Suppose that one has a portfolio of
assets.
The price of asset
at time
is denoted as
.
The return from asset
in a single time period (day, month, year etc.) is:
We observe the vectors
(i.e., the returns of the assets which are contained in the portfolio)
over several time periods.
We stack these observations into a data matrix
consisting of observations of a random variable
The return of the portfolio is the weighted sum of the
returns of the
assets:
 |
(17.1) |
where
(with
) denotes
the proportions of the assets in the portfolio.
The mean return of the portfolio is given by the expected value of
,
which is
. The risk or volatility of the portfolio
is given by the variance of
(Theorem 4.6),
which is equal to two times
 |
(17.2) |
The reason for taking half of the variance of
is merely technical.
The optimization of (17.2) with respect to
is of course equivalent
to minimizing
.
Our aim is to maximize the portfolio returns (17.1) given
a bound on the volatility (17.2) or vice versa to minimize risk given
a (desired) mean return of the portfolio.
Summary

-
Given a matrix of returns
from
assets
in
time periods, and that
the underlying distribution is stationary, i.e.,
, then the (theoretical) return of the portfolio
is a weighted sum of the returns of the
assets, namely
.

-
The expected value of
is
. For technical reasons one considers
optimizing
. The risk or
volatility is
.

-
The portfolio choice, i.e., the selection of
, is such that the return
is maximized for a given risk bound.