The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 477 Wednesday, February 4, 2004 1:09 PM

Portfolio Selection Using Mean-Variance Analysis 477

Consider the mean-variance plane, that is, a two-dimensional Carte-

sian plane whose coordinates are mean and variance. In this plane, each

portfolio is represented by a point. Consider now the set of all efficient

portfolios with all possible efficient mean-variance pairs. This set is

what we referred to earlier as the efficient frontier. Later in this chapter

we show actual efficient frontiers.

CAPITAL MARKET LINE

As demonstrated by William Sharpe,^4 James Tobin,^5 and John Lintner^6

the efficient set of portfolios available to investors who employ M-V anal-

ysis in the absence of a risk-free asset is inferior to that available when

there is a risk-free asset.^7 We present this formulation in this section.^8

Assume a risk-free asset with a risk-free return denoted by Rf. The

investor has to choose a combination of the N risky assets plus the risk-

free asset. The weights wR = {wi}R do not have to sum to 1 as the remain-

ing part (1 – wR′ι ) can be invested in the risk-free asset. Note that this

portion of investment can be positive or negative if we allow risk-free

borrowing and lending. The portfolio’s expected return and variance are:

μa = wR ′μμμμ+ ( 1 – wR ′ιιι)Rf

σ

2

a = wR ′ΣΣΣΣwR

The portfolio variance is the same expression as before because the

risk-free asset has zero variance and zero covariances with the risky assets.

(^4) William F. Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium Under
Conditions of Risk,” Journal of Finance (September 1964), pp. 425–442.
(^5) James Tobin, “Liquidity Preference as a Behavior Towards Risk,” Review of Eco-
nomic Studies (February 1958), pp. 65–86.
(^6) John Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments
in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics (Feb-
ruary 1965), pp. 13–37.
(^7) The portfolio selection model was further extended by Fischer Black in the case of
a restriction on short selling. See “Capital Market Equilibrium with Restricted Bor-
rowings,” Journal of Business (July 1972), pp. 444–455.
(^8) For a comprehensive discussion of these models and computational issues, see Har-
ry M. Markowitz (with a chapter and program by Peter Todd), Mean-Variance Anal-
ysis in Portfolio Choice and Capital Markets (New Hope, PA: Frank J. Fabozzi
Associates, 2000, originally published in 1987).