Ralph Vince - Portfolio Mathematics

244 THE HANDBOOK OF PORTFOLIO MATHEMATICS

Likewise:

∂T/∂X 2 =X 1 ∗COV2, 1+X 2 ∗COV2, 2+X 3 ∗COV2, 3+X 4 ∗COV2, 4

+. 5 ∗L 1 ∗U 2 +. 5 ∗L 2 = 0

∂T/∂X 3 =X 1 ∗COV3, 1+X 2 ∗COV3, 2+X 3 ∗COV3, 3+X 4 ∗COV3, 4

+. 5 ∗L 1 ∗U 3 +. 5 ∗L 2 = 0

∂T/∂X 4 =X 1 ∗COV4, 1+X 2 ∗COV4, 2+X 3 ∗COV4, 3+X 4 ∗COV4, 4

+. 5 ∗L 1 ∗U 4 +. 5 ∗L 2 = 0

And we already have∂T/∂L 1 as Equation (7.09) and∂T/∂L 2 as Equation
(7.10).
Thus, the problem of minimizing V for a given E can be expressed in
the N-component case as N+2 equations involving N+2 unknowns. For
the four-component case, the generalized form is:

X 1 ∗U 1 +X 2 ∗U 2 +X 3 ∗U 3 +X 4 ∗U 4 =E
X 1 +X 2 +X 3 +X 4 = 1
X 1 ∗COV1, 1+X 2 ∗COV1, 2+X 3 ∗COV1, 3+X 4 ∗COV1, 4+. 5 ∗L 1 ∗U 1 +. 5 ∗L 2 = 0
X 1 ∗COV2, 1+X 2 ∗COV2, 2+X 3 ∗COV2, 3+X 4 ∗COV2, 4+. 5 ∗L 1 ∗U 2 +. 5 ∗L 2 = 0
X 1 ∗COV3, 1+X 2 ∗COV3, 2+X 3 ∗COV3, 3+X 4 ∗COV3, 4+. 5 ∗L 1 ∗U 3 +. 5 ∗L 2 = 0
X 1 ∗COV4, 1+X 2 ∗COV4, 2+X 3 ∗COV4, 3+X 4 ∗COV4, 4+. 5 ∗L 1 ∗U 4 +. 5 ∗L 2 = 0

where: E=The expected return of the portfolio.

Xi=The percentege weighting of the ith security.

Ui=The expected return of the ith security.

COVA, B=The covariance between securities A and B.

L 1 =The first Lagrangian multiplier.

L 2 =The second Lagrangian multiplier.

This is the generalized form, and you use this basic form for any number
of components. For example, if we were working with the case of three
components (i.e., N=3), the generalized form would be:

X 1 ∗U 1 +X 2 ∗U 2 +X 3 ∗U 3 =E

X 1 +X 2 +X 3 = 1

X 1 ∗COV1, 1+X 2 ∗COV1, 2+X 3 ∗COV1, 3+. 5 ∗L 1 ∗U 1 +. 5 ∗L 2 = 0

X 1 ∗COV2, 1+X 2 ∗COV2, 2+X 3 ∗COV2, 3+. 5 ∗L 1 ∗U 2 +. 5 ∗L 2 = 0

X 1 ∗COV3, 1+X 2 ∗COV3, 2+X 3 ∗COV3, 3+. 5 ∗L 1 ∗U 3 +. 5 ∗L 2 = 0

You need to decide on a level of expected return (E) to solve for, and your
solution will be that combination of weightings which yields that E with the
least variance. Once you have decided on E, you now have all of the input
variables needed to construct the coefficients matrix.