Classical Portfolio Construction 245
The E on the right-hand side of the first equation is the E you have
decided you want to solve for (i.e., it is a given by you). The first line simply
states that the sum of all of the expected returns times their weightings
must equal the given E. The second line simply states that the sum of the
weights must equal 1. Shown here is the matrix for a three-security case,
but you can use the general form when solving for N securities. However,
these first two lines are always the same. The next N lines then follow the
prescribed form.
Now, using our expected returns and covariances (from the covariance
table we constructed earlier), we plug the coefficients into the generalized
form. We thus create a matrix that represents the coefficients of the gen-
eralized form. In our four-component case (N=4), we thus have six rows
(N+2):
X 1 X 2 X 3 X 4 L 1 L 2 | Answer.095 .13 .21 .085 | E
1111 | 1
.1 −.0237 .01 0 .095 1 | 0
−.0237 .25 .079 0 .13 1 | 0
.01 .079 .4 0 .21 1 | 0
0 0 0 0 .085 1 | 0Note that the expected returns arenotexpressed in the matrix as HPRs;
rather, they are expressed in their “raw” decimal state.
Notice that we also have six columns of coefficients. Adding the answer
portion of each equation onto the right, and separating it from the coeffi-
cients with a|creates what is known as anaugmented matrix, which is
constructed by fusing the coefficients matrix and the answer column, which
is also known as theright-hand side vector.
Notice that the coefficients in the matrix correspond to our generalized
form of the problem:
X 1 ∗U 1 +X 2 ∗U 2 +X 3 ∗U 3 +X 4 ∗U 4 =E
X 1 +X 2 +X 3 +X 4 = 1
X 1 ∗COV1, 1+X 2 ∗COV1, 2+X 3 ∗COV1, 3+X 4 ∗COV1, 4+. 5 ∗L 1 ∗U 1 +. 5 ∗L 2 = 0
X 1 ∗COV2, 1+X 2 ∗COV2, 2+X 3 ∗COV2, 3+X 4 ∗COV2, 4+. 5 ∗L 1 ∗U 2 +. 5 ∗L 2 = 0
X 1 ∗COV3, 1+X 2 ∗COV3, 2+X 3 ∗COV3, 3+X 4 ∗COV3, 4+. 5 ∗L 1 ∗U 3 +. 5 ∗L 2 = 0
X 1 ∗COV4, 1+X 2 ∗COV4, 2+X 3 ∗COV4, 3+X 4 ∗COV4, 4+. 5 ∗L 1 ∗U 4 +. 5 ∗L 2 = 0
The matrix is simply a representation of these equations. To solve for the
matrix, you must decide upon a level for E that you want to solve for. Once
the matrix is solved, the resultant answers will be the optimal weightings
required to minimize the variance in the portfolio as a whole for our specified
level of E.