Compound Interest

In one sentence: 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.

$A = P \left(1 + \frac{r}{n}\right)^{n t}$

Read it out loud: “A equals P times the quantity one plus r over n, all raised to the power n times t. A is the final amount, P is the starting amount, r is the yearly rate as a decimal, n is how many times a year the interest is added, and t is the number of years.”

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, and the sign % means divide by 100. So 6% is $\frac{6}{100} = 0.06$ . Every rate in the formula must be a decimal before you use it. More practice with this is on the math basics page.
Exponents, or powers: A small raised number is an exponent, which says how many times to multiply a number by itself. So $1.005^{3}$ means $1.005 \times 1.005 \times 1.005$ . An exponent of 60 means multiplying by the same number 60 times in a row. Calculators have a power key for this, often marked with a caret or with x^y.
Compounding period: A compounding period is one time the bank adds interest to the balance. Common periods are yearly, quarterly (4 times a year), monthly (12 times a year) and daily (about 365 times a year). The number of periods per year is written n, and it changes how fast the balance grows.
Order of operations: Brackets come first, then powers, then multiplying and dividing, and last adding and subtracting. In $P \left(1 + \frac{r}{n}\right)^{n t}$ you divide r by n and add 1 inside the brackets, raise the result to the power n t, and only then multiply by P. Doing the steps in another order gives a wrong balance.

What each symbol means

Symbol	Meaning
$A$	Final amount, the balance after t years, measured in dollars. It includes the starting amount and all the interest. It is the number the formula gives you, and the interest alone is A minus P.
$P$	Principal, the starting amount, measured in dollars. It is the sum put in on day one, before any interest is added to the balance.
$r$	Annual interest rate, written as a decimal per year. A rate of 6% is 0.06. Because it is a rate per year, it must be shared out into equal parts when interest is added more than once a year.
$n$	Compounding periods per year, a count with no unit. Yearly compounding is n = 1, twice a year is n = 2, monthly is n = 12 and daily is about n = 365. It says how often the interest is added to the balance.
$t$	Time, the number of years the money stays in the account, measured in years so that it matches the yearly rate. Three years and six months is t = 3.5.

The formula piece by piece

$r$: The yearly rate as a decimal, for example 0.06 for 6%. It is the share of the balance paid as interest over one year if the bank added it only once. It is always per year, so it must be split into periods.
$\frac{r}{n}$: The rate for one period: the yearly rate shared out equally. With 12 periods a year, 0.06 divided by 12 is 0.005, which is half a percent a month. Dividing by n is what lets one formula handle any compounding schedule.
$1 + \frac{r}{n}$: 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 6%, that factor is 1.005, so 1000 dollars becomes 1005 dollars after one month.
$\left(1 + \frac{r}{n}\right)^{n t}$: The growth factor for the whole time. The power $n t$ counts all the periods: periods per year times years. Raising the factor to this power multiplies it by itself once per period, so 1.005 to the power 60 is about 1.348850 for 5 years of monthly compounding.
$P \left(1 + \frac{r}{n}\right)^{n t}$: The starting amount times the growth over the whole time. This gives the balance $A$ with the starting amount included, so the interest alone is this number minus $P$ .
$A = P \left(1 + \frac{r}{n}\right)^{n t}$: The whole formula in one sentence: the final balance equals the starting amount times one plus the yearly rate divided by the periods per year, raised to the number of periods over the whole time.

What the formula is doing, and why

Compound interest is the rule that makes savings grow faster as time passes. Each time interest is added to the balance, the next round of interest is worked out on the bigger total. The formula $A = P \left(1 + \frac{r}{n}\right)^{n t}$ gives the balance at the end of $t$ years, and it shows exactly how interest builds on interest.

Why divide the rate by $n$ ? The yearly rate is shared out between the periods of the year. With monthly compounding, one month earns one twelfth of the yearly rate: 0.06 divided by 12 is 0.005, or half a percent a month. The bank adds this small share each month, so the rate for one period is $\frac{r}{n}$ .

Why add 1? Adding the period rate to 1 turns it into a growth factor. A balance multiplied by 1.005 is the old balance plus half a percent of it. So $1 + \frac{r}{n}$ is the number you multiply by once per period.

Why the power $n t$ ? The total number of periods is the number of periods per year, $n$ , times the number of years, $t$ . Every period multiplies the balance by the same factor, so the factor is raised to the total number of periods. Two periods means the factor is multiplied by itself twice, and the power does that counting for you.

Why more periods give a little more. Each extra period starts from a slightly larger balance, so its interest is slightly larger too. Monthly compounding of 6% gives a bit more than yearly compounding, and the gap widens as the years pass. The gap between compounding and simple interest is the interest earned on earlier interest.

An everyday picture

Think of a snowball rolling down a long hill. At each roll it picks up snow, and the bigger it gets, the more snow it picks up on the next roll. Simple interest would be a snowball that always picks up the same amount of snow, no matter how big it has become. In compound interest, each interest payment joins the balance and makes the next payment larger.

Worked example: Savings for five years with monthly interest

A savings account starts with 1000 dollars and pays 6% per year, with interest added every month. The money stays for 5 years, and we want the balance at the end. In the formula, $P = 1000$ dollars, $r = 0.06$ per year, $n = 12$ times a year and $t = 5$ years.

Write the formula and put in the numbers. We replace each letter with its value, so the balance is $A = 1000 \times \left(1 + \frac{0.06}{12}\right)^{12 \times 5}$ , with the result in dollars. Every letter is now a number, so the formula can be worked out with a calculator.
Find the rate for one month. The yearly rate of 0.06 is shared across 12 months, so each month earns $\frac{0.06}{12} = 0.005$ . That is half a percent a month, the share the bank adds to the balance at the end of each month.
Find the growth factor for one month. Adding 1 to the monthly rate gives $1 + 0.005 = 1.005$ . A balance multiplied by 1.005 grows by half a percent, so this is the number the balance is multiplied by once a month.
Count the number of months. There are 12 months in each year and 5 years, so the number of compounding periods is $12 \times 5 = 60$ . The power 60 says the monthly factor is applied 60 times in a row, once for each month in the five years.
Raise the growth factor to the power 60. This means multiplying 1.005 by itself 60 times in a row, which gives $1.005^{60} \approx 1.348850$ . A calculator does this in one step, and the result tells us the money grows by about 35 percent in total.
Multiply by the starting amount. The balance at the end is the starting 1000 dollars times this growth factor: $1000 \times 1.348850 = 1348.85$ dollars. This is the final amount $A$ , the number the formula was written to find.
Find the interest earned. The interest is the final amount minus the starting amount: $1348.85 - 1000 = 348.85$ dollars. Over five years, a single deposit of 1000 dollars earns 348.85 dollars of interest.
Compare with simple interest. Simple interest on the same numbers would give $1000 \times 0.06 \times 5 = 300$ dollars, so the balance would be 1300 dollars. The extra 48.85 dollars comes from interest earned on earlier interest, which is what compounding adds.

Answer: After 5 years the account holds about 1348.85 dollars, which is 348.85 dollars of interest. Simple interest would have added only 300 dollars, so monthly compounding adds about 48.85 dollars more on this 1000-dollar deposit.

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

Common mistakes

Using the yearly rate every month. If you add 6% each month, the balance grows far too fast. Divide the yearly rate by n first, so monthly compounding uses 0.06 divided by 12, which is 0.005.
Forgetting to turn the percent into a decimal. Write 6% as 0.06. With 6 in the formula, the monthly factor becomes 1.5 instead of 1.005, which would make the savings grow by half every month.
Mixing up n and t. The letter n counts periods per year and t counts years. The power is n times t, which for monthly compounding over 5 years is 60, not 5 and not 12.
Stopping at the growth factor. The formula gives the final amount A, which includes the starting amount P. The interest earned is A minus P.

Where you will meet it

Savings accounts and certificates of deposit, where the bank adds interest every month, every quarter or every day.
Credit card balances, where unpaid interest is added to the balance and then earns interest in the next period.
Long-term savings projections, which show how a single deposit grows over decades.
Unpaid debts with interest added each period, such as an overdue loan whose balance keeps growing.

Questions people ask

What is the difference between simple and compound interest?

Simple interest is always worked out on the starting amount, so each year adds the same sum. Compound interest is worked out on the growing balance, so each year adds a little more. Over 5 years, 1000 dollars at 6% gives 300 dollars of simple interest but about 348.85 dollars with monthly compounding.

Does it matter how often interest is added?

Yes, but the gain shrinks as the periods get shorter. Adding interest more often gives a slightly higher balance, and the step from yearly to monthly matters more than the step from daily to hourly. The continuous compounding page covers the limit as the number of periods grows without bound.

Can I use this formula if I add money every month?

Not with this formula. It works for one deposit that stays in place. Regular deposits need a different formula, called the future value of an annuity, which has its own page.

Simple Interest: Simple interest says the interest is the starting amount times the yearly rate times the number of years, and interest never earns interest of its own.
Effective Annual Rate: 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.
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.
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.