APR vs APY: the difference that quietly costs (or earns) you money

Two products advertise "5%." One is a savings account, the other a loan. You'd assume the 5% means the same thing in both places. It doesn't. The savings account is almost certainly quoting APY, and the loan is quoting APR, and on the exact same nominal rate those two numbers describe different amounts of money. The difference is compounding, and once you see how it works you'll never read a rate the same way again.

The core distinction in one sentence

• APR (Annual Percentage Rate) is the nominal yearly rate without accounting for compounding within the year.
• APY (Annual Percentage Yield), also called EAR (Effective Annual Rate), is the rate with compounding folded in — the actual amount you earn or pay over a full year.

If interest compounded exactly once a year, APR and APY would be identical. But interest almost never compounds annually. Savings accounts compound daily or monthly. Credit cards compound daily. The more often interest compounds, the more APY pulls ahead of APR.

Why compounding creates the gap

Compounding means interest earns interest. If you have a 12% APR compounded monthly, you don't get 12% once at year-end. You get 1% each month (12% divided by 12), and each month's 1% is calculated on a balance that already includes the previous months' interest. Those tiny monthly boosts stack, and by December you've earned slightly more than a flat 12%.

That "slightly more" is the entire story of APY. It's the honest, all-in annual figure.

The conversion formula

To turn a nominal APR into the true APY, use:

APY = (1 + APR/n)^n - 1

Where:

• APR = the nominal annual rate, as a decimal (12% = 0.12)
• n = the number of compounding periods per year (12 for monthly, 365 for daily)

The reverse — going from a known APY back to the nominal APR — is:

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

You rarely need to do this by hand. The Compound Interest Calculator lets you set the compounding frequency and shows the real growth, which is APY in action. But let's work the formula directly so the gap is concrete.

Worked example 1: 12% APR, monthly vs daily

Take a 12% nominal APR. Here's what the APY becomes at different compounding frequencies.

Compounded monthly (n = 12):

APY = (1 + 0.12/12)^12 - 1
    = (1 + 0.01)^12 - 1
    = 1.126825 - 1
    = 0.126825  ->  12.68%

Compounded daily (n = 365):

APY = (1 + 0.12/365)^365 - 1
    = (1.00032877)^365 - 1
    = 1.127475 - 1
    = 0.127475  ->  12.75%

So a "12%" rate is really 12.68% when compounded monthly and 12.75% when compounded daily. On a $10,000 balance, that's the difference between earning (or owing) $1,268 versus $1,275 in a year — and versus the $1,200 the naive "12%" suggests. The more frequent the compounding, the higher the effective figure, though you can see the gains shrink: going from monthly to daily only adds about $7 here, because there's a mathematical ceiling (continuous compounding) that frequent compounding approaches but never exceeds.

Worked example 2: 5% on savings

Now the friendlier direction. A savings account advertises a 5% rate compounded daily. What's the APY the bank is allowed to print?

APY = (1 + 0.05/365)^365 - 1
    = (1.00013699)^365 - 1
    = 1.051267 - 1
    = 0.051267  ->  5.13%

The bank quotes 5.13% APY even though the underlying nominal rate is 5%. They're not lying — that's the genuine yearly yield once daily compounding is included. On $20,000, you earn about $1,025 instead of a flat $1,000. The extra $25 is compounding working in your favor.

Why lenders quote APR and banks quote APY

This is not an accident. It's marketing physics.

• Lenders (credit cards, loans) quote APR because it's the lower-looking number. A card that charges 24.99% APR is actually costing you more than 24.99% once daily compounding is applied — closer to 28.3% APY. Quoting APR makes the cost look smaller.
• Banks (savings, CDs) quote APY because it's the higher-looking number. APY flatters a deposit by including the compounding upside.

In both cases the institution picks the framing that makes their offer look better. The lesson: when you're borrowing, mentally convert the APR upward to see what it really costs. When you're saving, the APY is already the honest figure, so compare APYs across banks directly. To compare two loans fairly, run both through the Loan Calculator, which works from the periodic rate and shows the true total interest.

The credit-card daily periodic rate

Credit cards make this tangible. Your statement lists an APR, but interest is charged using the daily periodic rate (DPR):

DPR = APR / 365

For a 24.99% APR card:

DPR = 0.2499 / 365 = 0.0006847  (about 0.06847% per day)

Each day, the card multiplies your balance by that DPR and adds it on. Tomorrow's interest is charged on a balance that includes today's interest — daily compounding, every day of the month. That's why the effective annual cost of a 24.99% card is over 28%:

APY = (1 + 0.2499/365)^365 - 1
    = (1.0006847)^365 - 1
    = 0.2838  ->  28.38%

A card advertised at 24.99% effectively charges 28.38% a year on a carried balance. The roughly 3.4 percentage point gap is pure compounding, hidden inside the friendlier APR headline. On a $5,000 balance carried for a year, that's about $1,419 in interest, not the $1,250 the APR implies.

A quick sanity-check habit

When you see a rate, ask two questions: Is this APR or APY? and How often does it compound? Those two facts fully determine the real number. A few rules of thumb:

• If two rates have the same APR, the one compounding more frequently has the higher APY — better for saving, worse for borrowing.
• APY is always greater than or equal to APR. They're equal only when compounding is annual.
• For borrowing decisions, treat APR as the floor of what you'll actually pay.
• For saving decisions, compare APYs, never nominal rates, across institutions.

Plug your own figures into the Compound Interest Calculator and toggle the compounding frequency to watch APY rise. Then use the Loan Calculator to see what a given APR truly costs over a full repayment term. The four-character difference between APR and APY is small on paper and large in your bank balance.