Convert an Annual Percentage Rate (APR) into the Effective Annual Yield (APY) for any compounding frequency. Visualize how compound interest accelerates growth, compare APY across different compounding periods, and understand the true annual return on savings or the real cost of borrowing.
When comparing financial products — whether savings accounts, certificates of deposit, credit cards, or mortgages — you will encounter two critical rates: the Annual Percentage Rate (APR) and the Annual Percentage Yield (APY). While they sound similar, they represent fundamentally different measures of cost and return, and confusing them can lead to costly financial mistakes.
APR is the simple annual interest rate that does not account for the effect of compounding. It is the nominal rate quoted by lenders and is often used for loans and credit products. APY, on the other hand, reflects the effective annual rate after compounding is applied. For savers, APY shows the true return on deposits; for borrowers, it reveals the actual annual cost of a loan when interest is compounded.
APY = (1 + APR / n)n − 1
where n = number of compounding periods per year.
For continuous compounding: APY = eAPR − 1
The frequency with which interest is compounded has a profound impact on the effective yield. The more frequently interest is compounded, the higher the APY for a given APR. This is because interest is earned on previously accumulated interest — the classic "interest on interest" effect that Albert Einstein reportedly called the "eighth wonder of the world."
For example, a 5% APR compounded annually yields exactly 5% APY. But if that same 5% APR is compounded daily, the APY rises to about 5.13% — a noticeable difference over time. For a $10,000 investment over 10 years, that 0.13% difference translates to over $140 in additional earnings. The gap widens with higher rates and longer terms.
Our APR to APY calculator uses the standard compound interest formula to convert any nominal APR into its effective annual yield. The algorithm handles both discrete compounding (annual, semi-annual, quarterly, monthly, weekly, daily) and continuous compounding using the exponential function.
Step 1: You provide the APR (as a percentage), the compounding frequency, an optional investment term, and an optional principal amount.
Step 2: The calculator computes the APY using the formula above. For continuous compounding, it applies er − 1 where r is the APR expressed as a decimal.
Step 3: Using the principal and term, it projects the growth of your investment under both compound and simple interest, displaying the difference visually.
Step 4: A comparison table shows the APY for the same APR across all common compounding frequencies, so you can see at a glance how much frequency matters.
Sarah is comparing two high-yield savings accounts. Bank A offers a 4.50% APR compounded monthly. Bank B offers a 4.45% APR compounded daily. At first glance, Bank A seems better because 4.50% > 4.45%. But using this calculator, Sarah discovers that Bank A's APY is 4.594% while Bank B's APY is 4.552%. Bank A still wins, but the margin is smaller than she expected. By calculating the APY, she makes an informed decision and maximizes her returns.
James has a credit card with an APR of 21.99% compounded daily. He wants to know the true annual cost of carrying a balance. Using the calculator, he finds that the APY is 24.61% — nearly 2.6 percentage points higher than the quoted APR. This eye-opening result motivates him to pay off his balance faster and avoid the compounding trap that credit card issuers rely on.
A homebuyer is choosing between two mortgage offers: Loan A at 6.25% APR with monthly compounding, and Loan B at 6.30% APR with semi-annual compounding. The calculator shows that Loan A has an APY of 6.432% while Loan B has an APY of 6.399% — surprisingly, Loan B is actually cheaper in effective annual terms despite the higher APR, because less frequent compounding reduces the effective rate. This insight saves the buyer thousands over the life of the loan.
In the United States, the Truth in Savings Act (TISA) and Truth in Lending Act (TILA) mandate that financial institutions clearly disclose both APR and APY to consumers. The Consumer Financial Protection Bureau (CFPB) enforces these rules to ensure transparency and protect consumers from deceptive marketing.
Regulation DD (TISA) Regulation Z (TILA) CFPB Oversight
Under these regulations, APY must be disclosed for deposit accounts, and APR must be disclosed for credit products. This calculator helps you verify those disclosures and understand the true economics of any financial product, empowering you to be a more informed consumer.
The relationship between APR and APY is rooted in the concept of compound interest. If a principal P is invested at an annual nominal rate r (APR, expressed as a decimal) compounded n times per year, the future value after t years is:
FV = P · (1 + r/n)n·t
The effective annual yield (APY) is the rate that would produce the same future value if compounding occurred only once per year. Setting t = 1 and equating:
P · (1 + APY) = P · (1 + r/n)n → APY = (1 + r/n)n − 1
For continuous compounding, we take the limit as n → ∞, which yields APY = er − 1. This derivation is standard in financial mathematics and appears in textbooks such as Principles of Corporate Finance by Brealey, Myers, and Allen, and Options, Futures, and Other Derivatives by Hull.