Portfolio Optimization

Chapter 7

Portfolio Optimization with

Constraints

In all the theoretical developments so far, we have used the model

problem

min{ − tµ

′

x +

1

2

x

′

Σx | l

′

x = 1}, (7.1)

or its extension to include a risk free asset. The parameter t ≥ 0 quan-

tifies t he risk aversion of the investor. Although (7 .1 ) is very useful

for developing the basic concepts for po r t folio optimization, it is not

particularly suitable in practice. There are two main reasons for this.

First, the solution o f (7.1) may result in excessive long and short sell-

ing. An example of this is an optimal solution of (7.1) with x

1

= 1000,

x

2

= −1000, x

3

= 1, x

4

= 0, . . . , x

n

= 0. This means that