The ins and outs of full-scale optimization1
Page, senior managing director, State Street Associates / State
Street Global Markets
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.
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.
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
•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.
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.
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.