ReckonBoxHow the APY calculator works
Annual percentage yield is the real yearly return on a deposit once compounding is counted, so it is the number that actually tells you how fast your money grows. This calculator takes a nominal rate and how often it compounds, then returns the APY along with the interest and ending balance for your deposit and term. A stated rate on its own hides the effect of compounding, and the APY is what puts it back in.
Take $10,000 at a 5% nominal rate compounded monthly for a year. The APY works out to 5.116%, the balance grows to about $10,511.62, and you earn about $511.62 in interest. The nominal rate said 5%, and monthly compounding quietly lifted the real return by 0.116 of a point.
The APY formula
APY is calculated as (1 + r/n) raised to the power n, then minus 1, where r is the nominal annual rate as a decimal and n is the number of times it compounds per year. For 5% compounded monthly that is (1 + 0.05/12) to the 12th power minus 1, which is 0.05116, or 5.116%. Compound the same 5% only once a year and the APY is exactly 5%, since with annual compounding the nominal rate and the yield are the same number.
The more often interest is added, the more it earns on itself, so the APY climbs with frequency. Each step up is smaller than the last, as the interest on a $10,000 deposit over a year shows.
| Compounding | APY | Interest on $10,000 (1 yr) |
|---|---|---|
| Annually | 5.000% | $500.00 |
| Semiannually | 5.0625% | $506.25 |
| Quarterly | 5.095% | $509.45 |
| Monthly | 5.116% | $511.62 |
| Daily | 5.127% | $512.67 |
| Continuously | 5.127% | $512.71 |
The formula US banks are required to use
Annual percentage yield is not only a piece of arithmetic in the United States, it is a disclosure defined in law by the Truth in Savings Act of 1991 and Regulation DD. The regulation prescribes its own formula, and it is not the one above:
APY = 100 [(1 + Interest/Principal)^(365/Days in term) - 1]
Three definitions carry it, and all three are worth reading slowly:
- Principal is the amount assumed to have been deposited at the start of the account.
- Interest is the total number of dollars earned on that principal over the term.
- Days in term is the actual day count, not a notional year.
The regulation works a NOW account paying $61.68 on $1,000 over 365 days, which comes to 6.17%. This calculator reproduces that to 6.168%.
Both formulas agree on a tidy one-year account, which is exactly the intent. What differs is what they are defined on. Ours starts from a rate and a frequency; the law starts from dollars and days. That is what lets you set a 6-month CD, an 18-month CD and a savings account side by side when none of them shares a compounding convention.
None of the five calculators ranking for this term mentions Regulation DD or the Truth in Savings Act. One credit union page names the Act and then prints the formula as APY = (1 + r/n) - 1, with the exponent missing altogether, which is a fair reminder to check any formula you are handed against a number you can verify.
The APY on your statement is a different number
Regulation DD defines a second figure, "annual percentage yield earned", which is what appears on a periodic statement, and it is computed differently from the APY in an advertisement.
APY Earned = 100 [(1 + Interest earned/Balance)^(365/Days in period) - 1]
The advertised APY is a forward projection on an assumed balance held for a full term. APY Earned looks backwards at the interest actually credited and the average daily balance across the statement period.
They come apart whenever the balance moved, which for a savings account is nearly always. Pay money in halfway through a quarter and your average daily balance is below the closing balance, so the APY Earned lands under the advertised rate even though the bank has done nothing wrong. The two numbers answer different questions, and only one of them is about your account.
Continuous compounding, the ceiling
Continuous compounding is the limit of compounding infinitely often, and it gives APY = e to the power r, minus 1. At a 5% nominal rate that is e to the 0.05 minus 1, which is 5.127%, essentially the same as daily. This is the highest yield any nominal rate can reach, no matter how the interest is sliced, and most APY calculators leave it out.
The practical lesson hides in how flat the top of that curve is. Going from monthly to daily compounding on a $10,000 balance at 5% adds about a dollar of interest over a year, and going from daily to continuous adds a few cents. Past monthly, the compounding frequency barely matters.
APR compared with APY
APR is the nominal rate without compounding, while APY is the effective rate with compounding folded in, so for the same rate the APY is always at least the APR. Banks quote savings accounts and CDs in APY because it is the higher, more flattering number, and they quote loans and credit cards in APR. When you compare two savings accounts, the APY is the fair comparison, since it already accounts for how often each one compounds.
The gap between them is entirely the compounding. A 5% APR compounded monthly is a 5.116% APY, and the 0.116 point difference is the interest your interest earned during the year. The APR calculator works the other side of that comparison, and the compound interest calculator shows how the same effect plays out over many years on a single deposit.
Interest earned from an APY
Over exactly one year the interest earned equals your deposit times the APY, which is the cleanest way to read the number. So $10,000 at a 5.116% APY earns $511.62 in the first year. That identity is why APY is useful: it is the one-year growth rate stated as a percentage, with the compounding already baked in.
Over a longer term the balance is your principal times (1 + r/n) to the power of n times the number of years, and the interest is that balance minus what you put in. At 5% monthly, $10,000 grows to about $12,834 after 5 years, for $2,834 of interest. The APY itself does not change with the term, since it is always an annual figure.
What this does not promise
The APY here assumes a fixed rate and a steady compounding schedule, which is how most savings accounts and CDs work. A variable-rate account can change its rate at any time, and Regulation DD says an advertised APY on one is computed from the opening rate alone, so the headline figure is a snapshot of the day it was published and carries no promise past it. Stepped-rate accounts instead use the total interest across every period, and tiered accounts have two permitted methods, so two banks can advertise honestly and still not be quoting the same thing. Taxes on the interest are not included, and a promotional or introductory rate may not last the full term. This is a math tool for comparing rates, not financial advice, so a licensed adviser can help you weigh accounts against your own situation and tax position. The last thing worth remembering is that beyond monthly compounding, chasing a higher frequency wins pennies, so the rate itself matters far more.
Frequently asked questions
What is an APY calculator? An APY calculator turns a nominal annual rate and a compounding frequency into the annual percentage yield, the true yearly return once compounding is counted. It uses APY = (1 + r/n) to the power n, minus 1, and also shows the interest and ending balance your deposit earns.
What is the APY formula? The APY formula is APY = (1 + r/n) raised to the power n, then minus 1, where r is the nominal annual rate as a decimal and n is the number of compounding periods per year. For continuous compounding the formula is APY = e to the power r, minus 1.
What is the difference between APR and APY? APR is the nominal annual rate without compounding, while APY is the effective rate once compounding is included. For the same nominal rate, APY is always at least APR and grows as the compounding frequency rises, so 5% compounded monthly is a 5.116% APY.
How much does compounding frequency change the APY? At a 5% nominal rate the APY is 5.000% compounded annually, 5.116% monthly, 5.127% daily, and 5.127% continuously. The jumps shrink quickly, so moving from monthly to daily on a 10,000 dollar balance adds only about one dollar of interest a year.
What is continuous compounding? Continuous compounding is the mathematical limit of compounding infinitely often, giving APY = e to the power r, minus 1. It is the highest APY any nominal rate can reach, and at 5% it is 5.127%, barely above daily compounding.
How do I calculate interest earned from APY? Over exactly one year the interest earned equals your deposit times the APY, so 10,000 dollars at a 5.116% APY earns 511.62 dollars. Over several years the balance is principal times (1 + r/n) to the power n times years, and the interest is that balance minus your deposit.
Which APY formula do US banks have to use? Regulation DD, made under the Truth in Savings Act of 1991, prescribes APY = 100 [(1 + Interest/Principal)^(365/Days in term) - 1], where Interest is the dollars earned over the term and Days in term is the actual day count. It agrees with the textbook (1 + r/n)^n - 1 on a clean one-year account, but being defined on dollars and days is what makes APY comparable across accounts with different terms and compounding.
Why is the APY on my statement lower than the advertised rate? Because they are two different regulated figures. The advertised APY projects forward on a balance assumed held for the full term. The "annual percentage yield earned" on a statement looks backwards at the interest actually credited against your average daily balance for that period. If your balance rose partway through, the average is below the closing figure and the APY Earned comes in under the advertised rate without anything being wrong.
Is a higher APY always better for savings? For a savings account or CD a higher APY means more interest for the same deposit, so comparing accounts by APY rather than the nominal rate is the fair comparison. APY already folds in the compounding frequency, which the nominal rate alone hides.