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Monthly vs continuous compounding

Monthly compounding is a finite schedule. Continuous compounding is the ceiling that schedule approaches. At 8 percent, monthly is 8.3000 percent ($2,074.99 on $25,000) and continuous is 8.3287 percent ($2,082.18).

 Monthly compoundingContinuous compounding
Formula(1+r/12)12−1(1 + r/12)^{12} - 1. Twelve credits a year.er−1e^{r} - 1. The limit as the number of credits goes to infinity.
Teaching sheet at 8 percentYield 8.3000 percent. $25,000 earns $2,074.99.Yield 8.3287 percent. $25,000 earns $2,082.18.
At 15 percentA monthly schedule on the cousin calculator, not this sheet's published dollars.16.1834 percent. $20,000 earns $3,236.68. The extra over a single annual credit is $236.68.
What it is silent onThe ceiling. 8.3000 percent is not 8.3287 percent.Whether any account actually credits every instant. Almost none do.
When the calendar is already dailyDaily at 8 percent is 8.3278 percent, already next to the ceiling.The remaining sliver. Marketing that sells continuous as a different product is selling that sliver.
One rateThe 8 percent sheet. Do not paste this 8.3000 percent onto the 5 percent APR page.The same 8 percent sheet. Do not paste this 8.3287 percent onto the 6 percent compound-interest table.
On this page

A schedule against a ceiling

Monthly compounding is one finite nn in the effective-rate identity:

(1+r12)12−1\left(1 + \frac{r}{12}\right)^{12} - 1

At 8 percent that is 8.3000 percent. On $25,000 the interest is $2,074.99.

Continuous compounding is the same identity with nn sent to infinity:

er−1e^{r} - 1

At 8 percent that is 8.3287 percent. On the same $25,000 the interest is $2,082.18.

How continuous compounding works owns the 8.3287 percent. How APR and APY work owns the finite conversion on a 5 percent sheet. The APR against APY calculator prints both rows.

The two numbers are close because a monthly calendar has already done most of the work. Daily compounding at 8 percent is 8.3278 percent, which sits between them.

The rate opens the gap, not the calendar

On $20,000 at 15 percent, continuous compounding is 16.1834 percent and the interest is $3,236.68. The extra over a single annual credit of $3,000 is $236.68. That extra is a larger share of the balance than the 8 percent sheet's $82.18, because er−1−re^{r} - 1 - r widens as rr rises.

A quarter of a point on the quoted rate still beats the jump from monthly to continuous at ordinary deposit rates. This is educational material, not financial advice.

Worked examples

8 percent on \$25,000, monthly

A nominal 8 percent is compounded monthly. What does $25,000 earn in a year?

  1. Effective rate: (1+0.08/12)12−1=0.083000(1 + 0.08/12)^{12} - 1 = 0.083000, which is 8.3000 percent.
  2. Interest: 25000×0.083000=2074.9925000 \times 0.083000 = 2074.99, so $2,074.99.

The yield is 8.3000 percent. $25,000 earns $2,074.99.

8 percent on \$25,000, continuously

The same 8 percent is compounded continuously. What does $25,000 earn in a year?

  1. Effective rate: e0.08−1=0.083287e^{0.08} - 1 = 0.083287, which is 8.3287 percent.
  2. Interest: 25000×0.083287=2082.1825000 \times 0.083287 = 2082.18, so $2,082.18.
  3. A single annual credit would pay $2,000. The extra is $82.18.

The yield is 8.3287 percent. $25,000 earns $2,082.18 rather than $2,000. The extra is $82.18.

15 percent on \$20,000, continuously

A nominal 15 percent is compounded continuously. What does $20,000 earn in a year?

  1. Effective rate: e0.15−1=0.161834e^{0.15} - 1 = 0.161834, which is 16.1834 percent.
  2. Interest: 20000×0.161834=3236.6820000 \times 0.161834 = 3236.68, so $3,236.68.
  3. A single annual credit would pay $3,000. The extra is $236.68.

The yield is 16.1834 percent. $20,000 earns $3,236.68 rather than $3,000. The extra is $236.68.

Common questions

Which one pays more?

Continuous compounding, for any positive rate. At 8 percent the gap on $25,000 is the difference between $2,082.18 and $2,074.99. Daily compounding has already captured most of it.

Is continuous compounding a real product?

Almost never as a credit calendar. It is the ceiling a quoted rate approaches. A published APY already contains the schedule the account uses.

Can I convert a monthly APY into a continuous rate?

Not usefully, and not by applying er−1e^{r} - 1 to the APY. That would compound a finished effective rate a second time. Convert from the nominal rate, or leave the APY alone.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.