The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 487 Wednesday, February 4, 2004 1:09 PM

Portfolio Selection Using Mean-Variance Analysis 487

Awa = c

and

wai ≥ bi

where the equation Awa = c constrains sector exposure. This is a qua-

dratic programming problem of the type described in Chapter 7.

In addition to the above, managers might want to impose turnover

or tradability constraints in the sense that assets can only be traded in

given lots. As observed in Chapter 7, these constraints result in a mixed-

integer programming problem, which is generally more difficult to solve

than quadratic programming problems.

The technology of optimization is presently available on desktop

computers. Mathematical software such as Matlab routinely solves qua-

dratic portfolio optimization problems of the type described above.

However special care is still needed in applying optimization technol-

ogy. In fact, optimization is sensitive to expected return forecasts that

are themselves typically unreliable.^11

A SECOND LOOK AT PORTFOLIO CHOICE

The mean-variance framework suggested by Markowitz is based on util-

ity functions defined on expected returns and variance. We now have to

generalize the optimization framework proposed by Markowitz in a

fully probabilistic setting. This generalization allows the consideration

of nonnormal distributions and paves the way for multiperiod portfolio

choice. The three key ingredients in a portfolio optimization methodol-

ogy are (1) a return forecast, (2) a utility function, and (3) an optimizer.

The Return Forecast

The return forecast has to be intended as a probabilistic forecast. This

means that models supply a joint pdf of all the assets that might contrib-

ute to forming the optimal portfolio. A return forecast implies a process

dynamics.

The first, and simplest, dynamics is the assumption that returns are

independent and identical normal (IIN) variables and, therefore, price

(^11) See, for example, Peter Muller, “Empirical Tests of Biases in Equity Portfolio Op-
timization,” in Stavros Zenios (ed.), Financial Optimization (Cambridge, MA, Cam-
bridge University Press, 1993).