Effective Interest Rate Calculator
Convert any nominal interest rate (APR) to its Effective Annual Rate (EAR/APY) for any compounding frequency. The EAR is the true annual cost of a loan or yield on an investment after compounding is accounted for. Compare all compounding frequencies side by side including continuous compounding.
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How It Works
Divide the nominal annual rate by the number of compounding periods per year to get the periodic rate, then compound it over n periods. The result minus 1 is the effective annual rate.
EAR = (1 + r/n)^n - 112% APR monthly: (1+0.12/12)^12 - 1 = 12.683% EARAs compounding frequency approaches infinity, the EAR approaches e^r - 1 where e is Euler's number. This represents the theoretical maximum compounding for a given nominal rate.
EAR = e^r - 112% APR continuous: e^0.12 - 1 = 12.750% EARThe table shows EAR for every standard frequency at the same nominal rate. Daily compounding always exceeds annual compounding. The percentage difference grows with the nominal rate.
Gap = (1+r/365)^365 - (1+r)5% APR: Annual 5.000% vs Daily 5.127% (+0.127%)Credit cards quoted at 24% APR with monthly compounding have an EAR of 26.82%. This is the true annual cost if you carry a balance for a full year. The difference seems small but on a large balance is significant.
24% monthly: (1+0.24/12)^12 - 1 = 26.82%$10K balance: 24% APR = $2,400; 26.82% EAR = $2,682Savings accounts advertise APY (same as EAR) because it's the higher, more attractive number. Daily compounding at 4.5% APR gives 4.603% APY — and that's what the bank advertises. This is why APY is used for savings.
APY = EAR = (1 + APR/n)^n - 14.5% APR daily = 4.603% APY (what bank advertises)Canadian mortgages compound semi-annually (n=2). US mortgages compound monthly (n=12). A 6% mortgage has EAR of 6.09% in Canada and 6.168% in the US — a small but real difference for large mortgage amounts.
Canada: (1+0.06/2)^2-1=6.09%; US: (1+0.06/12)^12-1=6.168%$500K mortgage: US costs $390 more in interest annuallyQuick Reference
Common examples — verify instantly above.
12% APR monthly
Effective annual rate
12.683% EAR
24% APR monthly
Credit card EAR
26.824% EAR
5% APR daily
Savings APY
5.127% APY
5% APR annual
EAR
5.000% EAR (same)
6% continuous
EAR
6.184% EAR
5% APR all freqs
Daily vs annual gap
5.127% vs 5.000% (0.127%)
4.5% APR daily
Bank-advertised APY
4.603% APY
20% APR monthly
EAR
21.939% EAR
Tips & Shortcuts
Always use EAR/APY when comparing products across different compounding frequencies. APR comparisons can be misleading when frequencies differ.
Check the compounding frequency on any savings account or CD. Daily compounding is slightly better than monthly compounding at the same APR.
Credit cards use monthly compounding. A 21% APR credit card has a 23.14% EAR — the true annual cost if you carry the balance all year.
When computing the effective monthly rate from an annual rate, use: Monthly Rate = (1 + EAR)^(1/12) - 1. Dividing APR by 12 gives the nominal monthly rate, which is slightly different.
For investment analysis, use the EAR for discounting cash flows and NPV calculations. Mixing APR and EAR in financial models produces errors.
Continuous compounding is the upper bound. In practice, no product compounds faster than daily. The difference between daily and continuous is negligible for most applications.
Common Mistakes to Avoid
Dividing APR by 12 to get the monthly rate for compounding
Dividing APR by 12 gives the nominal monthly rate. The true monthly effective rate is (1 + EAR)^(1/12) - 1. For high rates, the difference matters in financial modeling.
Comparing a monthly-compounding loan APR with a daily-compounding savings APY
These are not apples-to-apples comparisons. Convert both to EAR before comparing. A loan at 5% APR monthly and savings at 5% APY daily have the same effective rate.
Assuming all mortgages use the same compounding convention
Canadian mortgages compound semi-annually by law. US mortgages compound monthly. The same stated rate has a different EAR depending on the convention.
Not recognizing that advertised savings APY and loan APR differ by design
Banks advertise APY for savings (the higher number) and APR for loans (the lower number). They are not directly comparable. Convert both to EAR to see true cost and yield.
Ignoring compounding frequency on credit card interest
A 0% APR offer that transitions to 22% monthly compounding has a 24.36% EAR. Always check the compounding frequency on credit cards, especially deferred interest promotions.
Treating continuous compounding as practically different from daily
The difference between daily and continuous compounding on 5% APR is 0.003 percentage points. For practical purposes, they are identical. Only in theoretical finance models does continuous compounding have significant implications.
Frequently Asked Questions
The Effective Annual Rate, also called the Annual Percentage Yield (APY) or Effective Annual Percentage Rate (EAPR), is the actual annual rate earned or paid after compounding within the year is factored in. It is always equal to or higher than the nominal APR when compounding occurs more than once per year.
Use the formula EAR = (1 + APR/n)^n - 1, where n is the number of compounding periods per year. For daily compounding (n=365) at 6% APR: EAR = (1 + 0.06/365)^365 - 1 = 6.183%. For continuous compounding: EAR = e^r - 1.
Continuous compounding is the mathematical limit of compounding as the frequency approaches infinity. The formula is EAR = e^r - 1 where e is Euler's number (~2.718). It produces the highest possible EAR for a given APR. Some financial instruments use continuous compounding as a theoretical benchmark.
Credit cards typically charge 1.5% to 2% per month, which corresponds to an APR of 18% to 24%. The effective annual rate at monthly compounding is 19.56% to 26.82% — significantly higher than the advertised APR.
Two mortgages quoted at the same APR can have different effective costs if they compound differently. Canadian mortgages compound semi-annually (by law) while US mortgages compound monthly, making direct APR comparison misleading.
Frequency matters most for high rates and long periods. For a 2% savings rate, daily versus monthly compounding adds only a few dollars per year. For a 20% credit card rate or a 20-year investment, the frequency difference is substantial.
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