Teasing Out Tail Risk or, Quantum Mechanics to the Rescue : A Practical Method for Characterizing Downside Risk in Non-Gaussian Return Distributions

Using equations that arise in quantum mechanics, this paper describes a way to more accurately and efficiently represent non-Gaussian return distributions than the standard method of invoking skewness and kurtosis. Then, it provides a new single intuitive number, defined here as the “crash volatility”, to characterize the true left-tail risk as an alternative to the usual downside deviation. The crash volatility can be fed into a typical mean-variance optimizer, allowing the optimizer to finally “see” the risk effect of the non-Gaussian distribution. An example using Amaranth's returns before it lost -71% in September, 2006 illustrates how these new techniques caught a much higher level of risk lurking in the data