The rate you really earn.
The advertised rate isn't what you get. APY adjusts for compounding frequency. Real return adjusts for inflation. Two flavors of the same insight.
Two ads catch your eye while you scroll. Bank A: 4.85% APR, daily compounding. Bank B: 4.90% APR, monthly compounding. Bank B's headline number is higher. But which one actually pays more in a year?
The honest comparison is APY, not APR. Run the formula and Bank A's APY comes out to 4.97%; Bank B's lands at 5.01%. Bank B still wins, but barely — and not by what you'd guess from the headlines. APR alone hides the compounding difference. APY shows it.
Two questions an account balance never answers.
Same account, same headline rate, three honest answers to "what am I actually earning?" APY moves the rate up; inflation moves it down. Both matter.
The effective annual yield (APY) is the actual annual return on an account, accounting for how often interest compounds: Y = (1 + r/n)n − 1. The real return is the nominal return minus the inflation rate, in percentage points.
Vocabulary you'll see in word problems
- APY (Effective Annual Yield) What an account actually pays in a year, with compounding factored in. Always ≥ APR for any n > 1.
- APR (Nominal rate) The advertised annual rate, before compounding effect. Also called the stated or nominal rate.
- CPI (Consumer Price Index) The U.S. Bureau of Labor Statistics index tracking the cost of a basket of household goods. The data source for inflation.
- Inflation The annual percent increase in prices. Computed from CPI: (CPInew / CPIold) − 1.
- Nominal vs. real return Nominal is the headline rate. Real is what's left after subtracting inflation. Real is what your purchasing power actually did.
Two banks at "5%". Which actually wins?
"Bank A offers 5% APR compounded annually. Bank B offers 5% APR compounded daily. Compute each account's APY and decide which one actually pays more in a year."
Identify r and n for each.
Both have r = 0.05. Bank A: n = 1 (annual). Bank B: n = 365 (daily).
Compute Bank A's APY.
With n = 1, the formula collapses: APY equals the APR exactly.
Compute Bank B's APY.
With n = 365, the compounding effect bumps the actual yield slightly above the headline 5%.
Compare and pick a winner.
Bank B pays 5.13% APY vs. Bank A's 5.00% APY. Same advertised 5%, but daily compounding buys you an extra 0.13 percentage points per year. On a $10,000 deposit, that's an extra $13 the first year.
The pattern: APY is always ≥ APR, and the gap grows with n. Annual: APY = APR. Continuous: APY = er − 1. Everything else lives in between.
Three problems. APY, real return, head-to-head compare.
Compute the APY for an account paying 6% APR compounded monthly. Type just the number (e.g. 5 for 5%).
Your savings earned 5% nominal last year. Inflation was 3%. Approximate your real return.
Bank A: 4% APR daily. Bank B: 4.05% APR annually. Bank A's APY is...
Three fast questions before you move on.
Q1. For any account with compounding more often than once a year, the APY is...
Q2. Your savings paid 5% nominal and inflation was 3%. Your real return is approximately...
Q3. CPI stands for...
Picking a savings account, honestly.
The next time a bank ad catches your eye, run two checks before believing the headline rate:
Check 1 — APY: Is this the APY or the APR? Federal regulation (Truth in Savings) actually requires banks to show APY for deposit accounts, but plenty still lead with APR in the ad copy. The APY formula Y = (1 + r/n)^n − 1 tells you what the account actually pays per year. That's the apples-to-apples number across banks.
Check 2 — real return: Subtract the recent inflation rate from the APY. If your savings APY is 5% and inflation is 3%, you're only really gaining 2% in purchasing power. If APY is 0.5% (the going rate at most big banks for years) and inflation is 3%, your real return is negative: your dollars are growing in count but losing purchasing power.
Next: what happens when you keep adding to the pile. Lump sum vs. annuity — one deposit growing alone vs. a stream of deposits each growing on its own clock.
Continue to Lesson 05Different angle? Need another rep? These are optional — tap any that look helpful.
Annual Percentage Rate (APR) and effective APR | Finance & Capital Markets | Khan Academy
The 'effective APR' Sal computes here is exactly our APY formula, Y = (1+r/n)^n − 1, applied to a credit card example. Watching it from the borrowing side first actually clarifies the savings side: same math, one number is what they pay you, the other is what you pay them.
Something's Eating Your Money!
PBS-quality 6-minute explainer of why a 5% APY isn't really 5% if inflation is 3%. Hits the cost-push vs demand-pull distinction without getting jargon-heavy, and lands the purchasing-power point that motivates our real-vs-nominal section.