104 Optimizing Optimization
For the general n-return normal series case, d t is distributed as a Chi-Square
distribution with n degrees of freedom. Under this assumption, if we define
an outlier as falling beyond the outer 25% of the distribution and we have
12 return series, our tolerance boundary is a Chi-Square score of 14.84. Using
Equation (4.1) , we calculate the Chi-Square score for each vector in our series.
If the observed score is greater than 14.84, that vector is an outlier.
4.4.3 Summary
Returns are not typically generated by a single distribution or regime, but
instead by distributions associated with quiet and turbulent regimes. The risk
parameters that prevail in these separate regimes are often dramatically differ-
ent than their full sample averages. The WPA includes a procedure for parti-
tioning historical returns into quiet and turbulent subsamples, which can be
used to provide better guidance as to a portfolio’s exposure to loss during tur-
bulent episodes, and to structure portfolios that are more resilient to turbu-
lence whenever it may occur.
4.5 Full -scale optimization
4.5.1 The problem
Many assets and portfolios have return distributions that display significantly
nonnormal 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.
4.5.2 The WPA solution
The WPA employs a computationally efficient algorithm called full-scale opti-
mization that enables investors to derive optimal portfolios from the full set
of returns given a broad range of realistic investor preference functions. The
full-scale approach calculates a portfolio’s utility for every period in the return
sample considering as many asset mixes as necessary in order to identify the
weights that yield the highest expected utility.^6
Suppose , for example, the aim is to find the optimal blend between two
funds whose returns are displayed in Exhibit 10, assuming the investor has
log-wealth utility. Log-wealth utility assumes utility is equal to the natural
logarithm of wealth, and resides in a broader family of utility functions called
power utility. Power utility defines utility as [(1/ γ ) Wealth^ γ^ ]. A log-wealth
utility function is a special case of power utility. As γ approaches 0, utility
6 See Cremers et al (2005) for a discussion of this methodology.