Ralph Vince - Portfolio Mathematics

258 THE HANDBOOK OF PORTFOLIO MATHEMATICS

To use this method, we need to first discern the inverse matrix to our co-
efficients matrix. To do this, rather than start by augmenting the right-hand-
side vector onto the coefficients matrix, we augment the identity matrix
itself onto the coefficients matrix. For our four-stock example:

Starting Augmented Matrix

X 1 X 2 X 3 X 4 L 1 L 2 | Identity Matrix

0.095 0.13 0.21 0.085 0 0 | 100000
1 11100| 010000
0.1 −0.023 0.01 0 0.095 1 | 001000
−0.023 0.25 0.079 0 0.13 1 | 000100
0.01 0.079 0.4 0 0.21 1 | 000010
0 0 0 0 0.085 1 | 000001

Now we proceed using row operations to transform the coefficients
matrix to an identity matrix. In the process, since every row operation per-
formed on the left is also performed on the right, we will have transformed
the identity matrix on the right-hand side into the inverse matrixC−^1 ,of
the coefficients matrixC. In our example, the result of the row operations
yields:

C | C−^1

100000 | 2.2527 −0.1915 10.1049 0.9127 −1.1370 −9.8806
010000 | 2.3248 −0.1976 0.9127 4.1654 −1.5726 −3.5056
001000 | 6.9829 −0.5935 −1.1370 −1.5726 0.6571 2.0524
000100 |−11.5603 1.9826 −9.8806 −3.5056 2.0524 11.3337
000010 |−23.9957 2.0396 2.2526 2.3248 6.9829 −11.5603
000001 | 2.0396 −0.1734 −0.1915 −0.1976 −0.5935 1.9826

Now we can take the inverse matrix,C−^1 , and multiply it by our original
right-hand-side vector. Recall that our right-hand-side vector is:

E S 0 0 0 0