Extreme Value Theory and Financial risk management.pdf

Extreme Value Theory and financial risk management

∗

Mandira Sarma

†

March 21, 2002

∗

An earlier draft of this paper was presented at the the Fifth Capital Markets Conference at UTI Institute

of Capital Markets, Mumbai, India. I thank the participants of the conference for helpful comments and

suggestions. All errors are mine.

Value-at-Risk (VaR) is widely used as a tool for measuring the market risk of asset portfolios.

It quantifies in monetary terms the exposure of a portfolio to the market fluctuations. It is

defined as the maximum monetary loss of a portfolio such that the likelihood of experiencing

a loss exceeding that amount, due to its exposure to the market movements, over a specified

risk horizon is equal to a pre-specified tolerance level.

Extreme value theory (EVT) deals with the study of the asymptotic behaviour of extreme

(maxima and minima) observations of a random variable. Financial risk management is all

about understanding the large movements in the values of asset portfolios. It essentially

deals with the analysis of the tail regions of the distribution of changes in the market value of

the portfolio. Extreme value theory, by dealing with only extreme observations, can provide

estimated.

Extreme value theory, on the other hand, focuses primarily on analysing the extreme obser-

vations rather than the observations in the central region of the distribution. The theory

provides robust tools for estimating only the tails by making use of the available data. Tail

quantiles like VaR can be estimated more accurately by using EVT than the conventional

approaches.

Another appealing aspect of EVT is that it does not require to make a priori assumption

about the return distribution. The fundamental result of extreme value theory, known as the

“extremal types theorem”, identifies the possible classes of distributions for movements of

the extreme returns irrespective of the actual underlying return distribution. This extremely

This paper uses the recent developments of extreme value theory to analyse the tails of

the innovation distribution of the Nifty returns. Using the “Peaks-Over-Threshold” (POT)

model (McNeil and Frey, 1999) we estimate the tail regions of the innovation distribution of

the Nifty returns. We find that the lower tail of the Nifty innovation behaves very much like

a Gaussian tail whereas the upper tail behaves significantly different from that of a Gaussian

tail. The upper tail is found to exhibit significant “tail thickness” which indicates existence of

asymmetry in the innovation distribution. The existence of asymmetry and the tail thickness

in the innovation distribution provides valuable information while estimating risk measures

based on tail quantiles.

The rest of this paper is organised as follows. Sections 2 and 3 present an overview of extreme

The classical Extreme Value theory (EVT) deals with the study of the asymptotic behaviour

of extreme observations (maxima or minima of n random realisations).

= max{X

The extreme value theory deals with the distributional properties of Y

and Z

as n becomes

− a

and

− a

is non-degenerate.

A brief description about the minima can be found in Leadbetter et al. (1983)