r is the nominal annual rate as a decimal and n is the number of compounding periods in a year. The result is the single annual rate that, applied once, gives the same growth as compounding r ÷ n a total of n times.
A nominal rate is the headline annual figure before compounding. When interest compounds more than once a year, each period's interest earns interest in later periods of the same year, so money grows by more than the headline. The effective annual rate (EAR) measures that real one-year growth and puts quotes with different frequencies on one scale.
The formula worked at 12% for four frequencies
| Compounding | n | Rate per period | Growth factor (1 + r ÷ n)^n | EAR | Balance after one year |
|---|---|---|---|---|---|
| Annual | 1 | 12% | 1.120000 | 12.0000% | 11,200.00 |
| Quarterly | 4 | 3% | 1.125509 | 12.5509% | 11,255.09 |
| Monthly | 12 | 1% | 1.126825 | 12.6825% | 11,268.25 |
| Daily | 365 | 0.032877% | 1.127475 | 12.7475% | 11,274.75 |
Where the extra 68.25 comes from
With monthly compounding at 12%, the first month adds 1% of 10,000, which is 100. The second month adds 1% of 10,100, which is 101. The third adds 1% of 10,201, which is 102.01. Each month's interest is 1% of a slightly larger balance, and over twelve months these small increments add 68.25 to the 1,200 that a single annual credit would pay. That 68.25 is interest earned on interest within the year, and it is the whole difference between the 12% nominal rate and the 12.6825% effective rate.
The quarterly row is 1.03 × 1.03 × 1.03 × 1.03 = 1.125509, so 10,000 becomes 11,255.09 and the EAR is 12.5509%. As compounding becomes continuous, the EAR approaches e^0.12 − 1 = 12.7497%, only 0.0022 points above daily compounding.
You can reproduce every balance with the compound-interest calculator: principal 10,000, one year, the chosen frequency. Divide the interest earned by 100 and you have the EAR in percent.
Compound Interest CalculatorSet principal to 10,000 and years to 1, then switch frequency to read each effective rate.Higher rates make frequency matter more
At low rates the frequency barely matters. At high rates it adds several points, which is why it matters most for expensive borrowing.
| Nominal rate | Annual | Quarterly | Monthly | Daily |
|---|---|---|---|---|
| 6% | 6.0000% | 6.1364% | 6.1678% | 6.1831% |
| 12% | 12.0000% | 12.5509% | 12.6825% | 12.7475% |
| 24% | 24.0000% | 26.2477% | 26.8242% | 27.1149% |
In money, the gap is easy to underestimate. On a 500,000 balance left to accumulate for a year at 24%, monthly compounding charges or pays 14,120.90 more than annual compounding at the same headline rate.
Doubling the nominal rate from 12% to 24% roughly quadruples the compounding premium. Moving from monthly to daily, by contrast, adds little at any rate: 0.0153 points at 6% and 0.29 points at 24%.
Comparing two quotes on one scale
Suppose one deposit offers 11.9% compounded monthly and another offers 12.2% compounded annually. The second looks better. Converted, the first is (1 + 0.119 ÷ 12)^12 − 1 = 12.57%, so 10,000 grows to 11,257.10 against 11,220.00. The lower headline rate pays more because it compounds twelve times.
Borrowing works the same way in reverse. OpenStax's credit-card example charges 1.5% a month, quoted as 18% a year; compounded monthly, the effective cost is 19.56%.
Finding the frequency your account actually uses
Product terms often mention two schedules: how often interest is calculated (accrued) and how often it is added to the balance (credited, or capitalized). Compounding happens only at crediting, because accrued interest that has not yet been added cannot earn interest of its own. An account that accrues daily but credits monthly behaves like monthly compounding for the EAR formula.
If the terms are unclear, ask the provider two questions: on which dates is interest added to my balance, and is the quoted rate nominal or already effective? With those answers you can pick n, or skip the formula entirely if the rate is already effective. A profit-paying account that transfers each payout to a separate account does not compound at all on the original balance.
Going back from an effective rate to a nominal rate
Use this when you know the effective rate you want and need the nominal rate to enter for a given frequency. An EAR of 12% corresponds to 11.3866% compounded monthly, or 11.6601% compounded semiannually.
Bond yields use this logic too. A yield to maturity on a semiannual-coupon bond is usually quoted as a nominal rate, so a 7.4351% YTM is 7.5733% effective; the current yield and YTM guide uses that example.
Where an EAR comparison stops being enough
- EAR assumes interest stays in the balance. If interest is paid out, growth follows simple interest on the principal; see compound versus simple interest.
- Fees, minimum-balance tiers, withholding tax and promotional rates that expire are not in the formula, yet they can outweigh a fraction of a point of compounding.
- Day-count conventions differ. The calculator divides by 365 for daily compounding; some products use 360 days, which raises the effective rate slightly.
- A variable rate has no single EAR for the year ahead; the formula applies to each rate while it lasts.
- Many deposit advertisements already quote an annual percentage yield or effective rate that includes compounding. Applying the formula again would double count it.