Interest, APR & APY

Tier 1 · Debt & Credit · beginner · ~4 min read · interestaprapycompoundingcredit-cards

One rate, two names

Every interest rate you’ll ever be offered comes in one of two flavors, and the difference is just compound interest wearing a disguise.

APR (annual percentage rate) is the nominal rate — the sticker number. It says: “we charge 2% a month, and 2% × 12 = 24% a year.” Simple multiplication, no compounding.

APY (annual percentage yield, also called the effective annual rate) is what actually happens to a balance over a year, because interest charged in February starts earning — or costing — interest itself in March. A 24% APR compounded monthly works out to

(1 + 24%/12)¹² − 1 = 26.82%

That 2.82-point gap is not a fee or a trick. It’s the same engine from the credit-card lesson, measured honestly: on a $10,000 balance, “24%” suggests $2,400 of interest a year — compounding actually delivers $2,682.

Why both numbers exist

If APY is the truth, why does APR survive? Because each side of the counter gets to pick the flattering label, and both are perfectly legal:

• Lenders quote APR. Your credit card’s “24.99% APR” compounds daily in practice — closer to 28% effective. The small number is on the brochure; the big one is on your balance.
• Banks advertise savings in APY. A savings account paying 4.89% APR compounded daily gets marketed as “5.01% APY” — the same account, described with the bigger number, because here the bigger number sells.

Same math, opposite marketing. The skill this lesson teaches is a single reflex: before comparing two offers, convert both to APY. Comparing an APR against an APY is comparing meters against feet.

Watch the quote drift from reality

The dashed line is the quoted APR — what the brochure says. The teal curve is the APY — what your balance experiences. The amber wedge between them is what the quote hides, and it widens as rates climb: barely visible at savings-account rates, a chasm at credit-card rates.

Things worth trying

• Slide Compounding frequency from yearly to monthly. At yearly, APR and APY are the same number — the quote is the whole story. One notch of frequency and a 24% quote becomes a 26.82% reality.
• Keep going: monthly → daily → continuously. 26.82% → 27.11% → 27.12%. The first steps do almost all the damage; past daily, even continuous compounding — the mathematical ceiling — adds a rounding error. Frequency is powerful, then immediately exhausted.
• Drop the APR to 6%. The wedge nearly vanishes: 6% monthly is 6.17% — a 0.17-point gap. Now push it back to 30%: the gap is 4.49 points. The higher the rate, the more the quote understates it — which is exactly why the gap matters most on credit cards.
• Raise On a balance of to $100,000. At 24% compounded monthly, the quote now hides about $2,824 a year. The wedge isn’t abstract; it’s denominated in dollars.

How to actually use this

1. Convert before you compare. Two loan offers, one quoted as APR and one as APY, cannot be compared as printed. Put both in APY (the simulator is the converter) and the cheaper one is simply the smaller number.
2. On debt, expect the truth to be worse than the sticker. Cards compound daily — a “22% APR” card is really about 24.6%. When you estimate what a balance costs you, use the APY.
3. On savings, check which number you were shown. If an account advertises a rate without the letters “APY” next to it, ask. The gap is small at today’s savings rates, but the habit of asking is free.
4. Ignore frequency wars in marketing. “Compounded daily!” sounds generous, but you now know daily vs monthly moves the real rate by hundredths of a point. The rate is what matters; frequency is nearly saturated past monthly.

The vocabulary, precisely

The relationship is one line of math:

APY = (1 + APR ÷ n)ⁿ − 1, where n is how many times a year interest compounds.

Three things fall out of it, and the simulator shows all three: with n = 1 the two rates are identical; APY rises with n but flattens fast toward a hard ceiling (e^APR − 1, continuous compounding); and the APR-vs-APY gap grows superlinearly with the rate itself. Small print, fully decoded.

Key terms

• APR (annual percentage rate) — the nominal yearly rate: the per-period rate multiplied by the number of periods, ignoring compounding.
• APY (annual percentage yield) — the effective yearly rate: what a balance actually grows or costs over a year once compounding is included. Also called effective annual rate (EAR).
• Compounding frequency — how often interest is added to the balance (n in the formula). More frequent means a higher APY for the same APR, with sharply diminishing returns.
• Continuous compounding — the limit of compounding infinitely often: APY = e^APR − 1. The ceiling no real account quite reaches, and daily compounding nearly touches.

Next in Debt & Credit: debt payoff strategies — avalanche vs snowball, and what ordering your debts actually costs or saves.