Effective Annual Rate
With nominal annual rate 6 %, on a balance of 10000 USD, effective annual rate comes to 6.1678 % — effective annual rate. It is reached in 4 steps, the last of which is 0 * ((exp(6 / 100) - 1) * 100) + (1 - 0) * 6.1677812, and each one is printed on the page with its numbers filled in. The formula is the one published by 12 CFR Part 1030 (Regulation DD), not an approximation fitted to it.
The effective annual rate a nominal rate becomes once compounding is counted, and the gap that makes two accounts quoting the same rate pay different amounts.
Formula and sources checked · How we check
Nominal annual rate 6, Compounded Monthly — twelve times, On a balance of 10000
6.1678 %
Effective annual rate for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Effective annual rate
0 * ((exp(6 / 100) - 1) * 100) + (1 - 0) * 6.16778126.168 %- Added by compounding
6.1677812 - 60.168 %- A year of interest on that balance
10000 * 6.1677812 / 100616.778 USD- More than simple interest would pay
10000 * (6.1677812 - 6) / 10016.778 USD
Worked example
6% compounded monthly is an effective 6.1678%. On $10,000 that is $616.78 rather than $600 — the extra $16.78 is interest earned on interest, and it is the whole difference between the two numbers a bank may quote.
How to work it out yourself
- 1.Enter the nominal rate as advertised, then say how often it compounds. The nominal rate alone does not determine what you earn.
- 2.Compare accounts on the effective rate. A US savings account quotes APY, which is already effective; a loan quotes APR, which is nominal — so the two are not comparable as printed.
- 3.Continuous compounding is the ceiling. 6% continuous is 6.1837%, only two hundredths above monthly, so the difference between daily and continuous is almost never worth anything.
- 4.For a loan, the effective rate is what you pay, and it is higher than the APR whenever interest is charged more often than annually.
Effective rate from a monthly-compounded nominal rate
| Nominal annual rate (%) | Effective annual rate | Added by compounding |
|---|---|---|
| 1 | 1.0046 % | 0 % |
| 2 | 2.0184 % | 0.02 % |
| 3 | 3.0416 % | 0.04 % |
| 4 | 4.0742 % | 0.07 % |
| 5 | 5.1162 % | 0.12 % |
| 6 | 6.1678 % | 0.17 % |
| 8 | 8.3000 % | 0.3 % |
| 10 | 10.4713 % | 0.47 % |
| 12 | 12.6825 % | 0.68 % |
| 15 | 16.0755 % | 1.08 % |
| 18 | 19.5618 % | 1.56 % |
| 24 | 26.8242 % | 2.82 % |
The gap grows with the square of the rate, roughly. At 1% nominal monthly compounding adds 0.005 points; at 24% it adds 2.8. That is why credit card APRs and the rate you actually pay diverge so far.
The formula
- Effective annual rate
0 * ((exp(6 / 100) - 1) * 100) + (1 - 0) * 6.1677812 - Added by compounding
6.1677812 - 6 - A year of interest on that balance
10000 * 6.1677812 / 100 - More than simple interest would pay
10000 * (6.1677812 - 6) / 100
Source: 12 CFR Part 1030 (Regulation DD) — how APY must be calculated and disclosed, 12 CFR Part 1026 (Regulation Z) — APR disclosure on credit, FDIC — comparing deposit account rates
Questions people actually ask
- What is the difference between APR and APY?
- APR is the nominal rate with no compounding assumed; APY is the effective rate with compounding included. Regulation DD requires deposit accounts to advertise APY and Regulation Z requires credit to advertise APR, so a 6% savings account and a 6% loan are not offering the same rate.
- What is the effective annual rate formula?
- (1 + r/n)^n − 1, where r is the nominal rate and n the compounding periods per year. For continuous compounding it is e^r − 1.
- Why does compounding frequency matter so little?
- At ordinary rates it barely does. 6% compounded monthly is 6.1678% and compounded daily is 6.1831% — a 15-basis-point difference. It matters at high rates: at 24%, monthly gives 26.82% and daily gives 27.11%.
- What is the highest an effective rate can go?
- The continuous limit, e^r − 1. A 100% nominal rate compounded continuously is 171.83%, not infinite. That ceiling is where the number e comes from — it is the limit of (1 + 1/n)^n.
- Does this apply to credit cards?
- Yes, and it is where the gap bites. A card advertising 24.99% APR compounds daily, giving an effective 28.38%. Carrying $5,000 costs $1,419 a year rather than the $1,250 the headline implies.
Related
- Compound Interest CalculatorFuture value of a starting balance plus monthly contributions, separating what you put in from what the interest earned.
- APY CalculatorAnnual percentage yield from a rate and its compounding frequency, and what it earns on a balance in dollars.
- Doubling TimeHow long a quantity growing at a fixed rate takes to double, with the Rule of 72 shortcut beside the exact figure and the error between them.
- Savings Goal CalculatorThe monthly contribution needed to reach a target by a date, given what you have saved already.
- ROI CalculatorReturn on investment as a total percentage and as an annualised rate, so holdings of different lengths compare fairly.
- Interest CalculatorWhat a balance grows to under simple and compound interest, with the gap between the two shown as its own figure.
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