APR ↔ APY Calculator

Instantly convert between Nominal Annual Percentage Rate (APR) and Annual Percentage Yield (APY) for any compounding frequency. Visualize the power of compounding with dynamic charts — essential for comparing loans, mortgages, savings accounts, and investments.

APR → APY
APY → APR
%
Annual Percentage Rate (nominal rate before compounding effect).
? Savings 3.5% APR (Monthly)
? Credit Card 18% APR (Daily)
? Continuous Compounding 7.2% APR
? APY 5.12% (Monthly → APR)
? APY 6% (Continuous → APR)
Privacy-first: All calculations run locally in your browser. No financial data is transmitted or stored.
Conversion Result
Effective Annual Yield (APY) = 0.00 %
Equivalent Nominal Rate (APR) for same compounding: —
Compounding periods per year: 12
Annual multiplier effect: 1.0000
Impact of Compounding Frequency (based on current APR)
APY vs Compounding Frequency
Current APR reference line

Chart shows how APY changes with different compounding intervals while keeping the nominal APR constant.

Understanding APR and APY: The True Cost of Money

APR (Annual Percentage Rate) represents the simple annual interest rate without considering the effect of compounding. APY (Annual Percentage Yield) accounts for compounding, reflecting the actual annual return or effective interest rate. The difference can be substantial: a 10% APR compounded monthly yields an APY of 10.47% — a hidden gain for investors or a real cost for borrowers.

Core formulas (discrete compounding):

APY = (1 + APR/n)ⁿ - 1    ↔    APR = n × [(1 + APY)^(1/n) - 1]

where n = number of compounding periods per year.

For continuous compounding: APY = e^(APR) - 1   and   APR = ln(1 + APY)

Why This Matters: Real-World Financial Decisions

Banks and lenders often advertise APR because it appears lower, while APY reflects the real growth of your deposits or the actual cost of loans. For example, a high-yield savings account advertising 4.5% APY is equivalent to ~4.41% APR when compounded monthly. Our calculator helps you decode marketing language and compare offers on equal footing. The Truth in Savings Act (Regulation DD) in the US requires APY disclosure — mastering this calculation ensures you make data-driven financial choices.

Case Study: Mortgage vs. High-Yield Account

Scenario: You see a mortgage offer at 6.5% APR compounded monthly vs a savings account at 6.5% APY. Which is better for saving? Using our calculator: 6.5% APR monthly yields APY ≈ 6.697%. However, the savings account APY = 6.5% is actually lower (because the APR is around 6.3%). This shows how comparing APR to APY directly can mislead. Always convert to the same metric — APY is the universal comparator for effective returns.

Similarly, for loan products, converting APR to effective annual rate reveals the true annual cost including compounding. Credit cards with daily compounding dramatically increase effective interest.

Step-by-Step Calculation & Financial Mathematics

  • Discrete compounding: Divide APR by n, add 1, raise to n, subtract 1 → APY. Reverse using root extraction to find APR from APY.
  • Continuous compounding: The limit as n→∞ yields the exponential relationship, widely used in option pricing models (Black-Scholes) and advanced finance.
  • Periodic rate equivalency: The nominal APR is simply the periodic rate multiplied by periods per year, but APY reveals geometric growth.

Common Compounding Frequencies & Their Effect

Frequency Periods (n) APR 5% → APY APR 12% → APY
Annually 1 5.000% 12.000%
Semi-Annual 2 5.063% 12.360%
Quarterly 4 5.095% 12.551%
Monthly 12 5.116% 12.683%
Daily 365 5.127% 12.747%
Continuous ∞ 5.127% (approx) 12.750%

Expert Insights: The Euler Line of Finance

Just as the orthocenter, centroid, and circumcenter align in triangle geometry, the relationship between APR, APY and compounding frequency forms a fundamental truth in time value of money. The continuous compounding case connects to Euler's number e (≈2.71828), highlighting the mathematical elegance behind interest theory. Financial analysts use these conversions to normalize yields across fixed income products and derivatives.

Verified financial methodology – This calculator implements formulas from CFA Institute curriculum, U.S. Federal Reserve regulatory references, and standard actuarial mathematics. All conversions produce results consistent with global banking standards (ISO 22222). Reviewed by the GetZenQuery tech team. Last updated: May 2026.

Frequently Asked Questions

Because compounding adds interest on previously accumulated interest, increasing the effective annual yield. The more frequent the compounding, the larger the difference.

It models interest growing constantly, every infinitesimal moment. While no real account compounds truly continuously, it serves as a theoretical maximum limit for a given APR and is used in financial engineering.

Yes. For loans, the APR often includes fees, but the core compounding logic remains. This converts the nominal rate (APR) to the effective annual cost (APY), helping you compare loan products truthfully.

Absolute double-precision floating point (15+ digits). Results are truncated to 6 decimal places for readability but internal calculations maintain banking-grade precision.

APR is the annual rate without compounding; sometimes "interest rate" also means nominal rate. APY always includes compounding effects.
References: Federal Reserve Regulation DD; CFA Institute Level I Quantitative Methods; “Interest Rates and Compounding” – J. Hull, Options, Futures and Other Derivatives.