finance
APR vs APY, and Why Compounding Frequency Matters Less Than You Think
APY includes compounding, APR does not, and loan APR adds fees. At 5%, daily beats monthly compounding by about one basis point. Rate and time matter more.
A savings account advertises “4.88% interest, 5.00% APY.” A credit card says “24.99% APR.” A mortgage quote lists a rate and a slightly higher APR. That’s three ways of putting a number on interest, and they don’t mean the same thing. APY counts compounding. APR, by definition, does not, and on a loan it also includes certain fees. Once you know which is which, comparing offers is straightforward. One more thing to know before you start comparing: compounding frequency, the detail ads like to highlight, moves the result far less than the rate itself or how long the money stays put. At 5%, switching from monthly to daily compounding is worth about one basis point a year.
This post covers the conversion formula, the US rules that decide which number you see, and why “compounded daily!” rarely deserves much attention. It explains the terms only. It is not financial advice.
Nominal rate vs effective rate
A nominal annual rate is the per-period rate multiplied by the number of periods in a year. A bank paying 0.4167% a month quotes a 5% nominal rate. An effective annual rate is what you actually earn over a year once interest earns interest. To convert a nominal rate r compounded n times a year:
APY = (1 + r/n)^n − 1As n goes up without limit, this approaches continuous compounding:
APY = e^r − 1Here is a 5% nominal rate at each common frequency:
| Compounding | n | Effective annual rate (APY) | Gain over annual |
|---|---|---|---|
| Annually | 1 | 5.0000% | — |
| Quarterly | 4 | 5.0945% | +9.5 bp |
| Monthly | 12 | 5.1162% | +11.6 bp |
| Daily | 365 | 5.1267% | +12.7 bp |
| Continuously | ∞ | 5.1271% | +12.7 bp |
(A basis point, bp, is 0.01 percentage point.) Going from annual to monthly compounding adds 11.6 bp. Going from monthly to daily adds about 1.05 bp, roughly $1.05 a year on $10,000. Going from daily to continuous adds almost nothing. The curve flattens fast, so once interest compounds monthly, more frequent compounding barely changes the result.
That’s how a bank arrives at a pair like “4.88% interest rate, 5.00% APY.” A 4.88% nominal rate compounded daily is a 5.0007% effective rate, which rounds to 5.00%.
Which number US law makes them show you
In the US, the number you’re shown depends on whether you are lending money to a bank or borrowing from one.
Deposits show APY. The Truth in Savings Act is implemented by Regulation DD, whose stated purpose is to let consumers “make meaningful comparisons among depository institutions” (12 CFR 1030.1). It defines annual percentage yield as “a percentage rate reflecting the total amount of interest paid on an account, based on the interest rate and the frequency of compounding for a 365-day period.” It defines the interest rate as the annual rate “which does not reflect compounding” (12 CFR 1030.2). If a deposit ad states a rate of return, it must state it as an “annual percentage yield.” The interest rate can appear alongside it but cannot be more prominent (12 CFR 1030.8). The official formula in Appendix A works from the actual interest paid on a 365-day basis, APY = 100 × [(1 + Interest/Principal)^(365/Days in term) − 1], and for a one-year term it reduces to the compounding formula above.
Loans and cards show APR. The Truth in Lending Act is implemented by Regulation Z, which aims to “promote the informed use of consumer credit by requiring disclosures about its terms and cost” (12 CFR 1026.1). Two features of APR surprise people:
- APR is not compounded. For open-end credit such as credit cards, the APR is determined by “multiplying each periodic rate by the number of periods in a year” (12 CFR 1026.14). That’s a nominal rate.
- Loan APR includes finance charges, not just interest. The finance charge is “the cost of consumer credit as a dollar amount,” and Regulation Z’s list includes points, loan fees, finder’s fees, appraisal and credit report fees, and mortgage broker fees (12 CFR 1026.4). The CFPB describes APR as “a broader measure of the cost of borrowing money than the interest rate” (CFPB).
So on a mortgage or personal loan, the APR is not the interest rate your payment is calculated from. Our post on why your first mortgage payment is mostly interest explains why you should enter the note rate, not the APR, into a payment calculator.
How credit card interest actually compounds
According to the CFPB, many issuers “calculate the interest you owe daily, based on your average daily account balance” (CFPB). The daily rate is the daily periodic rate, which is the APR divided by 360 or 365 depending on the issuer. Each day’s interest is added to the balance, “which means that interest is compounding on a daily basis” (CFPB).
For a 24.99% APR card on a 365-day basis:
daily periodic rate = 24.99% / 365 = 0.06847% per day
effective rate if a balance sat untouched all year = (1 + 0.2499/365)^365 − 1 ≈ 28.38%The 28.38% is just an illustration of what daily compounding does to a nominal APR. Real cards have grace periods, payments and changing balances. The CFPB notes that with a grace period “you can avoid paying interest on purchases if you pay off your balance in full by the due date each month.” The point here is narrower: a card’s APR understates the effective cost of a balance you carry, in the same way a savings account’s interest rate understates its APY.
Rate and time beat compounding frequency
Now compare compounding frequency with the factors that matter more. Start with $10,000 and add nothing:
| Scenario | 10 years | 30 years |
|---|---|---|
| 4%, compounded monthly | $14,908.33 | $33,134.98 |
| 4%, compounded daily | $14,917.92 | $33,198.99 |
| 5%, compounded monthly | $16,470.09 | $44,677.44 |
| 5%, compounded daily | $16,486.65 | $44,812.29 |
Read across the rows and down the columns:
- Monthly to daily at 5%: +$16.56 over 10 years, +$134.85 over 30.
- 4% to 5%, both monthly: +$1,561.76 over 10 years, +$11,542.46 over 30.
- 10 years to 30 years at 5% monthly: $16,470 becomes $44,677.
One percentage point of rate is worth roughly 85 to 95 times as much as the jump from monthly to daily compounding, and time multiplies everything. If one account offers a higher APY with monthly compounding and another offers a lower APY with daily compounding, the higher APY wins, because the APY already includes the compounding.
The Rule of 72, and where it drifts
To estimate how long money takes to double, divide 72 by the annual rate in percent. The exact answer for an annual effective rate i is ln 2 / ln(1 + i):
| Rate | Rule of 72 | Exact (annual compounding) | Error |
|---|---|---|---|
| 2% | 36.0 yrs | 35.0 yrs | +1.0 |
| 5% | 14.4 yrs | 14.2 yrs | +0.2 |
| 8% | 9.0 yrs | 9.0 yrs | ≈ 0 |
| 12% | 6.0 yrs | 6.1 yrs | −0.1 |
| 18% | 4.0 yrs | 4.2 yrs | −0.2 |
The rule is very accurate around 8%. It runs a little long at low rates and a little short at high ones, and it assumes annual compounding. At 5% compounded daily, the exact doubling time is 13.9 years. Our compound interest calculator shows the exact doubling time from the effective rate, not the rule-of-thumb estimate.
Which number to compare
- Savings accounts, CDs, money market accounts: compare APY to APY. It already includes compounding, so you can ignore “compounded daily” claims when comparing.
- Loans: compare APR to APR for the same loan amount and term. That captures fees that the interest rate alone hides. Use the interest rate to work out the payment.
- Credit cards: the APR is nominal and interest is charged daily on the average daily balance. Paying in full within the grace period matters far more than the APR’s decimals.
- Promotional and teaser rates: check how long the rate lasts and what it changes to. A 6-month promo APY followed by a lower rate is not a one-year yield, and a card’s 0% intro APR ends on a set date.
- Watch the label: “rate,” “interest rate,” “APR,” and “APY” are different numbers. If two offers quote different ones, convert one to the other before comparing.
What our calculators do
The compound interest calculator takes a nominal annual rate and lets you choose daily (365 per year), monthly, quarterly, semiannual, annual or continuous compounding. It shows the resulting effective annual rate next to the final balance. You can add a monthly or yearly contribution, made at the start or end of each period, plus extra months and an optional inflation adjustment. When contributions and compounding run on different schedules, such as monthly deposits with daily compounding, it converts the nominal rate to the effective annual rate and grows the balance by the equivalent monthly factor, which is exact at each month boundary. If your bank quotes only an APY, enter that APY with annual compounding to reproduce it exactly. Entering an APY with daily compounding would count the compounding twice.
On the borrowing side, the loan calculator uses a fixed interest rate with monthly payments (rate ÷ 12 per month) and doesn’t model fees. Give it the interest rate, not the APR.
Try it
Open the compound interest calculator, enter $10,000 at 5% for 30 years, and switch the compounding from monthly to daily. The balance changes by about $135. Then change the rate to 4% and watch it drop by more than $11,000. Estimates assume a constant rate, which real accounts rarely have, and don’t account for taxes or fees.