A Portfolio Performance Index

Fund managers may sensibly be averse to earning a time-averaged portfolio return that is less than the average return of some designated benchmark. When a portfolio is expected to earn a higher average return than the benchmark return, the probability that it will not approaches zero asymptotically at a computable exponential decay rate. The probability decay rate is thus proposed here as a new portfolio “performance index.” In the widely analyzed special case in which returns are normally distributed, the new performance-index-maximizing portfolio is the same as the popular Sharpe-ratio-maximizing portfolio. The results of the two approaches generally differ, however, because of nonnormal levels of skewness and/or kurtosis in the portfolio attributable to large asymmetrical economic shocks or investments in options and other derivative securities. An illustrative example will show that the new index is easy to implement and, consistent with empirical evidence on portfolio choice, favors investments with positively skewed returns. Analysts and portfolio managers find important uses for conventional portfolio optimization tools. The estimated mean-variance-efficient frontier and the Sharpe-ratio-maximizing portfolio on it provide some information about the risk-return trade-offs inherent in asset allocation and in decisions concerning specific asset classes—for example, whether currency exposures should be hedged. However, many problems limit the practical applicability of these tools. Two of those problems may be remedied by use of the “performance index” method suggested in this article.One problem is that the mean-variance tools are designed to aid decisions about assets whose returns are normally distributed, but relevant nonnormalities arise in asset returns from at least two sources. First, a well-documented finding is that equity portfolio returns are often negatively skewed because of sporadic market crashes. Second, the application of option-like strategies introduces nonnormalities.Another problem is that mean-variance analytical tools must be modified to explicitly incorporate the influence of quantitative benchmarks that pension or endowment funds give to their portfolio managers. Such benchmarks are pervasive in the investment management industry today.The approaches that have been devised to cope with nonnormality in the past are either inherently ad hoc or require additional information that is not readily available. An example of an ad hoc approach is the use of the semivariance in place of the variance. An example of an additional requirement is the values of utility function coefficients needed to calculate a portfolio's expected utility. In addition to requiring utility function coefficients, the expected utility approach has never been given a concrete statistical interpretation.I develop an extension of modern portfolio theory that addresses these problems. It allows managers to specify a benchmark and then estimate the effect of various asset allocations on the probability that they will underperform the designated benchmark's average performance over the long span of time relevant to pension and endowment funds. The manager can then identify the allocation that minimizes the probability of underperformance and, therefore, maximizes the probability of outperforming the benchmark's average return.When portfolio returns are normally distributed and the designated benchmark is the riskless rate of interest, this new performance index ranks portfolios in the same order as the Sharpe ratio. When returns are not normally distributed, the performance index provides a different ranking that favorably weights positively skewed returns. The new ranking is equivalent to that given by a benchmark-modified expected utility index, with a specific and easily estimated risk-aversion coefficient. Thus, the performance index provides a concrete statistical interpretation for what is usually considered to be a subjective expected utility index.The article contains an example consisting of 23 randomly chosen stocks that demonstrates that the performance index, in contrast to the Sharpe ratio, favors positively skewed returns. The example illustrates how the performance index can be implemented by using garden-variety spreadsheet calculations.