| # 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 ).