Ralph Vince - Portfolio Mathematics

254 THE HANDBOOK OF PORTFOLIO MATHEMATICS

identity matrix is obtained before we can interpret them. This is in accor-
dance with the generalized form of our problem. The L 1 variable equals
−δV/δE. This means that L 1 represents the marginal variance in expected
returns. In the case of our example, where L 1 =− 2 .6394, we can state that
V is changing at a rate of−L 1 ,or−(−2.6394), or 2.6394 units for every unit
in E instantaneously at E=.14.
To interpret the L 2 variable requires that the problem first be restated.
Rather than havingi=1, we will state thati=M, where M equals
the dollar amount of funds to be invested. Then L 2 =δV/δM. In other
words, L 2 represents the marginal risk of increased or decreased invest-
ment.
Returning now to what the variance of the entire portfolio is, we can
use Equation (7.06) to discern the variance. Although we could use any
variation of Equation (7.06a) through (7.06d), here we will use variation a:

V=

∑N

i= 1

∑N

j= 1

Xi∗Xj∗COVi, j

Plugging in the values and performing Equation (7.06a) gives:

Xi Xj COVi,j

0.12391 ∗ 0.12391 ∗ 0.1 = 0.0015353688

0.12391 ∗ 0.12787 ∗−0.0237 =−0.0003755116

0.12391 ∗ 0.38407 ∗ 0.01 = 0.0004759011

0.12391 ∗ 0.36424 ∗ 0 = 0

0.12787 ∗ 0.12391 ∗−0.0237 =−0.0003755116

0.12787 ∗ 0.12787 ∗ 0.25 = 0.0040876842

0.12787 ∗ 0.38407 ∗ 0.079 = 0.0038797714

0.12787 ∗ 0.36424 ∗ 0 = 0

0.38407 ∗ 0.12391 ∗ 0.01 = 0.0004759011

0.38407 ∗ 0.12787 ∗ 0.079 = 0.0038797714

0.38407 ∗ 0.38407 ∗ 0.4 = 0.059003906

0.38407 ∗ 0.36424 ∗ 0 = 0

0.36424 ∗ 0.12391 ∗ 0 = 0

0.36424 ∗ 0.12787 ∗ 0 = 0

0.36424 ∗ 0.38407 ∗ 0 = 0

0.36424 ∗ 0.36424 ∗ 0 = 0

.0725872809

Thus, we see that at the value of E=.14, the lowest value for V is
obtained at V=.0725872809.