The effective annual interest rate is the true yearly cost of a loan, or the true yearly return on a deposit, once compounding is factored in. A 12% nominal rate that compounds monthly actually costs 12.68% over a full year, because each month’s interest becomes part of the base the next month’s interest is calculated on. That gap between the quoted rate and the real rate is why the effective annual interest rate, often abbreviated EAR, is the only reliable way to compare products with different compounding schedules.
The Formula
EAR = (1 + r/n)n − 1
Two inputs drive it:
- r is the nominal annual interest rate expressed as a decimal, so 12% becomes 0.12.
- n is the number of compounding periods per year: 1 for annual, 2 for semi-annual, 4 for quarterly, 12 for monthly, 365 for daily.
The result is a decimal. Multiply by 100 to read it as a percentage.
Working Through an Example
Take a loan at a 12% nominal rate that compounds monthly. Convert 12% to 0.12. Divide by 12 compounding periods to get 0.01, the periodic rate applied each month. Add 1 to get 1.01, which represents your balance growing by 1% each cycle. Raise 1.01 to the 12th power: 1.0112 = 1.12683. Subtract 1 and multiply by 100. The EAR is approximately 12.68%.
That extra 0.68 percentage points above the quoted 12% is the pure cost of compounding, money owed only because interest keeps building on itself. If the same 12% rate compounded daily instead, you would divide 0.12 by 365, add 1, and raise the result to the 365th power. The EAR climbs to about 12.75%.
How Compounding Frequency Changes the Answer
Each time a lender calculates interest and adds it to your balance, the base for the next calculation grows. More frequent compounding stacks more of these small increases. At a 12% nominal rate:
- Annual (n = 1): EAR = 12.00%, no compounding effect at all.
- Semi-annual (n = 2): EAR = 12.36%.
- Quarterly (n = 4): EAR = 12.55%.
- Monthly (n = 12): EAR = 12.68%.
- Daily (n = 365): EAR = 12.75%.
The jump from annual to monthly compounding is roughly two-thirds of a percentage point. That sounds small until it lands on a $300,000 mortgage or a large student loan balance over decades. On $100,000 at a 12% nominal rate, the difference between annual and daily compounding is about $750 in extra interest per year. For savers, the same dynamic works in your favor: a deposit that compounds daily grows slightly faster than one compounding monthly at the same quoted rate.
Continuous Compounding
Push the number of periods toward infinity and you reach continuous compounding, where interest accrues every instant. The formula simplifies to EAR = er − 1, where e is Euler’s number (approximately 2.71828). At a 12% nominal rate, continuous compounding produces an EAR of about 12.750%, barely higher than daily compounding’s 12.747%. The practical difference is negligible, but the math sets a ceiling: no compounding schedule can produce an EAR higher than er − 1 for a given nominal rate.
EAR vs. APR vs. the Nominal Rate
Three numbers get thrown around on loan paperwork, and they answer different questions.
The nominal rate, sometimes called the stated rate, is the raw annual percentage before any adjustment for compounding or fees. It is the r that goes into the EAR formula.
The APR is defined by the Truth in Lending Act and spreads the total finance charge over the life of the loan.1Office of the Law Revision Counsel. 15 USC 1606 – Determination of Annual Percentage Rate That finance charge includes service charges, points, origination fees, and certain required insurance premiums.2Consumer Financial Protection Bureau. 12 CFR Part 1026 Regulation Z – 1026.4 Finance Charge A mortgage with a 6.5% interest rate might carry a 6.8% APR once points and lender fees are folded in.
The EAR captures compounding but usually does not include non-interest fees. It answers a narrower question: given a nominal rate and a compounding schedule, what is the real annual rate of interest growth? For credit cards, where the disclosed APR is essentially a nominal rate compounded daily, the EAR is always higher than the stated APR. A card advertising a 24% APR compounded daily produces an EAR of about 27.1%. On a carried balance, that gap is real money.
So which number to use depends on the comparison. When two loan offers have different fee structures, APR gives the cleaner apples-to-apples view of total cost. When two products have the same fees but different compounding schedules, EAR is sharper. Neither number alone tells the whole story.
APY: The Same Math on the Savings Side
Deposits use a different label for the same idea. The Truth in Savings Act and Regulation DD require banks to express returns as an Annual Percentage Yield. APY is essentially EAR from the saver’s point of view: total interest earned on a deposit over one year, including compounding, stated as a percentage.3Cornell Law Institute. Appendix A to Part 1030 – Annual Percentage Yield Calculation If a bank advertises any rate of return, it must state the APY using that term, and no other rate in the advertisement can appear more prominently.4eCFR. 12 CFR 1030.8 – Advertising The rule exists because a bank could otherwise advertise a nice-sounding nominal rate in large print and bury the lower APY in footnotes.
When Your Calculation Doesn’t Match the Disclosure
If you run the EAR yourself and the figure does not match what your lender or bank disclosed, start by confirming the inputs. Loan documents specify the exact compounding frequency and, for some commercial loans, a day-count convention that divides by 360 rather than 365. Many apparent discrepancies come from that mismatch, or from overlooking fees the APR includes but the EAR does not.
If the numbers still do not line up, Regulation Z sets tight tolerances on the APR a lender may disclose, and creditors who violate Truth in Lending Act disclosure rules face real liability.5eCFR. 12 CFR 1026.22 – Determination of Annual Percentage Rate For open-end credit like credit cards, a borrower can recover twice the finance charge, with a floor of $500 and a ceiling of $5,000. For closed-end credit secured by a home, damages range from $400 to $4,000. Successful plaintiffs also recover attorney’s fees.6Office of the Law Revision Counsel. 15 USC 1640 – Civil Liability Complaints can be filed with the Consumer Financial Protection Bureau online or by calling (855) 411-2372, and companies generally respond within 15 days.7Consumer Financial Protection Bureau. Submit a Complaint