The Mathematics of Financial Modelingand Investment Management

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82 The Mathematics of Financial Modeling and Investment Management

Markowitz was interested in the investment decision-making pro-

cess. Along the lines set forth by Pareto 60 years earlier, Markowitz

assumed that investors order their preferences according to a utility

index, with utility as a convex function that takes into account inves-

tors’ risk-return preferences. Markowitz assumed that stock returns are

jointly normal. As a consequence, the return of any portfolio is a nor-

mal distribution, which can be characterized by two parameters: the

mean and the variance. Utility functions are therefore defined on two

variables—mean and variance—and the Markowitz framework for

portfolio selection is commonly referred to as mean-variance analysis.

The mean and variance of portfolio returns are in turn a function of a

portfolio’s weights. Given the variance-covariance matrix, utility is a

function of portfolio weights. The investment decision-making process

involves maximizing utility in the space of portfolio weights.

After writing his seminal article, Markowitz joined the Rand Corpo-

ration, where he met George Dantzig. Dantzig introduced Markowitz to

computer-based optimization technology.^9 The latter was quick to appre-

ciate the role that computers would have in bringing mathematics to bear

on business problems. Optimization and simulation were on the way to

becoming the tools of the future, replacing the quest for closed-form solu-

tions of mathematical problems.

In the following years, Markowitz developed a full theory of the invest-

ment management process based on optimization. His optimization theory

had the merit of being applicable to practical problems, even outside of the

realm of finance. With the progressive diffusion of high-speed computers,

the practice of financial optimization has found broad application.^10

(^9) The inputs to the mean-variance analysis include expected returns, variance of re-
turns, and either covariance or correlation of returns between each pair of securities.
For example, an analysis that allows 200 securities as possible candidates for port-
folio selection requires 200 expected returns, 200 variances of return, and 19,900
correlations or covariances. An investment team tracking 200 securities may reason-
ably be expected to summarize their analyses in terms of 200 means and variances,
but it is clearly unreasonable for them to produce 19,900 carefully considered corre-
lation coefficients or covariances. It was clear to Markowitz that some kind of model
of the covariance structure was needed for the practical application of the model. He
did little more than point out the problem and suggest some possible models of co-
variance for research to large portfolios. In 1963, William Sharpe suggested the sin-
gle index market model as a proxy for the covariance structure of security returns
(“A Simplified Model for Portfolio Analysis,” Management Science (January 1963),
pp. 277–293).
(^10) In Chapter 16 we illustrate one application. For a more detailed discussion, see
Frank J. Fabozzi, Francis Gupta, and Harry M. Markowitz, “The Legacy of Modern
Portfolio Theory,” Journal of Investing (Summer 2002), pp. 7–22.