The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 476 Wednesday, February 4, 2004 1:09 PM

476 The Mathematics of Financial Modeling and Investment Management

and its variance is

σ

2

wa 1 wa 2

σ 11 σ 12 w 1

a =

σ 21 σ 22 w 2

= {wa 1 σ 11 + wa 1 σ 21 wa 2 σ 12 + wa 2 σ 22 }

w 1

w 2

= w^2 a 1 σ 11 + wa^2 2 σ 22 + 2 wa 1 wa 2 σ 12

= w^2 σ^22

1 + w

(^2) σ
a 1 a 2 2 +^2 wa 1 wa 2 σ 12
By choosing the portfolio’s weights, an investor chooses among the
available mean-variance pairs. Following Markowitz, the investor’s
problem is a constrained minimization problem in the sense that the
investor must seek
min () σ^2 a = min(wa′ΣΣΣΣwa )
subject to the constraints
μa = wa ′μμμμ
wa ′ιιιι= 1 , ιιιι′ = [ 11 ,,..., 1 ]
This is a constrained optimization problem which can be solved
with the method of Lagrange multipliers. Recall from Chapter 7 that
this method transforms a constrained optimization problem into an
unconstrained optimization problem by forming the Lagrangian, that is,
the sum of the function to be optimized and a linear combination of the
constraints. In this case, the Lagrangian is
L = wa′ΣΣΣΣwa + δ 1 (μa – wa ′μμμ) + δ 2 ( 1 – wa ′ιιι)
The original optimization problem becomes the problem of uncon-
strained maximization of the Lagrangian. To solve this problem, it is
sufficient to set to zero the partial derivatives of the Lagrangian. Solving
yields
wa = gh+ μa
where gand hare two vectors which are functions of μμμμand ΣΣΣΣ.