Continuously Compounded Interest

You don't have to be Einstein to appreciate compound interest.

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A popular quote dubiously attributed to Albert Einstein refers to compound interest as the "most powerful force in the universe." With compound interest, you not only earn interest on money you've invested, but you also earn interest on your interest, making your money grow faster. With "continuously compounded interest," your money is growing at all times. Every second, every possible fraction of a second, you're getting richer. Sounds fantastic, in theory. In practice, it's not as exciting.

Compound Interest

With compound interest, the interest you earn gets added to your principal balance, so that there's more money there to earn interest next time around. Say you put $100 in an account that pays you 4 percent annual interest, compounded yearly. After one year, you've earned $4, and that money goes into your account. Now you have $104. After another year, you earn $4.16 -- 4 percent of $104 -- and that goes into your account, making your balance $108.16. The next year you earn $4.33, and so on, with your interest earnings increasing each year even though you haven't put any more money in the account.

Compounding Schedules

The more frequently your interest compounds, the sooner you start earning interest on interest. However, when compounding occurs more than once a year, you earn only a fraction of your annual rate in each compounding period. For example, if you are earning 4 percent compounded semiannually (twice a year), you'll actually earn 2 percent every six months. If it's compounded monthly, you'll earn 0.33 percent a month (4 percent divided by 12 months). With daily compounding, you earn about 0.011 percent a day (4 percent divided by 365).

Continuous Compounding

If you keep slicing the annual rate thin enough, you can compound once an hour, once a minute, once a second, and even further down. Which ultimately brings you to continuous compounding -- interest that compounds every single instant. In continuous compounding, the number of compounding periods becomes infinite. At the same time, though, the fractional rate of interest you earn in each one of those periods becomes infinitesimal, or infinitely small.


As it happens, when the number of compounding periods approaches infinity and the periodic interest rate approaches (but never reaches) zero, the total amount of interest you can earn converges toward a single amount. That convergence involves a number familiar to mathematicians: Euler's number, designated as "e," and equal to about 2.71828. It's one of those constants, like pi, that pops up all over the place. If you invest a principal amount, designated as "P," for "Y" years, continuously compounded at an annual rate "r," the amount you'll wind up with is given by the formula: P*e^(Y*r).


Say you again invest $100 in an interest-bearing account paying 4 percent a year, but this time the interest is continuously compounded. According to the formula, after one year you'll wind up with: $100*2.71828^(1*0.04), which comes out to $104.08. So in this instance, the difference between compounding once a year and an infinite number of times a year was just 8 cents. In fact, the difference between daily compounding (which is fairly common) and continuous compounding works out to less than three-hundredths of 1 cent. Even if you invested $1 million, the difference between daily and continuous compounding would be just pennies a year.