The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 491 Wednesday, February 4, 2004 1:09 PM

Portfolio Selection Using Mean-Variance Analysis 491

ity distribution of returns p(x) defined over the vector of returns r. For

each vector of portfolio weights wa the portfolio return will be wa ′r. The

portfolio’s utility will be a stochastic variable U with a pdf that can be

computed from the joint pdf of returns. For instance, if returns are

jointly normal, the portfolio pdf will be normal. The portfolio selection

problem is to maximize the expected value of this stochastic utility in

function of portfolio weights:

arg maxEU[ (rw, a )]

Portfolio optimization is a relatively mature technology, though its

formal implementation is not yet widespread in the industry. The prob-

lem is one of sensitivity to forecasts. Practitioners who have imple-

mented the optimization technology typically report a great sensitivity

of the optimization to forecast errors. Because the optimizer looks for

the best opportunities within the pdf that has been fed to it, any mistake

in the estimation of the pdf is magnified by the optimizer. This has led

some in the industry to refer to optimization as “error maximization.”^12

RELAXING THE ASSUMPTION OF NORMALITY

We can relax the assumption that returns are jointly normally distrib-

uted. It is a well known fact that returns are not normally distributed at

short-time horizons of the order of days. As we saw in Chapter 13, fat-

tailed distributions were proposed to represent returns at such short

time horizons. At the longer time horizons typical of portfolio manage-

ment, the assumption of normality is more plausible empirically speak-

ing. However, deviations from normality exist, either because of rare

large price movements or because of the importance of moments of

order higher than variance.

The general utility maximization framework discussed above is very

general and can be applied, in principle, to arbitrary distribution func-

tions provided that the maxima exist. Henrik Dahl, Alexander Meeraus,

and Stavros Zenios^13 argue that most financial engineering problems

can be cast into an optimization framework. However practical statisti-

cal and computational problems arise when there is the need to estimate

moments of high order in a multivariate environment. Extreme Value

(^12) Muller, “Empirical Tests of Biases in Equity Portfolio Optimization.”
(^13) Henrik Dahl, Alexander Meeraus, and Stavros Zenios, “Some Financial Optimi -
zation Models: I and II,” in Financial Optimization.