The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 480 Wednesday, February 4, 2004 1:09 PM

480 The Mathematics of Financial Modeling and Investment Management

example, compare portfolio PA, which is on the Markowitz efficient

frontier, with portfolio PB, which is on the CML and therefore some

combination of the risk-free asset and the efficient portfolio M. Notice

that for the same risk the expected return is greater for PB than for PA.

By Assumption 2, a risk-averse investor will prefer PB to PA. That is, PB

will dominate PA. In fact, this is true for all but one portfolio on the

CML, portfolio M, which is on the Markowitz efficient frontier. With

the introduction of the risk-free asset, we can now say that an investor

will select a portfolio on the CML that represents a combination of bor-

rowing or lending at the risk-free rate and the efficient portfolio M.

We can derive a formula for the CML algebraically. Based on the

assumption of homogeneous expectations regarding the inputs in the

portfolio construction process, all investors can create an efficient port-

folio consisting of wf placed in the risk-free asset and wM in the tan-

gency portfolio, portfolio M, where w represents the corresponding

percentage (weight) of the portfolio allocated to each asset.

Thus, wf + wM = 1 or wf = 1 – wm. The expected return is equal to

the weighted average of the expected returns of the two assets. There-

fore, the expected portfolio return, E(Rp), is equal to

E(Rp) = wf Rf + wM E(RM)

Since we know that wf = 1 – wM, we can rewrite E(Rp) as follows:

E(Rp) = (1 − wM) Rf + wM E(RM)

This can be simplified as follows:

E(Rp) = Rf + wM [E(RM) − Rf]

Earlier in this chapter we derived the variance of a portfolio con-

taining only two assets. The variance of the portfolio consisting of the

risk-free asset and portfolio M is

2 2

var(Rp) = wf var(Rf) + wM var(RM) + 2wf wM cov(Rf , RM)

We know that the variance of the risk-free asset, var(Rf), is equal to

zero. This is because there is no possible variation in the return since the

future return is known. The covariance between the risk-free asset and

portfolio M, cov(Rf,RM), is zero. This is because the risk-free asset has

no variability and therefore does not move at all with the return on

portfolio M which is a risky portfolio. Substituting these two values into

the formula for the portfolio’s variance, we get