Mean Time
The ins and outs of full-scale optimization¹

Sébastien Page, senior managing director, State Street Associates / State Street Global Markets

Many assets and portfolios have return distributions that display significantly non-normal skewness and kurtosis. Hedge fund returns, for example, are often negatively skewed and fat-tailed. Because mean-variance optimization ignores skewness and kurtosis, it misallocates assets for investors who are sensitive to these features of return distributions. Computational efficiency now allows us to perform full-scale optimization as an alternative to mean-variance optimization. With this approach we calculate a portfolio’s utility for every period in our sample, considering as many asset mixes as necessary in order to identify the weights that yield the highest expected utility, given a variety of investor preferences.

Investor behaviour
Full-scale optimization is especially appealing for investors who embrace behavioural finance, which holds that investors avoid risk when faced with gains but seek risk when faced with losses. These preferences are represented graphically by an S-shaped function. For investors with S-shaped preferences, skewness and kurtosis really matter, which invalidates the use of mean-variance optimization. Full-scale optimization is also appealing to investors who face specific loss thresholds. A bilinear utility function changes abruptly at a particular wealth or return level and is relevant for investors who are concerned with breaching a threshold. Consider, for example, a situation in which an investor requires a minimum level of wealth to maintain a certain standard of living. The investor’s lifestyle might change drastically if she penetrates this threshold. Or she may be faced with a situation in which she will become insolvent if her wealth breaches some threshold, or a particular decline in wealth may breach a covenant on a loan. In these and similar situations, a bilinear utility function is more likely to describe one’s attitude toward risk.

Full-scale optimization implicitly takes into account all the features of the return distribution, including skewness, kurtosis, and any other return peculiarities. Whereas mean-variance optimization yields an approximate optimal portfolio based only on mean and variance, full-scale optimization identifies the truly optimal portfolio based on the entire return distribution. Our analysis yields the following insights:
•Mean-variance optimization performs extremely well for investors with log wealth utility. This result prevails even though the distributions of the component hedge fund returns are significantly non-normal.
•Moreover, much of this non-normality survives into the mean-variance efficient portfolios, which implies that log wealth utility is fairly insensitive to higher moments.
•Mean-variance optimization performs poorly for investors with bilinear utility or S-shaped value functions. Full-scale optimization reduces kurtosis and negative skewness for investors with bilinear utility, and it maximizes the success rate for investors with S-shaped value functions.
•Negative skewness is not problematic, given S-shaped value functions, because it raises the success rate and because extreme negative results are not severely penalized.
•Kurtosis is also not problematic for investors with S-shaped value functions, again because these functions do not severely penalize extreme negative results.

Different parameters
These insights, of course, depend on the particular parameters chosen for the bilinear utility functions and S-shaped value functions. Different parameters may lead to different conclusions about the sensitivity of utility to skewness and kurtosis. In any event, it is safe to assume that mean-variance optimization is suitable for investors with log wealth utility and other variations of power utility, even when allocating among funds with significantly non-normal distributions. However, we strongly recommend that investors with bilinear utility or S-shaped value functions employ full-scale optimization when forming portfolios that include hedge funds or other assets with non-normal distributions.

The bottom line is that full-scale optimization is more effective than mean-variance optimization for identifying optimal allocations of assets whose returns display non-normal features for investors who are sensitive to skewness and kurtosis.

Endnote:
1. This presentation summary is comprised of abstracts from a product profile, and from the article “Optimal Hedge Fund Allocations,” by Mark Kritzman, Jan-Hein Cremers, and Sébastien Page, which was published in the Journal of Portfolio Management, Spring 2005.