APY Calculator
With nominal interest rate 4.5 %, balance 10000 usd, days held 365 days, apy comes to 4.602 % — apy. It is reached in 7 steps, the last of which is (pow(1 + 0.0001233, 365) - 1) * 100, and each one is printed on the page with its numbers filled in. The formula is the one published by 12 CFR Part 1030 (Regulation DD), Appendix A, not an approximation fitted to it.
Annual percentage yield from a rate and its compounding frequency, and what it earns on a balance in dollars.
Formula and sources checked · How we check
Nominal interest rate 4.5, Compounded Daily — 365 times a year, Balance 10000, Days held 365 days
4.602 %
APY for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Annual percentage yield
(pow(1 + 0.0001233, 365) - 1) * 1004.602 %- Balance after the term
10000 * pow(1 + 0.0001233, 365 * 365 / 365)10,460.25- Interest earned
10460.25 - 10000$460.25- APY by the Regulation DD formula
(pow(1 + 460.24958 / 10000, 365 / 365) - 1) * 1004.602 %- Difference between the two routes
4.6024958 - 4.60249580 %- Interest if it did not compound
10000 * 4.5 / 100 * 365 / 365450- Gained purely from compounding
460.24958 - 450$10.25
Ask about this in the chatAPY by Interest Rate Chart
Worked example
4.5% compounded daily is an APY of 4.602%. On $10,000 held a year that is $460.25, of which $10.25 came from compounding alone — the gap between the rate a bank advertises and the yield it actually pays.
How to work it out yourself
- 1.Enter the nominal rate and how often it compounds. Daily is the usual answer for a savings account, quarterly for many CDs.
- 2.APY is what the rate becomes once interest starts earning interest. It is always at least the nominal rate, and equal only when compounding is annual.
- 3.Compare APYs rather than rates when shopping. Two accounts at the same nominal rate pay differently if one compounds daily and the other quarterly.
APY by nominal rate, compounded daily
| Nominal interest rate (%) | APY | Interest earned | Gained purely from compounding |
|---|---|---|---|
| 0.5 | 0.501 % | $50 | $0 |
| 1 | 1.005 % | $101 | $1 |
| 1.5 | 1.511 % | $151 | $1 |
| 2 | 2.020 % | $202 | $2 |
| 2.5 | 2.531 % | $253 | $3 |
| 3 | 3.045 % | $305 | $5 |
| 3.5 | 3.562 % | $356 | $6 |
| 4 | 4.081 % | $408 | $8 |
| 4.5 | 4.602 % | $460 | $10 |
| 5 | 5.127 % | $513 | $13 |
| 5.5 | 5.654 % | $565 | $15 |
| 6 | 6.183 % | $618 | $18 |
The gap between rate and yield widens with the rate: at 1% daily compounding adds half a basis point of yield, at 6% it adds eighteen.
The formula
- Annual percentage yield
(pow(1 + 0.0001233, 365) - 1) * 100 - Balance after the term
10000 * pow(1 + 0.0001233, 365 * 365 / 365) - Interest earned
10460.25 - 10000 - APY by the Regulation DD formula
(pow(1 + 460.24958 / 10000, 365 / 365) - 1) * 100 - Difference between the two routes
4.6024958 - 4.6024958 - Interest if it did not compound
10000 * 4.5 / 100 * 365 / 365 - Gained purely from compounding
460.24958 - 450
Source: 12 CFR Part 1030 (Regulation DD), Appendix A — annual percentage yield calculation, US Consumer Financial Protection Bureau — what is APY
Questions people actually ask
- What is the difference between APY and APR?
- APY is what you earn and includes compounding; APR is what you are charged and does not. A 4.5% rate compounded daily is a 4.602% APY, so the two numbers describe the same rate from opposite sides of the ledger. Deposit accounts must be advertised in APY under Regulation DD precisely so they can be compared to each other.
- Why do two routes to APY appear?
- Because the definition everyone prints — (1 + r/n)^n − 1 — is the textbook one, while the legal definition in Regulation DD Appendix A works from the interest actually earned over the term the money was held. They agree exactly at 365 days and diverge on any other term. Both are shown here, with the gap, so the derivation checks itself rather than asking you to trust one of them.
- How much does compounding frequency matter?
- Less than the rate, and more than nothing. At 4.5% the difference between compounding daily and annually is about 10 basis points of yield — roughly $10 a year on $10,000. It is worth a tiebreak between similar offers and never worth chasing a lower rate for.
- Is APY guaranteed?
- Only on a fixed-term product like a CD. A savings account rate is variable, so its APY is what the current rate would yield if it held for a year, not a promise about the next twelve months. Banks change these rates without notice.
Related
- Compound Interest CalculatorFuture value of a starting balance plus monthly contributions, separating what you put in from what the interest earned.
- Savings Goal CalculatorThe monthly contribution needed to reach a target by a date, given what you have saved already.
- Deferred Payment and Bond CalculatorWhat a loan repaid in one lump sum at maturity grows to, or what a known amount due at maturity is worth today.
- CD CalculatorWhat a certificate of deposit is worth at maturity, and what breaking it early costs. Includes the early-withdrawal penalty and the after-tax yield.
- CAGR CalculatorCompound annual growth rate from a start, an end and a span — the one rate that would have produced the same result, beside the average that would not.
- Effective Annual RateThe effective annual rate a nominal rate becomes once compounding is counted, and the gap that makes two accounts quoting the same rate pay different amounts.
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