APR, APY and why the gap matters
Two savings accounts advertise 5%. One compounds monthly, the other daily. Over a year they pay different amounts, and the difference is exactly the compounding effect that APR ignores. APR (annual percentage rate) is the nominal figure: the periodic rate multiplied by the number of periods — 1% a month is "12% APR". APY (annual percentage yield) is what you actually earn in a year once each period's interest starts earning interest of its own, and it is always the larger of the two.
The formulas
- APY = (1 + APR ÷ n)ⁿ − 1, where n is the number of compounding periods a year: 365 daily, 52 weekly, 12 monthly, 4 quarterly, 2 semiannual, 1 annual.
- APR = n × ((1 + APY)^(1/n) − 1) — the same relationship rearranged.
- Continuous compounding: APY = e^APR − 1 and APR = ln(1 + APY). This is the mathematical limit as n goes to infinity, and every other frequency sits below it.
A worked example
A 12% APR compounded monthly: APY = (1 + 0.12 ÷ 12)^12 − 1 = (1.01)^12 − 1 = 12.6825%. On a $10,000 balance that is $1,268.25 of interest in a year, while simple interest at the same nominal rate would pay $1,200 — the compounding is worth $68.25. Compounded daily the same 12% APR produces 12.7475%, and continuous compounding 12.7497%, so the difference between daily and continuous is under three cents per thousand dollars.
Where the gap is biggest
| Compounding | APY for a 12% APR | Extra over nominal |
|---|---|---|
| Annual | 12.0000% | 0 |
| Semiannual | 12.3600% | 0.36 points |
| Quarterly | 12.5509% | 0.55 points |
| Monthly | 12.6825% | 0.68 points |
| Weekly | 12.7341% | 0.73 points |
| Daily | 12.7475% | 0.75 points |
| Continuous | 12.7497% | 0.75 points |
The pattern is worth internalising: the jump from annual to monthly compounding is worth about 0.68 percentage points on a 12% rate, but everything beyond monthly adds only a few hundredths. When a bank advertises "daily compounding", the real advantage is usually paid as a lower headline rate rather than a bigger APY.
APR, APY, AER and EAR: the naming trap
| Region | Term | What it means |
|---|---|---|
| US | APR | Nominal annual rate (periodic rate × periods), before compounding effects |
| US | APY | Effective annual rate with compounding — what savings actually pay |
| UK | AER | Annual equivalent rate on savings; the same thing as US APY |
| UK | APR | Effective annual rate on credit, including fees — much closer to APY than to US APR |
| EU | EAR / EAM | Effective annual rate under the EBA definition, equivalent to APY |
So the same phrase means different things on either side of the Atlantic: a UK credit APR of 20% is an effective rate including fees, while a US APR of 20% is nominal and could be 21.9% effective if interest compounds monthly. In comparisons, always convert to the effective annual figure — that is what this converter is for.
Where it bites hardest: credit cards
Cards quote a monthly periodic rate, often 1.5–2%. Multiplied by twelve that is an 18–24% nominal APR; the effective APY is 19.56–26.82%. Worse, interest on purchases usually accrues daily from the transaction date, and cash advances start immediately with no grace period at all. The numbers here let you see both views of the same debt.
Frequently asked questions
What is the difference between APR and APY?
APR is the nominal annual rate — the periodic rate multiplied by the number of periods — while APY is what you actually earn or pay in a year once compounding is included. A 12% APR is 12.6825% APY with monthly compounding, so APY is always the larger number on a savings account.
How do I convert APR to APY?
APY = (1 + APR ÷ n)ⁿ − 1, where n is the compounding periods per year: 12 for monthly, 365 for daily. For 12% APR monthly: (1 + 0.01)^12 − 1 = 12.6825%. For continuous compounding use APY = e^APR − 1, which gives 12.7497%.
How do I convert APY back to APR?
APR = n × ((1 + APY)^(1/n) − 1), or APR = ln(1 + APY) for continuous compounding. An APY of 5% with daily compounding is an APR of 4.879% — the same deal expressed the way a bank would advertise it.
Does compounding frequency really matter?
It matters, but far less than the headline rate. On a 12% nominal rate, moving from annual to monthly compounding adds 0.68 percentage points; daily adds only 0.065 points more than monthly, and continuous another 0.002. Compare effective rates, not compounding schedules.
Why is my credit card APR lower than what I pay?
Because the quoted APR is nominal and usually excludes fees. A card charging 1.5% a month is 18% APR but 19.56% APY, and interest on purchases typically accrues daily from the transaction date, with cash advances starting immediately and no grace period at all.
Is UK APR the same as US APR?
No. The UK APR for credit is an effective annual rate that includes compulsory fees, so it is much closer to the US APY. US APR is nominal and excludes compounding. The UK savings equivalent of US APY is called AER, and the EU equivalent is EAR or EAM.