APR vs APY: Simple vs Compound Interest Explained with Examples
Two accounts can both advertise "5%" and still pay different amounts. Two loans can have the same headline rate and cost different sums. The difference usually comes down to how interest is compounded and which rate is being quoted. This guide explains APR, APY and the maths behind them with numbers you can check.
Simple interest
Simple interest is paid only on the original amount (the principal):
Interest = principal × rate × years
Deposit 10,000 at 5% simple interest for 10 years and you earn 10,000 × 0.05 × 10 = 5,000, ending with 15,000. Each year's interest is the same 500, because it never earns interest itself. Simple interest is used for some short-term loans and for bond coupon payments.
Compound interest
With compound interest, interest is added to the balance and then earns interest itself:
Final amount = principal × (1 + r/n)n × t
where r is the annual rate as a decimal, n is how many times per year interest is compounded, and t is the number of years. The same 10,000 at 5% for 10 years gives:
| Method | Final amount | Interest earned |
|---|---|---|
| Simple interest | 15,000.00 | 5,000.00 |
| Compounded yearly | 16,288.95 | 6,288.95 |
| Compounded monthly | 16,470.09 | 6,470.09 |
| Compounded daily | 16,486.65 | 6,486.65 |
Compounding adds about 1,290 over simple interest in this example, and the gap widens dramatically over longer periods. More frequent compounding helps, but the jump from monthly to daily is small. Try your own figures in the compound interest calculator.
What APR means
APR (annual percentage rate) is the yearly rate before the effect of compounding within the year. A credit card with 24% APR charges 2% per month (24 ÷ 12), or about 0.0658% per day (24 ÷ 365), depending on how the issuer calculates it.
For loans, the legal definition of APR in many countries (for example under the US Truth in Lending Act and EU consumer credit rules) also folds in certain mandatory fees, so that loans can be compared on a like-for-like basis. That is why a mortgage's APR is often a little higher than its interest rate.
What APY (or AER) means
APY (annual percentage yield), called AER (annual equivalent rate) in the UK, is the rate after compounding: what you actually earn over a full year.
APY = (1 + r/n)n − 1
| Nominal rate (APR) | Compounded | Effective yearly rate (APY) |
|---|---|---|
| 5% | monthly | 5.12% |
| 5% | daily | 5.13% |
| 18% | monthly | 19.56% |
| 24% | monthly | 26.82% |
The effect is small at savings-account rates and large at credit-card rates. A card with 24% APR effectively costs almost 27% a year if the balance is left to compound.
Why banks quote different numbers
Lenders like to show the APR on loans because it is the lower-looking number. Savings accounts like to show the APY because it is the higher-looking number. Neither is misleading as long as you compare like with like: APR with APR, APY with APY.
The Rule of 72
To estimate how long money takes to double with compound interest, divide 72 by the yearly rate in percent:
- At 5%: 72 ÷ 5 ≈ 14.4 years (exact answer: 14.2 years).
- At 8%: 72 ÷ 8 = 9 years (exact: 9.0 years).
The rule works in reverse for debt and inflation too. At 3% inflation, prices double in about 24 years. The exact formula is years = ln(2) ÷ ln(1 + r).
Putting it to use
- Saving: compare accounts by APY/AER, and remember that regular deposits matter more than small rate differences. The savings goal calculator shows how much to set aside each month.
- Borrowing: compare loans by APR, which includes most fees, and look at the total amount repaid in the loan calculator.
- Credit cards: because the APR is high and compounds monthly, paying more than the minimum has a huge effect. See how credit card interest is calculated.
Common mistakes
- Comparing one product's APR with another's APY.
- Forgetting that the rate must be divided by the number of periods (a 5% annual rate is about 0.4167% per month, not 5%).
- Ignoring fees, taxes on interest and inflation, which all reduce the real return.
- Assuming "introductory" rates last; check the rate that applies after the offer ends.
This guide explains general principles and is not financial advice. Terms vary by country and lender, so check the exact conditions of any account or loan.