The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 482 Wednesday, February 4, 2004 1:09 PM

482 The Mathematics of Financial Modeling and Investment Management

Risk Premium in the CML

With homogeneous expectations, SD(RM) and SD(Rp) are the market’s

consensus for the expected return distributions for portfolio M and

portfolio p. The risk premium for the CML is

ER( M) – Rf

------------------------------ SD(Rp)

SD(RM)

Let’s examine the economic meaning of the risk premium.

The numerator of the first term is the expected return from investing

in the market beyond the risk-free return. It is a measure of the reward

for holding the risky market portfolio rather than the risk-free asset. The

denominator is the market risk of the market portfolio. Thus, the first

term measures the reward per unit of market risk. Since the CML repre-

sents the return offered to compensate for a perceived level of risk, each

point on the CML is a balanced market condition, or equilibrium. The

slope of the CML (i.e., the first term) determines the additional return

needed to compensate for a unit change in risk. That is why the slope of

the CML is also referred to as the equilibrium market price of risk.

The CML says that the expected return on a portfolio is equal to the

risk-free rate plus a risk premium equal to the market price of risk (as mea-

sured by the reward per unit of market risk) times the quantity of risk for the

portfolio (as measured by the standard deviation of the portfolio). That is,

ERp = Rf + market price of risk × quantity of risk

THE CML AND THE OPTIMAL PORTFOLIO

Given that the new efficient frontier is the CML, how does one select the

optimal portfolio? That is, how does one determine the optimal combi-

nation of the market portfolio and the risk-free asset in which to invest?

This depends on the preferences of the investors. To understand this, we

must introduce the notion of utility functions and indifference curves.

Utility Functions and Indifference Curves

In life there are many situations where entities (i.e., individuals and

firms) face two or more choices. The economic “theory of choice” uses

the concept of a utility function to describe the way entities make deci-

sions when faced with a set of choices. A utility function assigns a

(numeric) value to all possible choices faced by the entity. The utility