Ralph Vince - Portfolio Mathematics

Classical Portfolio Construction 243

and

(N

∑

i= 1

Xi

)

− 1 = 0 (7.10)

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

E=The expected return of the portfolio.

Xi=The percentage weighting of the ith security.

Ui=The expected return of the ith security.

The minimization of a restricted multivariable function can be han-
dled by introducing these Lagrangian multipliers and differentiating par-
tially with respect to each variable. Therefore, we express our problem in
terms of a Lagrangian function, which we call T. Let:

T=V+L 1 ∗

((

∑N

i= 1

Xi∗Ui

)

−E

)

+L 2 ∗

((

∑N

i= 1

Xi

)

− 1

)

(7.11)

where: V=The variance in the expected returns of the portfolio, from

Equation (7.06).

N=The number of securities comprising the portfolio.

E=The expected return of the portfolio.

Xi=The percentage weighting of the ith security.

Ui=The expected return of the ith security.

L 1 =The first Lagrangian multiplier.

L 2 =The second Lagrangian multiplier.

The minimum variance (risk) portfolio is found by setting the first-order
partial derivatives of T with respect to all variables equal to zero.
Let us again assume that we are looking at four possible investment
alternatives: Toxico, Incubeast Corp., LA Garb, and a savings account. If we
take the first-order partial derivative of T with respect to X 1 we obtain:

∂T/∂X 1 = 2 ∗X 1 ∗COV1, 1+ 2 ∗X 2 ∗COV1, 2+ 2 ∗X 3 ∗COV1, 3

+ 2 ∗X 4 ∗COV1, 4+L 1 ∗U 1 +L 2 (7.12)

Setting this equation equal to zero and dividing both sides by 2 yields:

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