How continuous compounding works

A quoted rate that is added $n$ times a year becomes an effective annual rate:

$$\left(1 + \frac{r}{n}\right)^{n} - 1$$

Send $n$ toward infinity and the expression converges on

$$e^{r} - 1$$

That is continuous compounding. It is not a fourth frequency next to monthly and daily. It is the limit those frequencies approach.

At 8 percent, $e^{0.08} - 1 = 0.083287$, which is 8.3287 percent. On $25,000 that is $2,082.18 of interest in a year. A single annual credit of 8 percent would pay $2,000. The extra $82.18 is interest that was itself earning interest, taken to the mathematical ceiling.

The APR against APY calculator on this page prints that ceiling as the last row of the frequency table. How APR and APY work is the finite conversion, on a 5 percent sheet. This page is the limit.

Keep the 8 percent and the $25,000. Compound monthly and the yield is 8.3000 percent, which is $2,074.99. Compound daily and the yield is 8.3278 percent, already next to the 8.3287 percent ceiling.

Daily compounding has already captured almost the whole ceiling. The remaining gap to 8.3287 percent is hundredths of a point. Marketing that treats continuous compounding as a different product, rather than as the limit of a daily calendar, is selling the last sliver of a curve that has already flattened.

Monthly against continuous compounding is that pair in a table. The APR APY explorer is the finite conversion as two bars.

Hold a $20,000 balance and raise the nominal rate to 15 percent. Continuous compounding is 16.1834 percent, which is $3,236.68 of interest. A single annual credit would pay $3,000. The extra is $236.68.

The 8 percent sheet's extra was $82.18 on $25,000. The 15 percent sheet's extra is larger as a share of the balance, because $e^{r} - 1 - r$ widens as $r$ rises. Frequency still has a ceiling. The rate is what opened the gap.

A quarter of a point on the quoted rate beats the entire jump from yearly compounding to continuous compounding at ordinary deposit rates. Chasing a continuous schedule on an 8 percent account, rather than an 8.25 percent account that credits monthly, is chasing hundredths while leaving tenths on the table.

It is not a promise that a bank credits interest every instant. Almost no deposit account does. The formula answers what the quoted rate would become if it did.

It is not a US APY by itself. APY is the effective annual rate a deposit account is required to quote. That quote already contains whatever schedule the account actually uses. Do not take a published APY and then apply $e^{r} - 1$ to it: that would compound a second time.

It is not the 5.127 percent ceiling on the 5 percent APR page, and it is not the 6.1837 percent line on the compound-interest table. Those pages own those rates. This page owns 8.3287 percent on $25,000 and 16.1834 percent on $20,000.

In actuarial notation the continuously compounded rate is the force of interest, $\delta$, and $e^{\delta} - 1$ is the effective annual rate. This page writes $r$ for the same nominal input the rest of the site uses, so the calculator above and the identity stay on one convention.

Nominal rate is the quoted $r$. Effective annual rate is what a year actually produces. Continuous compounding is the largest effective rate a given $r$ can become.

It is not a ranking of banks, not a claim that continuous compounding is available on a given account, and not a second pass over the 5 percent or 6 percent teaching sheets. The two sheets here are 8 percent on $25,000 (8.3287 percent, $2,082.18) and 15 percent on $20,000 (16.1834 percent, $3,236.68). This is educational material, not financial advice.