The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 474 Wednesday, February 4, 2004 1:09 PM

474 The Mathematics of Financial Modeling and Investment Management

EXHIBIT 16.1 The MPT Investment Process

Source: Exhibit 2 in Frank J. Fabozzi, Francis Gupta, and Harry M. Markowitz,

“The Legacy of Modern Portfolio Theory,” Journal of Investing (Fall 2002), p. 8.

constraints) it is possible to perform an optimization that results in the

risk-return or mean-variance efficient frontier.^3 This frontier is efficient

because underlying every point on this frontier is a portfolio that results

in the greatest possible return for that level of risk, or results in the

smallest possible risk for that level of return. The portfolios that lie on

the frontier make up the set of efficient portfolios.

When the efficient frontier is constructed using the M-V formula-

tion developed by Markowitz, they are referred to as Markowitz effi-

cient portfolios and the set or frontier of these portfolios is called the

Markowitz efficient frontier. Exhibit 16.2 provides a graphical depiction

of the Markowitz efficient frontier based on the feasible portfolios that

can be constructed. The Markowitz efficient frontier is the upper por-

tion of the curve from II to III.

MARKOWITZ’S MEAN-VARIANCE ANALYSIS

Let’s now place the above in a formal mathematical context developing

the analysis of mean-variance optimization. Suppose first that an inves-

tor has to choose a portfolio formed of N risky assets. The investor’s

choice is embodied in an N-vector w = {wi} of weights where each

weight i represents the percentage of the i-th asset held in the portfolio.

Suppose assets’ returns are jointly normally distributed with an N-vec-

(^3) In practice this optimization is performed using an off-the-shelf asset allocation
package.