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 $S_{rm end} := (1+r)S$ at the end. That is:

$r = frac{S_{rm end}-S}{S}.$

Log-returns

Often, the rates of return are approximated, especially if the period length is small. If r << 1 , then

$r = frac{S_{rm end}}{S} - 1 approx y:= log frac{S_{rm end}}{S},$

with the latter quantity known as log-return.

Rate of return of a portfolio

For assets, we can define the vector $r in mbox{bf R}^n$ , with r_i 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 x_i (in ) in the -th asset. Here $x in mathbf{R}^n$ is a non-negative vector which sums to one. Then the portfolio we constituted this way will earn

$S_{rm end} := sum_{i=1}^n (1 + r_i) x_i S .$

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

$frac{S_{rm end} - S}{S} = sum_{i=1}^n (1 + r_i) x_i - 1 = sum_{i=1}^n x_i - 1 + sum_{i=1}^n r_ix_i = r^Tx.$

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 ).