Interest is just a percentage applied over time. The confusion comes from how often it is applied. Here is APR vs APY, simple vs compound, and why the same nominal rate can pay you noticeably different amounts.
Understand APR vs APY, simple vs compound interest, and how compounding frequency (daily, monthly, annually) changes your real return — plus how percent yield works in chemistry.
See also: Percentage calculator · Formula cheat sheet
APR (Annual Percentage Rate) is the plain, stated yearly rate before compounding. APY (Annual Percentage Yield) is what you actually earn once compounding is included. If interest compounds more than once a year, APY is always higher than APR. Lenders like to quote APR (it looks smaller); savings accounts like to quote APY (it looks bigger). Same math, opposite marketing.
Simple interest is charged only on the original principal:
Interest = P × r × t
$1,000 at 5% simple for 3 years earns 1,000 × 0.05 × 3 = $150, ending at $1,150.
Compound interest is charged on principal plus previously earned interest:
A = P × (1 + r/n)^(n·t)
where P = principal, r = annual rate (decimal), n = compounding periods per year, and t = years. The same $1,000 at 5% compounded monthly for 3 years grows to $1,161.47 — about $11 more than simple interest, and the gap widens fast over longer horizons.
Take a single nominal rate — 12% APR — and change only how often it compounds. The effective yield (APY) climbs as the frequency rises:
| Compounding | Periods/yr (n) | APY on 12% APR | $1,000 after 1 year |
|---|---|---|---|
| Annually | 1 | 12.0000% | $1,120.00 |
| Semi-annually | 2 | 12.3600% | $1,123.60 |
| Quarterly | 4 | 12.5509% | $1,125.51 |
| Monthly | 12 | 12.6825% | $1,126.83 |
| Daily | 365 | 12.7475% | $1,127.47 |
| Continuously | ∞ | 12.7497% | $1,127.50 |
Notice the returns are diminishing: going from annual to monthly adds about 68 cents per $100, but going from daily to continuous adds almost nothing. Past monthly compounding, frequency barely matters.
To turn a nominal APR into its effective APY, use:
APY = (1 + r/n)^n − 1
For 12% compounded monthly: (1 + 0.12/12)^12 − 1 = 1.01^12 − 1 = 0.126825 = 12.6825%. To go the other way (you know APY, want the equivalent nominal rate), rearrange: r = n × [(1 + APY)^(1/n) − 1].
Students searching for "percent yield" are doing the exact same operation as a financial yield — comparing what you actually got against the maximum that was possible:
| Financial yield | Chemical percent yield | |
|---|---|---|
| Formula | Actual return ÷ Principal × 100 | Actual yield ÷ Theoretical yield × 100 |
| "What you got" | Interest earned | Mass of product isolated |
| "What was possible" | Principal invested | Mass predicted by stoichiometry |
| Example | $127.50 on $1,000 = 12.75% | 8.5 g of 10 g theoretical = 85% |
Both are a "part ÷ whole × 100" percentage. The financial version can exceed 100% (you can more than double your money); a chemical yield cannot exceed 100%, because you can't make more product than the reactants allow.
You compare two savings accounts. Account A advertises "5.00% APR, compounded daily." Account B advertises "5.05% APY." Which is better? Convert A to APY: (1 + 0.05/365)^365 − 1 = 5.127%. Account A's true yield (5.127%) beats Account B's 5.05%, even though A's headline number looked smaller. Always compare on APY, never on APR.
If interest compounds more than once per year, APY is always higher than APR, because APY includes interest earned on interest. They are only equal when compounding happens exactly once a year.
A lower number looks better on a loan and a higher number looks better on savings. APR (before compounding) is the smaller figure, so it's used for loans; APY (after compounding) is larger, so it's used to advertise deposit accounts.
Only barely. On a 12% rate, daily compounding yields 12.7475% versus 12.6825% monthly — about 6 cents more per $100 per year. The returns from higher frequency shrink quickly and are negligible past monthly.
Structurally yes: both divide what you actually obtained by the maximum that was possible, then multiply by 100. The difference is that a financial yield can exceed 100% while a chemical percent yield cannot.
Links point to primary sources and standards bodies. Tax rates and official formulas change over time — verify against the source for current figures.