Rate of return of a financial portfolio

Rate of return of a single asset

The rate of return  (or, return) of a financial asset over a given period (say, a year, or a day) is the interest obtained at the end of the period by investing in it. In other words, if, at the beginning of the period, we invest a sum in the asset, we will earn  at the end. That is:

Log-returns

Often, the rates of return are approximated, especially if the period length is small. If , then

with the latter quantity known as log-return.

Rate of return of a portfolio

For  assets, we can define the vector , with the rate of return of the -th asset.

Assume that at the beginning of the period, we invest a sum in all the assets, allocating a fraction (in ) in the -th asset. Here  is a non-negative vector which sums to one. Then the portfolio we constituted this way will earn

The rate of return of the porfolio is the relative increase in wealth:

The rate of return is thus the scalar product between the vector of individual returns and of the portfolio allocation weights .

Note that, in practice, rates of return are never known in advance, and they can be negative (although, by construction, they are never less than ).