How compounding frequency works
Daily, monthly, quarterly or annual compounding, what the difference is actually worth, and why the effective annual rate is the only figure worth comparing between products.
Try different frequenciesNominal and effective rates
A product quotes a nominal annual rate and a compounding frequency. Those two together produce an effective annual rate, which is what you actually earn over a year: (1 + r/n)^n − 1, where r is the nominal rate and n the number of compounding periods.
10% compounded monthly is an effective 10.47%. Compounded daily it is 10.52%. Compounded annually it is exactly 10%. The effective rate is the number to compare, and it is the number regulators in most countries require to be shown - as the APY on a deposit or the APR on a loan.
The diminishing return
Each step up in frequency adds less than the one before, and the sequence converges. Continuous compounding - the theoretical limit - gives 10.517% for a nominal 10%, which is barely above daily.
| Frequency | Effective annual rate |
|---|---|
| Annually | 10.000% |
| Half-yearly | 10.250% |
| Quarterly | 10.381% |
| Monthly | 10.471% |
| Daily | 10.516% |
Why the sequence converges
Each extra compounding period earns interest on a slightly larger balance, but it also earns it for a shorter time, and the two effects almost cancel. Going from annual to half-yearly adds a quarter of a percentage point; going from monthly to daily adds four hundredths. Past monthly, the frequency has effectively stopped mattering.
The limit is continuous compounding, where the effective rate is e^r − 1. For a nominal 10% that is 10.517%, which is barely more than daily. There is no frequency at which a 10% nominal rate becomes an 11% effective one, which is the intuition worth having when a product advertises how often it compounds.
It works the same way on debt
A credit card quoting a monthly rate of 2% is not charging 24% a year. Compounded monthly, 2% a month is an effective 26.8%, and the gap widens the higher the rate goes. The same arithmetic that makes frequent compounding barely matter on a 5% deposit makes it matter a great deal on a 30% balance.
Regulators generally require lenders to publish the effective figure - as an APR or an equivalent - for exactly this reason. If you are comparing a monthly rate with an annual one, convert first.
What to do with this
Convert every product you are comparing to its effective annual rate and then ignore the frequency entirely. A quarter of a percentage point on the headline rate is worth more than any change in compounding frequency, and a marketing page emphasising daily compounding is usually drawing attention away from the rate it is attached to.
The one exception is where compounding frequency interacts with when you pay in or take out. A deposit that compounds daily and lets you withdraw at any time is genuinely different from one that compounds annually and forfeits the year's interest on early withdrawal - but that difference is about the withdrawal terms, not about the compounding.