APR ↔ APY Converter & CAGR Calculator
Calculator
Formulas: $$\text{APY}=(1+\tfrac{\text{APR}}{m})^{m}-1 \quad;\quad \text{APR}=m\left((1+\text{APY})^{1/m}-1\right)$$ Continuous: $$\text{APY}=e^{\text{APR}}-1 \quad;\quad \text{APR}=\ln(1+\text{APY})$$
Formula: $$\text{CAGR}=\left(\frac{\text{Ending}}{\text{Beginning}}\right)^{1/n}-1$$
APR, APY/AER, and CAGR — Quick Guide
APR is a nominal yearly rate without intra-year compounding. APY/AER is the effective yearly rate that includes compounding. With m periods/year, \( \text{APY} = (1 + \text{APR}/m)^m - 1 \).
CAGR is the smoothed multi-period growth rate: \( \left(\frac{\text{Ending}}{\text{Beginning}}\right)^{1/n} - 1 \).
- Daily compounding: Choose 365 or 360 based on convention.
- Continuous compounding: \( \text{APY} = e^{\text{APR}} - 1 \).
- CAGR caveat: Ignores interim volatility and cash flows.
APR, APY/AER, and CAGR — What They Mean (and When to Use Each)
APR (Annual Percentage Rate) is a nominal yearly rate that does not include compounding within the year. Lenders and many savings products quote APR to describe the base rate. Because it is nominal, APR by itself can understate the true, compounded return (or cost) unless compounding happens only once per year.
APY (Annual Percentage Yield, also called AER—Annual Equivalent Rate—in the UK and EU) is the effective yearly rate that does include the effect of compounding. If compounding occurs more than once per year, APY/AER will be higher than the APR. The relationship with m compounding periods per year is: $$ \text{APY} = \left(1 + \frac{\text{APR}}{m}\right)^m - 1. $$
CAGR (Compound Annual Growth Rate) is a smoothed growth measure used for investments over multiple years. It answers: “If my value grew steadily each year to reach the final amount, what single annual rate would match that path?” The formula is: $$ \text{CAGR} = \left(\frac{\text{Ending}}{\text{Beginning}}\right)^{1/n} - 1, $$ where n is the number of years (or other periods) in your measurement window.
Why compounding frequency matters
The more frequently interest compounds, the higher the effective annual rate for a given APR. Common frequencies include annual (m=1), semi-annual (2), quarterly (4), monthly (12), weekly (52), and daily (commonly 365 or 360). Some disclosures also reference continuous compounding, the mathematical limit of very frequent compounding, where $$ \text{APY} = e^{\text{APR}} - 1 \quad\text{and}\quad \text{APR} = \ln(1 + \text{APY}). $$
Regional terms and day-count conventions
In the UK and much of Europe, AER is the standard consumer term (equivalent to APY). In the US, you’ll most often see APY. Daily compounding may use a 365-day or 360-day basis depending on the bank or product (you might also see ACT/365, ACT/360, or 30/360 on bonds and savings). This tool lets you choose daily-365 or daily-360 so the conversion matches the convention you’re comparing.
Quick examples
- APR → APY: 12% APR with monthly compounding (m=12) gives \( (1 + 0.12/12)^{12} - 1 \approx 12.6825\% \) APY.
- APY → APR: 5% APY with quarterly compounding implies an APR of \( 4 \big((1+0.05)^{1/4} - 1\big) \approx 4.9088\% \).
- CAGR: Growing from £10,000 to £15,000 over 3 years yields \( (15000/10000)^{1/3} - 1 \approx 14.472\% \) per year.
When to use which metric
- Comparing savings accounts or CDs/term deposits: Use APY/AER to compare true annual yield.
- Reading loan or credit card quotes: APR shows the nominal rate; check compounding to estimate the effective cost.
- Evaluating multi-year investments: Use CAGR to summarise long-run growth, bearing in mind it smooths volatility and ignores interim cash flows.
Helpful tips
Match the compounding setting in this calculator to the product’s actual terms. For apples-to-apples comparisons across banks, convert everything to APY/AER. For performance reporting, pair CAGR with context (fees, deposits/withdrawals, and risk). This page is for education only and is not financial advice.