Deriving the Capital Market Line

To derive the Capital Market Line (CML), we begin with the efficient frontier. In the absence of a risk-free asset, Markowitz efficient portfolios can be constructed as a constrained minimum problem based on expected return and variance, with the optimal portfolio being the one portfolio selected based on the investor's preference (which later we will see is quantified by the investor's utility function). The efficient frontier changes, however, once a risk-free asset is introduced and assuming that investors can borrow and lend at the risk-free rate. This is illustrated in Exhibit 16.3.

Every combination of the risk-free asset and the efficient portfolio M, which we referred to as the tangency portfolio in the previous section, is shown on the line drawn from the vertical axis at the risk-free rate tangent to the Markowitz efficient frontier. All the portfolios on the line are feasible for the investor to construct. Portfolios to the left of portfolio M represent combinations of risky assets and the risk-free asset. Portfolios to the right of portfolio M include purchases of risky assets made with funds borrowed at the risk-free rate. Such a portfolio is called a leveraged portfolio because it involves the use of borrowed funds. The line from the risk-free rate that is tangent to the efficient frontier of risky assets is called the capital market line (CML).

Let's compare a portfolio on the CML to a portfolio on the Markowitz efficient frontier with the same risk in Exhibit 16.3. For

EXHIBIT 16.3 Capital Market Line and the Markowitz Efficient Frontier

Capital -market line

Markowitz ft,

Markowitz

SD(/y example, compare portfolio Pa, which is on the Markowitz efficient frontier, with portfolio Pg, 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 Pg than for Pa. By Assumption 2, a risk-averse investor will prefer Pg to Pa. That is, Pg 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 borrowing 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 portfolio 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. Therefore, 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:

This can be simplified as follows:

Earlier in this chapter we derived the variance of a portfolio containing only two assets. The variance of the portfolio consisting of the risk-free asset and portfolio M is

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 var(Rp) = wM var(RM)

In other words, the variance of the portfolio is represented by the weighted variance of portfolio M. We can solve for the weight of portfolio M by substituting standard deviations for variances. Since the standard deviation is the square root of the variance, we can write

and therefore wM =

If we substitute the above result and rearrange terms we get the CML: