The geometric mean rate of return

Consider a horizon of disjoint periods labelled $1$ through $N$. On the entire horizon, an asset with a rate of return $r_n$ on the $n$-th period has a geometric mean rate of return of \begin{equation} \overline{r} = \left( \prod_{n = 1}^N \left(1 + r_n\right) \right)^{1 / N} - 1. \end{equation} Note in particular that \begin{equation} \left(1 + \overline{r}\right)^N = \prod_{n = 1}^N \left(1 + r_n\right). \end{equation} In other words, a hypothetical asset whose rate of return is equal to $\overline{r}$ on all $N$ periods yields the same return (over the horizon) as the original asset.