3.1b CAGR

Compound Annual Growth Rate

Compound Annual Growth Rate (also called compound return) is the cumulative growth of an investment over multiple periods, where the net simple return $R_t$ from each period $t$ is reinvested and earns returns in subsequent periods. Formally, the CAGR between time $t_0$ and $t$, given $n$-period returns, is given as
$1+R_{t_0,t}={\left[{\prod }_{i=1}^t(1+R_{t_0+i})\right]}^{\frac{1}{nt}}.$

The compounded return over $T$ years, paid $n$ times per year and given an APR $r$ can generally be calculated as
$1+R_T={\left(1+\frac{r}{n}\right)}^{nT}.$

Letting $n\to \infty$ yields continuous returns, such that
$1+R_T={\lim }_{n\to \infty }\left[{\left(1+\frac{r}{n}\right)}^{nt}\right]=e^{rT},$
which yields the log return $\ln (1+R_T)=rT$.

Compound Return Excludes the Initial Period $t$ Period $t$ isn’t included in the compound return formula because it’s the the subsequent returns that measure the growth or changes in value over the previous period, not the initial period itself.