Compound Interest Calculator

What is compound interest?

Compound interest is interest that earns interest. At the end of each period, the interest due is added to the balance, and the next period's interest is calculated on that larger amount. Over time the balance grows along a curve rather than a straight line. Simple interest, by contrast, is always figured on the original amount, so it grows by the same dollar figure every period.

The effect is easy to see over long stretches. $1,000 at 6% simple interest becomes $1,600 after 10 years. Compounded once a year it becomes about $1,791, and compounded monthly about $1,819. The longer the time and the higher the rate, the wider that gap becomes, which is why compounding works in a saver's favor and against a borrower carrying a balance.

Different compounding frequencies

Interest can be credited yearly, twice a year, quarterly, monthly, twice a month, every two weeks, weekly, daily or, in theory, continuously. For the same quoted rate, more frequent compounding produces more interest, because each credit starts earning sooner. The gains shrink as frequency rises, though: 6% compounded monthly is worth 6.1678% a year, daily 6.1831%, and continuously 6.1837%.

Lenders and banks quote rates in different ways, which makes direct comparison tricky. Loans in the US usually quote an APR, a nominal rate typically compounded monthly, while savings products quote an APY, the effective annual rate after compounding. A mortgage at 6% APR costs 6.1678% a year in effective terms. Converting both offers to the same basis, which is what this calculator does, shows which one actually pays or costs more.

Choose Annually (APY) as the output to turn any rate into its effective annual yield. Choose Monthly (APR) to go the other way and find the monthly-compounded rate that matches an APY.

Compound interest formulas

The standard formula grows a principal P at nominal rate r, compounded n times a year, for t years. The effective annual rate isolates one year of that growth, and continuous compounding is the limit as n grows without bound.

For example, 6% compounded monthly has an APY of (1 + 0.06/12)^12 − 1 = 6.1678%. The equivalent quarterly rate is 4 × [(1.061678)^(1/4) − 1] = 6.0300%, and the equivalent continuous rate is ln(1.061678) = 5.9851%. All three describe exactly the same growth.

A = P(1 + r/n)ⁿᵗ · APY = (1 + r/n)ⁿ − 1 · A = Peʳᵗ (continuous)

A

ending amount

P

principal

r

nominal annual rate as a decimal

n

compounding periods per year

t

years

e

Euler's number, about 2.71828

To convert a rate, the calculator finds its APY with the second formula, then solves the same formula for the nominal rate at the output frequency: r = n[(1 + APY)^(1/n) − 1], or r = ln(1 + APY) for continuous compounding.

History of Compound Interest

Charging interest on unpaid interest is ancient. Clay tablets from Mesopotamia record loans of grain and silver whose debts grew when interest went unpaid, and Roman law limited the practice, which it called anatocism. Many legal and religious traditions treated it with suspicion for centuries because debts could balloon beyond a borrower's ability to repay.

The mathematics took shape in early modern Europe. Merchants and bankers published compound interest tables in the 1500s and 1600s to price loans and annuities. In 1683 Jacob Bernoulli asked what happens if interest is compounded ever more often, and the answer led to the constant e, later named and studied by Leonhard Euler. Today compound interest underlies savings accounts, mortgages, bonds and retirement planning, and the law governing it varies by country and by type of loan.

Frequently asked questions

Sources

• Investor.gov – Compound Interest Calculator
• CFPB – What is the difference between APR and APY?