Choosing from among the portfolios on the minimum risk frontier requires some consideration of the relative importance of the mean return and risk of a portfolio. In this section, we consider the simplest analysis of this type, based on the belief that only risk is important when evaluating a portfolio. Under this assumption, the portfolio with minimum return variance is optimal; we will call this portfolio theminimum-variance portfolio.
As in the N= 2 case discussed in the previous chapter, the minimum- variance portfolio also provides a useful reference point for evaluating the mean and standard deviation of the returns on portfolios, and it will play a role in several results in this chapter.
Letwmv denote the weight vector of the minimum-variance portfolio and letRmv=wTmvR denote the corresponding portfolio return. Then
wmvT Σwmv≤wTΣw for any w∈ N such that wT1= 1.
The following proposition gives a useful characterization of the minimum- variance portfolio, stating that the covariance of the return on the minimum- variance portfolio and any other portfolio is constant, not depending on the portfolio under consideration. The proof proceeds by showing that, if this were not the case, then we could construct a portfolio with variance smaller than that of the minimum-variance portfolio.
Proposition 5.2. An asset with returnRˆ is the minimum-variance portfolio if and only if
Cov( ˆR, Rp) =Var( ˆR) (5.10) for Rp=wTR, for any weight vectorw.
Proof. First suppose that ˆRis the return on the minimum-variance portfolio so that ˆR=Rmv. Let w be the weight vector corresponding to a portfolio with returnRp. For a given real numberz, consider the portfolio with weight vectorwmv+z(w−wmv), which has return Rmv+z(Rp−Rmv). Define
f(z) = Var (Rmv+z(Rp−Rmv)), −∞< z <∞.
Because Rmv is the return on the minimum-variance portfolio, f(z) is mini- mized atz= 0. Note thatf(z) is a quadratic function ofz; hence,f(0) = 0.
By expanding the variance used to definef(z),
f(z) = Var(Rmv) + 2zCov(Rmv, Rp−Rmv) +z2Var(Rp−Rmv).
It follows that
f(0) = 2Cov(Rmv, Rp−Rmv) so that
Cov(Rmv, Rp−Rmv) = 0;
using properties of covariance,
Cov(Rmv, Rp−Rmv) = Cov(Rmv, Rp)−Cov(Rmv, Rmv)
= Cov(Rmv, Rp)−Var(Rmv) so that
Cov(Rmv, Rp) = Var(Rmv), as stated in the proposition.
Now suppose that Cov( ˆR, Rp) = Var( ˆR) holds for any portfolio returnRp. Because
Var(Rp) = Var
Rˆ+ (Rp−R)ˆ
= Var( ˆR) + 2Cov( ˆR, Rp−R) + Var(Rˆ p−R)ˆ
= Var( ˆR) + 2
Cov( ˆR, Rp)−Var( ˆR)
+ Var(Rp−R)ˆ
= Var( ˆR) + Var(Rp−R),ˆ it follows that
Var(Rp)≥Var( ˆR).
Because this holds for any portfolio return Rp, ˆR must be the return on
the minimum-variance portfolio.
One consequence of the Proposition 5.2 is that the return on the minimum- variance portfolio is uncorrelated with the return on any zero-investment portfolio. Note that, if there were a zero-investment portfolio with weight vectorv such thatRmv and R0=vTR are correlated, then we could find a constantcsuch that the portfolio with returnRmv+cR0has a smaller return variance than doesRmv.
The following corollary gives a formal statement of this result; the proof is left as an exercise.
Corollary 5.4. LetRmv denote the return on the minimum-variance portfo- lio and letR0 denote the return on a zero-investment portfolio. Then
Cov(Rmv, R0) = 0.
The characterization of the minimum-variance portfolio given in Propo- sition 5.2 can be used to suggest a form for wmv, the weight vector of the minimum-variance portfolio.
Let R1, R2, . . . , RN denote the returns on the N assets under consider- ation. Then, treating each asset as a portfolio in Proposition 5.2, for any j= 1,2, . . . , N
Cov(Rmv, Rj) = Cov(Rj, Rmv) = Cov(Rj,wmvT R) =eTjΣwmv=c (5.11) for some constantc, whereej denotes thejth column of theN×N identity matrix; that is, it is the vector in N consisting of all zeros, except for the jth element, which is 1. Of course, by Proposition 5.2, the constantcmust be Var(Rmv); however, that fact is not needed here.
Therefore, for eachj= 1,2, . . . , N, eTjΣwmv=c and combining these, it follows that
IΣwmv=c1 so that
Σwmv=c1.
It follows that
wmv=cΣ−11. (5.12)
Because the weights inwmvmust sum to 1,c must be 1
1TΣ−11.
The following result uses the Cauchy–Schwarz inequality to show directly that the weight vector of the minimum-variance portfolio is of the form given in (5.12).
Proposition 5.3. Let R denote the return vector for a set of assets and let Σ denote the covariance matrix of R. Then the weight vector of the minimum-variance portfolio is given by
wmv= Σ−11 1TΣ−11.
Proof. The variance of the return on a portfolio based on weight vectorw is given bywTΣw, which can be written
(Σ12w)T(Σ12w).
Using the Cauchy–Schwarz inequality withx=Σ1/2wand y=Σ−1/21,
(Σ12w)T(Σ−121) 2
≤
(Σ12w)T(Σ12w) (Σ−121)T(Σ−121)
(5.13) with equality ifΣ12w=cΣ−121for some scalarc.
Note that (5.13) may be written (wT1)2≤
wTΣw 1TΣ−11 and becausewT1= 1,
wTΣw≥ 1 1TΣ−11
with equality if w=cΣ−11; that is, the weight vector of the minimum- variance portfolio must be of the form cΣ−11 for some constant c. Since the weights must sum to 1, we must have
c= 1
1TΣ−11,
proving the result.
Example 5.6 Consider a set of three assets, with mean returns 0.25,0.125, and 0.3, respectively, and suppose that the returns have covariance matrix
Σ=
⎛
⎝0.25 0.1 0.24 0.1 0.16 0.096 0.24 0.096 0.36
⎞
⎠. (5.14)
Hence, the asset returns have standard deviations 0.5,0.4, and 0.6, respec- tively, and their correlation matrix is
⎛
⎝1 0.5 0.8 0.5 1 0.4 0.8 0.4 1
⎞
⎠ (5.15)
The mean vector and covariance matrix may be entered into R using the commands
> mu<-c(0.25, 0.125, 0.3)
> Sig<-matrix(c(0.25, 0.1, 0.24, 0.1, 0.16, 0.096, 0.24, 0.096, + 0.36),3,3)
> Sig
[,1] [,2] [,3]
[1,] 0.25 0.100 0.240 [2,] 0.10 0.160 0.096 [3,] 0.24 0.096 0.360
To calculate the weights of the minimum-variance portfolio, we may use the solvefunction. Let A denote an m×m invertible matrix and let b denote an m×1 vector; let A and b denote the corresponding R variables. Then solve(A, b)returnsA−1b; the function with the second argument omitted, that is,solve(A), returnsA−1.
Thus,wmvmay be calculated by
> w0<-solve(Sig, c(1,1,1))
> w_mv<-w0/sum(w0)
> w_mv
[1] 0.243 0.713 0.044
Example 5.7 Consider a set ofN assets with covariance matrix of the form
Σ=σ2
⎛
⎜⎜
⎜⎜
⎜⎝
1 ρ . . . . . . ρ ρ 1 ρ . . . ρ ... . .. . .. . .. ... ρ . . . ρ 1 ρ ρ . . . . . . ρ 1
⎞
⎟⎟
⎟⎟
⎟⎠
(5.16)
where 0≤ρ<1. Recall that, under this condition,Σ is positive-definite; see Example 5.4. Thus, for this covariance matrix, all asset returns have standard deviationσand the correlation between any two returns isρ.
To calculate the weight vector of the minimum-variance portfolio we need Σ−11. Recall that in Example 5.4 it was shown that1is an eigenvector ofΣ, with corresponding eigenvalue 1 + (N−1)ρ. It follows that
1=Σ−1Σ1= (1 + (N−1)ρ)Σ−11 and, hence, that
Σ−11= 1
1 + (N−1)ρ1.
Because
1TΣ−11= N 1 + (N−1)ρ,
it follows that the weight vector of the minimum-variance portfolio is given by Σ−11
1TΣ−11 = 1 N1
so that the equally weighted portfolio is the minimum-variance portfolio.
To find the variance of the minimum-variance portfolio, we use the fact that
1 N1
T Σ
1 N1
= σ2
N2(N+N(N−1)ρ) = σ2 N +
1− 1
N
ρσ2.
This is the minimum possible variance for a portfolio based on a return vector
with a covariance matrix of the form (5.16).