Consider two assets, with returns R1 and R2, respectively; for j = 1,2, let Rj have mean and standard deviation μj and σj, respectively, and let ρ12
denote the correlation ofR1 andR2. The portfolio based on portfolio weights w1=w and w2= 1−w has return Rp=wR1+ (1−w)R2. We summarize the properties of the portfolio with return Rp by the mean and standard deviation of the return, μp(w) and σp(w), respectively, viewed as functions ofw,−∞< w <∞. Recall that
μp(w) =wμ1+ (1−w)μ2
and
σ2p(w) =w2σ21+ (1−w)2σ22+ 2w(1−w)ρ12σ1σ2. In this section, we consider the problem of choosing the value ofw.
As w varies, μp(w) andσp(w) vary; we may view these values as points (σp(w), μp(w)) in the “risk-return space.” A plot of (σp(w), μp(w)) as w varies is a useful way to understand the relationship between the expected return and risk of a portfolio.
Example 4.6 Suppose that μ1= 0.2, σ1= 0.1, μ2= 0.1, σ2= 0.05, and ρ12= 0.25. Thenμp(w) =w(0.2) + (1−w)(0.1) = 0.1 + 0.1wand
σ2p(w) =w2(0.1)2+ (1−w)2(0.05)2+ 2w(1−w)(0.25)(0.1)(0.05)
= 0.01w2−0.0025w+ 0.0025.
Figure 4.1 contains a plot of (σp(w),μp(w)) aswvaries.
The curve in Figure 4.1 represents the set of all (σp(w),μp(w)) pairs that are available to the investor; it is known as the opportunity set. This term
0.04 0.07 0.10 0.13 0.16
−0.05 0.01 0.07 0.13 0.19 0.25
σp μp
FIGURE 4.1
Expected return and risk for different portfolios in Example 4.6.
will also be used to describe the corresponding portfolios; for instance, in the previous example, a portfolio in the opportunity set has a value of (σp,μp) on the curve in Figure 4.1.
Note that, unlessμ1=μ2, for a given valuemthere is exactly one value of w such that μp(w) =m. On the other hand, for a given value s >0, there may be zero, one, or two values ofw such thatσp(w) =s, depending on the number of solutions to the quadratic equationσ2p(w)−s2= 0.
Example 4.7 Consider the assets described in Example 4.6. Note that, for a given value ofμp, say m, the portfolio with that expected return may be found by solving
μp(w) = 0.1 + 0.1w=m
forw, yieldingw= 10m−1. On the other hand, for a given value of σp, say s, there may be zero, one, or two portfolios with that standard deviation, depending on the solutions to the quadratic equation
σ2p(w) = 0.01w2−0.0025w+ 0.0025 =s2.
For instance, there are no portfolios with σp(w) = 0.0375 because there are no (real) solutions to the equation
0.01w2−0.0025w+ 0.0025 = (0.0375)2;
there are two portfolios with σp(w) = 0.0625, corresponding to the two solutions to the equation
0.01w2−0.0025w+ 0.0025 = (0.0625)2, w= (1 +√
10)/8 .
= 0.52 andw= (1−√ 10)/8 .
=−0.27; there is one portfolio withσp(w) =√
15/80, corresponding to the one solution to the equation
0.01w2−0.0025w+ 0.0025 = √
15 80
2
= 15 6400,
w= 1/8. These three possibilities are illustrated in Figure 4.2.
Efficient Portfolios
For the case in which two portfolios have the same return standard deviation, only the one with the larger return mean is of interest. For example, in Fig- ure 4.1, each portfolio with a (σp(w), μp(w)) pair on the lower half of the curve is dominated by a portfolio with a (σp(w), μp(w)) pair on the upper half of the curve. Thus, the portfolios corresponding to the upper half of the curve are known asefficient portfolios.
0.04 0.07 0.10 0.13 0.16
−0.05 0.01 0.07 0.13 0.19 0.25
σp μp
FIGURE 4.2
Three possibilities for the solutions toσ2p(w) =sin Example 4.7.
0.04 0.07 0.10 0.13 0.16
−0.05 0.01 0.07 0.13 0.19 0.25
σp μp
FIGURE 4.3
Efficient frontier in Example 4.7.
That is, for an efficient portfolio, it is not possible to have a larger expected return without increasing the risk or, conversely, it is not possible to have lower risk without decreasing the expected return. This upper half of the opportu- nity set is known as the efficient frontier; the efficient frontier for the assets described in Example 4.7 is given in Figure 4.3. The term “efficient frontier”
will also be used to be describe the portfolios with a (σp(w), μp(w)) pair on the efficient frontier.
Each portfolio on the efficient frontier has the largest possible expected return for a given level of risk and the lowest possible risk for a given expected return. Therefore, there is no objective way to choose from among the effi- cient portfolios; such a choice depends on an investor’s view of the relative importance of a portfolio’s expected return and risk.
The Minimum-Variance Portfolio
Suppose that our goal is simply to minimize risk, without consideration of the expected return of the portfolio. Then we choosew to minimize the return standard deviation σp(w) or, equivalently, to minimize the return variance σ2p(w). The resulting portfolio is known as the minimum-variance portfolio.
To find the minimum-variance portfolio, we need to find the value of w that minimizes σ2p(w). To solve this minimization problem, we may use the approach used in calculus. Note that
dσ2p(w)
dw = 2wσ21−2(1−w)σ22+ 2(1−2w)ρ12σ1σ2
and
d2σ2p(w)
dw2 = 2σ21+ 2σ22−4ρ12σ1σ2. Clearly,
d2σ2p(w)
dw2 ≥2σ21+ 2σ22−4σ1σ2
= 2(σ1−σ2)2≥0 and, provided thatρ12<1,
d2σ2p(w) dw2 >0.
Hence,σ2p(w) can be minimized by solving
wσ21−(1−w)σ22+ (1−2w)ρ12σ1σ2= 0 forw, yielding the solution
w=wmv≡ σ22−ρ12σ1σ2
σ21+σ22−2ρ12σ1σ2
and
1−wmv= σ21−ρ12σ1σ2 σ21+σ22−2ρ12σ1σ2
.
Note thatσ21+σ22−2ρ12σ1σ2= Var(R1−R2). Using the fact that Var(R1−R2) =σ21+σ22−2ρ12σ1σ2= (σ1−σ2)2+ 2(1−ρ12)σ1σ2, it follows that if ρ12<1, then Var(R1−R2)>0. Therefore, provided that ρ12<1, the denominator in the expression forwmv is nonzero.
Example 4.8 Consider Example 4.6 in whichσ1= 0.1,σ2= 0.05, andρ12= 0.25. Then
wmv= (0.05)2−(0.25)(0.1)(0.05)
(0.1)2+ (0.05)2−2(0.25)(0.1)(0.05)=1 8.
Recall that in Example 4.6 we saw that the quadratic equation σ2p(w)− 15/6400 = 0 has a single root, atw= 1/8. Such a single root always occurs at the point of minimum risk; see Figure 4.2.
Therefore, risk is minimized by placing 1/8th of our investment in asset 1.
Here, μ1= 0.2 and μ2= 0.1, so that the minimum-variance portfolio has expected return
(1/8)μ1+ (7/8)μ2= 0.1125
and, using the result in Example 4.6, the standard deviation of the return is (0.01w2−0.0025w+ 0.0025)21
w=1/8
= 0.0484..
This gives the portfolio with minimum risk. However, it is only the opti- mal choice if our goal is to minimize risk. For example, suppose that we are willing to increase the standard deviation of the portfolio to 0.05; the solu- tions to 0.01w2−0.0025w+ 0.0025 = (0.05)2 are w= 0.25 and w= 0. Only the first of these corresponds to a portfolio on the efficient frontier (why?), and its expected return is 0.1 + 0.1(0.25) = 0.125. Hence, a 3.3% increase in risk (i.e., 0.0484 to 0.05) yields a 11% increase in expected return (i.e., 0.1125
to 0.125).
The Risk-Aversion Criterion
The minimum-variance portfolio completely ignores the expected return of the portfolio. An alternative, and usually preferable, approach is to use a criterion that takes into account both the risk and the expected return of the portfolio.
Because of risk aversion, investors are generally willing to accept a lower expected return if that lower expected return corresponds to lower risk as well. Conversely, high portfolio risk may be tolerable if the portfolio has a high expected return.
Consider the portfolio placing weight w on asset 1; let μp(w) andσp(w) denote the mean and standard deviation, respectively, of the return on that portfolio. We might consider evaluating this portfolio by
fλ(w) =μp(w)−λ 2σ2p(w)
whereλ>0 is a given parameter, known as therisk aversion parameter.
The function fλ is a type of penalized mean return, with a penalty based on the return variance; the magnitude of the penalty is controlled by the parameter λ. Thus, larger values of fλ(ã) correspond to portfolios with a greater expected return relative to the portfolio risk, with the tradeoff between expected return and risk controlled byλ. For a given value ofλ, letwλ denote the value ofwthat maximizesfλ(w).
To find wλ, we may use an approach similar to the one used to find the weights of the minimum-variance portfolio. Note that
fλ(w) =μp(w)−λσp(w)σp(w)
= (μ1−μ2)−λ(wσ21−(1−w)σ22+ (1−2w)ρ12σ1σ2) and
fλ(w) =−λ(2σ21+ 2σ22−4ρ12σ1σ2)
=−2λVar(R1−R2)<0
provided thatρ12<1. Hence, we can maximizefλ(w) by solvingfλ(w) = 0 forw, yielding the solution
wλ =σ22−ρ12σ1σ2+ (μ1−μ2)/λ σ21+σ22−2ρ12σ1σ2
; (4.1)
it follows that
1−wλ=σ21−ρ12σ1σ2−(μ1−μ2)/λ σ21+σ22−2ρ12σ1σ2
.
Example 4.9 Consider two assets such thatμ1= 0.04,μ2= 0.02,σ1= 0.2, σ2= 0.1, andρ12= 0.25. Then, using (4.1),
w=wλ≡1 8+ 1
2λ, λ>0.
Figure 4.4 contains plots ofμp(wλ) andσp(wλ) asλvaries. Note that, for large values ofλ, there is a large penalty on the variance of the return; hence, the optimal portfolio has small risk. To achieve this, the portfolio must also have small expected return. Conversely, for a smallλ, the variance of the return is largely irrelevant; hence, the optimal portfolio has large risk. As a reward for the large risk incurred, the portfolio also has a large expected return.
Whenλis small, the weight on asset 1 is large; hence, the weight on asset 2 is negative. That is, in order to achieve a large expected return, we must borrow asset 2 (which has a low expected return) in order to buy asset 1 (which has a large expected return). For instance, if λ= 0.25, the optimal portfolio places weight 2.125 on asset 1 and weight−1.125 on asset 2.
0.03 0.04 0.05 0.06 0.07
Expected return
1 2 3 4 5
0.1 0.2 0.3 0.4 0.5
λ
1 2 3 4 5
λ
Risk
FIGURE 4.4
Properties of the optimal portfolio as a function ofλin Example 4.9.