Programming Project One: Portfolio Allocation
(Due Wednesday April 30, 2008)
The object of this project is to setup and solve a nontrivial linear programming problem. The
particular problem we will examine comes from Chapter 13, Section 1 in your book and you are
encouraged to read the discussion found there.
We will use Matlab (which is available on the math department computers) to solve this problem.
Part of the goal of this project is to become familar with solving linear programs on a computer. I
will provide a short tutorial on the use of the Matlab command “linprog” on the course webpage
and possibly discuss it in class.
Introduction
We will solve the problem of selecting an appropriate mix of assets to include in an investment
portfolio. Supp ose we are given a selection of npotential investments. We will let Rjdenote the
return on investment jin the next investment period. Of course, we do not know a priori the values
of the Rj. They represent the future return on an investment made now. We can, however, try to
estimate the behavior of the Rjusing historical data.
Random Variables
In particular, we treat the Rjas random variables — unknown quanitities associated with a prob-
ability distribution. We might not be able to say that Rjwill have a particular value in the next
time period, but we are able to estimate the probability that it will have a certain value. For our
purposes, we will only be interested in a single quantity associated with a random variable, the
expected value.
Given a random variable Rand a set of observed values x1, . . . , xnof R, we can estimate the expected
value of Ras
(1) ER≈1
nX
j
xj.
Let us now consider the particular case of an investment Rj. For each investment j, we will be given
a set
(2) Rj(1), Rj(2), . . . , Rj(T)
for Tprevious observations of the return on investment j. This is the historical data available for
each investment. Each observation is the return on that investment over one time interval in the
past. We will then estimate the expected value of the random variable Rjas
(3) ERj≈1
TX
t
Rj(t).
We interpret this quanitity as the amount of money we expect to make in the next investment
interval if we invest one dollar in the jth investment.
Portfolios
A portfolio is determined by specifying what fraction of one’s assets are put into each asset. That
is, a portfolio is a sequence of nonnegative real numbers x1, . . . , xnsuch that
(4) x1+. . . +xn= 1.
The return on a given portfolio in the next time period would then be
(5) R=X
j
xjRj.
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