Skewness, sampling risk, and the importance of diversification

Abstract

Recent papers have extended portfolio theory to include skewness along with mean return and variance to explain security preferences. Because the positive skewness typically present in individual assets such as stocks and call options is rapidly reduced through diversification, several authors have suggested that a preference for positive skewness can lead to antidiversification as investors attempt to capture the greatest amount of positive skew. These analyses ignore the sampling risk (the probability that the skewness of a particular portfolio of size n will be near its expected value) that exists when less than fully diversified portfolios are held. This analysis presents skewness statistics for three security populations over three market periods to illustrate the relationship between sampling risk and diversification. Small portfolios of stocks or call options are shown to possess greater expected skewness than larger portfolios for each sample period examined, but small portfolios also exhibit the greatest sampling risk for skewness. Depending upon an investor's skewness preference, some diversification in these assets appears appropriate to increase the confidence regarding the actual skewness that will be observed for a particular portfolio chosen. Expected skewness for covered call portfolios is negative for all portfolio sizes and sample periods and declines as portfolio size inreases. The elimination of sampling risk and the reduction of negative expected skewness can be accomplished through complete diversification in these assets.