Interest

Simple interest is interest calculated only on the initial principal, without compounding.

Compound interest is hence interest calculated based on the initial principal and the accumulated interest from previous periods. Related to the Compound Effect.

Compound interest grows at exponential rates while simple interest grows at linear rates. This is because compound interest includes interest on previously earned interest, while simple interest is only applied on the principal.

Annual Percentage Rate

APR, or $r_{A P R}$ represents an annual interest for a loan or investment compounded over a shorter time period.

To convert an APR interest rate to a per-period interest rate:

r = \frac{r_{A P R}}{m}

For instance:

A credit card interest rate of 18% per year compounded monthly
$r$ = 1.5% per month

Use the per-period interest rate when compounding.

The FV of $100 after 3 years with 10% semi-annual compounding:
$F V = $ 100 (1 + \frac{0.10}{2})^{2 \times 3} = $ 100 (1.05)^{6}$

Effective Annual Rate

EAR, or $r_{E F F}$ is the annual interest rate with annual compounding that produces the same Future Value as $r_{A P R}$ compounded over a shorter time period.

In other ways, EAR is the annual interest rate that accounts for compounding:

E A R = (1 + \frac{r_{A P R}}{m})^{m} - 1

In continuous compounding i.e. $m$ tends to $\infty$ , $r_{E F F} = e^{r_{A} P R} - 1$