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: