Good and bad news on capital market return ellipticity.(ANTHOLOGY)

Since the seminal paper of Owen and Rabinovitch (1983; "On the Class of Elliptical Distributions and Their Application to the Theory of Portfolio Choice." Journal of Finance 38 (3), 745-752) the key role played by elliptical distributions in financial modelling is well-known. The main property of this class of distributions is that they can be completely specified by the mean, the variance, and the density generator g that qualifies the specific family. Elliptical distributions include as special cases the normal (for g(u) = [ce.sup.-u/2] , c > 0), the t Student (g(u)=c(v, p)(1 + u/v)-(v+p)/2, v>0, c>0), and the logistic (g(u) = [e.sup.-[square root of u]]/[(1 - [e.sup.-[square root of u]]).sup.2]) among other distributions. Thanks to the property that a linear transformation of ellipticals of the same family is still elliptical of the same family, the mean-variance analysis can be generalized to non-normal, fat-tailed portfolios. Specifically, the Capital Asset Pricing Model maintains its validity under elliptical (symmetric) distributions (see the recent alternative proof to Owen and Rabinovitch (1983) given by Hamada and Valdez (2008), "CAPM and Option Pricing with Elliptically Contoured Distributions," Journal of Risk & Insurance, 75 (2), 387-409). From the practitioner's point of view, two spontaneous questions arise: (1) are empirical fund returns elliptically distributed? (2) In the affirmative case, which are most representative families?

In recent years, ellipticity in returns has been empirical tested. Eling (2008; "Does the Measure Matter in the Mutual Fund Industry?," Financial Analysts Journal, 64 (3), 54-66), considering the 38,954 worldwide investment funds belonging to seven asset classes, showed that most of these funds belong to the group of ellipticals. Same findings were drawn by Galea, Diaz-Garcia, and Vilca (2008; "Influence diagnostics in the capital asset pricing model under elliptical distributions," Journal of Applied Statistics, 35 (2), 179-192) considering monthly returns of stocks. However, these authors only consider four stocks and the small Chilean stock market. The aim of our analysis is double: at first we test the ellipticity in stock returns for a much broader set of stocks and considering the US market, which is the largest stock market in the world. Secondly, we split up the data in order to find out which ones are the most represented elliptical families.

In our empirical analysis, we consider returns of the 500 stocks listed in the S&P 500 index. We consider those stocks listed in the index by December 2004; we have daily return data available for the period January 1990 to December 2004 from the Datastream database. A number of fit tests are carried out to model the empirical distributions of daily rates of return using distribution-fitting software Best Fit. Via a chi-square goodness-of-fit test, we estimate which one of 22 given statistical distributions best fits the empirical returns.

Results show both good and bad news. The good news is that the best fitting distribution of the daily rates of return turns out to be the (asymmetrical) log-logistic (58.40%), the second best the (elliptical) logistic (39.00%), whereas the normal fits well only for 1.40% of funds. It is worthwhile noting that if the daily rate is log-logistic, its equivalent continuous rate is logistic. So, ellipticity is found for daily rates in 40.40% of cases (logistic plus normal cases) and for continuous rates in another 58.60% of cases (log-logistic plus log-normal cases).

Let us turn to the bad news. As already mentioned, a key property for preserving mean-variance results is that returns composing the portfolio be ellipticals of the same family. Vice versa, no portfolio ellipticity is guaranteed if elliptical returns belong to different families or not all returns are ellipticals. In other terms, a portfolio composed of a logistic and a normal daily rate of return may be even asymmetrical. The same is true for a portfolio composed of a logistic and a (asymmetrical) log-logistic daily rate. Unfortunately, our findings have spotlighted that funds are spread into a number of different families. Consequently, to preserve standard CAPM results, the choice of the funds to insert into the portfolio must be confined to those sub-sets of funds that exhibit the same distribution.

Published online: 6 March 2009

JEL G10 * G11 * G12 * G15

M. Eling

Institute of Insurance Economics, University of St. Gallen, St. Gallen, Switzerland

e-mail: martin.eling@unisg.ch

L. Tibiletti (mail])

Dipartimento di Statistica e Matematica, Universita degli Studi di Torino, Torino, Italy

e-mail: luisa.tibiletti@unito.it

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