PRINCIPLES BOX: DEFINITION OF THE EFFICIENT FRONTIER THEOREM
6.5 INTRODUCING A RISK-FREE ASSET
Up to this point we have been dealing with portfolios that contain only risky assets.
Every candidate asset has positive variance, σ2 >0. If we assume the existence
of the risk-free asset, σ2 =0, that one small additional assumption introduces the possibilities of borrowing and lending at a risk-free interest rate. For example, borrowing at a risk-free interest rate might involve going to an FDIC-insured bank and taking out a short-term loan at a low interest rate. Borrowing can also be accomplished by selling a risk-free asset short and collecting the cash proceeds from the short sale, while delivering aborrowedrisk-free asset to the buyer. Short sales are more complex beause they involve selling something not owned. Lending at some risk-free interest rate can be as simple as going to an FDIC-insured bank and making a deposit at some low fixed rate of interest. Making the bank deposit is economically equivalent to investing in a riskless asset and holding it in a long position.
Consider what happens when a risk-free security is held in a long position in conjunction with a long position in a risky asset. Denote the proportions held in the risky asset and the risk-free asset asw1 and 1−w1, respectively. The expected return of this portfolio is defined by the unambiguously linear equation (6.9).
E rp
= 1−w1
rf+w1E r1
(6.9) whererf denotes the risk-free rate of return andE
r1
denotes the expected return on the risky asset 1. The variance of this two-asset portfolio is defined by equation (6.10).
σp2=w21σ12+ 1−w21
σf2+2w1 1−w1
σ1f (6.10)
whereσf2is the variance of the risk-free asset’s rate of return, andσ1fis the covariance between returns of the risky asset and the risk-free asset. By definition,σf2=0. The covariance also equals zero, σ1f =0, because there is no association (correlation) between a series of constant (riskless) returns and a series of fluctuating (risky) returns. The preceding sentence also follows from the fact (shown in Theorem A6 of the appendix at the end of this book) that the correlation (or covariance) of a constant (zero variance asset) with a random variable (positive variance) is zero.
Reconsidering equation (6.10) in view of these facts allows us to equivalently rewrite equation (6.10) to become (6.10a) below.
σp2=w21σ12+
1−w12
σf2+2w1 1−w1
σ1f (6.10)
=w21σ12+
1−w12
(0)+2w1 1−w1
(0)
σp2=w21σ12 (6.10a)
Taking the square root of equation (6.10a) reduces it to the simple linear equation (6.11).
σp=w1σ1 (6.11)
Solving equation (6.11) forw1 yields w1=σp/σ1. Substituting this expression above forw1into equation (6.9) and simplifying results in
E rp
=rf + E
r1
−rf σ1
σp (6.12)
Lending
Borrowing Security 1
sp s1
E (rp)
rf E (r1)
FIGURE 6.9 Opportunity Line from Combining a Risky Asset with the Risk-free Asset
Equation (6.12) shows that E rp
is linearly related to σp. This linear rela- tionship is depicted in Figure 6.9. These examples generalize to mean that any portfolio formed from the risk-free asset and any risky asset (like security 1) will lie on a straight line connecting those two points, the exact location depending on the weights in the two securities. If w1 = 0, then from equations (6.9) and (6.11) it can be seen that σp=0 and E
r1
=rf. The resulting portfolio is the risk-free asset itself. Similarly, if w1=0, then σp=σ1 and E
rp
=E r1
, which is risky asset 1. Assuming that rf <E
r1
, if 0<w1<1, then 0< σp< σ1 and rf <E
rp
<E r1
.
Depositing some money in an FDIC-insured bank and some more money in risky security 1 results in a two-asset portfolio located somewhere on the solid line segment between rf and asset 1 in Figure 6.9. This line segment is labeled lending, because wf >0, where wf =1−w1. Next, assume rf <E
r1
, ifw1>1, thenσp> σ1andE
rp
>E r1
. This corresponds to a point located somewhere on the dotted line segment above asset 1 in Figure 6.9, which is labeled borrowing, becausewf <0. The more borrowing that is done, the farther out on this dashed line segment the investor’s portfolio will lie.
Because asset 1 can be viewed as a portfolio just as easily as a single asset, this analysis can be extended to a general case of combining any risky portfolio in the opportunity set with the risk-free asset. In other words, combining the risk-free asset with any risky portfolio will result in a new portfolio somewhere on the straight line connecting the two. Next, consider how the efficient frontier is changed when the risk-free asset is introduced into an opportunity set comprised of risky assets.
Figure 6.10 shows the combination of the risk-free asset with various risky assets in the opportunity set. Combinations of portfolioBwith the risk-free asset along the line rfBare more desirable than (dominant) combinations of portfolioA with the risk-free asset along the linerfA, because portfolios along the linerfBhave the higher expected returns in the same risk class than portfolios along the linerfA. However, combinations of portfolioBwith the risk-free asset along the linerfBare dominated
A B
H
m
Capital Market Line (CML)
s E (r)
rf
FIGURE 6.10 Combining the Risk-free Asset with Various Risky Portfolios
by combinations of portfoliomwith the risk-free asset along the linerfmH. In fact, in Figure 6.10 the portfolios along the linerfmHhave the highest returns at every level of risk.
Within Figure 6.10, any ray out of the riskless rate,rf, that has a smaller slope than the tangency ray (linerfmH) will be dominated by the tangency ray because it represents a less favorable risk-return trade-off than the rayrfmH. Any ray with a slope greater than the tangency ray is infeasible, because such a line would lie entirely outside the opportunity set of assets that exist. Thus, over all the levels of risk illustrated in Figure 6.10, the tangency ray (line rfmH) represents dominant portfolios.
The result is that the new efficient frontier is a straight line where all efficient portfolios are simply linear combinations of the risk-free asset and the tangency portfoliom.5 Because this efficient frontier is a straight line, it can be described by the linear equation (6.13) where m is the tangency portfolio [the role previously assumed by security 1 in both equation (6.12) and in Figure 6.9].
E rp
=rf +
μm−rf σm
σp (6.13)
In this case, and in general, efficient portfolios betweenrfandminvolve lending at the risk-free interest rate, and efficient portfolios above m involve borrowing at the risk-free interest rate. This new linear efficient frontier is called the capital market line(CML).
Given a linear efficient frontier and convex indifference curves for a risk- averse investor, the efficient frontier theorem still holds and the investor optimally will choose a portfolio on this new linear efficient frontier. Figure 6.11 displays the portfolio chosen by a risk-averse investor. This investor chooses portfolio
m I2 I3
O*
I1 E (r)
s
FIGURE 6.11 Identifying the Optimal Port- folio on the Straight Line Efficient Frontier
H
m m′
E (r)
s rf
r′f
FIGURE 6.12 Tangency Portfolios with Two Different Risk-free Interest Rates
O* because it will give the maximum utility, I2. This optimal portfolio O* in Figure 6.11 involves positive investments in both the risk-free asset and the tangency portfoliom.
The tangency portfolios,min Figures (6.10) and (6.11), change their location as the level of the risk-free interest rate changes. As seen in Figure 6.12, as the risk-free rate of return increases fromrf torf, the tangency portfolio moves fromm tom. Figure 6.12 also represents the shape of the efficient frontier when investors borrowing at the risk-free interest rate r,f pay an interest rate that exceeds the risk-free lending raterf. If investors can lend atrfbut not borrow at this low interest rate, the efficient frontier becomes the discontinuous linerfmm(first linear and then curved). If the investor can lend at the lower raterf but must borrow at the higher raterf, the efficient frontier becomes rfmmH (linear, curved, and linear with less slope).