August 07, 2008
POTholding the movement in the tails of financial return series: The story of Peaks Over Threshold (POT) Method
Unraveling the volatilities in the financial markets has always been an undecipherable mystery like intrigue smile of Monalisa. When conventional VaR methods fail to decipher the heightened movement in the tails of the financial returns series, one can do it: The Extreme Value Theory (EVT).
While Basel II stipulates for VaR based capital charge under Internal Model Approach, it is important to make the estimate accurate or else may lead to sub-optimal allocation. EVT-based VaR could be a good candidate.
The Extreme Value Theory is a branch of statistics, dealing with extreme deviation from the median of probability distribution. There are two classes of theorem based on data generation process:
Theorem I :
a)Fisher and Tippett (1928)( Fisher, R. A. and L. H. C. Tippett (1928): “Limiting forms of the frequency distribution of the largest or smallest member of a sample”, Proceeding of Cambridge Philoshophical Society,24, 180-190)
(b)Gnedenko (1943)(Gnedenko , B. (1943): “Sur la distribution limite du terme maximum d’une s´erie al´eatoire”, Annals of Mathematics, 44, 423-453).
Theorem II :
(a)Pickands (1975)(Pickands, J. (1975): Statistical inference using extreme order statistics, Annals of Statistics, 3, 119–131)
(b)Balkema and de Haan (1974)(Balkema, A. A. and L. de Haan (1974): “Residual lifetime at great age”, Annals of Probability, 2, 792–804)
In POT, the excess distribution is taken. It is also called Generalized Pareto Distribution(GPD). The GPD takes three parameters viz. tail parameter, location parameter, and location parameter.
The equation of GPD is
Download file
The challenge arises when the distribution is non-linear. The non-linearity could be ascertained by using a graphical exploration tool like QQ Plot.
A QQ plot shall look like this for non-linear data i.e. tending towards straight line.
The sample mean excess plot could be used.
Next step is to find a threshold and GPD is fitted to the exceedences above the threshold.
Then VaR can be estimated by POT-ML estimation for upper bound, point estimation and lower bound.
The EVIM package of MATLAB is very useful input for EVT analysis though I use another version , modified by me. EVIM can be downloaded from link (http://www.bilkent.edu.tr/~faruk/evim.htm) i.e. Faruk's webpage along with an amazing documentation by them.
During my research in my academic capacity, I have empirically found that there are huge understated VaR, computed by conventional measures. Those VaR does not depict true movement in tails and Basel II regulatory multiplier is a straight forward quick fix for the Banks without going into reasons .
From my professional perspective, I strongly believe there is necessity for Basel to specify the methodologies for VaR with a framework to monitor the movement in the tail specially while laying the ICAAP framework. I am unable to justify with facts and figures as I am bound by non-disclosure agreement in my professional capacity.
Only I believe Basel must direct the banks to POThold the movement in the tails of the returns series to compute a prudent estimate of VaR and subsequent correct computation of regulatory and economic capital.We can not afford another subprime crisis.
Posted by ddutta7 at 04:31 PM
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