Ralph Vince - Portfolio Mathematics

240 THE HANDBOOK OF PORTFOLIO MATHEMATICS

We can now create a table of covariances for our example of four in-
vestment alternatives:

TI LS

T.1 −.0237 .01 0

I −.0237 .25 .079 0

L .01 .079 .4 0

S0 0 0 0

We now have compiled the basic parametric information, and we can
begin to state the basic problem formally. First, the sum of the weights of
the securities comprising the portfolio must be equal to 1, since this is being
done in a cash account and each security is paid for in full:

∑N

i= 1

Xi= 1 (7.04)

where: N=The number of securities comprising the portfolio.

Xi=The percentage weighting of the ith security.

It is important to note that in Equation (7.04) each Ximust be nonnegative.
That is, each Ximust be zero or positive.
The next equation defining what we are trying to do regards the ex-
pected return of the entire portfolio. This is the E in E–V theory. Essentially,
what it says is that the expected return of the portfolio is the sum of the
returns of its components times their respective weightings:

∑N

i= 1

Ui∗Xi=E (7.05)

where: E=The expected return of the portfolio.

N=The number of securities comprising the portfolio.

Xi=The percentage weighting of the ith security.

Ui=The expected return of the ith security.

Finally, we come to the V portion of E–V theory, the variance in expected
returns. This is the sum of the variances contributed by each security in the
portfolio plus the sum of all the possible covariances in the portfolio:

V=

∑N

i= 1

∑N

j= 1

Xi∗Xj∗COVi, j (7.06a)