# APY vs. APR: The Critical Difference in Interest Rate Calculations Published: 2026-01-20 Author: Warren Team URL: https://www.heywarren.com/blog/apy-vs-apr --- APY and APR sound similar, but they measure different things, and confusing them can cost you money or make you think you're earning more than you really are. Here's the precise difference, with formulas and real-world examples. ## The Core Definitions **APR (Annual Percentage Rate)** is the simple annualized interest rate, without accounting for compounding within the year. ![How monthly compounding widens the gap between a 5% APR and its effective APY.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20800%20210%22%20width%3D%22800%22%20height%3D%22210%22%20role%3D%22img%22%3E%3Ctitle%3EComparison%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Ctext%20x%3D%22230%22%20y%3D%2257.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EAPR%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%2225%22%20width%3D%22439.7967161845192%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%232563eb%22%2F%3E%3Ctext%20x%3D%22691.7967161845193%22%20y%3D%2257.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%232563eb%22%3E%255%3C%2Ftext%3E%3Ctext%20x%3D%22230%22%20y%3D%22152.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EAPY%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%22120%22%20width%3D%22450%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%237c3aed%22%2F%3E%3Ctext%20x%3D%22702%22%20y%3D%22152.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%237c3aed%22%3E%255.116%3C%2Ftext%3E%3C%2Fsvg%3E) *How monthly compounding widens the gap between a 5% APR and its effective APY.* **APY (Annual Percentage Yield)** is the effective annual rate, accounting for the effect of compounding within the year. **The rule**: For any interest rate that compounds more than once per year, APY will be greater than APR. The more frequent the compounding, the larger the gap. ## The Formulas **APR → APY conversion**: **APY = (1 + APR/n)^n − 1** Where: - APR = stated annual rate (as a decimal) - n = number of compounding periods per year **Example, 5% APR compounded monthly (n=12)**: - APY = (1 + 0.05/12)^12 − 1 - APY = (1.004167)^12 − 1 - APY = 1.05116 − 1 - **APY = 5.116%** The compounding effect adds 11.6 [basis points](/blog/basis-points). On a $100,000 balance, that's $116/year in additional interest. **Common compounding scenarios**: | APR | Compounding | APY | |---|---|---| | 5% | Annually | 5.000% | | 5% | Semi-annually | 5.063% | | 5% | Quarterly | 5.095% | | 5% | Monthly | 5.116% | | 5% | Daily | 5.127% | | 5% | Continuously | 5.127% | Notice: daily and [continuous compounding](/blog/compound-interest-formula-compounded-continuously) converge, going from daily to continuous adds virtually nothing because daily is already close to the theoretical maximum. ## Why APR Is Used for Loans and APY for Savings **Loans quote APR**: Banks legally must disclose APR on loans (per the Truth in Lending Act) because APR tends to understate the true cost of borrowing (ignoring intra-year compounding). Regulators chose APR as a standardized disclosure. **Savings quote APY**: Banks must disclose APY on deposit accounts (per Truth in Savings Act) because APY is the higher number, and regulators wanted consumers to compare apples to apples across different compounding frequencies. This asymmetry, APR on loans, APY on savings, is sometimes criticized as favoring banks: - Loans look cheaper (APR) than they truly are - Savings look more attractive (APY) than the stated rate alone **When comparing loans with different compounding**: Always compute the APY-equivalent for each to make them truly comparable. ## APR on Loans: What It Includes For loans (mortgages, auto loans, credit cards), the **APR is supposed to reflect not just interest but also loan costs spread over the loan term**: ![APR bundles the stated interest rate with select upfront loan costs, but excludes others like appraisal and title fees.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20760%20211%22%20width%3D%22760%22%20height%3D%22211%22%20role%3D%22img%22%3E%3Ctitle%3EHierarchy%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Crect%20x%3D%22300%22%20y%3D%2220%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22%232563eb%22%2F%3E%3Ctext%20x%3D%22380%22%20y%3D%2254%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22white%22%3ELoan%20APR%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20110%20105.5%20L%20110%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%2230%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22110%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EInterest%20Rate%3C%2Ftext%3E%3Ctext%20x%3D%22110%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3Estated%20rate%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20290%20105.5%20L%20290%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22210%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22290%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EPoints%3C%2Ftext%3E%3Ctext%20x%3D%22290%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3Epaid%20at%20closing%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20470%20105.5%20L%20470%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22390%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22470%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EOrigination%20Fees%3C%2Ftext%3E%3Ctext%20x%3D%22470%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3Elender%20costs%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20650%20105.5%20L%20650%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22570%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22650%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EMortgage%20Insurance%3C%2Ftext%3E%3Ctext%20x%3D%22650%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3Eif%20applicable%3C%2Ftext%3E%3C%2Fsvg%3E) *APR bundles the stated interest rate with select upfront loan costs, but excludes others like appraisal and title fees.* **APR components**: - Stated interest rate - Points paid at closing - Origination fees - Mortgage insurance (for mortgages) - Some other loan costs **APR does NOT include**: - Appraisal fees - Title insurance (in many cases) - Escrow fees - Real estate taxes **Example, Mortgage APR**: - Loan: $400,000, 30-year fixed at 6.5% stated rate - Points: 1 point = $4,000 paid at closing - Origination fee: $2,000 The APR will be slightly higher than 6.5% (perhaps 6.62%) because those upfront costs are spread across the loan term, reducing the effective "amount borrowed" and increasing the effective rate. **Why this matters**: When comparing two mortgage offers, compare APR, not stated rates. A 6.5% rate with 2 points may have a higher APR (thus higher true cost) than a 6.75% rate with zero points. ## APY on Savings Products **High-yield savings accounts, CDs, money market accounts**, all typically quote APY: **Example comparison**: - Bank A: 4.80% APR compounded monthly → APY = 4.907% - Bank B: 4.85% APR compounded quarterly → APY = 4.939% - Bank C: 4.90% APR compounded annually → APY = 4.900% Despite Bank C having the highest stated rate, Bank B actually pays the highest effective yield because of the compounding differential. Since banks are required to disclose APY, consumers can compare APY numbers directly, the rate that actually lands in their account. ## Credit Card APR: A Special Case Credit card APRs have unusual characteristics: ![Daily compounding on an 18% APR card produces an effective APY of 19.72%, a 172 basis-point gap.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20800%20210%22%20width%3D%22800%22%20height%3D%22210%22%20role%3D%22img%22%3E%3Ctitle%3EComparison%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Ctext%20x%3D%22230%22%20y%3D%2257.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EStated%20APR%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%2225%22%20width%3D%22410.8338405356056%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%232563eb%22%2F%3E%3Ctext%20x%3D%22662.8338405356055%22%20y%3D%2257.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%232563eb%22%3E%2518%3C%2Ftext%3E%3Ctext%20x%3D%22230%22%20y%3D%22152.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EEffective%20APY%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%22120%22%20width%3D%22450%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%237c3aed%22%2F%3E%3Ctext%20x%3D%22702%22%20y%3D%22152.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%237c3aed%22%3E%2520%3C%2Ftext%3E%3C%2Fsvg%3E) *Daily compounding on an 18% APR card produces an effective APY of 19.72%, a 172 basis-point gap.* **Daily compounding**: Most credit cards use daily periodic rate (DPR) and compound daily: - Stated APR 18% → Daily rate = 18% / 365 = 0.0493% - Effective APY = (1 + 0.0493%)^365 − 1 = **19.716%** The gap between stated APR (18%) and effective APY (19.72%) is substantial, 172 basis points. **Grace period**: Credit cards typically have a grace period, if you pay the balance in full each month, no interest accrues. In this case, stated APR doesn't directly apply. **Revolving balances**: If you carry a balance, APR compounds daily on the average daily balance, and APY is the right measure of actual cost. **Cash advances and balance transfers**: Often have different (higher) APRs and no grace period, interest accrues from day one. ## Mortgage APR Caveat Mortgage APR can be misleading because of the assumption built into the calculation: **APR assumes you hold the mortgage for the full term** (typically 30 years). If you sell or refinance in 5-7 years (most mortgages are paid off or refinanced within this timeframe), the effective APR is much higher than the quoted APR, because the upfront costs are amortized over a much shorter period than 30 years. **Practical implication**: For short-term mortgage holders, minimizing upfront costs (points, fees) often beats paying upfront for a lower rate, even if the latter has a better stated APR. ## Variable vs. Fixed Rates APR and APY both apply to fixed-rate products. For variable-rate products: **Introductory vs. Ongoing**: Promotional 0% APR credit cards revert to a standard rate after 6-18 months. The ongoing APR is what matters long-term. **Index + margin**: ARM mortgages and many credit cards use "SOFR + 3%" or similar formulas. The APR changes as the index changes. **Rate caps**: Most variable-rate products have caps (e.g., lifetime cap on ARM). Understanding caps is crucial for risk assessment. ## Continuous Compounding: The Theoretical Limit The mathematical limit of infinite compounding is continuous compounding: **APY (continuous) = e^APR − 1** Where e ≈ 2.71828 is Euler's number. **5% APR continuously compounded**: e^0.05 − 1 = 5.127% In practice, daily compounding (5.126% in the example) is virtually identical to continuous compounding, banks often use daily compounding as a proxy for continuous. ## Summary Table: When Each Is Used | Product | Typical Disclosure | Why | |---|---|---| | Credit cards | APR (also some APY) | Legal requirement; daily compounding | | Mortgages | APR (with points, fees) | Truth in Lending Act | | Auto loans | APR | Truth in Lending Act | | Student loans | APR | Truth in Lending Act | | Savings accounts | APY | Truth in Savings Act | | CDs | APY | Truth in Savings Act | | Money market accounts | APY | Truth in Savings Act | | Bonds | [Yield to maturity](/blog/formula-of-ytm) (similar to APY) | Industry convention | ## Authoritative Sources For deeper background and primary-source data on this topic, the following authoritative sources are useful starting points: - [IRS](https://www.irs.gov/) - [SEC](https://www.sec.gov/) ## Conclusion The APR vs. APY distinction comes down to compounding: APR ignores intra-year compounding, APY includes it. For a quick mental model: APY is always greater than (or equal to) APR for any product that compounds more than once per year. On loans, you pay the APY cost even though only the APR is disclosed; on savings, you receive the APY benefit, which is what banks emphasize. For precise comparison of financial products, always convert to APY. For precise comparison of loans with different fee structures, use APR. For related financial concepts, see our guides on [CAGR formula](/blog/cagr-formula), [accrued interest](/blog/accrued-interest), and [certificate of deposit](/blog/certificate-of-deposit). Warren at [heywarren.com](https://heywarren.com) helps you compare savings, credit, and investment yields on a consistent basis, so you can optimize your portfolio and borrowing decisions. --- ## Related Reading **More from Warren**: - [Cox-Ingersoll-Ross (CIR) Model: The Interest Rate Model Explained](/blog/cox-ingersoll-ross) - [Profit and Loss Statement (P&L): How to Read One and What Every Line Means](/blog/profit-and-loss-statement) - [Rental Return: How to Calculate ROI on Rental Property and Whether It's Worth Investing](/blog/rental-return) **Authoritative sources**: - [SEC EDGAR, Company Filings](https://www.sec.gov/edgar/searchedgar/companysearch) - [Federal Reserve Economic Data (FRED)](https://fred.stlouisfed.org/) - [Bureau of Labor Statistics, Data Tools](https://www.bls.gov/data/)