The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 488 Wednesday, February 4, 2004 1:09 PM

488 The Mathematics of Financial Modeling and Investment Management

processes are random walks. This assumption entails that the expected

returns of each asset are known constants. Later in this chapter we will

consider autoregressive linear models and nonlinear models that follow

a more complex dynamics than the assumption of IID variables.

The Utility Function

In the mean-variance framework, utility functions are defined on

expected returns and variances. The probability structure of returns is

summarized by returns and variances. Utility functions express the

trade-off between risk and return preferred by the investor or by the

asset manager. By choosing a utility function, an investor decides how

much return he or she wants to be compensated for taking more risk.

The choice of utility functions is dictated by (1) a question of mathemat-

ical and computational tractability and (2) the risk-return preferences of

the investor.

In the one-period framework of Markowitz, utility is a function of

two variables: mean and variance. In this way, the problem of portfolio

choice becomes a problem of finding the return-variance pair with the

maximum utility:

arg maxU( w ⁄ μμμμ, ΣΣΣΣ)

where “arg” is shorthand to denote “argument” and with the con-

straints

μ a = wa ′ μμμμ

w′ι = 1 , ιιιι′ = [ 11 ,,..., 1 ]

This is a problem of constrained maximum. Additional constraints

might be imposed, for instance, that weights are all positive and/or that

weights are within given intervals. The first condition precludes short

selling; the second condition ensures that no asset has a weight either

too big or too small.

In a more general probabilistic setting, utility functions are defined on

the variables of interest, be they returns or consumption. The investor’s risk

preference is represented by the shape of the utility function. A linear func-

tion corresponds to risk neutrality. A concave function, that is, a function

with negative second derivative, expresses risk aversion in so far as utility

grows less rapidly than the variable.

A formal measure of absolute risk aversion is defined as