Aus with Gauss

10 May 2010 By Martin Hutchinson

The credit crisis would have been less bad if risk had been managed better. Unfortunately, many financial firms and watchdogs use mathematical models that grossly underestimate tail risk. It can seem an arcane debate, but it matters and improvements are fairly easy to make.

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The Value at Risk (VaR) risk management system, invented by RiskMetrics, then part of JPMorgan Chase, in 199093 and endorsed by the Basel Committee in 1996, involves calculating the maximum loss on a position in 95 percent or 99 percent of trading periods, given the position's historic volatility. Users of VaR typically multiply the 99 percent VaR over a 10day period by 3 or 4, to get a measure of the bank's overall exposure.

The VaR system rests on the central assumption that the "tails" of the risk distribution will be neither "fat" nor "long" in other words that there will not be circumstances in which the risk of exceeding the VaR limit greatly exceeds one percent, nor circumstances in which the loss in unlikely but possible cases on one percent of trading days is a large multiple of the VaR limit. The VaR system does not strictly require price movements to follow a Gaussian "random walk" but it does require the deviations from "Gaussianity" to be modest.

The Gaussian distribution is one of a family of "stable Paretian" distributions in which the probability of the outcome exceeding a value x declines proportionately by x to the power of (1+a) where the parameter a (or alpha), between 0 and 2, determines the nature of the distribution. If a=2 the distribution is the Gaussian, for a<2 the "tails" are fatter and the standard deviation of the distribution cannot be calculated. Three stable Paretian distributions are mathematically tractable: the Gaussian, the Cauchy distribution (a=0) and the ParetoLevy distribution (a=0.5). The ParetoLevy distribution is used by the Options Clearing Corporation in calculating counterparty risk in options trades.

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