MulticalcEffective Annual Rate (EAR) Calculator
Convert nominal rates to effective annual yieldEnter the nominal rate and compounding frequency to calculate the true annual yield (EAR).
Results
Enter a nominal rate and compounding frequency to compute the Effective Annual Rate.
About Effective Annual Rate (EAR)
The Effective Annual Rate (EAR) is the interest rate that reflects compounding over a year — it shows the true annual return when interest compounds more frequently than annually. Use EAR to compare financial products that use different compounding schedules.
Key Concepts
- Nominal Rate (APR): The stated annual interest rate, not accounting for intra-year compounding.
- Compounding Frequency: How often interest is applied (monthly, quarterly, daily, or continuously).
- EAR formula (periodic): EAR = (1 + r/n)n - 1, where r = nominal rate (decimal), n = periods per year.
- Continuous compounding: EAR = er - 1 when interest compounds continuously.
Practical Uses
Compare savings accounts, loans, mortgages, credit cards, and investment products using EAR — especially when one product compounds monthly and another compounds daily or continuously.
Frequently Asked Questions
The Ultimate Guide to Effective Annual Rate (EAR)
In the world of corporate finance and consumer banking, the interest rate you are quoted is rarely the interest rate you actually pay. The Effective Annual Rate (EAR) strips away marketing deception and mathematical technicalities to reveal the true cost of a loan or the true yield of an investment. This guide explores the deep mathematics of compounding interest.
1. The Deception of the "Nominal Rate"
Financial institutions operate on a simple marketing principle regarding interest rates: When you borrow money, they want the rate to look as low as possible. When you deposit money, they want the rate to look as high as possible.
Borrowing (Loans & Credit Cards)
When you borrow money, banks quote the Annual Percentage Rate (APR) or Nominal Rate. The APR completely ignores the effect of intra-year compounding. A credit card might advertise a "24% APR", but because they compound interest daily, your actual Effective Annual Rate (the true cost) is over 27.11%.
Investing (Savings & CDs)
When you deposit money, banks suddenly switch terminology. They quote the Annual Percentage Yield (APY), which is mathematically identical to the EAR. They want to show you the highest possible number. A savings account with a 5.0% Nominal Rate compounded daily will be advertised as having a 5.13% APY (EAR).
2. The Mathematics of Compounding Frequencies
Albert Einstein is famously (though perhaps apocryphally) quoted as calling compound interest the "Eighth Wonder of the World." Compounding is simply earning interest on your interest. The more frequently it happens, the higher the EAR.
| Compounding Frequency ($n$) | Periods per Year | Common Financial Instruments | EAR at 10% Nominal |
|---|---|---|---|
| Annual | 1 | Simple Bonds, Some Corporate Dividends | 10.000% |
| Semi-Annual | 2 | US Treasury Bonds, Canadian Mortgages | 10.250% |
| Monthly | 12 | US Mortgages, Auto Loans, Student Loans | 10.471% |
| Daily | 365 | Credit Cards, High-Yield Savings Accounts | 10.516% |
| Continuous | $\infty$ | Theoretical Finance, Options Pricing (Black-Scholes) | 10.517% |
3. The Formula Breakdown
$EAR = \left(1 + \frac{i}{n}\right)^n - 1$
- $EAR$ = Effective Annual Rate
- $i$ = Stated Nominal Interest Rate (as a decimal)
- $n$ = Number of compounding periods per year (e.g., 12 for monthly)
Example: A 12% nominal rate compounded monthly ($i=0.12, n=12$).
$EAR = (1 + 0.12/12)^{12} - 1$
$EAR = (1.01)^{12} - 1$
$EAR = 1.1268 - 1 = 12.68\%$
What happens if interest compounds every hour? Every minute? Every second? As $n$ approaches infinity, we reach the mathematical limit of compounding, calculated using Euler's number ($e \approx 2.71828$).
$EAR = e^r - 1$
- $e$ = Euler's Number (constant)
- $r$ = Stated Nominal Interest Rate (as a decimal)
Note: Continuous compounding is rarely used in consumer banking, but it is the foundational assumption in institutional quantitative finance, such as the Black-Scholes options pricing model.
4. The Danger of Credit Cards (Daily Compounding)
Credit cards are arguably the most dangerous financial product available to consumers, largely because of their compounding frequency. While mortgages compound monthly, almost all credit cards compound daily.
The Daily Trap:
If you carry a balance on a credit card with an advertised 29.99% APR, the bank takes that rate and divides it by 365 to get your Daily Periodic Rate (DPR) of roughly 0.082%.
Every single night at midnight, the bank charges you 0.082% interest on your balance. The next night, they charge you interest on your balance plus the interest they charged you yesterday.
Because of daily compounding, a 29.99% APR credit card has a true Effective Annual Rate (EAR) of 34.96%. You are paying nearly 5% more per year than the advertised rate implies.
5. International Banking Oddities (US vs. Canadian Mortgages)
The calculation of EAR is heavily influenced by national banking laws.
US Mortgages (Monthly)
By standard convention, mortgages in the United States compound monthly ($n=12$). If a US bank quotes a 6.00% mortgage, the true EAR you are paying is 6.167%.
Canadian Mortgages (Semi-Annual)
Under the Canadian Interest Act, fixed-rate mortgages are legally required to be compounded semi-annually ($n=2$), even though payments are made monthly. A Canadian 6.00% mortgage has an EAR of only 6.090%. Therefore, a 6% mortgage in Canada is mathematically cheaper than a 6% mortgage in the US!