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APR vs. APY Converter

Instantly convert between the nominal Annual Percentage Rate (APR) and the effective Annual Percentage Yield (APY) based on compounding frequency.

Conversion Setup

%

Converted APY Rate

5.116%
True annual cost/yield including compounding
Math Breakdown:
Starting Rate:5%
Compounding Periods (n):12
Final APY:5.116%
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What is APR (Annual Percentage Rate) vs APY (Annual Percentage Yield)?

Both APR and APY measure interest, but banks use them strategically to their advantage. APR is the simple interest rate without factoring in compounding. APY is the "effective" rate that includes the snowball effect of compounding over the year.

Mathematical Foundation

APY=(1+APRn)n−1APY = \left(1 + \frac{APR}{n}\right)^n - 1
nn= Number of compounding periods per year (e.g., 12 for monthly, 365 for daily)
APR=n⋅((1+APY)1n−1)APR = n \cdot \left((1 + APY)^{\frac{1}{n}} - 1\right)

Laws & Principles

  • The Marketing Trick (Lending): When a bank lends you money (like a mortgage or credit card), they will legally advertise the APR, not the APY. Why? Because the APR is mathematically lower, making the debt look cheaper to the consumer. But because credit cards compound daily, you are actually paying the higher APY rate.
  • The Marketing Trick (Saving): When a bank wants you to deposit money in their High-Yield Savings Account, they will heavily advertise the APY, not the APR. Why? Because compounding makes the APY look higher, making the returns look more attractive.

Step-by-Step Example Walkthrough

" A credit card advertises a 24.00% APR. The fine print says interest compounds daily (365 times a year). "

  1. APR: 24.00%
  2. n = 365
  3. Math: (1 + 0.24/365)^365 - 1
  4. Math: (1.000657)^365 - 1 = 27.11%
Final Result: Even though the big font on the envelope said 24%, the true interest rate you will pay on a carried balance over a year is 27.11% APY.

Quick Answer: What is the difference between APR and APY?

APR (Annual Percentage Rate) is the nominal interest rate — the stated rate that does not account for the effect of intra-year compounding. APY (Annual Percentage Yield) is the effective rate that reflects how much you actually earn or pay after compounding within the year. The conversion formula is: APY = (1 + APR ÷ n)n − 1, where n is the number of compounding periods per year. Example: a savings account with a 5.00% APR compounded monthly produces an APY of (1 + 0.05/12)12 − 1 = 5.116%. The gap between APR and APY grows with both the rate level and compounding frequency — a 12% APR daily-compounded produces an APY of 12.747%, while the same 12% compounded annually stays at exactly 12.000%.

APR ↔ APY Conversion Formulas

APR → APY (Nominal to Effective)

APY = (1 + APR ÷ n)n − 1

APY → APR (Effective to Nominal)

APR = n × [(1 + APY)1/n − 1]

Continuous Compounding Limit (n → ∞)

APYcontinuous = eAPR − 1

  • APR— Annual Percentage Rate (nominal, as a decimal). The stated interest rate used in the formula. Banks advertise APR on loans (regulated by TILA — Truth in Lending Act) and APY on savings accounts (regulated by the Truth in Savings Act). A 5% APR = 0.05 in the formula. APR for the same effective rate decreases as compounding frequency increases: a 5.116% APY requires only 5.00% APR if compounding monthly, vs. 5.095% APR if compounding daily.
  • n— Number of compounding periods per year. Common values: daily = 365 (some institutions use 360), monthly = 12, quarterly = 4, semi-annually = 2, annually = 1. Credit cards typically compound daily (n=365). CDs and savings accounts commonly compound monthly (n=12) or daily. Mortgages generally compound monthly despite APR being stated annually. Bonds typically compound semi-annually (n=2).

APR vs APY at Different Compounding Frequencies

Compounding n APY at 5% APR APY at 12% APR APY at 20% APR
Annually 1 5.000% 12.000% 20.000%
Semi-annually 2 5.063% 12.360% 21.000%
Quarterly 4 5.095% 12.551% 21.551%
Monthly 12 5.116% 12.683% 21.939%
Daily 365 5.127% 12.747% 22.134%
Continuous (∞) ∞ 5.127% 12.750% 22.140%
Continuous compounding is the mathematical upper limit. Daily and continuous APYs converge to within 0.003% at 5% APR — beyond daily compounding, diminishing returns are negligible for practical banking.

APR vs APY for Savers vs Borrowers

Savings & Investments — APY is What Matters

  • Disclosed as: APY (required by Truth in Savings Act for US deposit accounts)
  • Why APY wins: Banks compound interest on your balance, so APY is the actual growth rate of your money. A 5% APR compounded daily = 5.127% APY — you earn 5.127% on every $1,000 deposited.
  • How to use this calculator: Convert the advertised APY back to APR to compare accounts that compound at different frequencies on an apples-to-apples nominal basis. High-yield savings accounts (HYSA) often advertise APY; CDs may vary.
  • Rule: Always maximize APY when comparing savings products — higher APY means more money in your pocket regardless of compounding frequency.

Loans & Credit — APR is What's Disclosed, APY is What You Pay

  • Disclosed as: APR (required by Truth in Lending Act / TILA for loans, credit cards, mortgages)
  • The trap: A credit card advertising 24.99% APR compounding daily has an actual APY of 28.37% — you pay 28.37% effectively on any balance carried. Banks quote APR on credit cards because it’s lower than the effective rate.
  • Mortgage APR: Includes fees (origination, points) amortized into the rate — useful for comparing loan costs but still not the effective compounding rate.
  • Rule: Always minimize APY when comparing debt products — not the advertised APR. Use this converter to find the true effective rate you will pay.

Pro Tips & Critical APR/APY Mistakes

Do This

  • ✓When comparing savings accounts, always compare APY, not APR. Even if two accounts advertise the same APR (e.g., 5.00%), one compounded daily (APY 5.127%) will outperform one compounded monthly (APY 5.116%) by $11 annually on a $100,000 deposit. The difference is small at normal savings rates but becomes substantial on large balances or at high compounding rates. Many online banks advertise APY directly — make sure you’re comparing the same metric across all accounts in your search.
  • ✓For credit cards, convert the stated APR to APY to know your true annual cost on revolving balances. A typical credit card at 22.99% APR compounding daily has an effective APY of 25.81%. On a $5,000 carried balance for a full year, this means $1,290 in interest at the APY rate versus $1,150 at the face APR — a $140 difference that compounds against you every month you delay payoff. The APY is the rate you should use when modeling how long it truly takes to pay off a revolving balance.

Avoid This

  • ✗Don't confuse APR for mortgages with the simple interest rate. Federal TILA requires lenders to disclose mortgage APR inclusive of origination fees, mortgage points, and some closing costs amortized over the loan term. A mortgage with a 6.75% interest rate and $8,000 in fees on a $300,000 30-year loan may have an APR of 6.95% once fees are factored in. This is useful for comparing total loan costs, but the 6.95% is NOT the compounding rate — to find the actual effective yield, use the interest rate (6.75%) and monthly compounding (n=12) in this converter, not the TILA APR.
  • ✗Don't ignore teaser rate vs. ongoing APR distinction for credit cards. Many credit cards offer 0% APR promotional periods (12–21 months). The APY conversion is technically 0% during this window. However, after the promotional period, rates often jump to 24–29.99% APR. If you carry a balance from month 13 onward, that full high APY applies to the entire remaining balance — missing the payoff deadline by even one billing cycle often triggers deferred interest on the full original promotional balance under specific card agreements. Always model the post-promotional APY separately.

Frequently Asked Questions

Why do banks advertise APR for loans but APY for savings accounts?

Banks quote whichever metric is more marketing-favorable. For loans, APR is always lower than APY (because APR = nominal rate, APY = effective compounded rate), so advertising APR makes the borrowing cost look smaller. For savings accounts, APY is higher than APR — advertising APY makes yields look larger and more attractive to depositors. US law enforces this asymmetry: the Truth in Lending Act (TILA) requires APR disclosure for credit products, while the Truth in Savings Act requires APY disclosure for deposit accounts. The practical result: always convert both to the same metric (APY) before making side-by-side comparisons of rates from different product types.

What is continuous compounding and how does it compare to daily compounding?

Continuous compounding is the mathematical limit of increasing compounding frequency infinitely: APY = eAPR − 1, where e = 2.71828… (Euler’s number). At 5% APR: continuous APY = e0.05 − 1 = 5.127%, versus daily APY = 5.127%. At 5%, the difference between daily and continuous is only 0.0003% — economically negligible. At 20% APR: continuous = 22.140%, daily = 22.134%, difference = 0.006%. Continuous compounding is used in pricing theoretical derivatives (Black-Scholes uses continuous compounding) and in financial mathematics proofs, but no retail bank product actually compounds continuously. The practical takeaway: daily compounding captures essentially all the benefit of continuous compounding at any real-world interest rate.

Does APY include fees, or just compounding?

For savings accounts, the disclosed APY under the Truth in Savings Act reflects only the interest rate and compounding frequency — monthly maintenance fees, minimum balance fees, and inactivity fees are excluded. These fees reduce your effective net yield below the advertised APY. A HYSA advertising 5.0% APY but charging a $5/month maintenance fee on a $10,000 balance effectively yields only 4.4% net. For loans (TILA APR), the disclosed APR does include certain fees — origination points, broker fees, and other upfront financing charges are incorporated. However, third-party fees (title insurance, appraisals, attorney fees) are typically excluded from TILA APR. Always read the loan disclosure documents for the full fee schedule before accepting the APR as your total cost of borrowing.

How does APR/APY apply to crypto staking and DeFi yield rates?

In crypto, APR is the simple annualized yield without reinvestment, and APY assumes yield is continuously reinvested (auto-compounded into the staking position). A DeFi protocol advertising 50% APR with auto-compounding produces APY = e0.50 − 1 = 64.87% — a massive difference at high rates not seen in traditional banking. However, crypto APR/APY figures carry additional risks not reflected in the rate: smart contract exploit risk, liquidity risk, impermanent loss (for LP positions), protocol insolvency, and token price volatility mean the advertised APY can disappear overnight. Unlike FDIC-insured savings accounts where APY is a reliable metric for comparing products, DeFi APY must be evaluated alongside these protocol-specific risk factors. Always verify whether the displayed rate is APR or APY on any crypto yield platform before comparing to traditional savings rates.

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