Extreme value theory – an introduction

Value at risk generally assumes that returns are normally or log-normally distributed
and largely ignores the fat tails of financial return series. The assumption of normality
works well enough when markets are themselves behaving normally. As already
pointed out above, however, risk managers care far more about extreme events than
about the 1.645 or 2.33 standard deviation price changes (95% or 99% confidence)
given by standard VaR.
If measuring VaR with 99% confidence it is clear that, on average, a portfolio value
change will be experienced one day in every hundred that will exceed VaR. By how
much, is the key question. Clearly, the bank must have enough capital available to
cover extreme events – how much does it need?
Extreme value theory (EVT) is a branch of statistics that deals with the analysis
and interpretation of extreme events – i.e. fat tails. EVT has been used in engineering
to help assess whether a particular construction will be able to withstand extremes
(e.g. a hurricane hitting a bridge) and has also been used in the insurance industry
to investigate the risk of extreme claims, i.e. their size and frequency. The idea of
using EVT in finance and specifically risk management is a recent development
which holds out the promise of a better understanding of extreme market events and
how to ensure a bank can survive them.
EVT risk measures
There are two key measures of risk that EVT helps quantify:
Ω The magnitude of an ‘X’ year return. Assume that senior management in a
bank had defined its extreme appetite for risk as the loss that could be suffered
from an event that occurs only once in twenty years – i.e. a twenty-year return.
EVT allows the size of the twenty-year return to be estimated, based on an analysis
of past extreme returns. We can express the quantity of the X year return, RX,
where:
P(r[RX)ó1ñF(RX)
Or in words; the probability that a return will exceed RX can be drawn from the
distribution function, F. Unfortunately F is not known and must be estimated
distribution functions used in EVT are discussed below. For a twenty-year return
P(r[RX) is the probability that an event, r, occurring that is greater than RX will
only happen, on average, once in twenty years.
Ω The excess loss given VaR. This is an estimate of the size of loss that may be
suffered given that the return exceeds VaR. As with VaR this measure comes with
a confidence interval – which can be very wide, depending on the distribution of
the extreme values. The excess loss given VaR is sometimes called ‘Beyond VaR’,
B-VaR, and can be expressed as:
BñVaRóESrñVaR Dr[VaRT
In words; Beyond VaR is the expected loss (mean loss) over and above VaR given
(i.e. conditional on the fact) that VaR has been exceeded. Again a distribution
function of the excess losses is required.

where m, k and t are parameters which define the distribution, k is the location
parameter (analogous to the mean), t is the scale parameter and m, the most
important, is the shape parameter. The shape parameter defines the specific distribution
to be used. mó0 is called the Gumbel distribution, m\0 is known as the Weibull
and finally and most importantly for finance, m[0 is referred to as the Frechet
distribution. Most applications of EVT to finance use the Fre´chet distribution.
From Figure 8.5 the fat-tailed behavior of the Fre´chet distribution is clear. Also
notice that the distribution has unbounded support to the right. For a more formal
exposition of the theory of EVT see the appendix at the end of this chapter.by
fitting a fat-tailed distribution function to the extreme values of the series. Typical