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№ 054 · Savings cluster

APR vs APY Calculator

Two ways to quote the same rate — and they’re not equal. APR is the nominal rate; APY is what compounding actually earns. Convert either way at any frequency, and see the compounding ladder from annual all the way to the continuous limit, so you can compare any savings or loan offer on a fair basis.

Nominal ↔ effective rate

Locking in a fixed APY? See what it matures to in the CD Calculator, or watch a rate compound over decades in the Compound Interest Calculator.

Your rate01
%
Compounding02

Continuous compounding (the theoretical limit) is shown in the ladder below.

Effective annual yield (APY) · monthly

5.12%

5.00% APR compounded monthly = 5.12% APY

+0.12 pts from compoundingfrom 5.00% APR
Nominal (APR)5.00%
Effective (APY)5.12%
Compounding gain+0.12 pts
The compounding ladder
Effective APY at each compounding frequency for a 5.00% nominal rate — the more often it compounds, the more it yields.
CompoundingPeriods / yrAPY
Annually15.000%
Semiannually25.063%
Quarterly45.095%
Monthly125.116%
Daily3655.127%
Continuously∞5.127%

APR is the stated (nominal) rate; APY is what you actually earn or pay once compounding is counted. They are equal only when interest compounds once a year — compound more often and the APY pulls ahead, toward the continuous limit of eAPR − 1. Banks advertise APY on savings (the bigger number) and APR on loans (the smaller one), so converting to a common basis is the only fair way to compare offers.

How it works · methodology

The math, in the open.

APY = (1 + APR ÷ n)^n − 1          (n compounding periods per year)
APR = n × ((1 + APY)^(1/n) − 1)     (the inverse)

continuous limit:  APY = e^APR − 1,  APR = ln(1 + APY)
The only difference between APR and APY is compounding. With n = 1 (annual) they are equal; raise n and the APY pulls ahead, approaching e^APR − 1 as compounding becomes continuous. The conversion is exact and invertible.
APR
Nominal annual rate (the stated rate)
APY
Effective annual yield (after compounding)
n
Compounding periods per year (12 = monthly, 365 = daily)
e
Euler’s number, for the continuous-compounding limit

Assumptions & limits

  • A pure rate-to-yield conversion: it counts compounding only, not fees.
  • A loan’s regulatory APR may fold in certain fees, so it can exceed the note rate; that fee-inclusive figure is a separate calculation.
  • Daily compounding uses a 365-day year; some institutions use 360, which shifts the APY a hair.
  • Compare offers in APY terms — it is the basis that already reflects compounding.

Sources

By the MintNumber Editorial Team·Reviewed
Frequently asked

Questions about APR & APY

What is the difference between APR and APY?
APR (annual percentage rate) is the nominal rate — the stated rate before compounding is counted. APY (annual percentage yield) is the effective rate — what you actually earn or pay once interest compounds within the year. If a rate compounds more than once a year, the APY is always higher than the APR; the gap is the interest you earn on your interest. They are identical only when interest compounds exactly once a year.
How do you convert APR to APY?
APY = (1 + APR ÷ n)^n − 1, where n is the number of compounding periods per year. A 12% APR compounded monthly (n = 12) becomes (1 + 0.12/12)^12 − 1 = 12.68% APY. To go the other way, APR = n × ((1 + APY)^(1/n) − 1). As compounding approaches continuous, the APY reaches its ceiling of e^APR − 1 — for 12% that is 12.75%, only a hair above the monthly figure.
Why is APY higher than APR?
Because of compounding within the year. With monthly compounding, January’s interest itself earns interest in February, and so on — so by year-end you have earned slightly more than the flat nominal rate implies. The more frequently interest compounds, the larger this effect, which is why the APY climbs as you move from annual to semiannual to monthly to daily compounding. The jumps shrink each step, flattening out at the continuous limit.
Why do banks quote APY on savings but APR on loans?
Marketing. On a savings account or CD, the APY is the larger, more attractive number, so that is what gets advertised. On a loan or credit card, the APR is the smaller number, so lenders lead with that — even though the amount you actually pay reflects compounding and is closer to the APY. U.S. truth-in-lending and truth-in-savings rules require these disclosures precisely so you can compare on a consistent basis.
Which should I use to compare two offers?
Always compare on the same basis, and APY is the honest one because it already accounts for compounding. A 5.9% APR compounded daily can cost more than a 6% APR compounded annually, even though the headline rate looks lower — converting both to APY reveals the truth. When you are shopping savings accounts, CDs, loans, or credit cards, put every offer in APY terms before deciding.
Does the APR include fees?
On a deposit account, no — APR there is purely the compounding-basis rate. On a loan, the regulatory APR is meant to fold in certain fees (origination, points) on top of the interest rate, which is why a mortgage’s APR is usually a bit higher than its note rate. This calculator handles the pure rate-to-yield compounding conversion; for a loan’s fee-inclusive APR, see the personal loan calculator, which derives the effective APR from an origination fee.