Classical Portfolio Construction 247
When we have more than one linear equation that must be solved simul-
taneously we can use what is called themethod of row-equivalent matrices.
This technique is also often referred to as theGauss-Jordan procedureor
theGaussian elimination method.
To perform the technique, we first create the augmented matrix of the
problem by combining the coefficients matrix with the right-hand side vec-
tor as we have done. Next, we want to use what are calledelementary trans-
formationsto obtain what is known as theidentity matrix.An elementary
transformation is a method of processing a matrix to obtain a different but
equivalent matrix. Elementary transformations are accomplished by what
are calledrow operations.(We will cover row operations in a moment.)
An identity matrix is a square coefficients matrix where all of the el-
ements are zeros except for a diagonal line of ones starting in the upper
left corner. For a six-by-six coefficients matrix such as we are using in our
example, the identity matrix would appear as:
100000
010000
001000
000100
000010
000001This type of matrix, where the number of rows is equal to the number of
columns, is called asquare matrix.Fortunately, due to the generalized form
of our problem of minimizing V for a given E, we are always dealing with a
square coefficients matrix.
Once an identity matrix is obtained through row operations, it can be re-
garded as equivalent to the starting coefficients matrix. The answers then are
read from the right-hand-side vector. That is, in the first row of the identity
matrix, the 1 corresponds to the variable X 1 , so the answer in the right-hand
side vector for the first row is the answer for X 1. Likewise, the second row
of the right-hand side vector contains the answer for X 2 , since the 1 in the
second row corresponds to X 2. By using row operations we can make ele-
mentary transformations to our original matrix until we obtain the identity
matrix. From the identity matrix, we can discern the answers, the weights
X 1 ,...,XN, for the components in a portfolio. These weights will produce
the portfolio with the minimum variance, V, for a given level of expected
return, E.^3
(^3) That is, these weights will produce the portfolio with a minimum V for a given E only
to the extent that our inputs of E and V for each component and the linear correlation
coefficient of every possible pair of components are accurate and variance in returns
finite.