Effective Annual Rate

Q: What is the largest EAR I could get from a stated rate?

With continuous compounding the limit is $e^{r} - 1$. For a stated rate of 18%, that is about 19.72%, so monthly compounding at 19.56% is close to that ceiling. The continuous compounding page explains where the limit comes from.

In one sentence: The effective annual rate says what a stated yearly rate really adds up to over a year, once interest is added more than once and earns interest of its own.

$\text{EAR} = \left(1 + \frac{r}{n}\right)^{n} - 1$

Read it out loud: “EAR equals one plus r over n, all raised to the power n, then minus 1. EAR is the effective annual rate, r is the stated yearly rate as a decimal, and n is how many times a year the interest is added.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Percent and decimal: A percent is a number out of 100, so 18% means $\frac{18}{100} = 0.18$ . The formula takes the rate as a decimal, and its result is also a decimal, so 0.195618 is read as 19.5618%. More on this is on the math basics page.
Exponents, or powers: An exponent is a small raised number. $x^{12}$ means x multiplied by itself 12 times. Here the exponent 12 counts the months in a year, so the monthly growth factor is multiplied by itself 12 times to give the yearly growth factor.
Compounding period: A compounding period is one time that interest is added to the balance. Yearly, quarterly, monthly and daily are common choices, with 1, 4, 12 and 365 periods in a year. The more often interest is added, the more it can earn interest of its own.
Nominal rate: The nominal rate is the rate as stated, before the effect of compounding is counted. It is also called the stated or quoted yearly rate. It is the r in the formula, and it is not the same as the true yearly cost when interest is added more than once a year.

What each symbol means

Symbol	Meaning
$\text{EAR}$	Effective annual rate, the true yearly rate once compounding is included, written as a decimal such as 0.1956 or as a percent such as 19.56%. Savings accounts call it the annual percentage yield, or APY. It is the number the formula gives you.
$r$	Nominal annual rate, the stated yearly rate before compounding is counted, written as a decimal. A rate of 18% is 0.18. It is the figure often quoted as the annual percentage rate, or APR, on loans and cards.
$n$	Compounding periods per year, a count with no unit. Yearly interest is n = 1, monthly is n = 12 and daily is n = 365. It tells the formula how often the interest is added during the year.

The formula piece by piece

$\frac{r}{n}$: The rate for one period: the stated yearly rate shared out equally across the periods of the year. For 18% a year compounded monthly, 0.18 divided by 12 gives 0.015, or 1.5% for each month.
$1 + \frac{r}{n}$: The growth factor for one period. Adding 1 to the period rate gives the number the balance is multiplied by in each period. For monthly compounding at 18%, that factor is 1.015.
$\left(1 + \frac{r}{n}\right)^{n}$: The growth factor for the whole year. Raising the period factor to the power n multiplies it by itself once for each period in the year. For 18% compounded monthly, 1.015 to the power 12 is about 1.195618.
$\left(1 + \frac{r}{n}\right)^{n} - 1$: The growth as a share of the starting balance. Subtracting 1 removes the original money, so 0.195618 is the interest earned as a fraction of what you started with.
$\text{EAR} = \left(1 + \frac{r}{n}\right)^{n} - 1$: The whole formula in one sentence: the effective annual rate equals one plus the stated rate divided by the number of periods, raised to the number of periods, and then minus one.

What the formula is doing, and why

A stated yearly rate can mislead when interest is added more than once a year. A card that says 18% per year, compounded monthly, really costs more than 18% over a whole year, because each month’s interest is added and then earns interest. The effective annual rate, EAR, is the true yearly figure. The formula $\text{EAR} = \left(1 + \frac{r}{n}\right)^{n} - 1$ turns the stated rate into that true yearly figure.

Why the power $n$ ? Over one year, the balance is multiplied by $1 + \frac{r}{n}$ once for each of the $n$ periods. Multiplying by the same factor $n$ times is the same as raising it to the power $n$ . The result is the growth factor from the start of the year to the end.

Why take away 1? The growth factor, such as 1.1956, is the balance at the end divided by the balance at the start, so it includes the original money. Subtracting 1 leaves only the growth, which is the interest earned as a share of the starting balance. That is why the rate is the factor minus 1.

Why the EAR is never lower than the stated rate. Adding interest more often can only raise the total, because each period’s interest earns interest in the later periods. With $n = 1$ , which means yearly compounding, the EAR equals the stated rate. With any larger $n$ it is higher. As $n$ grows without bound, the EAR approaches $e^{r} - 1$ , the continuous compounding limit explained on its own page.

Why it matters. Banks and card issuers can quote the same yearly rate with different compounding schedules. The EAR turns each offer into one yearly number, so they can be compared directly. For savings, a higher EAR means more money at the end of a year. For borrowing, a higher EAR means more cost over that year.

An everyday picture

Think of comparing the price per litre at a shop instead of the price printed on the bottle. The stated rate is the printed price, and the EAR is the price per litre. Once both products are measured the same way, you can see which one really costs more. The EAR does the same job for two interest offers that add interest at different times.

Worked example: The true yearly cost of an 18% card that compounds monthly

A credit card states a yearly rate of 18%, and interest is added to the balance each month. We want the effective annual rate, and what it means for a 1000-dollar balance left unpaid for one year. In the formula, $r = 0.18$ per year and $n = 12$ periods a year.

Write the formula and put in the numbers. The stated rate r is 18%, which is 0.18 as a decimal, and there are 12 monthly periods, so n is 12. The formula becomes $\text{EAR} = \left(1 + \frac{0.18}{12}\right)^{12} - 1$ . The rate printed on the statement is the starting point for the whole calculation.
Find the rate for one month. Dividing 0.18 by 12 gives $\frac{0.18}{12} = 0.015$ , which is 1.5% a month. The card adds this share of the balance each month, so each month’s rate is the yearly rate shared out equally.
Find the growth factor for one month. Adding 1 gives $1 + 0.015 = 1.015$ . Multiplying a balance by 1.015 adds 1.5% of it, so 1.015 is the factor for one month.
Raise the factor to the power 12. Twelve months means multiplying by 1.015 twelve times in a row, which gives $1.015^{12} \approx 1.195618$ . This is the growth factor for the whole year, and it includes the interest that was added in earlier months and then earned more interest later.
Subtract 1 to get the yearly rate. Taking 1 away leaves the growth as a share of the starting balance: $1.195618 - 1 = 0.195618$ . As a percent this is 19.56%, so the true yearly rate is about 19.56%, which is more than the stated 18%.
Compare with the stated rate. The difference is $19.56 - 18 = 1.56$ percentage points. The extra comes from interest that is added each month and then earns interest in the following months, which a single figure of 18% per year does not show.
Apply it to a 1000-dollar balance. If the 1000 dollars stays unpaid for one year in this illustration, the total interest is $1000 \times 0.195618 \approx 195.62$ dollars. The stated 18% would have given only 180 dollars, so the extra cost over the year is about 15.62 dollars.

Answer: The stated 18% per year, compounded monthly, has an effective annual rate of about 19.56%. On a 1000-dollar balance left for a year, that is about 195.62 dollars of interest, against 180 dollars at the stated rate. The EAR is the figure to compare when two offers compound differently.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Treating the stated rate as the true yearly cost when interest is added more than once a year. A stated 18% compounded monthly really comes to about 19.56% over the year. Use the EAR when you compare offers.
Stopping at the growth factor. The power gives 1.195618, which is the balance multiplier. Subtract 1 to get the rate, 0.195618, which is 19.56%.
Using the number of years for n. The letter n counts periods per year, so monthly compounding is n = 12 whatever the number of years. Daily compounding is n = 365.
Forgetting the decimal. Write 18% as 0.18 inside the formula. With 18 in place of 0.18, the monthly factor is 2.5 instead of 1.015, which makes the result far too large.

Where you will meet it

Comparing savings accounts and certificates of deposit, where banks quote an annual percentage yield (APY) that already includes compounding.
Comparing loans and credit cards with different compounding schedules, so that each one is put on the same yearly scale.
Working out the real yearly return on an investment that pays interest or income several times a year.
Textbook finance problems that ask for the annual rate equivalent to a monthly or quarterly rate.

Questions people ask

What is the difference between APR and APY?

APR is the stated yearly rate without compounding, which is the figure many loans and cards quote in the United States. APY is the yearly yield including compounding, which is the figure savings accounts quote. For the same stated rate, the APY is at least as large as the APR, and it is larger whenever interest is added more than once a year.

When does the EAR equal the stated rate?

Only when interest is added once a year, so n = 1. Then the formula gives $(1 + r) - 1 = r$ . For any more frequent compounding, the EAR is larger than the stated rate.

What is the largest EAR I could get from a stated rate?

With continuous compounding the limit is $e^{r} - 1$ . For a stated rate of 18%, that is about 19.72%, so monthly compounding at 19.56% is close to that ceiling. The continuous compounding page explains where the limit comes from.

Compound Interest: Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.
Continuous Compounding: Continuous compounding says money grows as if interest were added every instant, and the number e, about 2.718, sets the size of that growth.
Rule of 72: The rule of 72 says that dividing 72 by a yearly percentage rate gives a quick estimate of the number of years it takes money to double.