Higher moments and finance

Incorporating Higher Moments into Financial Data Analysis--James M. Sfiridis

Oh boy, here we go! Hold on tight and let's tag along for the ride. This could change some things!

Virtually everything we do with respect to risk and return centers on the assumption of normality (that is that the normal distribution correclt explains returns, or minimally the log of returns).

The problem is nearly perfectly laid out in the opening paragraph of the paper:

"There is ample empirical evidence that the distribution of short-term security returns, e.g., daily, weekly or monthly, is non-normal (Mandelbrot, 1963; Fama, 1965; Campbell, Lo and MacKinlay, 1997). Such data are inadequately described by linear models commonly used in financial economics that assume a normal data-generating process (DGP). For example, return outliers occur with much greater frequency than would be expected assuming normality, resulting in fat-tailed distributions described as leptokurtotic in the fourth moment. Additionally, returns may exhibit asymmetry, having the tendency for either positive or negative returns to persist for any number of reasons (Campbell, Lo and MacKinlay, 1997). This phenomenon of return “clustering” results in skewness in the third moment and has been studied by Conine and Tamarkin (1981) and Harvey and Siddique (2000)."

While I will leave the more rigorous math to the paper itself, but the basic idea is that author uses a Markov Chain Monte Carlo simulation to account for the higher moments needed to describe a non normal distribution. Using a simulation also allows for tests of whether these high moments matter.

The answer, in a word, is 'yes'.

In the paper's words:

"Bayesian sampling-based Markov chain Monte Carlo (MCMC) methods are used in this paper to estimate an extension of the normal model by using a modification of the normal likelihood function presented by Fernandez and Steele (1998), henceforth known as FS, that incorporates all four moments of the underlying return DGP."

Building on a great deal of past work in the field that examines higher moments, the author looks specifically at CAPM and equity risk premums.

The findings? I will allow Sfridis conclude:

"In this paper Bayesian-based Markov chain Monte Carlo (MCMC) analysis is used to estimate a non-normal likelihood function that effectively specifies the return DGP by the explicit incorporation of higher moments. The significant effect of higher moments on testing of the [Sharp-Lintner] CAPM has been illustrated. Results show that model testing requires incorporation of more complete return information as summarized by the higher statistical moments, leading to possibly contrary conclusions from those obtained under an assumption of normality. A second exercise considers specification of the non-normally distributed market risk premium. Incorporation of additional information contained in the higher moments can yield nearly equivalent point estimation to that from re-specification of the first two moments that implicitly account for excess skewness and kurtosis. However, a major advantage of accurately re-specifying the model is the greater precision of resulting parameter estimates."

Which definitely deserves a wow! While the temptation might be to overlook such a finding because of its complexity, it could be why so many of our theortcial models do not stand up to empirical tests. Which is obviously VERY important!

Cite:
James Sfiridis, "Incorporating Higher Moments into Financial Data Analysis"
(Upload date: Aug 16, 2005)