What is the difference between the interest rate and the annual percentage yield?

The interest rate is the stated rate. The annual percentage yield is what the account actually pays over a year once the compounding is taken into account, expressed as a rate. On 10,000 at 5 percent compounded monthly the interest rate is 5 percent and the annual percentage yield is 5.116 percent, because the interest credited in January earns interest for the rest of the year. The two are equal only when interest compounds once a year. Deposit disclosures in the United States are required to state both, using those terms.

How do I calculate the annual percentage yield?

Divide the nominal rate by the number of compounding periods in the year, add one, raise the result to that same number of periods, and subtract one. At 5 percent compounded monthly that is 1.0041667 raised to the twelfth power minus one, which is 0.051162, or 5.116 percent. The regulation writes the same idea in terms of dollars: 100 times the quantity one plus interest over principal, raised to 365 divided by days in term, minus one. Both give the same answer when the term is a year.

Which compounding frequency pays the most?

The most frequent one, and continuous compounding is the ceiling. At a nominal 5 percent, annual pays 500 on 10,000, quarterly pays 509.45, monthly pays 511.62, daily pays 512.67, and continuous pays 512.71. The gaps shrink as the frequency rises — the step from annual to quarterly is worth 9.45, and the step from daily to continuous is worth 4 cents. An account that compounds daily will always beat the same rate compounded monthly, but by very little, and a quarter of a point on the rate is worth far more than any frequency change.

Does the annual percentage yield include a sign-up bonus or a promotional rate?

No, and that is a deliberate limit in the regulation: the yield reflects interest only and does not include the value of a bonus, and interest that depends on circumstances which may or may not occur — a promotional rate that expires, a rate that steps up if you keep the account open — is not counted either. A 200 bonus on a 10,000 balance is worth two percentage points for one year, far more than the difference between any two compounding frequencies on this page, so it is worth adding to the comparison by hand rather than assuming the yield covers it.

What does continuous compounding mean?

It is the limit the growth factor approaches as the compounding period shrinks toward zero — at a nominal 5 percent, a year's growth of 1.051271, or a yield of 5.1271 percent. It is not a schedule any bank posts, but it is a useful ceiling: it is the most a nominal rate can ever be worth, so it tells you how much room is left above the frequency you are actually offered. At 5 percent, daily compounding is already within 0.0004 of a percentage point of it.

Why does this page ask for a deposit amount?

Because the yield is defined in terms of the interest actually paid on money actually on deposit, so the amount is part of the definition and not merely an input for scaling. It also gives you the second figure on the panel, the interest earned in dollars, which is the number most people actually want — a 5.116 percent yield means nothing until it is 511.62 on a 10,000 balance. The yield itself does not depend on the size of the deposit; 4.577 percent is 4.577 percent on 100 or on 500,000.

APY Calculator

Result

5.116%

Effective annual rate (APY, %)

Interest earned: 511.62
Total amount: 10,511.62

An APY calculator answers the question an advertised rate does not: what does this account actually pay over a year? Enter a deposit, a nominal annual rate and how often the interest is compounded, and the page returns the annual percentage yield, the interest earned over a year and the closing balance. The distinction it is built on is the one the United States Truth in Savings Act makes in Regulation DD: the interest rate is the stated rate, and the annual percentage yield is the rate that reflects the total amount of interest actually paid over a 365-day period given the frequency of compounding — a percentage rate, as the regulation puts it, based on the interest rate and the frequency of compounding, calculated under the rules in Appendix A. The two numbers are the same only when interest is compounded once a year. 10,000 at 5 percent compounded annually pays 500 and yields 5 percent. The same 10,000 at the same 5 percent compounded monthly pays 511.62 and yields 5.116 percent. Compounded daily it pays 512.67 and yields 5.127 percent. Compounded continuously — the limit as the period shrinks toward zero — it pays 512.71. Nothing about the rate changed in any of those cases; what changed is how often the interest that has already been credited starts earning interest of its own. That is the whole content of the difference, and it is why disclosure rules require the yield rather than the rate: two accounts advertising 5 percent do not pay the same amount. The reference table holds the deposit and the rate fixed and walks the six frequencies, so the compounding column does the only moving. Two things are worth knowing about the figure itself. It is computed on an assumed 365-day year — that is the convention in the regulation, and an institution may use 366 days in a leap year instead. And it reflects interest only: a sign-up bonus, a waived fee or a promotional rate that may not last is not part of it, which is exactly the kind of thing an advertised rate tends to leave out. Where this page and its neighbours differ is in what you have to know before you start. This one needs a balance, because an account's yield is a statement about money actually on deposit. Its closest relative, the effective interest rate page, computes the same annual figure from a rate and an arbitrary compounding count, which is the version to use when the count you have in mind — 26, 52 — is not one of the six a dropdown can offer. And the effective annual rate page starts from a rate charged per period instead of a nominal annual rate, which is the version to use when you are working from a statement rather than an advertisement.

10,000 at 5% a year, compounded six ways

Compounding frequency	Annual percentage yield	Interest earned	Closing balance
Annual — interest added once, at the end of the year	5	500	10500
Semiannual — interest added twice a year	5.062	506.25	10506.25
Quarterly — interest added four times a year	5.095	509.45	10509.45
Monthly — interest added twelve times a year	5.116	511.62	10511.62
Daily — interest added 365 times a year	5.127	512.67	10512.67
Continuous — compounded at every instant	5.127	512.71	10512.71

Every row has the same deposits and the same stated rate, so the third and fourth columns are the whole story: what the frequency is worth. Annual compounding pays 500, which is exactly 5 percent of 10,000, and the yield equals the rate — the row that shows what the other five rows are measured against. Semiannual pays 6.25 more, quarterly 9.45 more, monthly 11.62 more, daily 12.67 more, and continuous 12.71 more, which is the ceiling. The increments are worth reading twice, because they shrink fast: the second row gains 6.25 over the first, and the last row gains 4 cents over the one above it. Two practical rules follow. Compounding frequency matters most when the rate is high and the balance is large — on a million at 25 percent the difference between annual and daily compounding runs into tens of thousands. And when a bank advertises two accounts at the same rate, the one that compounds more often is strictly better, but the amount it is better by is usually small enough that fees and minimum balances will decide the comparison instead.

Formula

Interest = principal × [(1 + rate ÷ periods per year)^(periods per year) − 1] Annual percentage yield = interest ÷ principal × 100

Initial amount: What is on deposit at the start of the year, which the regulation treats as untouched for the whole term — no further deposits and no withdrawals, so that the yield describes the account rather than a saver's behaviour
Nominal annual rate: The advertised or stated rate, per year, before any compounding is applied — the number that is printed next to the yield on a disclosure and that is almost always the smaller of the two
Compounding frequency: How many times a year the interest is computed and added to the balance: annual, semiannual, quarterly, monthly, daily, or continuous. This is the only input that changes the answer when everything else is held fixed, and the reference table walks all six
Interest earned: The dollars credited over the year — the figure that actually lands in the account, and the one to compare when two accounts quote the same rate with different compounding
Annual percentage yield: The interest earned divided by the amount deposited, expressed as a yearly rate. On a 365-day term it is the same as the effective annual rate; the name is the one deposit disclosures use, and it is computed for a 365-day period under Appendix A to Regulation DD

Use it when you are looking at a deposit account rather than a loan, and when you want the number that describes what a year in that account pays. The clearest case is comparing two accounts that advertise the same rate: 5 percent compounded monthly beats 5 percent compounded quarterly beats 5 percent compounded annually, and the yields are 5.116, 5.095 and 5.000 percent, which is a difference you can see in a year's interest without waiting a year. The second case is checking a disclosure. The Truth in Savings rules require account disclosures to state both the interest rate and the annual percentage yield, using those terms, along with the frequency with which interest is compounded and credited, so a bank that shows 5 percent and 5.116 percent is telling you that the account compounds monthly — and a bank whose yield equals its rate is telling you it compounds annually, or that something in the disclosure is wrong. A third case is the one that catches people out at the top of the range: the lower the rate, the less the compounding matters, and at a nominal 50 percent compounded daily the year comes to 64.816 percent, where monthly compounding gives 63.21 and annual gives 50. Choose a frequency only if the account really uses it: continuous compounding is a mathematical limit, not a product feature, and it is here because the ceiling it describes is the right way to see how much room is left between daily compounding and the limit — at 5 percent, daily compounding is 0.0004 of a percentage point below it.

Worked examples

10,000 at 5% compounded monthly
1. Periodic rate: 5% ÷ 12 = 0.41667% a month
2. Growth over the year: 1.0041667¹² = 1.051162
3. Interest earned: 10,000 × 0.051162 = 511.62
4. Closing balance: 10,000 + 511.62 = 10,511.62
5. Annual percentage yield: 511.62 ÷ 10,000 × 100 = 5.116%
The default case. Monthly compounding adds 11.62 more than the 500 that the rate alone would suggest, which is 0.116 of a percentage point of yield — small, but it is the difference between the advertised rate and what the account pays, and it is exactly the gap the disclosure rules exist to expose.
The same account compounded daily
1. Periodic rate: 5% ÷ 365 = 0.0137% a day
2. Growth over the year: 1.00013699³⁶⁵ = 1.051267
3. Interest earned: 10,000 × 0.051267 = 512.67
4. Closing balance: 10,000 + 512.67 = 10,512.67
5. Annual percentage yield: 512.67 ÷ 10,000 × 100 = 5.127%
Daily compounding gains 1.05 over monthly compounding on a 10,000 balance — 0.011 of a percentage point of yield. That is the shape of this whole page: the steps between frequencies get smaller as the frequency rises, and the last step, from daily to continuous, is worth 0.04 a year. Anyone choosing between two accounts should care about the difference between annual and monthly far more than the one between daily and continuous.
A quarter of a million at 4.5% compounded quarterly
1. Periodic rate: 4.5% ÷ 4 = 1.125% a quarter
2. Growth over the year: 1.01125⁴ = 1.045765
3. Interest earned: 500,000 × 0.045765 = 22,882.54
4. Closing balance: 500,000 + 22,882.54 = 522,882.54
5. Annual percentage yield: 22,882.54 ÷ 500,000 × 100 = 4.577%
A larger balance does not change the yield — 4.577 percent is a property of the rate and the frequency, and it would be the same on 100. What the balance changes is the dollars, which is why the yield is the figure to compare accounts by and the interest earned is the figure to compare outcomes by. At this size the 0.077 of a point that quarterly compounding adds over the stated rate is worth 385 a year.

Limitations

It assumes the balance sits untouched for the whole year. No further deposits, no withdrawals, no transfers in or out — the yield is a property of the account under that assumption, which is what the regulation specifies and what makes two accounts comparable, but it is not what most savers actually do. It computes interest only. A sign-up bonus, a waived monthly fee, a promotional rate that applies for three months, a tiered rate that pays more above a balance threshold — none of them are in this figure, and each of them can be worth more than the compounding difference the page is built to show. It assumes the rate holds for the whole year: on a variable-rate account the yield is a snapshot of today's rate, and the disclosure says as much, which is why variable-rate disclosures must state that the rate and the yield may change. It uses a 365-day year, the convention in the regulation; a bank may use 366 days in a leap year instead, and a bank computing on a 360-day basis — as banks in some countries do — will not match this page to the last digit. It does not model how interest is credited in the middle of the year being withdrawn: on an account that credits monthly but pays out the interest each month rather than leaving it to compound, the balance never grows and the yield collapses to the stated rate, and that is a different product from the one this page describes. Finally, it is a calculator rather than a disclosure: the annual percentage yield a depository institution must give you is computed under the regulation as written, including rules for stepped rates, tiered rates and time accounts longer than a year that this page does not implement.

Frequently asked questions

What is the difference between the interest rate and the annual percentage yield?: The interest rate is the stated rate. The annual percentage yield is what the account actually pays over a year once the compounding is taken into account, expressed as a rate. On 10,000 at 5 percent compounded monthly the interest rate is 5 percent and the annual percentage yield is 5.116 percent, because the interest credited in January earns interest for the rest of the year. The two are equal only when interest compounds once a year. Deposit disclosures in the United States are required to state both, using those terms.
How do I calculate the annual percentage yield?: Divide the nominal rate by the number of compounding periods in the year, add one, raise the result to that same number of periods, and subtract one. At 5 percent compounded monthly that is 1.0041667 raised to the twelfth power minus one, which is 0.051162, or 5.116 percent. The regulation writes the same idea in terms of dollars: 100 times the quantity one plus interest over principal, raised to 365 divided by days in term, minus one. Both give the same answer when the term is a year.
Which compounding frequency pays the most?: The most frequent one, and continuous compounding is the ceiling. At a nominal 5 percent, annual pays 500 on 10,000, quarterly pays 509.45, monthly pays 511.62, daily pays 512.67, and continuous pays 512.71. The gaps shrink as the frequency rises — the step from annual to quarterly is worth 9.45, and the step from daily to continuous is worth 4 cents. An account that compounds daily will always beat the same rate compounded monthly, but by very little, and a quarter of a point on the rate is worth far more than any frequency change.
Does the annual percentage yield include a sign-up bonus or a promotional rate?: No, and that is a deliberate limit in the regulation: the yield reflects interest only and does not include the value of a bonus, and interest that depends on circumstances which may or may not occur — a promotional rate that expires, a rate that steps up if you keep the account open — is not counted either. A 200 bonus on a 10,000 balance is worth two percentage points for one year, far more than the difference between any two compounding frequencies on this page, so it is worth adding to the comparison by hand rather than assuming the yield covers it.
What does continuous compounding mean?: It is the limit the growth factor approaches as the compounding period shrinks toward zero — at a nominal 5 percent, a year's growth of 1.051271, or a yield of 5.1271 percent. It is not a schedule any bank posts, but it is a useful ceiling: it is the most a nominal rate can ever be worth, so it tells you how much room is left above the frequency you are actually offered. At 5 percent, daily compounding is already within 0.0004 of a percentage point of it.
Why does this page ask for a deposit amount?: Because the yield is defined in terms of the interest actually paid on money actually on deposit, so the amount is part of the definition and not merely an input for scaling. It also gives you the second figure on the panel, the interest earned in dollars, which is the number most people actually want — a 5.116 percent yield means nothing until it is 511.62 on a 10,000 balance. The yield itself does not depend on the size of the deposit; 4.577 percent is 4.577 percent on 100 or on 500,000.

References

12 CFR 1030.2(c) — annual percentage yield: a percentage rate reflecting the total amount of interest paid on an account, based on the interest rate and the frequency of compounding for a 365-day period and calculated according to the rules in Appendix A — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)
12 CFR Part 1030, Appendix A (Annual Percentage Yield Calculation) — the formula APY = 100[(1 + Interest ÷ Principal)^(365 ÷ Days in term) − 1], the 365-day year, the 366-day election in a leap year, and the assumption that all principal and interest remain on deposit for the entire term — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)
12 CFR 1030.4(b) — account disclosures must state both the annual percentage yield and the interest rate using those terms, and the frequency with which interest is compounded and credited — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)

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