The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 489 Wednesday, February 4, 2004 1:09 PM

Portfolio Selection Using Mean-Variance Analysis 489

rA()x = –U′′()x ⁄U′() x

This measure expresses the intuitive fact that the more the utility func-

tion is curved, the more the investor is risk-averse. Listed below are

some examples of utility functions:

■ Linear utility function:

Ux()= abx + , U′()x = b , U′′()x = 0

The linear function is not concave; it represents a risk-neutral investor.

■ Power utility function:

1 – a

Ux()= ---------------------x –^1 - , U′()x = x –a , U′′()x = –ax – a –^1 < 0

1 – a

The power utility function is concave; it represents a risk-averse investor.

■ Logarithmic utility function:

Ux()= ln ()x , U′()x = 1 ⁄x , U′′()x = – 1 ⁄x^2 < 0

The logarithmic utility function is concave; it represents a risk-averse

investor.

■ Quadratic utility function:

Ux()= abx + – ---x^2 , U′()= bcx , U′′()= –c < 0

c

x – x

2

The quadratic utility function is concave; it represents a risk-averse

investor.

In a probabilistic setting, the utility function is a monotone function

of a random variable and is, therefore, a random variable itself. To opti-

mize, one single utility number must be defined for each portfolio

choice. Utility is therefore defined as the expected value of stochastic

utility: