A bank advertises a 4.5% mortgage and a 4.5% savings account. The mortgage rate is the APR; the savings rate is the APY. Both numbers look identical, but they describe different things. The mortgage is quoted before compounding, the savings account after. Plugging the same 4.5% into both sides of a financial calculator and assuming you understand your money is one of the most common, and most consequential, consumer-calculator mistakes.
The difference between APR (Annual Percentage Rate) and APY (Annual Percentage Yield) is small over short periods and large over long ones. On a $300,000 mortgage at 6% over 30 years, the gap between treating it as APR vs APY is about $11,000 in total interest. On a $10,000 savings account at 4.5% over 30 years, the gap is about $1,800 less in your account if you assumed APR when the bank quoted APY.
APR: defined by the Truth in Lending Act
APR is a regulatory term for consumer credit. The Truth in Lending Act (TILA), passed in 1968 and implemented through Regulation Z, requires lenders to quote the cost of credit as an "annual percentage rate" using a standardized calculation method. The intent was to make loans comparable: if you're shopping a 30-year mortgage from three banks, the APR should let you compare them on the same basis.
Reg Z defines APR as the simple-interest annualized cost of credit. For a mortgage, the calculation is the loan's nominal interest rate plus an amortization of certain upfront fees (origination, discount points, mortgage insurance), expressed as an annualized percentage of the unpaid balance over the loan term. The calculation method is standardized down to which fees are includible (12 CFR 1026.32) and which are not.
Three things matter for understanding APR specifically:
It does not account for compounding within the year. A loan quoted at 6% APR with monthly payments has a real effective annual rate that's slightly higher than 6% (around 6.17%) because each monthly payment of interest on the unpaid balance is calculated on a balance that itself accrued interest the previous month. Reg Z lets the lender quote the simple-rate APR and ignore the compounding effect, on the theory that the standardized APR is meant for cross-loan comparison rather than dollar accuracy.
It does include certain fees, which is why the APR on a mortgage is almost always slightly higher than the "interest rate." The interest rate is just the rate; the APR is the rate plus the amortized cost of the fees that get bundled into the loan. A mortgage with a 6.0% interest rate and $4,000 in points often has a 6.15% APR. Both numbers are honest; they just answer different questions.
It applies primarily to closed-end credit like mortgages and auto loans. Open-end credit (credit cards) uses a similar concept but the calculation method differs slightly. Credit card APR is the daily periodic rate multiplied by the number of days in a billing cycle and annualized, which is closer to a true periodic rate but still doesn't compound across cycles for disclosure purposes.
APY: defined by the Truth in Savings Act
APY is the deposit-side analog. The Truth in Savings Act (TISA), passed in 1991 and implemented through Regulation DD, requires depository institutions to quote interest on savings products using "annual percentage yield," explicitly defined to include the effect of compounding.
Reg DD's APY formula is:
APY = (1 + r/n)^n - 1Where r is the nominal interest rate and n is the number of compounding periods per year. For a savings account that pays 4.5% nominal interest compounded daily (n = 365), the APY is (1 + 0.045/365)^365 - 1 = 4.602%. The bank can advertise either 4.5% or 4.602% legally, but only the 4.602% APY can be presented as the "yield," which is the number consumers are supposed to use for comparison.
The asymmetry is that the law requires APY for deposits but allows simple-rate APR for most consumer loans. The lender gets to quote the lower-looking number; the depositor gets to see the higher-looking number. Both sides of the bank's books advertise rates that flatter the bank's offer.
Where the difference adds up: 30 years of compounding
The mathematical gap between APR and APY for a single year is small. A 4.5% APR with monthly compounding produces an effective annual yield of 4.594%, a difference of about 9 basis points. But compounding multiplies that gap over time. The relevant numbers for common consumer products:
| Product | Stated rate | Compounding | True annual yield | Gap (bps) |
|---|---|---|---|---|
| 30-year mortgage | 6.0% APR | Monthly | 6.168% | 17 |
| Credit card | 19.99% APR | Daily | 22.13% | 214 |
| Savings (Reg DD compliant) | 4.5% APY | Daily | 4.5% (already accounts for it) | 0 |
| Car loan | 7.0% APR | Monthly | 7.229% | 23 |
The credit card row is the one that surprises people. A "19.99% APR" credit card, if you carry a balance and the issuer compounds interest daily (which most do), is really charging you 22.13% in effective annual interest. The disclosed APR is honest under Reg Z. It's just not the number that determines what you actually pay.
A worked example for a 30-year savings comparison: $10,000 deposited at 4.5% nominal interest, compounded daily, with no additional contributions:
- After 1 year: $10,460 (the daily compounding turned the 4.5% nominal into a 4.602% effective yield)
- After 10 years: $15,683
- After 20 years: $24,594
- After 30 years: $38,569
If you'd treated that 4.5% as a simple-interest APR (no compounding within the year), the same $10,000 grows to roughly $36,749 over 30 years. The compounding gap on a single deposit over 30 years is $1,820, about 4.7% of the final balance. On regular contributions, the gap grows because each contribution gets compounded for less time, but the overall fraction the compounding effect contributes stays in the same range.
Where consumer calculators get this wrong
Many free online financial calculators don't distinguish the two. They take a single "interest rate" input and compute compound interest internally, regardless of whether the user typed in an APR or an APY. The results are off by 5-15% for long-horizon savings calculations and off by 5-25% for credit card debt projections.
The mistakes split into two patterns:
Treating a quoted APR as if it were already compounded. A user types in "6% APR" from their mortgage statement. The calculator runs compound-interest math on that 6%, which means it's compounding the rate twice: once because the math says to, and once implicitly because 6% is already being treated as a yield. The total interest calculation comes out about 17 basis points higher than reality on a 30-year mortgage, which is roughly $11,000 of phantom interest on a $300,000 loan.
Treating a quoted APY as if it were a nominal rate to compound. The opposite mistake on the savings side. A user types in "4.5% APY" and the calculator compounds it. The result undershoots the actual balance over time, because the 4.5% APY already includes the compounding the calculator is now adding on top.
The honest behavior for a consumer calculator is to ask which kind of rate the user is entering and adjust the math accordingly, or at minimum, label the input clearly so the user knows which to type. Most don't.
What you actually want to use, when
For comparing mortgage offers from different lenders: APR. That's what TILA standardized, and it's the only number where the fees are included.
For projecting how much a savings account will be worth in 20 years: APY (which is already the compounded number). Don't compound it again.
For figuring out the real cost of credit card debt: convert the APR to an effective annual rate using the compounding formula. The disclosed APR understates what you'll actually pay if you carry a balance.
For comparing two savings products with different compounding frequencies: APY. That's what Reg DD requires, and it's the only number that lets you compare a daily-compounded account to a monthly-compounded one on the same basis.
For consumer-loan products other than mortgages: APR is the disclosed number, but check whether the lender is offering daily or monthly compounding. Two loans with the same APR and different compounding frequencies have different actual costs.
Accruo takes both kinds of input and shows the year-by-year balance with the math visible: stated rate, compounding frequency, effective annual rate, and the ending balance per year. Useful when you'd rather not guess whether the calculator you're using is treating your input as APR or APY.