## Ian J. Wright and Dr. Keith P Vorkink, Business Management

In order to make optimal portfolio choices, individuals, firms and portfolio managers are interested in the probabilities attached to their receiving various returns in the stock market. These probabilities of returns on securities are described by statistical functions called return distributions. However, much uncertainty concerning such return distributions exists and over the years many models have been created that attempt to predict the mean and variance of return distributions, and recently some have even been put forth that attempt to predict the skewness of such distributions. The skewness of return distributions is of import to investors in particular because the degree (positive or negative) of skewness the return distribution of a security exhibits is an indication of the chances an investor has to “hit the lottery” (in the case of positive skewness), or, on the other hand, “hit the cellar” (in the case of negative skewness) by investing in that security. If a particular stock has a positively (negatively) skewed return distribution, investors have a higher probability of receiving very large positive (negative) returns on that stock than they would have if the return distribution was not positively (negatively) skewed.

As alluded to above, much work has been done examining how the mean and variance of return distributions are affected by particular variables. However, relatively little work has been done that investigates the impact that firm variables such as volatility, size, book-to-market ratios, momentum and turnover have on the skewness of return distributions. Due to this fact, I decided to investigate, using a framework not previously used, whether or not particular firm variables in one period affect the return distribution skewness of those firms’ securities in future periods.

I did this by modeling a return distribution’s skewness as a linear function of a constant term and the firm characteristics of lagged skewness, momentum, turnover, size, volatility and book-to-market ratios. That is, I used a regression model using the degree of skewness exhibited by the return distribution of security i in period t as the dependent variable, and the other variables stated above observed in period t-1 for firm i as explanatory variables in my model, as below:

I then hypothesized stock returns were distributed using the “Skew-Normal Probability Distribution” given by Sahu, Dey and Branco (2003) because this distribution is one of the few multivariate distributions to allow for parameterization of skewness explicitly, while also parameterizing mean and variance as well. As a Bayesian statistical framework requires, I additionally specified prior distributions on each of the skewness parameters in equation (1) above, as well as prior distributions on the mean vector and covariance matrix parameters of the skew-normal distribution, reflecting my a priori beliefs that each variable had no impact upon skewness, so as not to bias the results obtained in any way, and instead let the data determine the posterior distributions of the parameters.

At this point, rather than using the approach of maximum likelihood posited by frequentist statisticians that produces point estimates, standard errors and confidence intervals for the estimates, I followed the approach of Bayesian statisticians (see Gelman, Carlin, Stern and Rubin, 2003), due to the predictive and distributional underpinnings of Bayesian statistics. After deriving complete conditionals for each of the parameters and using the Metropolis Hastings algorithm to sample from them, I obtained distributions predicting how each firm characteristic is related to skewness and a distribution predicting what the stock returns should have been distributed as, given my assumptions. Preliminary results under this framework indicate that return distributions generally exhibit positive skewness, with high levels of momentum predicting additional positive skewness. Larger firms and firms with larger book-to-market ratios are predicted to have securities with more negatively skewed return distributions than usual, and volatility does not appear to influence the skewness of return distributions. The effect of share turnover of a security on skewness has not yet been determined at this point.

Although I was able to obtain these preliminary results using data on 20-30 firms and fixing non-skewness parameters of the skew-normal distribution (such as the mean vector and covariance matrix parameters), computational issues arose when I attempted to use data on 300-400 firms (to get more robust results) and no longer fix the mean vector and covariance matrix parameters, but use their prior distributions. After using Monte Carlo integration techniques to estimate a particular integral involved in the skew-normal distribution, I determined it would take roughly three years for the necessary computation to run on the computer. After spending a number of months attempting to find ways to speed up this computation, a statistics professor specializing in Bayesian methods also began working on this issue. For many recent months we have been working on this and I await our resolving of the computational issues before I proceed to complete the final computations, estimations and representations of results. Hopefully this will be completed within the next one to two years.

Interim works, including one 7-page document quantitatively summarizing my efforts on this project so far and one 11-page document performing computations using a simplified distribution (see Hansen, McDonald and Turley, 2004) are available from the author upon request, as is the associated code used for computation.