π EAR Calculator β Effective Annual Rate
Calculate the Effective Annual Rate (EAR) β the true annual interest rate accounting for compounding within the year.
EAR Formula
EAR is always β₯ APR. A 12% nominal rate compounded monthly = 12.68% EAR. This is why credit card rates quoted as monthly (e.g. 1.5%/month) must disclose the APR (18%) and EAR (~19.56%).
Why Nominal Rate and Effective Rate Are Never the Same
The nominal rate β also called the stated rate or APR β is the interest rate before compounding is applied. The Effective Annual Rate is what you actually pay or earn after accounting for compounding within the year. These two numbers are only equal in one scenario: when interest compounds once per year. The moment compounding becomes more frequent β monthly on a mortgage, daily on a savings account, or continuously on certain financial instruments β the EAR diverges upward from the nominal rate. The more frequently interest compounds, the higher the EAR relative to the stated rate. At 12% nominal, monthly compounding produces a 12.68% EAR. Daily compounding produces 12.747%. Continuous compounding (the mathematical limit as n approaches infinity) produces 12.750%. The gap seems small on a savings account, but on a $200,000 mortgage over 30 years, the difference between monthly and daily compounding adds up to thousands of dollars.
Where EAR Appears in Real Financial Products
Credit cards are the most common place consumers encounter the EAR gap without realizing it. A card advertising an 18% APR compounds daily in most cases, producing an EAR of approximately 19.72%. When you carry a balance, it's the EAR β not the APR β that determines your true annual interest cost. Savings accounts and high-yield accounts work the other way: a HYSA advertised at 5.00% APY (Annual Percentage Yield) is already the EAR β the bank is showing you the effective rate after compounding, which is why APY is always higher than the stated rate. Mortgages in the US compound monthly, so the difference between nominal and effective is smaller than with daily-compounding products. In Canada, mortgages are legally required to compound semi-annually, so the same stated rate produces a lower EAR than its US equivalent β something Canadian borrowers comparing products need to account for.
APR vs APY vs EAR: Clearing Up the Confusion
These three terms are used interchangeably in casual conversation but have precise and distinct meanings. APR (Annual Percentage Rate) is the nominal rate multiplied by compounding periods β it deliberately excludes the compounding effect and, in the US, is required by the Truth in Lending Act to be disclosed on loans. APY (Annual Percentage Yield) is the EAR expressed from the perspective of a depositor β it includes compounding and is required to be disclosed by banks on deposit products. EAR (Effective Annual Rate) is the same mathematical concept as APY but expressed from the borrower's perspective and used more commonly in academic finance and international contexts. The takeaway: when comparing borrowing costs, find the EAR. When comparing savings rates, find the APY. Both are the same calculation β just different names applied in different contexts.
Continuous Compounding: The Mathematical Ceiling
Continuous compounding is the theoretical limit where interest is calculated and added to the principal infinitely many times per year. The formula becomes EAR = eΚ³ β 1, where e is Euler's number (approximately 2.71828) and r is the nominal rate. At 10% nominal, continuous compounding produces an EAR of 10.517% versus 10.471% for daily compounding β the gap between daily and continuous is negligibly small in practice. Continuous compounding appears most often in options pricing models (Black-Scholes uses it), certain bond yield calculations, and theoretical finance. For practical consumer products, daily compounding is the most frequent you'll encounter, and the difference from continuous compounding is so small it has no material impact on real decisions.
People Also Ask
APR (Annual Percentage Rate) is the nominal interest rate β the stated rate before compounding effects are applied. EAR (Effective Annual Rate) is the true rate after accounting for how frequently interest compounds within the year. For any compounding frequency greater than once per year, EAR will always be higher than APR. The formula is EAR = (1 + APR/n)βΏ β 1, where n is the number of compounding periods. A 24% APR compounded monthly has an EAR of 26.82%. The APR understates your true annual cost whenever interest compounds more than once per year β which is the standard on virtually all consumer lending products.
If you're given a monthly rate (like a credit card's monthly periodic rate), convert to EAR using: EAR = (1 + monthly rate)ΒΉΒ² β 1. A 1.5% monthly rate gives EAR = (1.015)ΒΉΒ² β 1 = 19.56%. This is why credit cards that advertise a 1.5% monthly rate are legally required to also disclose their 18% APR (1.5% Γ 12) β even though the true annual cost is 19.56% EAR. The monthly-to-annual conversion always produces a higher EAR than simply multiplying the monthly rate by 12, because it accounts for interest-on-interest within the year.
It depends entirely on whether you're a borrower or a saver. For borrowers, a lower EAR is always better β it means you pay less interest over time. For savers and investors, a higher EAR is better β it means your money earns more. This is why banks advertise APY (the effective rate) on savings accounts β the higher number makes the product look more attractive β while advertising APR on loans β the lower number makes the cost seem smaller. A savvy consumer checks the EAR on any borrowing product and the APY on any savings product to make true apples-to-apples comparisons across different compounding frequencies.
On small balances over short periods, the difference between monthly and daily compounding is negligible. On large balances over long periods, it matters more but is still secondary to the stated rate itself. A 5.00% APR compounded monthly produces a 5.116% EAR. The same rate compounded daily produces 5.127% EAR β a difference of 0.011 percentage points. On a $10,000 savings balance over one year, that's about $1.10. On a $500,000 mortgage over 30 years, the accumulated difference is more meaningful but still dwarfed by even a 0.25% difference in the stated rate. The compounding frequency disclosure matters most for comparing products that appear to have the same rate but different compounding structures β like comparing a savings account quoting APR vs one quoting APY.