260 THE HANDBOOK OF PORTFOLIO MATHEMATICS
would expect to make on the declining stock, and from that you would then
need tosubtractthe dividend (however many dividends go ex-date over
the holding period you are calculating your E and V on) as a percentage.^4
Lastly, any linear correlation coefficients of which the stock you are looking
to short is a member must be multiplied by−1. Therefore, since the linear
correlation coefficient between Toxico and Incubeast is−.15, if you were
looking to short Toxico, you would multiply this by−1. In such a case
you would use−.15∗− 1 =.15 as the linear correlation coefficient. If you
were looking to short both of these stocks, the linear correlation coefficient
between the two would be−.15∗− 1 ∗–1=−.15. In other words, if you
are looking to short both stocks, the linear correlation coefficient between
them remains unchanged, as it would if you were looking to go long both
stocks.
Thus far we have sought to obtain the optimal portfolio, and its variance,
V, when we know the expected return, E, that we seek. We can also solve
for E when we know V. The simplest way to do this is by iteration using the
techniques discussed thus far in this chapter.
There is much more to matrix algebra than is presented in this chap-
ter. There are other matrix algebra techniques to solve systems of lin-
ear equations. Often, you will encounter reference to techniques such
asCramer’s Rule, theSimplex Method,ortheSimplex Tableau. These
are techniques similar to the ones described in this chapter, although
more involved. There are a multitude of applications in business and
science for matrix algebra, and the topic is considerably involved.
We have only etched the surface, just enough for what we need to ac-
complish. For a more detailed discussion of matrix algebra and its
applications in business and science, the reader is referred toSets,
Matrices, and Linear Programming, by Robert L. Childress.(^4) In this chapter we are assuming that all transactions are performed in a cash ac-
count. Thus, even though a short position is required to be performed in a margin
account as opposed to a cash account, we will not calculate interest on the margin.