Interest & Compounding

Interest that earns interest: from simple growth to the continuous limit.

The idea

Interest is what a lender is paid for parting with money for a time. Leave a principal $P$ in an account at an annual rate $r$, and a year later the account has earned $rP$. Everything else depends on what happens to that interest.

If the interest is withdrawn each year, so that only the original $P$ ever earns, the account gains the same $rP$ every year, and after $t$ years it holds

$A = P(1 + rt).$

This is simple interest, and it grows linearly. If instead the interest is left in the account, it earns interest itself: each year multiplies the whole balance by $1 + r$, so after $t$ years

$A = P(1 + r)^{t}.$

This is compound interest, and it grows exponentially.

A bank may also compound within the year. Cut the year into $m$ equal slices and pay $\frac{r}{m}$ on each; $t$ years contain $mt$ slices, so $A = P\left(1 + \frac{r}{m}\right)^{mt}$. Finer slicing always pays slightly more, but the balance does not grow without bound: as $m$ increases it approaches $A = Pe^{rt}$, which is continuous compounding.

Ways to work on it

• Walkthrough. Compare simple and compound interest on the same deposit, up to the continuous limit.
• Practice. Compute balances under simple, compound, and continuous compounding.
• Hardest. Recover an unknown rate or time from a target balance, in exact form.

Not sure where to start? Take the ten-question placement test.