1.2a Compounding

Compounding

Compounding is the process by which interest earned (or charged) on a principal $P$ also begins to earn additional interest in subsequent periods.

Cash flows can be compounded using the nominal interest rate over one compounding period per year, also known as the Annual Percentage Rate (APR), or using its geometric average, the Effective Annual Rate (EAR), over $n$ compounding periods per year, such that:
$\mathrm{EAR}={\left(1+\frac{r}{n}\right)}^n-1.$

Letting $n\to \infty$ gives continuous compounding:
${\lim }_{n\to \infty }\left[{\left(1+\frac{r}{n}\right)}^n-1\right]=e^r-1,$
for which $\ln (1+\text{EAR})=r$.