Now suppose that there are two risky assets available for investment, as in Section 4.4, plus a risk-free asset, as discussed in the previous section. Consider risky assets with returnsR1andR2, respectively. Forj= 1,2, letμj = E(Rj) and σ2j = Var(Rj) and let ρ12 denote the correlation of R1, R2; assume that
|ρ12|<1. LetRf denote the return on the risk-free asset, and letμf = E(Rf);
recall that, by definition, Var(Rf) = 0.
A portfolio consisting of the two risky assets together with the risk-free asset has a return of the form
w1R1+w2R2+wfRf (4.4) where w1+w2+wf = 1; assume that wf = 1 so that the portfolio contains one or both of the risky assets. Note that we can write (4.4) as
(1−wf)( ¯w1R1+ ¯w2R2) +wfRf where ¯wj=wj/(1−wf),j= 1,2. Hence, ¯w1+ ¯w2= 1.
That is, it is convenient to view the portfolio selection problem as having two stages. In the first stage, we construct a portfolio of the two risky assets, with weights ¯w1and ¯w2; in the second stage, that portfolio is combined with the risk-free asset by choosing the value ofwf. However, when choosing ¯w1 and ¯w2, it is important to keep in mind that the portfolio of the risky assets will be combined with a risk-free asset.
Including the possibility of investing in a risk-free asset has a large effect on the portfolio selection problem. Because we may always decrease risk by investing in the risk-free asset, there is a sense in which the risk in the portfolio of risky assets becomes less important.
LetRpdenote the return on the portfolio of risky assets; once the portfolio of risky assets has been selected, we are effectively back to the case of one risky asset together with the risk-free asset.
The return on the entire portfolio may be written (1−wf)Rp+wfRf.
This portfolio has expected return (1−wf)μp+wfμf, whereμp= E(Rp) and return variance
(1−wf)2σ2p.
For wf ≥0, a plot of the expected return of the portfolio versus its risk is simply a line segment starting at (0,μf) and passing through (σp,μp), similar to the efficient frontier in Example 4.11.
Example 4.13 Suppose two risky assets have returns with meansμ1= 0.05 and μ2= 0.15, respectively, standard deviations σ1=σ2= 0.25, and corre- lation ρ12= 0.125 and suppose that there is a risk-free asset with expected return μf = 0.025. Figure 4.6 gives a plot of the efficient frontier (the solid line) for portfolios of the two risky assets. Thus, in the first stage of the port- folio selection problem, we choose a portfolio from among those corresponding to the points on the efficient frontier.
In choosing such a portfolio, the fact that the portfolio will be combined with the risk-free asset plays an important role. Figure 4.7 contains a plot of the risk-expected-return pairs of the portfolio, consisting of the two risky assets plus the risk-free asset, for a particular choice of a portfolio of risky assets.
0 0.1 0.2 0.3 0.4 0.5 0.6
0 0.05 0.10 0.15 0.20
Risk
Expected return
FIGURE 4.6
Efficient frontier for portfolios of the two risky assets in Example 4.13.
(0, μf)
0 0.1 0.2 0.3 0.4 0.5 0.6
0 0.05 0.10 0.15 0.20
Risk
Expected return
FIGURE 4.7
Risk, mean–return pairs corresponding to a particular portfolio of risky assets in Example 4.13.
This plot illustrates a number of points regarding combining a portfolio of risky assets with a risk-free asset.
• The line segment connecting (0,μf) and the point on the curve corre- sponds to portfolios placing weightwf, 0≤wf ≤1, on the risk-free asset and weight 1−wfon the portfolio of risky assets corresponding to the point on the curve.
• The dashed line segment extending beyond the curve corresponds to portfolios withwf<0, for which the investor borrows at the risk-free rate in order to purchase the portfolio of risky assets. Note that such a portfolio has a smaller expected return than does a portfolio of risky assets alone with the same risk; such portfolios correspond to the points on the solid curve that lies above the dotted line segment.
• Thus, the portfolio problem consists of choosing a point on the curve (i.e., choosing a portfolio of risky assets) together with choosing a point on the half-line that starts at (0,μf) and passes through the
point on the curve (i.e., choosingwf).
Let (σp,μp) denote the risk, mean–return pair for a portfolio of risky assets; for example, in Figure 4.7, (σp,μp) is a point on the curve. Because all half-lines starting at (0,μf) and passing through (σp,μp) have the same starting point, the different possible half-lines may be described by their slopes,
μp−μf
σp .
0 0.1 0.2 0.3 0.4 0.5 0.6 0
0.05 0.10 0.15 0.20
Risk
Expected return
FIGURE 4.8
Comparison of two portfolios of risky assets in Example 4.13.
Therefore, choosing between two risky portfolios is essentially choosing between two possible slopes for such lines; see Figure 4.8 for such a comparison in the context of Example 4.13.
Note that the half-line with the larger slope (the dashed line) is preferred for any level of risk; that is, for any desired level of risk, the portfolio corre- sponding to the larger slope has a greater expected return. This suggests that the portfolio of risky assets with the value of (σp,μp) that yields the largest possible slope for the line connecting (0,μf) and (σp,μp) is optimal.
Sharpe Ratio
Consider a portfolio of risky assets with expected returnμpand riskσp. The slope of the line connecting (0,μf) and (σp,μp) is known as theSharpe ratio of the portfolio; hence, the Sharpe ratio is given by
SR = μp−μf
σp .
The Sharpe ratio has a useful interpretation as the expected excess return on the portfolio per unit of risk.
Note that the portfolio giving weightwf to the risk-free asset and weight 1−wf to the portfolio of risky assets has expected return
(1−wf)μp+wfμf
and return standard deviation|1−wf|σp.
Suppose that an investor would like to choose wf so that the expected return on the portfolio consisting of the two risky assets together with the risk-free asset ismfor some valuem >μf. This may be achieved by choosing wf to solve
(1−wf)μp+wfμf =m wf = μp−m
μp−μf
;
note here we may assume thatμp>μf because, otherwise, there would be no reason to invest in the portfolio of risky assets and that 0≤wf <1 provided thatμf < m≤μp. If m >μp, thenwf <0, indicating that the investor will need to borrow capital at the risk-free rate in order to attain an expected return ofm.
The resulting portfolio has risk 1− μp−m
μp−μf
σp= m−μf μp−μf
σp=m−μf SR ,
where SR denotes the Sharpe ratio of the portfolio of risky assets. Hence, the risk of the portfolio with expected return m is inversely proportional to the Sharpe ratio of the portfolio of risky assets. That is, we should choose the portfolio of risky assets to have the largest Sharpe ratio possible.
Conversely, if wf is chosen to achieve a given level of risk, s, then either wf = 1 +s/σp or wf = 1−s/σp. It is easy to show that the second of these yields the larger expected return,
1− s
σp
μf+ s
σpμp =μf+s(SR).
Hence, the expected excess return of the porfolio with risksis proportional to the Sharpe ratio of the portfolio of risky assets, showing again that we should choose the portfolio of risky assets to have the largest Sharpe ratio possible.
Thus, we have proven the following important result. Note the previous analysis is not limited to the case of two risky assets; it applies to any portfolio of risky assets.
Proposition 4.1. Consider a portfolio consisting of the risk-free asset and a portfolio of risky assets. Then the optimal portfolio of risky assets is the one with the largest Sharpe ratio.
Tangency Portfolio
Thus, to construct the optimal portfolio of risky assets with returnsR1 and R2, we find ¯w1,w¯2, ¯w1+ ¯w2= 1, so that the portfolio with return
¯
w1R1+ ¯w2R2 has the maximum possible Sharpe ratio.
Write ¯w1=w and ¯w2= 1−w. Our goal is to find the value of w that maximizes
f(w) = (μp(w)−μf)/σp(w) where
μp(w) =wμ1+ (1−w)μ2
and
σ2p(w) =w2σ21+ (1−w)2σ22+ 2w(1−w)ρ12σ1σ2.
We may maximize f(w) using standard results from calculus. Using the rule for taking the derivative of a ratio, it follows that
f(w) =μp(w)−f(w)σp(w) σp(w) ; hence, the solution tof(w) = 0 solves
μp(w)
σp(w) =f(w). (4.5)
It may be shown thatf(w) is maximized by
w=wT ≡ (μ1−μf)σ22−(μ2−μf)ρ12σ1σ2
(μ2−μf)σ21+ (μ1−μf)σ22−[(μ1−μf) + (μ2−μf)]ρ12σ1σ2
; (4.6) see Proposition 5.7 for a proof of this result in a more general setting.
Hence, the Sharpe ratio of the portfolio of the two risky assets is maximized by choosingw=wT, as given by (4.6). The portfolio corresponding tow=wT is known as the tangency portfolio, a term that is based on an important property of the line connecting (0,μf) and (σp(wT),μp(wT)).
Consider a curve (x(z), y(z)) parameterized by a real number z. The tan- gent vector to the curve at z is given by (x(z), y(z)) and the slope of the tangent vector is y(z)/x(z). Thus, μp(w)/σp(w) is the slope of the tan- gent vector to the curve (σp(w),μp(w)) atwand the fact that the first-order condition (4.5) is satisfied at w=wT may be interpreted as the condition that the slope of the line connecting (0,μf) and (σp(wT),μp(wT)) is equal to the slope of the tangent vector to the efficient frontier at w=wT. Since (σp(wT),μp(wT)) is on the efficient frontier, it follows that the line connecting (0,μf) and (σp(wT),μp(wT)) is tangent to the efficient frontier.
Example 4.14 As in Example 4.13, consider two risky assets with expected returns μ1= 0.05 and μ2= 0.15, respectively, return standard deviations σ1=σ2= 0.25, and correlation of the returns given by ρ12= 0.125, and sup- pose that there is a risk-free asset with expected return μf = 0.025. Then, using the formula (4.6), the weight given to asset 1 in the tangency portfolio is wT = 1/14 .
= 0.071. Therefore, the optimal portfolio of the risky assets is obtained by placing weight 1/14 on asset 1 and weight 13/14 on asset 2.
0 0.1 0.2 0.3 0.4 0.5 0.6 0
0.05 0.10 0.15 0.20
Risk
Expected return
FIGURE 4.9
Tangency portfolio in Example 4.14.
Figure 4.9 shows the efficient frontier in this example with the location of the tangency portfolio (the point shown on the efficient frontier) and a dashed line representing the risk-expected return pairs for portfolios constructed from the tangency portfolio and the risk-free asset.
Note that, since the efficient frontier of the two risky assets lies below the dashed line, for any desired level of risk, a portfolio based on the tangency portfolio and the risk-free asset will have an expected return at least as large (and usually larger) than that of any portfolio on the efficient frontier with
that level of risk.
Consider the problem of finding the best combination of risky assets, with returnsR1, andR2, and risk-free asset, with returnRf; that is, consider the problem of finding the optimal portfolio return of the form
wfRf+ (1−wf)[wR1+ (1−w)R2].
The previous results show that the optimal solution is to first take w=wT, corresponding to the tangency portfolio; this gives the optimal combination of risky assets. LetμT andσT denote the mean and standard deviation of the return on the tangency portfolio.
Given a desired level of riskswe then choosewf so thatwfRf+ (1−wf) RT has a standard deviation equal tos, that is, takewf =s/σT; alternatively, given a desired value for the expected return,m, we findwf so that
wfμf+ (1−μf)μT =m.
Note that, according to this theory, all investors should use the same com- bination of risky assets; only the proportion of the tangency portfolio versus the risk-free asset depends on the investor’s goals.