk management.

• Consequently, risk managers should focus attention

explicitly on modeling the tails of the returns distribution.

• Fortunately, a branch of statistics is devoted exactly to the

modeling of these extreme values.



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Extreme Value Theory (EVT)

• The central result in EVTstates that the extreme tail of a

wide range of distributions can approximately be

described by a relatively simple distribution, the socalled Generalized Pareto distribution.

• Virtually all results in Extreme Value Theory assumes

that returns are i.i.d. and are therefore not very useful

unless modified to the asset return environment.

• Asset returns appear to approach normality at long

horizons, thus EVT is more important at short horizons,

such as daily.



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



54



Extreme Value Theory (EVT)

• Unfortunately, the i.i.d assumption is the least

appropriate at short horizons due to the timevarying variance patterns.

• We therefore need to get rid of the variance

dynamics before applying EVT.

• Consider therefore again the standardized

portfolio returns



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



55



The Distribution of Extremes



56



• Define a threshold value u on the horizontal axis of the

histogram in Figure 6.1

• As you let the threshold u go to infinity, the distribution of

observations beyond the threshold (y) converge to the

Generalized Pareto Distribution,

where



• With β>0 and y≥u. Tail-index parameter ξ controls the

shape of the distribution tail

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Estimating Tail Index Parameter, ξ



57



• If we assume that the tail parameter, ξ, is strictly positive,

then we can use the Hill estimator to approximate the

GPD distribution



• for y > u and ξ> 0. Recall now the definition of a

conditional distribution

• Note that from the definition of F(y) we have



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Estimating Tail Index Parameter, ξ



58



• We can get the density function of y from F(y):



• The likelihood function for all observations yi larger than

the threshold, u,



• where Tu is the number of observations y larger than u



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Estimating Tail Index Parameter, ξ



59



• The log-likelihood function is therefore



• Taking the derivative with respect to ξ and setting it to zero

yields the Hill estimator of tail index parameter



Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



Estimating Tail Index Parameter,



60



We can estimate the c parameter by ensuring that the fraction

of observations beyond the threshold is accurately captured

by the density as in



• Solving this equation for c yields the estimate



• Cumulative density function for observations beyond u is



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Estimating Tail Index Parameter, ξ

• Notice that our estimates are available in closed form

• So far, we have implicitly referred to extreme returns as

being large gains. As risk managers, we are more

interested in extreme negative returns corresponding to

large losses

• We can simply do the EVT analysis on the negative of

returns (i.e. the losses) instead of returns themselves.



Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



61



Choosing the Threshold, u



62



When choosing u we must balance two evils: bias and

variance.

• If u is set too large, then only very few observations are

left in the tail and the estimate of the tail parameter, ξ,

will be very noisy.

• If on the other hand u is set too small, then the data to the

right of the threshold does not conform sufficiently well to

the Generalized Pareto Distribution to generate unbiased

estimates of ξ.



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Choosing the Threshold, u



63



• Simulation studies have shown that in typical data sets

with daily asset returns, a good rule of thumb is to set the

threshold so as to keep the largest 50 observations for

estimating ξ

• We set Tu = 50.

• Visually gauging the QQ plot can provide useful guidance

as well.

• Only those observations in the tail that are clearly

deviating from the 45-degree line indicating the normal

distribution should be used in the estimation of the tail

index parameter, ξ

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Figure 6.6: QQ Plot of S&P 500 GARCH Shocks against the Asymmetric t Distribution

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