APR vs APY (Annual Percentage Rate vs Annual Percentage Yield)

Definition: APR (Annual Percentage Rate) is the simple, non-compounded annual cost of borrowing, or a stated rate of return without compounding, while APY (Annual Percentage Yield) is the effective annual rate once compounding is factored in. The exact same underlying periodic rate can produce a meaningfully different APR and APY, depending purely on how often it compounds.

How It Works

What APR Measures

• APR expresses a rate as if it applied just once, linearly, over a year, with no compounding
• It’s the rate lenders are generally required to disclose on loans, mortgages, and credit cards, meant to give borrowers a standardized way to compare the cost of credit across different products
• Depending on the product and jurisdiction, APR can bundle in certain fees and closing costs alongside the base interest rate, spreading their cost across the loan term into one annualized figure
• Because it ignores compounding, APR will always understate the true annual cost of a loan that compounds more frequently than once a year

What APY Measures

• APY expresses the effective annual rate: what a stated periodic rate actually works out to once interest starts earning interest on itself within the year
• It directly mirrors the effective annual rate (EAR) concept from Compound Interest — APY and EAR are, for practical purposes, the same calculation applied to a savings or investment product
• Banks are generally required to disclose APY on savings accounts, money market accounts, and certificates of deposit, so depositors can compare products on a true, compounding-adjusted basis
• Because it factors in compounding, APY will always be equal to or higher than the underlying nominal rate it’s derived from, for any compounding frequency greater than once a year

Why the Same Rate Splits Into Two Numbers

• Both APR and APY start from the same underlying periodic rate — the rate charged or paid each compounding period, whether monthly or daily
• APR simply multiplies that periodic rate by the number of periods in a year, a linear, non-compounded projection
• APY compounds that periodic rate across all periods in the year, capturing the interest-on-interest effect
• The more frequently a rate compounds, the wider the gap between its APR and its APY becomes
• Because both numbers can be derived from each other with the formula below, neither one hides information from a sophisticated reader — the risk is entirely in a casual reader assuming the two are interchangeable

The Conversion Formula

APY is derived from APR using the same compounding logic as the effective annual rate:

$$APY = \left(1 + \frac{APR}{n}\right)^n - 1$$

where $APR$ is the stated annual rate (as a decimal) and $n$ is the number of compounding periods per year.

• When $n = 1$ (annual compounding), APY and APR are identical, since there’s no intra-year compounding to create a gap
• As $n$ increases — monthly ($n=12$), daily ($n=365$) — APY pulls further above APR for the same stated rate
• In the theoretical limit of continuous compounding, this converges to $APY = e^{APR} - 1$, the same continuous-compounding boundary used in Compound Interest

APR vs. APY at a Glance

	APR	APY
What it captures	Simple, non-compounded annual rate	Effective annual rate, including compounding
Typically advertised on	Loans, mortgages, credit cards	Savings accounts, CDs, money market accounts
Relative size	Always ≤ APY for the same periodic rate	Always ≥ APR for the same periodic rate
Flatters the advertiser because	It looks like a lower cost of borrowing	It looks like a higher return on savings
Includes compounding?	No	Yes

Why the mismatch isn’t an accident: lenders are incentivized to advertise the rate that looks smallest for something they’re charging you for (a loan), while savings products are incentivized to advertise the rate that looks largest for something they’re paying you for (a deposit). Regulators require disclosure of both in many contexts precisely because either figure, used alone, can make the same underlying periodic rate look more attractive than it truly is.

How Compounding Frequency Changes the Gap

Holding the nominal rate fixed at 10% shows exactly how much compounding frequency alone can move APY:

Compounding frequency	Periods per year ($n$)	APY on a 10% APR
Annual	1	10.00%
Semiannual	2	10.25%
Quarterly	4	10.38%
Monthly	12	10.47%
Daily	365	10.52%
Continuous	→ ∞	10.52% (the mathematical ceiling)

• Each step-up in compounding frequency pushes APY a little higher, but the increments shrink as frequency rises
• This shrinking-increment pattern is exactly what the continuous-compounding limit predicts: daily compounding already sits within a hundredth of a percentage point of the theoretical maximum for a 10% nominal rate, echoing the same limit described in Compound Interest
• The practical takeaway is that the jump from annual to monthly compounding matters far more than the jump from monthly to daily — most of the achievable gap is captured well before compounding becomes extremely frequent

Worked Example: Credit Card APR vs APY

A credit card advertises a 24% APR on carried balances, compounded daily ($n = 365$):

$$APY = \left(1 + \frac{0.24}{365}\right)^{365} - 1 \approx 27.1\%$$

A cardholder who carries a balance for a full year isn’t really paying 24% — they’re paying closer to 27.1% once daily compounding is factored in. The advertised “24% APR” is the number printed on the cardholder agreement, but 27.1% is the figure that actually describes the balance’s true annual growth if left unpaid.

Worked Example: Savings Account APR vs APY

A savings account advertises a 5% APY, compounded monthly ($n = 12$). Solving the formula in reverse for the underlying nominal rate:

$$0.05 = \left(1 + \frac{APR}{12}\right)^{12} - 1 \quad \Rightarrow \quad APR \approx 4.89\%$$

The bank markets the account using its 5% APY because it’s the larger, more attractive-looking number — and it also happens to be the mathematically correct figure for “how much will this actually grow,” which is exactly what a saver cares about. Advertising the underlying 4.89% APR instead would understate the account’s true value to a depositor.

What’s Included and Excluded

• APR’s treatment of fees is inconsistent across product types and jurisdictions — a mortgage APR often bundles in origination fees, discount points, and certain closing costs, while a credit card APR typically reflects only the interest rate, with fees like annual or late fees disclosed separately
• Because fee treatment varies, comparing two loans purely by APR can still be misleading if one lender bundles more fees into the figure than another
• APY, by contrast, is a purely mathematical conversion of a rate and compounding frequency — it doesn’t typically absorb fees, though account maintenance fees on a savings product can still erode the realized yield even when the stated APY is accurate
• Some products quote a plain nominal rate distinct from both — the stated periodic rate before either the fee-inclusive APR adjustment or the compounding-inclusive APY adjustment is applied
• Regulatory formulas for APR disclosure can also vary by product category within the same country, so two loans of different types (say, a mortgage and a personal line of credit) may not treat the same fee identically even under the same underlying law
• International disclosure standards differ as well: what one country’s regulator defines as APR may not perfectly match another country’s definition, which matters for anyone comparing cross-border financial products

Fixed vs. Variable Rates

• Both APR and APY can be quoted on either a fixed rate, locked for the life of the product, or a variable rate, tied to a benchmark (such as a central bank’s policy rate) and subject to change
• A variable-rate credit card’s APR can rise or fall over time as the underlying benchmark moves, and its APY recalculates right along with it
• Fixed-rate products give both borrowers and savers certainty about the APR/APY relationship for the life of the product, while a variable-rate product’s advertised APY is a snapshot, not a promise
• This distinction matters most for adjustable-rate mortgages and variable-rate savings accounts, where the rate quoted at signup is explicitly not guaranteed to hold for the full term
• A rate described as “promotional” or “introductory” is a special case worth watching closely — it is typically fixed only temporarily before reverting to a standard variable or fixed rate once the introductory period ends

Why It Matters

• Borrowers who compare loans using only the sticker interest rate, ignoring APR, can miss real cost differences created by fees or compounding
• Savers who don’t distinguish APY from APR can undervalue how much a frequently compounded savings product will actually grow over time
• The APR/APY distinction is a textbook case of how the same underlying number can be framed to look more or less favorable depending on which party is doing the advertising
• Consumer protection law in many countries specifically mandates disclosure of one or both figures because of this framing effect — a rare case where everyday financial terminology is directly shaped by rules designed to prevent misleading marketing
• Understanding the conversion formula lets a consumer sanity-check any advertised rate against a product quoted differently, rather than assuming two differently labeled numbers are directly comparable
• It reinforces the broader compounding lesson from Compound Interest: compounding frequency alone can change a product’s true cost or return without the headline rate changing at all
• It matters most precisely when compounding is frequent (daily or monthly) and the horizon is long, since that’s when the APR/APY gap is largest
• For large, long-term borrowing like mortgages, even a small gap between the advertised rate and the true effective rate compounds into a substantial dollar difference over a 15- or 30-year term

Common Pitfalls

• Comparing a loan’s APR directly to a savings product’s APY: these are different calculations by design; treating them as the same measure produces a misleading comparison
• Assuming APR already includes compounding: it deliberately does not — that omission is exactly what APY corrects for
• Assuming a lower APR always means a cheaper loan: if one lender’s APR excludes fees that another lender’s APR includes, the “lower” number can still represent the more expensive loan overall
• Ignoring compounding frequency when two products quote the same APR: a 20% APR compounded daily costs meaningfully more than a 20% APR compounded annually, even though both are advertised with the identical APR figure
• Treating an advertised APY as guaranteed for a full year: many high-yield savings and promotional rates are variable and can change with little notice, so the realized annual yield may differ from the APY quoted at account opening
• Overlooking fees that erode realized yield despite an accurate stated APY: monthly maintenance or minimum-balance fees can quietly reduce the return a saver actually receives below the advertised figure
• Forgetting that APR on a credit card can vary by transaction type: purchases, balance transfers, and cash advances often carry different APRs on the very same card, so “the card’s APR” is sometimes not a single number at all

APR and APY in Loan Shopping

• Disclosure rules like the U.S. Truth in Lending Act require lenders to present APR prominently specifically so a borrower comparing multiple offers has one standardized number, even when the underlying fee structures differ
• A loan with a lower interest rate but high upfront fees can carry a higher APR than a loan with a slightly higher rate but minimal fees — APR is designed to surface exactly this kind of tradeoff in a single figure
• Shorter-term loans amplify the impact of upfront fees on APR, since those fees are spread over fewer years; the same fee moves a 3-year personal loan’s APR far more than it moves a 30-year mortgage’s APR
• Comparing offers side by side on APR alone still has limits, since it doesn’t capture prepayment penalties, flexibility to refinance, or other non-rate terms that can matter just as much over the life of a loan
• Online loan comparison tools typically sort or filter offers by APR by default, which is one more reason lenders are especially motivated to keep their advertised APR as low as the underlying fee structure allows

Where These Disclosure Rules Come From

• In the United States, the Truth in Lending Act established APR as the standard disclosure for consumer credit, specifically to make comparing loan offers easier across lenders
• The Truth in Savings Act later established APY as the standard disclosure for deposit accounts, closing the equivalent gap on the savings side
• Many other countries have adopted broadly similar concepts under different names, such as an “effective annual rate” or “annual equivalent rate,” serving the same underlying disclosure purpose
• These rules exist specifically because, before standardized disclosure, financial institutions had wide latitude to advertise whichever rate calculation made a given product look most attractive

Related Terms

• Compound Interest
• Interest Rate
• Credit and Debt
• Real vs Nominal Value
• Opportunity Cost

Example

Consider a consumer shopping for both a loan and a place to park savings at the same time. The first product is a personal loan quoted at “9.5% APR,” compounded monthly. Converting to APY shows the true effective annual cost is about:

$$\left(1 + \frac{0.095}{12}\right)^{12} - 1 \approx 9.92\%$$

— a modest but real gap between the advertised figure and the true cost of carrying the loan for a full year. The second product is a high-yield savings account quoted at “4.75% APY.” Because the bank chose to advertise the APY rather than the underlying nominal rate, that number already reflects monthly compounding, and no further adjustment is needed to know the account will grow by 4.75% over a year, assuming the rate holds steady.

If this consumer mistakenly compared the loan’s headline 9.5% APR directly to the savings account’s 4.75% APY — treating both as apples-to-apples numbers — they might underestimate how much more expensive the loan truly is relative to what the savings account earns. Converting both figures onto the same basis shows the loan actually costs about 9.92% effectively, while the deposit earns 4.75% effectively — a gap of roughly 5.17 percentage points, wider than the 4.75-point gap the headline numbers alone suggested. Over a $20,000 loan balance carried for a year alongside a $20,000 savings balance, that difference in effective rates amounts to real money: roughly $1,984 in interest owed on the loan versus roughly $950 earned on the savings, a gap of about $1,034 that the raw headline percentages made harder to see clearly.