Classical Portfolio Construction 259
Whenever we multiply a matrix by a columnar vector (such as this) we
multiply all elements in the first column of the matrix by the first element in
the vector, all elements in the second column of the matrix by the second
element in the vector, and so on. If our vector were a row vector, we would
multiply all elements in the first row of the matrix by the first element in the
vector, all elements in the second row of the matrix by the second element
in the vector, and so on. Since our vector is columnar, and since the last four
elements are zeros, we need only multiply the first column of the inverse
matrix by E (the expected return for the portfolio) and the second column
of the inverse matrix by S, the sum of the weights. This yields the following
set of equations, which we can plug values for E and S into and obtain the
optimal weightings. In our example, this yields:
E∗ 2. 2527 +S∗− 0. 1915 =Optimal weight for first stock
E∗ 2. 3248 +S∗− 0. 1976 =Optimal weight for second stock
E∗ 6. 9829 +S∗− 0. 5935 =Optimal weight for third stock
E∗− 11. 5603 +S∗ 1. 9826 =Optimal weight for fourth stock
E∗− 23. 9957 +S∗ 2. 0396 =.5 of first Lagrangian
E∗ 2. 0396 +S∗− 0. 1734 =.5 of second LagrangianThus, to solve for an expected return of 14% (E=.14) with the sum of
the weights equal to 1:
. 14 ∗ 2. 2527 + 1 ∗− 0. 1915 =. 315378 −. 1915 =.1239 Toxico
. 14 ∗ 2. 3248 + 1 ∗− 0. 1976 =. 325472 −. 1976 =.1279 Incubeast
. 14 ∗ 6. 9829 + 1 ∗− 0. 5935 =. 977606 −. 5935 =.3841 LA Garb
. 14 ∗− 11. 5603 + 1 ∗ 1. 9826 =− 1. 618442 + 1. 9826 =.3641 Savings
. 14 ∗− 23. 9957 + 1 ∗ 2. 0396 =− 3. 359398 + 2. 0396 =− 1. 319798 ∗ 2
=− 2 .6395 L 1
. 14 ∗ 2. 0396 + 1 ∗− 0. 1734 =. 285544 −. 1734 =. 1121144 ∗ 2
=.2243 L 2
Once you have obtained the inverse to the coefficients matrix, you can
quickly solve for any value of E provided that your answers, the optimal
weights, are all positive. If not, again you must create the coefficients matrix
without that item, and obtain a new inverse matrix.
Thus far we have looked at investing in stocks from the long side only.
How can we consider short sale candidates in our analysis?
To begin with, you would be looking to sell short a stock if you expected
it would decline. Recall that the term “returns” means not only the dividends
in the underlying security, but any gains in the value of the security as well.
This figure is then specified as a percentage. Thus, in determining the returns
of a short position, you would have to estimate what percentage gain you