The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 486 Wednesday, February 4, 2004 1:09 PM

486 The Mathematics of Financial Modeling and Investment Management

Suppose asset returns are determined by a multifactor model (as

described in Chapter 18) so that the expected return of the i-th security

is a linear combination of p factors. We can then write

p

μi = αi + ∑βijfj , j = 1,2,..., p

j = 1

where μi are expected returns and fj are the expectations of factors.

Exposure to the j-th factor can be controlled by constraining the

beta βaj of portfolio a relative to that factor:

N

∑ waiβij = βaj

i = 1

where wai are the weights of portfolio a.

A portfolio manager might want to maximize a portfolio’s return

given a target level of risk. This problem would lead to maximizing a

linear function subject to quadratic constraints of the form

wa′ΣΣΣΣwa = wa

In practice, however, a portfolio manager prefers to minimize a

function of the type:

wa′ΣΣΣΣwa – λwa ′μμμμ

where μμμμis the vector of securities’ expected returns and λ is a risk-aver-

sion parameter. A function of this type implements a compromise

between risk and returns.

Finally, a portfolio manager needs to impose lower thresholds on

portfolio weights to avoid portfolios being made up of a large number

of small holdings. This implies the constraints wai ≥ bi. In practice,

therefore, mean-variance portfolio selection leads to a quadratic optimi-

zation problem of the following type:

Minimize

wa′ΣΣΣΣwa – λwa ′μμμμ

subject to