Performance Measurement When Return Distributions are Nonsymmetric
Alan & Stephens Utah State University
Dennis Proffitt(*) Grand Canyon University
This paper arugues that to ignore higher moments of utility one must do
so on the basis of economic arguments (normal returns or quadratic
utility) or empirical evidence demonstrating that higher moments are
not important to investors. Extending the work of Prakash and Bear
(1986), a generalized performance evaluation model is presented that
allows for multiple moments of utility. The evaluation models are
applied to the returns of internationally diversified mutual funds. The
results indicate that ignoring higher moments of utility has significant
impact upon the performance rankings of these funds.
Modern portfolio construction often involves attempts by portfolio managers to modify the return distributions of their portfolios. For example, Bookstaber and Clarke (1983) have shown that the returns on portfolios of common stocks can be modified substantially using a variety of option strategies. In addition, international funds with highly skewed return distributions offer their investors risk reduction possibilities through their low covariance of returns with domestic securities (Proffitt and Seitz, 1983).
If return distributions are nonsymmetrical and investors value skewness, as argued by Arditti (1967), then traditional performance measures are inadequate. To address the problem of defining performance in the face of nonsymmetric distributions, Prakash and Bear (1986) (hereafter PB) developed measures that recognized skewness of return distributions. The PB formulas were developed on the basis of the Kraus and Litzenberger (1976) (hereafter KL) skewness preference model.
This paper demonstrates that the KL three moment model is a special case of Rubinstein's (1973) parameter preference model. In the KL model, higher moments were ignored, as they were not behaviorally justified. Without entering this controversy of whether more than three moments are behaviorally justified, a performance measure based on n-moments is developed. This general performance measure has, as special cases, the Treynor and PB performance measures.
One of the desirable attributes of the general performance model is that it readily is adaptable to a form that is useful in empirical tests. (1) The application of the performance measures is illustrated with an analysis of internationally diversified mutual funds. Research concerning internationally diversified mutual funds continually encounters problems with the performance measurement of these funds relative to domestic funds using traditional techniques. Because internationally diversified funds have skewed and flattened return distributions, performance problems of international funds may be addressed using the general model of performance measurement. Performance ranking based on three or higher moments are essentially the same for internationally diversified funds, while those based on two moments differ significantly with the ranking achieved using PB measures. Thus, this paper not only empirically applies the PB measures to a set of internationally diversified funds, but also (albeit indirectly) tests the validity of the higher moments CAPM.
Development of the Model
By stating utility as a Taylor series expansion of an individual's expected wealth, Rubinstein (1973) developed the fundamental theorem of parameter preference security valuation. This model, which is a state-independent, state-preference model in which probability data are summarized by central moments, states: [Mathematical Expression Omitted]
where: [Mathematical Expression Omitted]
[U.sub.i] = an individual's continuously differentiable utility …