The Mathematics of Financial Modelingand Investment Management

17-Capital Asset Pricing Model Page 512 Wednesday, February 4, 2004 1:10 PM

512 The Mathematics of Financial Modeling and Investment Management

The subject of this chapter is the Capital Asset Pricing Model

(CAPM) formulated by William Sharpe, John Lintner, and Jan Mossin.^1

As explained in the previous chapter, portfolio selection based on mean-

variance analysis is a normative theory that describes the investment

behavior of market agents in constructing a portfolio. Given this invest-

ment behavior, the capital asset pricing model formalizes the relation-

ship that should exist between asset returns and risk.

CAPM ASSUMPTIONS

The CAPM is an equilibrium asset pricing model derived from a set of

assumptions. Here we demonstrate how the CAPM is derived.

The CAPM is an abstraction of the real world capital markets and,

as such, is based upon some assumptions. These assumptions simplify

matters a great deal, and some of them may even seem unrealistic. How-

ever, these assumptions make the CAPM more tractable from a mathe-

matical standpoint. The CAPM assumptions are as follows:

■ Assumption 1. Investors make investment decisions based on the

expected return and variance of returns.

■ Assumption 2. Investors are rational and risk averse.

■ Assumption 3. Investors subscribe to the Markowitz method of

portfolio diversification.

■ Assumption 4. Investors all invest for the same period of time.

■ Assumption 5. Investors have the same expectations about the

expected return and variance of all assets.

■ Assumption 6. There is a risk-free asset and investors can borrow

and lend any amount at the risk-free rate.

■ Assumption 7. Capital markets are completely competitive and fric-

tionless.

The first five assumptions deal with the way investors make deci-

sions. The last two assumptions relate to characteristics of the capital

market. Some of these assumptions require further explanation. As

explained in Chapter 16, in mean-variance analysis, it is assumed that

(^1) William F. Sharpe, “Capital Asset Prices,” Journal of Finance (September 1964),
pp. 425–442. Others who reached a similar conclusion regarding the pricing of risk
assets include: John Lintner, “The Valuation of Risk Assets and the Selection of
Risky Investments in Stock Portfolio and Capital Budgets,” Review of Economics
and Statistics (February 1965), pp. 13–37 and Jan Mossin, “Equilibrium in a Capital
Asset Market,” Econometrica (October 1966), pp. 768–783.