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