Convert nominal rate (APR) to effective annual rate (EAR/APY) for any compounding, or convert APY back to the required nominal rate. Compare loans and savings.
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What is the Effective Interest Rate?
The effective interest rate (also called the effective annual rate, EAR, or APY) is the actual annual rate of interest you earn on savings or pay on a loan after accounting for compounding within the year. It's almost always higher than the headline 'nominal' rate, because compounding earns interest on previously-earned interest. The faster the compounding (monthly vs. annually, daily vs. monthly), the larger the gap between nominal and effective.
Use effective rates to compare any two financial products honestly. A credit card quoting 18% APR compounded daily has an EAR of 19.72%. A savings account quoting 5% APR compounded monthly has an APY of 5.12%. Lenders and banks tell you whichever number sounds better for their pitch — comparing 'apples to apples' means converting everything to the effective annual rate first. This calculator handles all standard compounding intervals plus continuous compounding (the theoretical limit as periods approach infinity, where EAR = e^r − 1).
EAR = 6.17%, which is 0.17% higher than the 6% nominal rate. The period rate is 6% ÷ 12 = 0.50% per month.
Example 2: Daily Compounding (typical credit card)
Calculate the effective annual rate for a nominal rate of 18% compounded daily (typical credit card):
EAR = (1 + 0.18/365)^365 − 1 = 0.197164 = 19.72%
EAR = 19.72%, which is 1.72% higher than the headline 18% APR. Across a $5,000 balance held for a year, that's an extra $86 in interest the cardholder pays.
How Compounding Frequency Changes the Effective Rate
For a 6% nominal annual rate, here's how the effective rate scales with compounding frequency. Notice the diminishing returns — the gap between monthly and continuous is much smaller than between annual and monthly:
Annually (n=1): 6.0000%
Semi-annually (n=2): 6.0900%
Quarterly (n=4): 6.1364%
Monthly (n=12): 6.1678%
Weekly (n=52): 6.1800%
Daily (n=365): 6.1831%
Continuous (n→∞): 6.1837% (= e^0.06 − 1)
Common Applications
Comparing savings accounts that quote APR vs APY — the bank that quotes APY is the one being honest
Evaluating credit cards: a 22% APR compounded daily is an EAR of 24.6%, not 22%
Mortgage shopping: most US mortgages compound monthly, so the APR and EAR are usually close but not identical
CD (certificate of deposit) and money-market account comparisons across compounding schedules
Bond yield analysis: yield-to-maturity is quoted nominally, but holding-period yield is effective
International rate comparisons: European VAT-relevant 'AER' (Annual Equivalent Rate) is identical to US APY
Calculating true cost of payday loans (often disguised as period rates: 2% per 14 days = 67% EAR)
Bank treasury and corporate finance reconciliations
Effective Interest Rate Calculator
Why the Effective Rate Matters
True comparison: compare loans and savings with different compounding on equal footing
Regulatory disclosure: the Truth in Lending Act (US) requires APR but not always EAR; the EU often requires both
Decision quality: the wrong rate choice on a 30-year mortgage costs thousands; on retirement savings, tens of thousands
Compounding intuition: the bigger the n, the bigger the gap, but with diminishing returns toward continuous
Hidden cost detection: payday lenders and credit cards often quote period rates that look small but compound to enormous EARs
Investment performance: a 7% APR continuous return = 7.25% EAR — the difference compounds further over years
Tips for Using the Effective Interest Rate Calculator
Compare like with like — always convert both products to effective annual rate before deciding
More frequent compounding always gives a higher effective rate, but the marginal gain shrinks rapidly past monthly
For loans, look for the LOWER effective rate; for savings, the HIGHER one
Check whether the quoted rate is APR (nominal) or APY (effective); banks pick the more flattering number
Watch for fees not included in the rate — origination, late, transfer fees can dwarf the rate difference
Use the continuous compounding option (e^r) as a theoretical maximum; many advanced finance formulas use it
For Treasury bills and short-term commercial paper, the rate is sometimes quoted as 'discount yield' — convert before comparing
If you're refinancing, the closing costs amortized over the life of the new loan can change which EAR is actually lower
APR vs APY (vs EAR — they're all the same idea)
APR: Annual Percentage Rate. The 'simple' rate: nominal annual rate without compounding. Used in US loan disclosures by law (Truth in Lending Act). For loans with fees, APR can include fees, making it sometimes higher than the pure interest rate.
APY: Annual Percentage Yield. The 'effective' rate: includes the effect of compounding within the year. Used in US savings account disclosures by law (Truth in Savings Act). Identical mathematically to EAR (Effective Annual Rate) and the EU's AER (Annual Equivalent Rate).
Quick example: a savings account with 5% APR compounded monthly has an APY of 5.12%. The bank can legally advertise either number — APY is usually the larger one for savings (banks like big numbers there) and APR is usually advertised for loans (banks like small numbers there). When you see a single rate in marketing, always ask which one it is.
Frequently Asked Questions
APR is the nominal rate — what the bank quotes without compounding effects. APY is the effective rate — what you actually earn or pay after compounding. They're identical when compounding is annual (n=1), but diverge as n increases. For a 6% rate: APR = 6%, APY at monthly compounding = 6.17%, APY at daily compounding = 6.18%. US law requires loans to disclose APR (Truth in Lending Act, 1968) and savings to disclose APY (Truth in Savings Act, 1991) — different laws because Congress wanted consumers to compare correctly. The trap: a 'low' APR loan compounded daily can have a higher true cost than a higher APR loan compounded annually. Always convert to EAR before comparing two products with different compounding.
Because each compounding period earns interest on the previous period's interest, not just on the original principal. After month 1 of a 6%/year monthly account, you have $100 + $0.50 interest = $100.50. Month 2's interest is computed on $100.50, not $100 — so you earn $0.5025, not $0.50. The extra $0.0025 is tiny but accumulates: 12 months of compounding turns a 6% nominal into 6.17% effective. With daily compounding, each day's tiny extra rolls forward; with continuous compounding, the math takes the limit as n → ∞, yielding EAR = e^r − 1. The gap between monthly and continuous is small (6.17% → 6.18%) because compounding has diminishing returns — most of the benefit comes from going from annual to monthly, not from monthly to daily.
Continuous compounding is the theoretical limit where the number of compounding periods approaches infinity. As n → ∞, the formula (1 + r/n)^n → e^r, where e ≈ 2.71828 is Euler's number. The EAR for a nominal r under continuous compounding is e^r − 1. For 6%: e^0.06 − 1 = 6.1837%, only 0.0006 percentage points higher than daily compounding (6.1831%). In practice, no bank actually compounds continuously — they use daily at most. But continuous compounding shows up everywhere in finance theory: Black-Scholes option pricing, bond duration calculations, Treasury yield curves, present-value formulas for cash flows. The intuition: e^r captures the absolute maximum compounding effect for a given nominal rate — useful as a theoretical upper bound or for derivatives modeling.
Because the 18% APR is the nominal rate, but compounding makes the effective annual rate much higher. Credit cards compound daily — every day a $0.0049 of interest accrues on a $100 balance held at 18% APR. Over 365 days, that compounds to 19.72% EAR. Worse: credit card balances often include the previous month's interest as part of the new balance (this is called 'compounding on outstanding balance'), so the math runs full daily compounding. The $5,000 balance you owe at 18% APR will actually cost you $986 in interest over a year if you never pay it down — not $900. Lender disclosure rules now require both APR and total cost estimate on many credit card statements; check the 'finance charge over a year if minimum paid' figure to see your true effective rate.
Because US mortgages compound monthly, and at 12 compounding periods per year, the gap between APR and EAR is small but not zero. A 6% APR mortgage has an EAR of 6.17% — about 17 basis points. Over a 30-year, $300,000 loan, that 17 bp difference accumulates to about $9,500 of extra interest you pay vs. what the headline rate suggests. The bigger gap with mortgages is between APR (which includes loan fees like origination and points) and the pure interest rate. APR for mortgages by law (Truth in Lending Act) MUST include loan-related fees, so the disclosed APR is higher than just the interest rate. Don't confuse 'APR includes fees' (a US disclosure rule) with 'APR vs EAR' (a compounding math issue). For shopping mortgages, the APR-with-fees is what you actually compare; EAR rarely matters unless you're calculating early payoff savings.
The Rule of 72 (divide 72 by the rate to get years to double) works on EAR, not nominal APR. For a 6% EAR savings account, your money doubles in 72/6 = 12 years. For a 6% APR account that compounds monthly (EAR 6.17%), it actually doubles in 72/6.17 ≈ 11.67 years — about 4 months faster than the nominal number suggests. The Rule of 72 is accurate within 1% for rates between 5% and 15%. Below that range, use the Rule of 70; above, use the Rule of 76 or 78. For continuous compounding, use ln(2)/r ≈ 0.6931/r — slightly more accurate than 72/r. Practical use: this is why a high-interest savings account paying 5.0% APR compounded monthly (EAR ≈ 5.12%) is meaningfully better than 5.0% APR compounded annually — over 30 years, the difference is about $1,800 per $10,000 invested.
Only when compounding happens exactly once per year (n=1), or never compounds at all (simple interest only). Every other compounding frequency makes the effective rate higher than the nominal rate. Three specific cases: (1) A zero-coupon bond paying simple interest with no internal compounding — nominal = effective on a single-period basis but compounding matters across multiple bonds. (2) Some mortgages quote 'interest rate' that's actually the periodic rate already (you'll see 'rate per month' on Mexican and some Brazilian credit products) — convert manually before comparing. (3) Sub-period products like 30-day Treasury bills, where the discount yield is the simple interest for that one period and there's no within-period compounding. For most modern consumer products in the US/EU, the rate quoted is annual and at least some compounding occurs, so nominal < effective.
Two different concepts. Effective rate accounts for compounding within the year (covered above). Real rate accounts for INFLATION across years. A 5% effective rate at 3% inflation gives you a real rate of about 1.94% (Fisher equation: 1 + real = (1 + nominal) / (1 + inflation)). They stack together: a 5% APR account compounded monthly has 5.12% EAR; if inflation is 3%, the real return is 5.12% − 3% ≈ 2.06% (or more precisely, (1.0512/1.03) − 1 = 2.06%). When evaluating long-term savings, especially retirement, you want effective real return — both adjustments. Inflation-protected bonds (TIPS in the US, ILBs elsewhere) have their face value adjusted by CPI so the quoted rate IS the real rate, no Fisher equation needed. This is why TIPS are useful for retirement portfolios: the rate is already net of inflation, simplifying long-term planning.
Use the inverse direction of this calculator: switch 'Solve for' to 'Effective (EAR) → Nominal', enter your target APY, pick the compounding frequency, and read back the required nominal rate. The exact formula is nominal = n × ((1 + EAR)^(1/n) − 1), where n is the number of compounding periods per year. For continuous compounding it simplifies to nominal = ln(1 + EAR). Worked example: to advertise a 6.1678% APY compounded monthly (n=12), the bank must quote a 6.0000% nominal rate, since 12 × ((1.061678)^(1/12) − 1) = 0.06. This is exactly what banks do under the Truth in Savings Act: they decide the APY they want to advertise on a deposit product, then back-solve the nominal/periodic rate to publish on the rate card. Treasury and structured-finance analysts run the same inversion to turn a required effective yield into the equivalent nominal compounding rate for cash-flow and bond models. Because the function is an exact algebraic inverse of the forward EAR formula, the round trip is consistent to the basis point: convert nominal → APY and back, and you recover the original rate.
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