Compound Interest Formula & How to Calculate

Compound Interest Formula & How to Calculate
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Albert Einstein is supposed to have quipped that compound interest was the eighth wonder of the world. The reason for his enthusiasm may lie in the fact that compound interest can make you rich if you have patience, according to the Federal Deposit Insurance Corporation. Fortunately, you don’t have to be an Einstein to understand and calculate compound interest.

What Is Compound Interest?

Simply stated, compound interest is interest on interest. When you have a savings or investment account that reinvests interest, compounding adds each payment of accumulated interest to the original principal. As the principal grows, so does the interest you earn in each period.

The elements involved in calculating compound interest are:

  • Interest (I):‌ The total interest you earn from an investment (or pay on a loan)
  • Principal amount (P)‌: The initial amount you contribute to an investment (or the amount you borrow through a loan)
  • Interest rate (r):‌ The interest rate expressed as an annualized percentage
  • Time (t):‌ The number of years or the time period for the investment or loan
  • Compounding frequency (n):‌ The number of times interest is compounded per year. For example, n = 1 is annual compounding, n = 2 is semiannual, n = 12 is monthly, etc.

Simple Interest vs. Compound Interest

Simple interest does not increase the amount of an investment’s principal. Instead, you receive interest payments separately. The formula of simple interest is:

I = P x r x n

For example, suppose an ordinary five-year bond with a face value of $1,000 pays a 5 percent simple annual interest rate in two semiannual payments of 2.5 percent each. Every six months, you’ll receive a check for $25 so that after five years, you will have earned $250 in interest, and the bond will return your initial $1,000 investment:

I = $1,000 x 5 percent/year x 5 years = $250

If instead the bond compounded the interest by adding it to the principal, the payment you receive after five years would be the $1,000 original investment plus $280.08 in interest, or a total amount of $1,280.08, according to the compound interest calculator on Compounding beats simple interest by $30.08.

Types of Compound Interest

The two main types of compound interest are periodic and continuous. Periodic compounded interest is based on a discrete interval between each compounding event, such as one day, one month, six months or a year. The most popular compounding interval is one day since that’s how credit cards compute interest.

Continuous compounding uses infinitely small intervals so that there are theoretically an infinite number of compounding periods per year. While continuous compounding sounds extreme, the bottom line is that its results aren’t very different from daily compounding.

How to Calculate Compound Interest

You can employ an online or handheld calculator or an Excel or Google spreadsheet to compute compound interest using the following formulas.

Compound Interest Formulas

The formula for the periodic compound interest on an investment is:

I = P[(1+ r/n)nt - 1 ]

The formula for continuous compounding is:

I = Pe(rt)

where e is a constant, the base of the natural logarithm (approximately 2.718). In Excel, you can use the following:

I = PEXP(1)^(rt)

where the EXP function provides the value of e.

Examples of Compound Interest

To solve in Excel the earlier example of an investment paying compounded semiannual interest, where P = $1,000, r = 5 percent, n = 2, and t = 5:

I = P ((1+ r/n)nt – 1] = $1,000[(1 + 0.05 / 2)^(2*5)– 1) = $280.08

Monthly compounding generates the following result:

I = $1,000[(1 + 0.05 / 12)^(12*5)– 1) = $283.36

Daily compounding yields:

I = $1,000[(1 + 0.05 / 365)^(365*5)– 1) = $284.00

If the investment instead used continuous compounding, the result would be:

I = P(EXP(1)^(rt) -1) = $1,000(EXP(1)^(.055) -1) = $284.03, or $3.94 more than semiannual compounding.

Can Compound Interest Work Against You?

Compound interest investments are available through vehicles such as regular and high-yield savings accounts (daily compounding), certificates of deposit (varies), money market accounts (daily), zero coupon bonds (semiannually), savings bonds (semiannually), mutual funds (varies) and stocks that reinvest dividends (quarterly). Investments often quote interest as an annual percentage yield (APY), a measure that includes compounding.

Student loans charge simple interest, whereas mortgages compound monthly and credit card debt uses daily compounding. Typically, loans quote interest as an annual percentage return (APR), which ignores compounding and thus may understate the true cost.