Compound Interest Formula: Examples and How to Calculate

Money left in an account does not just wait. Each time interest is paid it joins the balance and starts earning too, and one formula captures that loop for any rate, any schedule and any number of years. Below you get the formula and a calculator, worked examples with every step shown, the versions for finding a starting amount, a rate or a time, and the difference between a quoted rate and the yield a bank must show. The arithmetic is the point. It does not tell you where to put money; for your own case, talk to a licensed adviser. The money section collects the neighboring formulas.

What compound interest is

Interest earned on the money you saved and on the interest you earn along the way is compound interest¹. Put shortly, it is the interest you earn on interest². A glossary entry from the same agency defines it as interest paid on principal and on accumulated interest³.

The FDIC lays out the vocabulary. The money you deposit in a savings account is called principal, and the money you earn on your principal is called interest⁴. Each time interest is earned it is added to the principal, so the next round of interest is figured on a larger number⁴. Earning money on your principal and interest over and over again is compound interest⁴.

Simple interest is the contrast: it pays only on the original amount. A Texas State University exercise invests $500 at a simple interest rate of 6% and finds $2.50 of interest in one month⁷. Compounding changes the bookkeeping. Financial institutions divide the year into a certain number of periods and add simple interest to an account after each period⁷.

The compound interest formula

For an account that compounds n times a year, the balance after t years is A = P(1 + r/n)^(nt)⁶⁷. The symbols are the same in every textbook, with small changes of letter.

Percent becomes a decimal by dividing by 100, and the yearly rate divided by n gives the rate for one period. For a rate compounded every six months, the Texas State notes find 12%/2 = 6% per period, and for quarterly compounding 12%/4 = 3%⁷. In their notation the periodic rate is r/m⁷.

Where the formula comes from

The key idea is that for each period that passes, the amount at the end of the period equals the amount at the beginning multiplied by (1 + i), where i is the periodic rate⁷. Repeat that for every period and the multiplications stack into a power.

The FDIC shows the stacking one month at a time. Its example deposits $100 and adds 3 percent: multiply the $100 principal by 3 percent, so $100 x .03 equals $3 in interest⁴, then do the same on $103, which gives $3.09⁴. After six months the deposit has earned $19.41⁴.

One thing to know if you check this table against its own wording. The problem is written as 3 percent interest compounded monthly⁴, but the table adds 3 percent every month, which is a monthly rate. Read as 3% a year compounded monthly, the same deposit would grow far less in six months.

Worked example (our arithmetic): the FDIC deposit under both readings. Inputs: $100, 3 percent, six months⁴.

How do you calculate compound interest?

The steps are the same for every problem.

1. Write down the inputs. The starting amount P, the yearly rate, the number of compounding periods per year n and the years t⁶.
2. Convert the rate to a decimal. Divide the percent by 100; the FDIC writes 3 percent as .03⁴.
3. Find the rate per period. Divide the yearly rate by n; with quarterly compounding the periodic rate is the yearly rate divided by four⁷.
4. Count the periods. Multiply n by t.
5. Raise (1 + periodic rate) to the number of periods. Use the exponent key on a calculator or spreadsheet⁶.
6. Multiply by P. The result is A, the balance including interest⁸.
7. Subtract P to see the interest alone: Ann's edge over Sally in a Texas State exercise is found by subtracting one total from the other⁷.

Worked example 1: $3,000 at 3% compounded quarterly

An OpenStax precalculus problem asks how much an account will be worth in 10 years if $3,000 earns 3% interest compounded quarterly⁶. Quarterly means n is 4. The textbook answer is about $4,045.05⁶.

Worked example (our arithmetic). Inputs from⁶: P = 3,000, r = 3%, n = 4, t = 10.

Try your own numbers

Enter n as 1 for annual, 12 for monthly, 52 for weekly or 365 for daily compounding¹⁰; quarterly is 4⁶. The calculator opens on the first worked example, so you can see it match the textbook. Change n and watch how little the answer moves compared with changing the rate.

Worked example 2: $1,500 at 4.3% compounded quarterly

A DePaul University study guide deposits $1,500.00 at an annual interest rate of 4.3%, compounded quarterly, and asks for the balance after 6 years. It comes to approximately $1,938.84⁸.

Worked example (our arithmetic). Inputs from⁸: P = 1,500, r = 4.3%, n = 4, t = 6.

Simple versus compound interest, side by side

Texas State sets two savers against each other. Sally invests $1,000 at 12% simple interest for three years, and Ann invests $1,000 at 12% compounded monthly for the same three years⁷.

Early on the two are close. Ann leads by $70.77 after three years and by $1,100.39 after ten, because her interest keeps being reinvested⁷. The chart above draws the whole ten years.

Investor.gov shows the same effect with small numbers. With $100 earning 5% a year, you have $105 at the end of the first year and $110.25 at the end of the second². The extra $0.25 is interest on the $5 of interest². With no further deposits, that $100 grows to more than $162 in 10 years and almost $340 in 25 years².

A classroom problem on the same page invests a single $730, the cost of a $2 slice of pizza every day for a year, at 5%: after five years it is $931.69 and after 30 years $3,155.02². Time does most of the work. A second exercise makes the point at a larger scale: $100 in a bank paying 5% compounded yearly for fifty years reaches $1146.74, while $1,000 left in a safe stays the same⁷.

How often interest compounds

Compounding more often pays more, but with shrinking returns. OpenStax invests $1,000 at 10% for one year and finds that the account value increases as the compounding frequency increases⁶.

The increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding⁶. Worked example (our arithmetic): annual to semiannual adds 1102.50 − 1100 = 2.50, while monthly to daily adds 1105.16 − 1104.71 = 0.45.

Continuous compounding and the number e

Push the frequency up without limit and you get continuous compounding, which can be regarded as letting the compounding period become infinitesimally small¹⁰. OpenStax calls the growth formula built on e the continuous compounding formula in business settings⁶. The number e is irrational, its decimal expansion going on forever without repeating⁶, and it was named by the Swiss mathematician Leonhard Euler (1707–1783)⁶.

A person who invested $1,000 in an account earning a nominal 10% per year compounded continuously has $1,105.17 after one year⁶. Daily compounding gave $1105.16⁶. The two are one cent apart, which is why most savers never need the continuous version.

Three accounts, one deposit

Another Texas State exercise gives a bank three accounts. Bronze earns 12% annual interest compounded monthly, Silver earns 12.2% compounded twice a year, and Gold earns 12.4% compounded once a year. Deposit $100 and see what you have at the end of the year⁷. The results are Bronze $112.68, Silver $112.57 and Gold $112.40⁷. The account with the lowest quoted rate wins, because it compounds most often. That is the reason regulators require a yield, covered below.

How to calculate compound interest on $10,000 at 10% for 3 years

Worked example (our arithmetic). Inputs: P = 10,000, r = 10%, t = 3, annual compounding so n = 1, formula from⁶.

What is the compound interest on $8,000 at 5% for 2 years?

Worked example (our arithmetic). Inputs: P = 8,000, r = 5%, t = 2, annual compounding so n = 1, formula from⁶.

What are the formulas for compound interest?

One formula, rearranged to solve for whichever quantity you do not know.

Solve for the starting amount

When the goal is known, divide instead of multiply. OpenStax’s college-savings problem wants $40,000 in 18 years from an account earning 6% compounded semi-annually, and the answer is that Lily needs to invest $13,801⁶. Texas State asks how much to deposit at 12% per year (1% per month) compounded monthly to have $500 in 3 years; the answer is 500/(1.01^36) = 349.46⁷.

Worked example (our arithmetic). Inputs from⁶: A = 40,000, r = 6%, n = 2, t = 18. P = A / (1 + r/n)^(nt) = 40,000 / 1.03^36 = 40,000 / 2.898278 = 13,801.30, which rounds to the textbook’s 13,801.

Solve for the rate

An OpenStax exercise asks readers to use properties of rational exponents to solve the compound interest formula for the interest rate⁶, with data for an account compounded semi-annually, with an initial deposit of $9,000, worth $13,373.53 after 10 years⁶. Taking the nt-th root gives r = n[(A/P)^(1/(nt)) − 1].

Worked example (our arithmetic). Inputs from⁶: P = 9,000, A = 13,373.53, n = 2, t = 10. A/P = 1.485948; (A/P)^(1/20) = 1.020000; r = 2 × (1.020000 − 1) = 0.040000, which is 4.00% a year.

Solve for the time

Take logarithms to bring the exponent down: t = ln(A/P) / (n × ln(1 + r/n)). The Texas State exercise for $1,000 at 12% compounded monthly reads the answer off a graph: about 10 months to reach $1,100⁷.

Worked example (our arithmetic). Inputs from⁷: P = 1,000, A = 1,100, 1% per month. Months = ln(1.10) / ln(1.01) = 0.095310 / 0.009950 = 9.58, so the balance passes $1,100 during the 10th month, in line with the graph estimate.

APR, APY and the effective rate

Savings are advertised with a yield. The annual percentage yield measures the total amount of interest paid on an account based on the interest rate and the frequency of compounding⁵. It is expressed as an annualized rate, based on a 365-day year⁵. For a typical savings account without a stated maturity, the calculation is based on an assumed term of 365 days⁵, assuming that all principal and interest remain on deposit and that no other transactions occur⁵. APY reflects only interest, so a sign-up bonus is not part of it⁵.

OpenStax gives the same idea under another name: when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate⁶. The effective annual rate is the total accumulated interest that would be payable up to the end of one year, divided by the principal sum¹⁰. Regulation DD, the federal rule behind APY, works examples: $61.68 in interest on $1,000 over a 365-day year is an APY of 6.17%⁵, and assuming daily compounding, the institution pays $53.90 in interest on a $1,000 deposit, an APY of 5.39%⁵.

Worked example (our arithmetic). APY = (1 + r/n)^n − 1⁵⁶.

The Rule of 72

To estimate how long money takes to double, divide 72 by the interest rate². At a 9% return that is about every 8 years². It is a quick check, not the formula.

Worked example (our arithmetic). Exact doubling time = ln 2 / ln(1 + r), compared with 72 / r².

The rule drifts at high rates, so use the formula when the answer matters.

Compound interest in a spreadsheet

Excel’s FV function calculates the future value of an investment based on a constant interest rate¹¹. Make sure you are consistent about the units you use for rate and nper¹¹: for monthly payments on a four-year loan at 12 percent annual interest, Microsoft uses 12%/12 for rate and 4*12 for nper¹¹. Deposits are entered as negative numbers and cash you receive as positive¹¹, and the lump sum goes in the pv argument¹¹.

Worked example (our arithmetic). Example 1 as a formula: =FV(3%/4, 4*10, 0, -3000) returns 4,045.05, matching⁶.

When compounding meets a late bill

Compounding is not limited to savings. The Bureau of the Fiscal Service calculates monthly compounding interest owed on payments made after the payment due date⁹. The first part of its equation calculates compounded monthly interest, and the second part calculates simple interest on any additional days beyond the number of months⁹. In its example the amount owed is $1,500, the due date is April 1, the agency does not pay until June 15, and the rate is 6%; the interest is $18.83⁹.

Worked example (our arithmetic). Inputs from⁹: 1,500 × (1 + 0.06/12)^2 × (1 + 0.06/360 × 15) − 1,500 = 1,500 × 1.010025 × 1.0025 − 1,500 = 18.83.

Mistakes that spoil the answer

• Using the yearly rate for every period. Divide the yearly rate by n first⁷. The FDIC’s 3 percent table is this mistake read the other way: a 3 percent monthly rate is a very different account from 3 percent a year⁴.
• Mixing units. If the rate is per month, count periods in months¹¹.
• Ranking accounts by the quoted rate. Bronze at 12% beats Gold at 12.4%⁷.
• Forgetting to subtract P. A is the balance; interest alone is A minus P⁷.
• Rounding the growth factor early. Rounding 1.348349 to 1.35 turns 4,045.05 into 4,050.00, nearly five dollars off (our arithmetic from⁶).
• Assuming withdrawals fit the formula. APY assumes no other deposits or withdrawals⁵.

Where the idea came from

Compound interest was known to ancient civilisations, but the first traces of mathematicians analysing it come from the medieval era¹⁰. Charged by lenders, it was once regarded as the worst kind of usury and severely condemned by Roman law and the common laws of many other countries¹⁰. Today it is the basis of savings accounts, and of the constant e.

Next steps

Run a bank’s quoted rate through the formula, or the calculator above, and compare the result with the stated yield. A loan payment is this formula run in reverse, the subject of the loan payment formula page, and how long to pay off a loan solves it for time. Interest is a rate acting on a balance, the same idea as the cooling coffee in calculus in everyday life, and the rule for differentiating e^(rt) is on the derivative rules cheat sheet. For the textbook versions, see the OpenStax exponential functions chapter and the Investor.gov calculator, which also accepts monthly contributions¹².