APR vs. APY vs. interest rate: which number to compare

Compare APR with APR when you borrow, and APY with APY when you save. A loan’s APR is a yearly rate that adds certain fees to the interest rate but leaves out compounding. An APY is a yearly rate that adds the effect of compounding but leaves out fees. The plain interest rate has neither, so it lines up only with another interest rate compounded the same way.

What is the difference between APR and APY?

APR adjusts the interest rate for fees; APY adjusts it for compounding. Regulation Z calls the APR the cost of credit as a yearly rate, worked out from what you actually receive and what you repay, so fees counted as finance charges push it above the interest rate. Regulation DD defines the APY as the rate reflecting the total interest paid over a 365-day period, given the interest rate and how often it compounds.

Neither one does the other’s job. Appendix J to Regulation Z makes the APR a nominal rate: the rate per payment period multiplied by the number of periods in a year. A loan with no fees that charges 0.75% a month has a 9.00% APR, although 0.75% compounded for 12 months is 9.38%. The APY formula uses interest alone. Account fees appear elsewhere in the disclosures and do not reduce it.

Two labels are easy to misread:

• “APR” on a savings account. In deposit account disclosures a bank may also call the plain interest rate the “annual percentage rate” (§1030.2). Credit unions follow the NCUA’s version of the rule and call it the “dividend rate” (12 CFR 707.2). Either way it is the rate before compounding, not a fee-inclusive APR.
• Ads. A deposit ad that states a rate of return must state it as an APY and may show the interest rate only alongside it, no more prominently (§1030.8). A loan ad that states a rate of finance charge must state it as an APR (§1026.24).

Why is APY higher than the interest rate?

APY is higher because interest credited during the year earns interest itself. The more often interest compounds, the higher the APY for the same rate:

$$\text{APY} = \left(1 + \frac{r}{m}\right)^{m} - 1$$

Here $r$ is the interest rate as a decimal and $m$ is the number of compounding periods per year: 12 for monthly, 365 for daily. Continuous compounding is the limit, $e^{r} - 1$. With annual compounding, the APY equals the rate.

The same formula turns any nominal rate, a card APR included, into its effective annual rate: the rate actually earned or paid over a year once compounding is counted. An APY is the effective annual rate on a deposit.

Effective annual rate by compounding frequency for two example rates, 4.35% for a deposit and 21.99% for a card (illustrations, not current market rates):

Most of the effect is already there with monthly compounding: at 4.35%, annual to monthly adds 0.09 percentage point and monthly to daily only 0.01. The effect grows with the rate: at 21.99%, daily compounding adds 2.60 points.

Does a loan’s APR include compounding?

No. What lifts a loan’s APR above its interest rate is the fees in the finance charge. Under §1026.4 the finance charge covers interest plus charges such as points, loan fees, and appraisal and credit report fees. Late fees, a card’s annual fee and application fees charged to every applicant are left out. So are title, appraisal, credit-report and document fees on loans secured by real estate.

Worked example: one loan, two adjustments

Example assumptions: a $12,000 personal loan at a 9.00% interest rate, repaid in 36 monthly payments, with a $360 origination fee subtracted from the proceeds. So $11,640 reaches your account.

1. Payment. The lender charges 9.00% ÷ 12 = 0.75% a month on the $12,000 note: $\text{PMT} = \dfrac{12{,}000 \times 0.0075}{1 - 1.0075^{-36}} = 381.60$, so the payment is $381.60.
2. APR. Find the monthly rate $i$ at which 36 payments of $381.60 are worth exactly the $11,640 you received. That rate is $i = 0.9245\%$, and the APR is $12 \times i =$ 11.09%.
3. Compounding. The same monthly rate compounded over a year is $(1.009245)^{12} - 1 =$ 11.68%.

Each figure answers a different question. The 11.09% APR is the figure to set against another lender’s APR for the same amount and term; comparing loan offers walks through that. The compounded figures are effective annual rates, which put a loan on the same footing as an APY. Once the loan exists, the fee has already been paid. If you are weighing extra loan payments against cash in a 4.45% APY account, the like-for-like pair is 9.38% against 4.45%, before taxes.

The APR a lender discloses is the official figure: it is in the loan disclosures, and on a mortgage Loan Estimate it is on page 3 under “Comparisons”. It can differ a little from a calculation like this one. The first payment may fall more or less than a month after the loan starts, the finance charge may include other fees, and Regulation Z counts a disclosed APR on a regular loan as accurate within 1/8 of a percentage point (§1026.22).

Why does a credit card cost more than its APR?

A card’s APR is a simple multiplication, but the interest on a carried balance compounds. Under §1026.14 the APR is the periodic rate multiplied by the number of periods in a year. Some issuers charge a daily periodic rate: the APR divided by 365 or 360, depending on the issuer. Each day’s interest is added to the balance, so it compounds daily.

For an example APR of 21.99%:

• The daily periodic rate is 21.99% ÷ 365 = 0.0602%, or 0.0611% on a 360-day basis.
• A balance carried for a full year with interest compounding daily on the 365-day basis costs 24.59%, not 21.99%.
• As an illustration only (a real account requires minimum payments), $2,000 left for 12 months with no payments or new charges would build up $491.74 of interest, not the $439.80 that 21.99% of $2,000 suggests.

The compounded figure matters only for a balance you carry. On most cards you avoid interest on purchases by paying the full balance by the due date.

Which number should you compare for savings accounts and CDs?

Compare the APY. Account disclosures must state both the APY and the interest rate using those terms (§1030.4), and Appendix A to Regulation DD fixes how the APY is calculated:

$$\text{APY} = 100\left[\left(1 + \frac{\text{Interest}}{\text{Principal}}\right)^{365/\text{Days in term}} - 1\right]$$

For a 365-day term this reduces to interest ÷ principal, so an APY is the interest a deposit earns in a year for every $100. The calculation assumes the principal and interest stay in the account for the whole term, with no deposits or withdrawals. For a savings account without a maturity date, the term is taken to be 365 days. For a variable-rate account, the APY assumes today’s rate holds for the year. A savings account’s APY is therefore a snapshot. A fixed-rate CD’s APY holds for its term.

Banks must round the APY to two decimals, and a disclosed APY counts as accurate if it is within 0.05 percentage point of the exact figure (§1030.3). A difference of 0.01 or 0.02 point between two disclosures can come partly from rounding.

Worked example: a 4.35% rate or a 4.42% APY?

Example assumptions: $15,000 in a 12-month (365-day) CD, with interest left in the CD until maturity. The rates are illustrations, not current offers.

• CD A quotes a 4.35% interest rate, compounded daily.
• CD B quotes a 4.42% APY, compounded monthly.

1. A’s APY: $(1 + 0.0435/365)^{365} - 1 = 4.4457\%$, disclosed as 4.45%.
2. B’s interest rate: $12 \times \left[(1.0442)^{1/12} - 1\right] = 4.3329\%$, disclosed as 4.33%.
3. A at maturity: $15{,}000 \times (1 + 0.0435/365)^{365} =$ $15,666.86.
4. B at maturity: $15{,}000 \times 1.0442 =$ $15,663.00.

Put A’s rate beside B’s APY and B looks 0.07 point better. On the same basis, A pays $3.86 more over the year. The Regulation DD check agrees: $666.86 ÷ $15,000 = 4.4457%. A gap this small is easily outweighed by other terms, such as the early-withdrawal penalty, the minimum deposit and whether the rate is fixed.

How do you convert an APY back to an interest rate?

Reverse the formula for the compounding frequency you need:

$$r = m\left[(1 + \text{APY})^{1/m} - 1\right]$$

CD B’s 4.42% APY is a 4.3329% rate compounded monthly, or 4.3254% compounded daily. You need this when a calculator asks for an interest rate and a compounding frequency but the account quotes only an APY. If you enter 4.42 as a monthly-compounded rate instead, the calculator treats it as a 4.5107% APY. A year’s interest on $15,000 then comes out at $676.60 instead of $663.00, $13.60 too high.

What APR and APY leave out

• How long you keep the loan. APR spreads upfront fees over the full term, so APRs compare fairly only between loans of the same amount and term that run to the end. Repay the example loan after 12 months and the $360 fee is spread over one year instead of three; the same calculation then gives about 12.71%.
• Future rate changes. An adjustable-rate mortgage’s APR does not reflect the highest rate the loan could reach. A variable savings APY is today’s rate held constant.
• Charges outside the finance charge. Late fees, card annual fees and the other exclusions in §1026.4 are real costs that no APR shows. A home equity line’s APR does not include fees, so it should not be set against a closed-end loan’s fee-inclusive APR.
• Deposit fees, taxes and inflation. A monthly maintenance fee lowers what you keep but not the APY. Neither APR nor APY accounts for taxes or inflation.

Try it

• APY calculator: choose interest rate to APY and enter 4.35 with daily compounding: 4.4457%, and the frequency table matches the 4.35% column above (enter 21.99 for the other column). Switch to APY to interest rate and enter 4.42 with monthly compounding: 4.3329%.
• CD calculator: enter a deposit of 15,000 and a term of 12 months. For CD A, choose an interest rate of 4.35 compounded daily: $15,666.86. For CD B, choose an APY of 4.42: $15,663.00.
• APR calculator: enter a loan amount of 12,000, an interest rate of 9, a term of 36 months and fees of 360 paid at closing (deducted from the amount you receive). Leave days to first payment at the default. The payment is $381.60 and the APR is 11.09%.
• Credit card payoff calculator: enter your balance, an APR of 21.99 and your monthly payment to see what the rate costs on a balance you pay down. It charges APR ÷ 12 a month, which compounds to 24.35% a year rather than the daily 24.59%.