APR vs APY: The Rate on the Label vs the Rate You Get

Every rate has two faces. There’s the number printed in big friendly font on the ad — and there’s the number that actually shows up in your balance at the end of the year. Banks love quoting whichever face flatters them: a savings account brags about the bigger number, a credit card whispers the smaller one. But here’s the punchline — once interest compounds during the year, the rate on the label is not the rate you get. The gap between them has a name, a formula, and a habit of costing inattentive people real money. Let’s drag both faces into the light.

Two rates, one account

You already know from the last lesson that how often interest compounds changes how much you end up with. Now we name the two rates that fall out of that fact.

The APR — Annual Percentage Rate, also called the nominal rate — is the headline rate that gets quoted. It’s just the per-period rate multiplied back up to a year, and crucially it ignores intra-year compounding. Think of it as the sticker price of borrowing or lending: simple, advertised, and slightly fictional. If a loan charges 1% per month, its APR is just $1\% \times 12 = 12\%$. No compounding baked in.

The APY — Annual Percentage Yield, also called the effective annual rate (EAR) — is what you actually earn or pay over a full year with compounding included. It’s the real, all-in yearly rate after every intra-year interest payment has had its chance to earn interest on itself. Think of it as the out-the-door price: what your balance truly reflects after a year.

The key relationship: APY is always greater than or equal to APR. They’re only equal in one special case (compounding once a year), and otherwise APY pulls strictly above APR. The headline never overstates what you get when interest compounds — it understates it.

From APR to APY

Here’s the analogy: APR is the recipe’s list of ingredients per batch, and APY is how much cake you actually end up with once you bake batch after batch all year, each new batch rising on top of the last. To convert one into the other, you compound the per-period slice across all the periods in the year.

If the APR is $r$ and interest compounds $n$ times a year, then each period earns $r/n$, and after $n$ periods:

$\text{APY} = \left(1 + \frac{r}{n}\right)^{n} - 1$

You divide the headline rate into $n$ little pieces, grow by each piece in turn (that’s the exponent), and subtract the 1 you started with to leave just the growth.

Worked example

Take the classic: 12% APR compounded monthly, so $r = 0.12$ and $n = 12$. Each month earns $0.12 / 12 = 1\%$:

$\text{APY} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = (1.01)^{12} - 1$

$= 1.126825 - 1 = 0.126825 \approx 12.68\%$

So a loan or account quoted at a tidy 12% actually runs at 12.68% once monthly compounding does its thing. That extra 0.68 percentage points is pure interest-on-interest sneaking in between the lines — and you never saw it on the label.

Drag the periods slider below and watch the blue APY bar climb above the grey APR bar. The APR bar never budges when you change the frequency — that’s the whole point. The headline stays put; the rate you actually get keeps rising:

When it matters

The conversion barely matters at tiny rates compounded annually — at 2% compounded yearly, APR and APY are both 2%. It matters enormously for high-rate, frequently-compounding products: credit cards, payday loans, and aggressively-marketed savings accounts. The higher the rate and the more often it compounds, the more the two faces diverge — which is exactly the next section.

More compounding = bigger gap

Hold the headline rate fixed at 12% APR and crank up only the compounding frequency. Watch the APY creep upward — and then notice where it stops:

Two patterns jump out. First, the gap widens as you compound more often — but with sharply diminishing returns. Going from annual to quarterly buys you 0.55 points; going from daily all the way to continuous (infinitely often) buys you a measly 0.003 points. Second, there’s a ceiling: no matter how often you compound, the APY can’t exceed the continuous limit, given by $e^{r} - 1$ (where $e \approx 2.71828$, the natural-growth constant). At 12% that ceiling is exactly 12.75%, and daily compounding has already gotten you 99.98% of the way there.

The other lever is the rate itself. The gap between APR and APY grows with $r$, and fast — because compounding feeds on a bigger base. At 6% APR monthly the gap is about 0.17 points; at 24% APR monthly it balloons to roughly 2.8 points. Double the rate and you far more than double the gap.

Who quotes which — and why it’s not neutral

Now the slightly cynical part. The choice of which face to advertise is never an accident — it’s marketing, and it’s regulated precisely because it’s marketing.

Savings, deposits, CDs → advertised as APY. When an institution is paying you, it wants the bigger number on the poster, so it quotes the compounded yield. “4.50% APY!” sounds better than “4.41% nominal,” even though they’re the same product. Quoting APY flatters the payer.

Loans, credit cards, mortgages → advertised as APR. When an institution is charging you, it wants the smaller number on the poster, so it quotes the nominal rate. “19.99% APR” reads more gently than “21.9% APY,” even though — surprise — the APY is what you actually pay. Quoting APR flatters the lender.

That asymmetry is the whole game: you’re shown the flattering face, and you experience the other one.

Worked example: the credit-card sticker shock

A card advertises 24.99% APR, and like almost all cards it compounds daily. What do you actually pay? Here $r = 0.2499$ and $n = 365$:

$\text{APY} = \left(1 + \frac{0.2499}{365}\right)^{365} - 1 \approx e^{0.2499} - 1 \approx 0.2840 = 28.4\%$

The “24.99% APR” on the offer is really about a 28.4% APY in your pocket — over 3.4 extra percentage points the headline conveniently omitted. On a $5,000 balance carried for a year, that’s roughly $1,420 in interest, not the $1,250 the 24.99% sticker might suggest. The label undersold the damage by about $170.

When it matters

To compare products fairly, always line them up on the same basis — convert everything to APY (the effective rate) and compare like with like. A “5.0% APR compounded daily” CD beats a “5.05% APY” CD only after you convert: $e^{0.05}-1 \approx 5.13\%$ versus 5.05%. The headline ranking can flip once you do the math.

When APR = APY

The two faces collapse into one in exactly one situation: when interest compounds only once a year ($n = 1$). Plug it in and the formula tells the whole story:

$\text{APY} = \left(1 + \frac{r}{1}\right)^{1} - 1 = (1 + r) - 1 = r = \text{APR}$

With annual compounding there’s no intra-year interest to earn interest on, so the headline rate and the effective rate are identical. The moment you compound more than once a year — semiannual, monthly, daily — APY pulls ahead and the two diverge. So if you ever see “APR = APY” quoted, it’s a quiet signal that the product compounds annually.

One footnote worth a single sentence and no more: on loans, the regulatory APR sometimes also folds in fees (origination, points, mortgage insurance) to give a more honest cost of borrowing — that’s a separate adjustment from compounding frequency, governed by lending-disclosure rules, and a rabbit hole we’ll leave for another day.

Putting it together

Two faces of one rate, joined by compounding. The APR is what they say; the APY is what you get. Chunk the whole idea into one picture:

A mixed recap — it pulls from everything above:

Key Takeaways