Consider a holding period consisting of two consecutive years. In year one, the return from investing in a given stock share is \(+10 \%\); in year two the return is \(10 \%\). What was the "average" return?

As it turns out, the question is stated in a very imprecise way. It might be tempting to say that, trivially, the average return was \(0 \%\), the familiar arithmetic mean of \(+10 \%\) and \(10 \%\). However, we cannot really add returns like this. Over the two years, the gain was

\[G=(1+010) \quad(1 \quad 010)=099\]

i.e., we have lost money, as the holding period return was \(1 \%\) [we may recall the rule \(\left.(1+x)\left(\begin{array}{lll}1 & x\end{array}\right)=1 \quad x^{2}\right]\). Indeed, the problem is that the very term "average" is ambiguous. If what we actually mean is the expected value of the annual return, which we may estimate by a sample mean, then we may say that the arithmetic average is, in fact

\[\bar{R}_{a}=\frac{010 \quad 010}{2}=0\]

But if we mean an average over time, we should deal with a sort of geometric average over two years:

\[(1+010) \quad(1 \quad 010)=\left(1+\bar{R}_{g}\right)^{2} \quad \bar{R}_{g}=05013 \%\]

We may also notice that, in this case, an average should refer to a standard time interval, usually one year. Indeed, we should not confuse the holding period return with an annual (rate of) return. We will need a way to annualize a generic holding period return.