5.5Finding the Efficient Frontierof Risky Assets

One can reasonably argue that no risk-free asset exists. While many default-free secu-

rities such as U.S. Treasury bills are available to investors, even a one-month U.S. T-bill

fluctuates in value unpredictably from day to day. Thus, when the investment horizon

is shorter than a month, this asset is definitely not “risk-free.” In addition, foreign

investors would not consider the U.S. Treasury bill a risk-free asset. An Italian investor,

for example, views the certain dollar payoff at the maturity of the T-bill as risky because

it must be translated into Italian lire at an uncertain exchange rate. Even to a U.S.

investor with a one-month horizon, the purchasing power of an asset, not just its nom-

inal value, is critical. Thus, the inflation-adjusted returns of Treasury bills are risky,

even when calculated to maturity. Also, in many settings there may be no risk-free asset.

For example, a variety of investment and corporate finance problems preclude invest-

ment in a risk-free asset. For these reasons, it is useful to learn how to compute all of

the portfolios on the (hyperbolic-shaped) boundary of the feasible set of risky invest-

ments, detailed in Exhibit 5.1. This section uses the insights from Section 5.4 to find

this boundary.

⁹Section 5.7 develops more intuition for equation (5.2) and (the equivalent) Result 5.3.

Grinblatt307Titman: Financial	II. Valuing Financial Assets	5. Mean307Variance Analysis	© The McGraw307Hill
Markets and Corporate		and the Capital Asset	Companies, 2002
Strategy, Second Edition		Pricing Model

144Part IIValuing Financial Assets

Because of two-fund separation, the identification of any two portfolios on the

boundary is enough to construct the entire set of risky portfolios that minimize variance

for a given mean return. Use the minimum variance portfolio of the risky assets as one

of the two portfolios since computing its weights is so easy. For the other portfolio, note

that (with one exception)¹⁰it is possible to draw a tangent line from every point on the

vertical axis of the mean-standard deviation diagram to the hyperbolic boundary. We will

refer to the point of tangency as the “hypothetical tangency portfolio.” Hence:

1.Select any return that is less than the expected return of the minimum-

variance portfolio.

2.Compute the hypothetical tangency portfolio by pretending that the return in

step 1 is the risk-free return, even if a risk-free asset does not exist.

3.Take weighted averages of the minimum variance portfolio and the

hypothetical tangency portfolio found in step 2 to generate the entire set of

mean-variance efficient portfolios. The weight on the minimum variance

portfolio must be less than 1 to be on the top half of the hyperbolic boundary.

Example 5.5 illustrates this three-step technique.

Example 5.5:Finding the Efficient FrontierWhen No Risk-Free Asset Exists

Find the portfolios on the efficient frontier constructed from investment in the three fran-

chising projects in Example 5.3 (and Example 4.19).

Answer:Solve for the portfolio “weights”that make the portfolio’s covariance with each

stock equal to the stock’s risk premium.(In this example, “risk premium”refers to the

expected return less some hypothetical return that you select.) Then, rescale the weights so

that they sum to one.If the hypothetical return is 6 percent, the weights (unscaled) are given

by the simultaneous solution of the three equations

x.001x0x .15.06

.002

12 3

.001x.002x.001x .17.06

123

x.001x.002x .17.06

0

123

The solution to these equations, when rescaled, generate portfolio weights of

x .4 x .1 x .5

123

(Not surprisingly, with a hypothetical risk-free return in this example that is identical to the

risk-free return in Example 5.3, the weights in the two examples, .4, .1, and .5, match.Alter-

natively, instead of subtracting .06 from the expected returns on the right-hand side of the

first three equations, you could have subtracted other numbers (for example, .04 or zero).If

.04 had been used in lieu of .06, the right-hand side of the first three equations would be

.11, .13, and .13, respectively, instead of .09, .11, and .11.If zero had been used, the right-

hand side would be .15, .17, and .17, respectively.)

For the other portfolio, use the minimum variance portfolio (which was computed in Exam-

ple 4.19 to be)

x .5 x 0 x .5

123

Thus, the portfolios on the boundary of the feasible set are described by

x .4w.5(1w)

1

x .1w

2

x .5w.5(1w) .5

3

Those with w 0 are on the top half of the boundary and are mean-variance efficient.

¹⁰The

exception is at the expected return of the minimum variance portfolio, point Vin Exhibit 5.2.

Grinblatt309Titman: Financial	II. Valuing Financial Assets	5. Mean309Variance Analysis	© The McGraw309Hill
Markets and Corporate		and the Capital Asset	Companies, 2002
Strategy, Second Edition		Pricing Model

Chapter 5

Mean-Variance Analysis and the Capital Asset Pricing Model

145

Since the financial markets contain numerous risky investments available to form

a portfolio, finding the efficient frontier of risky investments in realistic settings is best

left to a computer. Examples in this chapter, like the one above, which are simplified

so that these calculations can be performed by hand, illustrate basic principles that you

can apply to solve more realistic problems.¹¹