Rate of return of a single asset

The rate of return [latex]r[/latex] (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 [latex]S[/latex] in the asset, we will earn [latex]S_{\text{end}}:=(1+r)S[/latex] at the end. That is:

[latex]\begin{align*} r &= \frac{S_{\text{end}} - S}{S}. \end{align*}[/latex]

Log-returns

Often, the rates of return are approximated, especially if the period length is small. If [latex]r \ll 1[/latex], then

[latex]\begin{align*} r &= \frac{S_{\text{end}}}{S} - 1 \approx y := \log\left(\frac{S_{\text{end}}}{S}\right), \end{align*}[/latex]

with the latter quantity known as log-return.

Rate of return of a portfolio

For [latex]n[/latex] assets, we can define the vector [latex]r \in \mathbb{R}^n[/latex], with [latex]r_i[/latex] the rate of return of the [latex]i[/latex]-th asset.

Assume that at the beginning of the period, we invest a sum [latex]S[/latex] in all the assets, allocating a fraction [latex]x_i[/latex] (in [latex]\%[/latex]) in the [latex]i[/latex]-th asset. Here [latex]x \in \mathbb{R}^n[/latex] is a non-negative vector which sums to one. Then the portfolio we constituted this way will earn

[latex]\begin{align*} S_{\text{end}} &= \sum\limits_{i=1}^{n} (1 + r_i)x_iS. \end{align*}[/latex]

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

[latex]\begin{align*} \frac{S_{\text{end}} - S}{S} &= \sum\limits_{i=1}^n (1 + r_i)x_i - 1 = \sum\limits_{i=1}^n x_i - 1 + \sum\limits_{i=1}^n r_i x_i = r^T x. \end{align*}[/latex]

The rate of return is thus the scalar product between the vector of individual returns [latex]r[/latex] and of the portfolio allocation weights [latex]x[/latex].

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 [latex]-1[/latex]).