Learning goals
- Explain what APR and APY each include and what they omit
- Convert a nominal APR and compounding frequency into APY
- Choose the right rate when comparing loans versus savings products
Two annual rate labels
APR (annual percentage rate) is a yearly rate that often excludes the effect of intra-year compounding in the headline number. On consumer loans, APR is meant to help compare credit costs and may fold in certain fees depending on local rules — always read the disclosure.
APY (annual percentage yield), sometimes called effective annual rate (EAR), answers: if I leave $1.00 for a year with compounding, what percent do I earn? Savings accounts advertise APY so daily or monthly compounding is already reflected.
Rule of thumb: when comparing how savings grow, prefer APY. When reading many loan offers, APR is the standard comparison label — but payment schedules still depend on the periodic rate and fees. Do not treat APR and APY as interchangeable numbers.
APY from nominal APR
APY = (1 + r/n)^n − 1
r is the nominal annual rate as a decimal; n is the number of compounding periods per year. Monthly compounding uses n = 12; daily often uses n = 365 (conventions vary).
Periodic rate from APR
periodic rate = APR ÷ n
Loan amortization usually applies APR/12 each month. That monthly rate compounds across the year, so the effective yearly cost can exceed the nominal APR when interest capitalizes.
Example 1 — Turn APR into APY
Given
A savings account quotes 4.80% APR compounded monthly. What is APY?
Steps
- r = 0.048; n = 12.
- APY = (1 + 0.048/12)^12 − 1 = (1.004)^12 − 1.
- (1.004)^12 ≈ 1.0491, so APY ≈ 0.0491 = 4.91%.
Answer
APY is about 4.91%.
Tip: More frequent compounding (daily vs monthly) nudges APY slightly higher for the same nominal APR.
Example 2 — Why the yield beats the nominal rate
Given
You deposit $1,000.00 at 4.80% nominal, compounded monthly, for one year with no withdrawals.
Steps
- Monthly rate = 0.048/12 = 0.004.
- Ending balance ≈ 1,000 × (1.004)^12 ≈ $1,049.14.
- Interest earned ≈ $49.14 → effective yield ≈ 4.91%, matching APY.
Answer
You earn about $49.14, not $48.00, because interest compounds.
Example 3 — Comparing a loan APR to a savings APY
Given
Loan offer 6.00% APR (monthly payments). Savings APY 4.90%. Can you “arbitrage” casually?
Steps
- The loan’s monthly rate is 0.06/12 = 0.005; interest accrues on the outstanding balance.
- The savings APY already includes compounding on deposits.
- Borrowing at ~6% to invest at ~4.9% loses money before taxes and risk — compare like with like, and remember loan APR disclosures may treat fees differently than deposit APY.
Answer
Do not assume a lower savings APY beats a higher loan APR in a casual side-by-side without matching compounding, fees, and risk.
Tip: For loans, also compare total interest and fees on an amortization schedule — not the rate label alone.
Treating APR and APY as the same number
Common mistake
“Both say about 5%, so the credit card and the savings account cost/earn the same.”
Better approach
Labels differ, compounding differs, and loan APR vs deposit APY answer different questions. Convert to a common effective basis before comparing.
Check your understanding
1.Nominal 6% compounded monthly. Approximate APY?
Answer: (1 + 0.06/12)^12 − 1 ≈ 6.17%.
2.APR 9%, monthly loan. What is the monthly rate r?
Answer: 0.09 / 12 = 0.0075.
3.If compounding is annual (n = 1), how do APR and APY relate?
Answer: They match: APY = (1 + r/1)^1 − 1 = r.
4.Which quote usually already includes compounding for savers?
Answer: APY (annual percentage yield).
Key takeaways
- APR is a nominal annual rate; APY/EAR includes compounding over a year.
- APY = (1 + r/n)^n − 1 for nominal rate r with n compounds per year.
- Savings ads often quote APY; many loans quote APR — they are not drop-in replacements.
- Compare products on a common effective basis and still check fees, payments, and risk.