As shown in Chapters 5 and 6, if a risk-free asset is included in the list of risky investment candidates, a linear efficient frontier results. This section omits the risk- free asset from the list of risky investment candidates. As a result of this omission, this section derives a convex efficient frontier.
7.1.1 A General Formulation
Consider how a portfolio on the efficient frontier can be identified. What is desired are weights for the portfolio that minimize the portfolio variance,σp2, at a given level of the expected return,E(rp). As mentioned in the previous paragraph, this section’s optimization problem omits the risk-free asset.
The problem involves finding the weights that minimize the portfolio variance.
Minimize σp2= n
i=1
n j=1
wiwjσij (7.1)
subject to two mathematical constraints that are explained in the next paragraphs.
As is common, we multiply the constant fraction 1/2 times the objective function in equation (7.1) to simplify the solution that follows. This is a cosmetic change. This multiplication does not change any of the conclusions that follow.
135
The first constraint on minimizing equation (7.1) requires that some desired (or target) level of expected return,E(rp), from the portfolio be achieved. This constraint is equivalent to requiring that equation (7.2) not be violated.
n i=1
wiE ri
=E(rp) (7.2)
The second constraint on minimizing equation (7.1) is the familiar requirement that the weights sum to 1.
n i=1
wi=1 (2.11)
Note that short sales (negative weights) are allowed.
Equations (2.11) and (7.2) are called Lagrangian constraints. Combining the three equations, (7.1), (7.2), and (2.11), allows us to form a Lagrangian objective functionfor the risk-minimization problem with two constraints.
MinimizeL= 1 2
n i=1
n j=1
wiwjσij+λ
E(rp)− n
i=1
wiE ri
+γ
1− n
i=1
wi
(7.3)
The lower-case Greek letters lambda and gamma, denotedλandγ in equation (7.3), are called Lagrangian multipliers because they are multiplied by the two Lagrangian constraints defined in equations (2.11) and (7.2).
The minimum-risk portfolio is found by setting the partial derivatives, denoted
∂L/∂wi, equal to zero, for all integer values of the subscriptiranging from one ton.
In other words, set∂L/∂wi=0 fori=1,. . .,n. The partial derivatives of the two Lagrangian constraints are also set equal to zero, thus,∂L/∂λ=0 and∂L/∂γ =0.
Thesen+2 partial derivatives result in the following system ofn+2 equations.
∂L
∂w1
=w1σ11+w2σ12+ · · · +wnσ1n−λE r1
−γ =0
∂L
∂w2
=w1σ21+w2σ22+ · · · +wnσ2n−λE r2
−γ =0
∂L ...
∂wn
=w1σn1+w2σn2+ · · · +wnσnn−λE rn
−γ =0
⎫⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎭
(7.4)
∂L
∂λ =w1E r1
+w2E r2
+ · · · +wnE rn
−E(rp)=0 (7.5)
∂L
∂γ =w1+w2+ · · · +wn−1=0 (7.6) The last two partial derivatives above are equations (7.2) and (2.11). They were obtained by taking the partial derivatives of the Lagrangian objective function (7.3) with respect to the two Lagrangian multipliers, λ and γ, which are included to provide the two constraints that are necessary for a rational solution.
The n +2 equations in equation system (7.4), (7.5), and (7.6) all happen to be linear equations. Stated differently, every weight variable in the equations has a superscript equal to one. To simplify writing out these n + 2 equations, the superscripts of one are not written explicitly; they are implicit.
The firstnterms in the firstn partial derivatives in equation system (7.4) were obtained by taking partial derivatives of the variance formula shown next with respect to each of thenweight variables.
σp2= n
i=1
n j=1
wiwjσij=
⎛
⎜⎜
⎜⎝
+w1w1σ11 +w1w2σ12 · · · +w1wnσ1n
+w2w1σ21 +w2w2σ22 · · · +w2wnσ2n
... ... ... ...
+wnw1σn1 +wnw2σn2 · · · +wnwnσnn
⎞
⎟⎟
⎟⎠ (2.19)
The preceding variance is the first term in the Lagrangian equation (7.3). The partial derivative of the variance with respect to thei-th weight variable is shown below in an expanded form.
∂σ2p
∂wi =
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
+w1σ1i
+w2σ2i
+wiσi1+wiσi2+ · · · +2w... iσii+ · · · +wiσiN
+...wNσNi
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
(7.7)
The preceding expanded first partial derivative may be equivalently rewrit- ten in more compact notation, after making an inconsequential (purely cosmetic) multiplication by one-half, as mentioned earlier.
∂σ2p
∂wi
=wiσi1+wiσi2+ · · · +wiσin=0 (7.8) The preceding equation differs from thenpartial derivatives in equation system (7.4) because it fails to include the last two terms with thendifferential equations.
These last two terms (omitted here) were obtained by taking the partial derivatives of equations (7.2) and (2.11) with respect to thei-th weight variable,wi, as the two Lagrangian constraints were encountered in differentiating the Lagrangian objective function (7.3).
Because the (n +2) differential equations in equation system (7.4), (7.5), and (7.6) are all linear equations, they can be reformulated into a matrix algebra format that is more concise. The (n + 2) first-order partial derivatives are restated as Jacobian matrixequation (7.9).
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎣
σ11 σ12
σ21 σ22 · · ·
· · · σ1n E(r1) 1 σ2n E(r2) 1 ... ... ... ... ... ...
σn1 σn2
E(r1) E(r2)
1 1
· · ·
· · ·
· · ·
σnn E(rn) 1 E(rn) 0 0
1 0 0
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎦
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎣ w1 w2 ...
wn
−λ−γ
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎦
=
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎣ 0 0...
0 E(rp)
1
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎦
(7.9)
Using matrix notation, we can equivalently rewrite equation (7.9) as follows.
C x=b (7.10)
where the coefficient matrixCis an (n+2) by (n+2) matrix, the weight vectorx is an (n+2) by 1 matrix, andbis an (n+2) by 1 matrix (vector) of constants. This system may be solved in several different ways. Using matrix notation, the inverse of the coefficients matrix (C−1) may be used to find the solution (weight) vector (x) as follows:
C x=b C−1C x=C−1b
Ix=C−1b
⇒ x=C−1b (7.11)
whereIis an identity matrix that has dimensions of (n+2) by (n+2).1The solution of equation (7.11) will give the n values for the weights, w1,w2,. . .,wn, which minimize the variance of the portfolio with a desired expected return (constrained solution),E(rp), and two values for the Lagrangian multipliersλandγ. The weights of the individual assets in the solution equation (7.11) will be expressed in terms of the independent variable,E(rp), because the solution for the weights in equation (7.11) is expressed as a function of E(rp). Exact mathematical expressions for the weights in terms ofE(rp) will be provided later in this chapter.
EXAMPLE 7.1
Consider three securities whose expected returns and variance-covariances are given as follows: ⎡
⎢⎣ E
r1 E
r2 E
r3
⎤
⎥⎦=
⎛
⎜⎝ 2.07 0.21 1.01
⎞
⎟⎠ and
⎛
⎝σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33
⎞
⎠=
⎛
⎝48.20 7.82 −2.65 7.82 16.34 0.99
−2.65 0.99 34.25
⎞
⎠
The information above is also available in Chapter 5’s Table 5.5. In order to find the weights minimizing the variance of an efficient portfolio that earns E(rp)=1.5 percent, we create the following matrix equation by substituting the preceding numerical values into equation (7.9):
⎡
⎢⎢
⎢⎢
⎣
48.20 7.82 −2.65 2.07 7.82 16.34 0.99 0.21
−2.65 0.99 34.25 1.01 2.07 0.21 1.01 0
1 1 1 0
1 1 1 0 0
⎤
⎥⎥
⎥⎥
⎦
⎡
⎢⎢
⎢⎢
⎣ w1 w2 w3
−λ−γ
⎤
⎥⎥
⎥⎥
⎦ =
⎡
⎢⎢
⎢⎢
⎣ 0 0 0 1.5
1
⎤
⎥⎥
⎥⎥
⎦ (7.12)
EXAMPLE 7.1
(Continued)Equation (7.12) is identical to equation (7.9); it is merely the specific case of equation (7.9) that exists when n=3. Equation (7.12) is solved by evaluating the following matrix equation:
x=
⎡
⎢⎢
⎢⎢
⎣ w1 w2 w3
−λ−γ
⎤
⎥⎥
⎥⎥
⎦=
⎡
⎢⎢
⎢⎢
⎣
48.20 7.82 −2.65 2.07 7.82 16.34 0.99 0.21
−2.65 0.99 34.25 1.01 2.07 0.21 1.01 0
1 1 1 0
1 1 1 0 0
⎤
⎥⎥
⎥⎥
⎦
−1 ⎡
⎢⎢
⎢⎢
⎣ 0 0 0 1.5
1
⎤
⎥⎥
⎥⎥
⎦ (7.13)
=
⎡
⎢⎢
⎢⎢
⎣
0.5094 0.0625 0.4281
−9.9350
−3.3416
⎤
⎥⎥
⎥⎥
⎦
The weights that minimize the variance of the portfolio whose expected return is constrained to be 1.5 percent are w1=0.5094, w2=0.0625, and w3= 0.4281, as shown in the top portion of the vector above. Not surprisingly, the portfolio’s constrained expected return of 1.5 percent is what the portfolio is attained in equation (7.2a):
E(rp)=w1E r1
+w2E r2
+w3E r3
=(0.5094) (2.07%)+(0.0625) (0.21%)+(0.4281) (1.01%)
=1.5% (7.2a)
The standard deviation of this efficient portfolio is σp=√
18.244= 4.27%, as shown below.
σp2=w21σ12+w22σ22+w23σ32+2w1w2σ12+2w1w3σ13+2w2w3σ23
=(0.5094)2(48.20)+(0.0625)2(16.34)+(0.4281)2(34.25) +2(0.5094) (0.0625) (7.82)+2(0.5094) (0.4281) (−2.65) +2(0.0625) (0.4281) (0.99)
=18.244 Thus,σp=√
18.244=4.27%.
For any desired value of the expected returnE(rp) (1.5% in this example), equation (7.13) gives the weights of the minimum-variance portfolio (MVP).
The weights of the portfolios in the efficient set are generated by varying the target expected returnE(rp) and evaluating the values of the weightswi’s that result.
7.1.2 Formulating with Concise Matrix Notation
This section solves the same portfolio problem that was solved in Section 7.1.1, except that a more specific mathematical expression for the optimal weights is derived in this section by using some vector and matrix differentiation.
To begin, we restate the objective function and the two constraints of equations (7.1), (7.2), and (2.10) in vector and matrix notation. A boldface letter or number indicates a vector or matrix. The two constraints of equations (7.2) and (2.10) can be rewritten as follows;
n i=1
wiE ri
=E(rp) ⇒ wE=E(rp) (7.2a)
n i=1
wi=1 ⇒ w1=1 (2.11a)
where w =
w1,w2,. . . ,wn
is the weight vector of n assets, E = E
r1 , E
r2
, . . .,E rn
is the expected return vector, and1=(1, 1,. . ., 1) is then-vector of ones. The objective function can be transformed as follows:
Minimize σp2= n
i=1
n j=1
wiwjσij ⇒ Minimize σp2=ww (7.1a) whereis the(n×n)variance-covariance matrix.
The problem formulation, using matrix notation, is therefore Minimize L=1
2ww (7.1a)
subject to wE=E(rp). (7.2a)
w1=1 (2.11a)
As mentioned previously, the constant 1/2 is multiplied in equation (7.1a) only for computational and writing simplicity.
Because the constraints are equalities, we use the Lagrangian multipliers and form the following optimization;
Minimize L=1
2ww+λ
E(rp)−wE +γ
1−w1
(7.3a) where λ and γ are Lagrangian multipliers. Equation (7.3a) is a matrix notation restatement of equation (7.3). First-order conditions for the solution to this problem require that the partial derivatives ofLwith respect to the three unknown variables, w,λ, andγ, be set equal to zero.2That is,
∂L
∂w=w−λE−γ1=0 (7.4a)
∂L
∂λ =E(rp)−wE=0 (7.5a)
∂L
∂γ =1−w1=0 (7.6a)
where 0is the n-vector of zeros. If the covariance matrix, , conforms to certain plausible assumptions (that are mathematically referred to as a positive definite covariance matrix), the first-order conditions are necessary and sufficient for a global optimum.3Explanations for vector and matrix differentiation may be found in a mathematical appendix at the end of this book.
Using equation (7.4a), we have
w=λ−1E+γ−11 (7.4b)
Combining equations (7.5a) and (7.4b) yields E(rp)=wE=Ew=λ
E−1E +γ
E−11
(7.5b) and combining equations (7.6a) and (7.4b) yields
1=w1=1w=λ
1−1E +γ
1−11
(7.6b) Solving the last two equations (7.5b) and (7.6b) with respect toλandγ yields
λ=C E(rp)−A
D (7.14)
γ =B−A E(rp)
D (7.15)
where
A=1−1E B=E−1E
C=1−11 D=BC−A2 (7.16)
The inverse of a positive definite matrix is also a positive definite matrix, therefore B>0 and C > 0.4 By the Cauchy-Schwarz inequality (or the corre- lation inequality)5, BC>A2. Thus, D>0. Plugging equations (7.14) and (7.15) into equation (7.4b) gives us the optimal weights for the mean-variance efficient portfolios, or efficient frontier portfolios, as
wp =
C E(rp)−A D
−1E+
B−A E(rp) D
−11
=g+hE(rp) (7.17)
where
g= 1 D
B−11−A−1E
(7.18) h= 1
D
C−1E−A−11
(7.19) Thus, the optimal weights of the mean-variance efficient portfolios are repre- sented as a linear function of the given level of the expected return of the portfolio,
E(rp), becausegandhare fixed constants. Note that the efficient portfolio represented by equation (7.17) has the expected return equal toE(rp); that is,wpE=E(rp). Any efficient portfolio can be represented by equation (7.17), and any portfolio that can be represented by equation (7.17) is an efficient portfolio. The variance of the efficient portfoliopis computed asσp2=wpwp.
EXAMPLE 7.2
Consider three securities: 1, 2, and 3. Their expected returns and covariance matrix are:
E=
⎡
⎣E(r1) E(r2) E(r3)
⎤
⎦=
⎛
⎝2.07 0.21 1.10
⎞
⎠ and =
⎛
⎝σ11 σ12 σ13
σ21 σ22 σ23 σ31 σ32 σ33
⎞
⎠=
⎛
⎝48.20 7.82 −2.65 7.82 16.34 0.99
−2.65 0.99 34.25
⎞
⎠
We will find an efficient portfolio whose expected return is 1.5 percent for these three securities. In Figure 7.1, portfolio phas the smallest variance among portfolios earning 1.5 percent return.
1
2
3
s p
E (r)
E (rp) = 1.5%
FIGURE 7.1 Identifying an Efficient Portfolio withE(rp)=1.5%
To find the optimal values of the three weights for the efficient portfoliop in equation (7.17), we first compute the values ofA,B,C, andDin equation (7.16).
A=1−1E=
1 1 1⎛
⎝48.20 7.82 −2.65 7.82 16.34 0.99
−2.65 0.99 34.25
⎞
⎠
−1⎛
⎝2.07 0.21 1.10
⎞
⎠=0.0686
B=E−1E=
2.07 0.21 1.01⎛
⎝48.20 7.82 −2.65 7.82 16.34 0.99
−2.65 0.99 34.25
⎞
⎠
−1⎛
⎝2.07 0.21 1.10
⎞
⎠=0.1279
EXAMPLE 7.2
(Continued)C=1−11=
1 1 1⎛
⎝48.20 7.82 −2.65 7.82 16.34 0.99
−2.65 0.99 34.25
⎞
⎠
−1⎛
⎝1 1 1
⎞
⎠
=0.0954
D=BC−A2=(0.1279) (0.0954)−(0.0686)2
=0.0075
Then, we compute the values ofgandhin equations (7.18) and (7.19).
g= 1 D
B−11−A−1E
=
⎛
⎝−0.1922 1.0078 0.1845
⎞
⎠ (7.18a)
h= 1 D
C−1E−A−11
=
⎛
⎝ 0.4678
−0.6302 0.1624
⎞
⎠ (7.19a)
Finally, we compute the optimal weights of the three securities for the efficient portfolio earning 1.5 percent return by using equation (7.17), as shown in equation (7.20).
wp=g+hE(rp)=
⎛
⎝−0.1922 1.0078 0.1845
⎞
⎠+
⎛
⎝ 0.4678
−0.6302 0.1624
⎞
⎠×(1.5)
=
⎛
⎝0.5094 0.0625 0.4281
⎞
⎠ (7.20)
These optimal weights of the three assets in equation (7.20) are the same as in Example 7.1. The minimum variance at the given level of the expected return equal to 1.5 percent is computed with these optimal weights as
σp2=wpwp
=
0.5094 0.0625 0.4281⎛
⎝48.20 7.82 −2.65 7.82 16.34 0.99
−2.65 0.99 34.25
⎞
⎠
⎛
⎝0.5094 0.0625 0.4281
⎞
⎠
=18.244
Thus, the standard deviation of the efficient portfolio is σp=√
18.244=4.27%
EXAMPLE 7.3
If the target expected return changes from 1.5 percent to 2.0 percent using the same securities in Examples 7.1 and 7.2, we can easily obtain the optimal weights for another efficient portfolioqby using equation (7.20) again. Note that g and h are fixed and independent of the given target expected return.
Thus, we simply replace the target expected return 1.5 percent in equation (7.20) with 2.0 percent:
wq =g+hE rq
=
⎛
⎝−0.1922 1.0078 0.1845
⎞
⎠+
⎛
⎝ 0.4678
−0.6302 0.1624
⎞
⎠×(2.0)=
⎛
⎝ 0.7433
−0.2526 0.5093
⎞
⎠
And the minimum variance at a given level of the expected return equal to 2.0 percent is computed as
σq2=wqwq
=
0.7433 −0.2526 0.5093⎛
⎝48.20 7.82 −2.65 7.82 16.34 0.99
−2.65 0.99 34.25
⎞
⎠
⎛
⎝ 0.7433
−0.2526 0.5093
⎞
⎠
=31.360
Thus, the standard deviation of the efficient portfolio is σq=√
31.360=5.60%
This solution technique can be repeated iteratively using different values for the desired expected return constraint and the risk-return statistics for all the efficient portfolios can be computed. Figure 7.2 illustrates the two efficient portfoliospandqidentified in Examples 7.2 and 7.3.
E (rq) = 2.0%
E (rp) = 1.5%
4.27 5.60 E (r)
1
3 q p
sp sq s
2
FIGURE 7.2 Two Efficient Portfolios with No Risk-free Asset
As seen in Examples 7.2 and 7.3, when the expected return of an efficient portfolio changes, its variance also changes in a functional way. In other words, a functional relationship between the expected return, E(rp), and variance, σ2p, of an efficient portfolio can be obtained. The equation representing the functional relationship betweenE(rp) andσpfor efficient portfolios is derived in this chapter’s Appendix A7.1. This equation represents the efficient frontier depicted in Figure 6.5 of Chapter 6 with no risk-free asset. A function to explain the covariance between an efficient portfolio and any other portfolio is also derived in Appendix A7.2.
7.1.3 The Two-Fund Separation Theorem
All portfolios on the mean-variance efficient frontier can be formed as a linear combination of any two portfolios (or funds) on the efficient frontier. Let portfolios p1 and p2 be on the efficient frontier EF in Figure 6.5 of Chapter 6 (or BD in Figure A7.1 of the appendix to this chapter), assuming the expected returns on these two portfolios are not equal,E
rp1
=E rp1
. Let a portfolio formed by these two efficient portfolios be denotedq. Then, there exists a constant,α, such that
E rq
=αE rp1
+(1−α)E rp2
(7.21) The investment weight for portfolioqis determined as
wq=αwp
1+(1−α)wp
2 (7.22)
Because the optimal weight vector for an efficient portfolio is expressed as equation (7.17),
wq=α
g+hE rp1
+(1−α)
g+hE rp2
(7.23) Rearranging equation (7.23) yields
wq=g+h αE
rp1
+(1−α)E rp2
=g+hE rq
(7.24) As seen in equation (7.24), portfolioqis also represented in the form of equation (7.17). Thus, portfolioqis an efficient portfolio.
We have shown that a portfolio formed by a linear combination of two efficient portfolios is also an efficient portfolio. In other words, if any two portfolios on the efficient frontier (i.e., any two points on the efficient frontier EFin Figure 6.5 of Chapter 6) are given, the entire efficient frontier can be generated by a linear combination of these two portfolios. If an investor prefers an efficient portfolio, the investor can simply hold a linear combination of two efficient portfolios or mutual funds. In that case, there exists a portfolio (or a linear combination of two efficient portfolios) the investor desires at least as much as the original portfolio.
This phenomenon is called thetwo-fund separation.
Some important implications arise from the property of the weights for an efficient portfolio represented in equation (7.22). First, the proportion invested in any particular asset changes monotonically as one moves up the efficient frontier
from the vertex. This can be seen by noting that α changes monotonically as one moves up the efficient frontier and that equation (7.22) can be rewritten as wp
2+α(wp1−wp
2). Thus, the weights will change monotonically with changes inα.
Second, if securityihas a nonzero weight in two efficient portfolios, then it will have a nonzero weight in all efficient portfolios except, at most, one efficient portfolio.
Lettingwi1andwi2denote the weight of securityiin portfolios 1 and 2, respectively, this can be seen by noting that ifwi2+α
wi1−wi2
is nonzero for at least one value ofα, then it is nonzero for all values except, at most, for one efficient portfolio when α= −wi2/
wi1−wi2
and w1=wi2k. Third, if wi1=wi2, then the proportion for securityiis the same everywhere on the efficient frontier becausewi2+α(wi1−wi2)
=nonzero constant.6Fourth, it follows that it is possible for no efficient portfolios to exist that havew>0 (that is, all positive weights).
7.1.4 Caveat about Negative Weights
The weights of the efficient portfolio can be negative. Such solutions are not always realistic. Some large public portfolios and some mutual funds are legally forbidden to use leverage. Also, securities having the same risk and return as the security with the negative weight may not be easy to create and issue. Furthermore, because of U.S. securities regulations, a mutual fund manager is required to hold no more than five percent of the portfolio in any given security.
When negative weights are not permitted, quadratic programming (QP) should be used to optimize equations (7.1a), (7.2a), and (2.11a) by adding nonnegative con- straints and upper bounds of five percent on any security entering the optimization system. Another way to optimize without using QP is to continue the optimization just described to the point where it begins to produce negative weights. At the point where the weight for an asset reaches zero (before becoming negative), stop the analysis. Remove the asset in the optimization system of equations (7.1a), (7.2a), and (2.11a) that corresponds to the security with the zero weight. The solution now has one less asset from which to select. Then, solve the new optimization system of equations for a new set of optimal weights. This set of efficient portfolios will intersect the original efficient set where the eliminated asset’s weight went to zero.
Thus, the analysis proceeds. Each time another asset’s weight reaches zero, that asset is eliminated, and the new smaller optimization system is solved. Of course, this recommended process provides an approximate solution, not an exact solution.