Risk and Return Measures for a Non-Gaussian World

We propose new measures of both risk and anticipated return that incorporate the effects of skewness and heavy tails from a financial return’s probability distribution. Our cosine-based analysis, which involves maximizing the marginal Shannon information associated with the Fourier transform of the distribution’s probability density function, also facilitates the use of Lévy-stable distributions for asset prices, as suggested by Mandelbrot (1963). The new measures generalize the concepts of standard deviation and mean in the sense that they simplify to constant multiples of these widely used parameters in the case of Gaussian returns.