Rethinking risk and return: part 2 – some felicitous Fourier frequencies

Abstract

PurposeThis editorial, the second of a two‐part series, proposes a new measure of risk for analyzing highly non‐normal (i.e. asymmetric and long‐tailed) random variables in the context of both investment and insurance portfolios. The proposed measure replaces the p‐norm‐based definition of “risk” – found wanting in Part 1 – with a cosine‐based alternative.Design/methodology/approachJust as p‐norm‐based risk measures were derived as generalizations of the standard deviation in Part 1, the paper now extend this approach to a cosine‐based risk measure. This involves computing the Fourier transform of the underlying random variable for a given frequency value. Methods for selecting an appropriate frequency are then discussed.FindingsThe cosine‐based risk measure provides an effective alternative to p‐norm‐based measures because the Fourier transform is always well defined, even for long‐tailed random variables. The frequency parameter necessary for the Fourier transform may be computed according to several interesting criteria, including the maximization of marginal Shannon information, as well as consideration of “Planck boundaries” in human cognition.Originality/valueThe editorial explores the use of cosine‐based measures in constructing a general definition of “risk” that is equally applicable to asymmetric and long‐tailed random variables as to normal random variables.