## Tuesday, July 3, 2007

### Interest Compounded Continuously

Have you heard of the concept of continuous compounding? This is a question that I posed a while back while commenting on a post at My Money Blog. In that post, the author, Jonathan asked the question if it matters whether interest is compounded monthly or daily. The answer was that it doesn't make much difference.

In Jonathan's post, he included the following example of a \$10,000, 1-Year CD paying 5% APR (Annual Percentage Rate). He then went on to calculate the amount of money you would end up with after one year of compounding.

At this point, some readers might like to review my post about how to calculate APY on a bank account, as the following math is similar. For an account that is compounded monthly, you will have the following after one year:

\$10,000 x (1 + .05/12)12 = \$10511.62

And if this amount is compounded daily, you would end of with this amount:

\$10,000 x (1 + .05/365)365 = \$10,512.67

In other words, you'd end up with \$1.05 more by compounding daily versus compounding monthly.

I have always held that it is best to compare the Annual Percentage Yield (APY) for deposit accounts, rather than Annual Percentage Rate (APR). And, the author agrees with this sentiment.

But getting back to the original question, what about continuous interest compounding? This is not compounded daily, hourly, every minute, or every second, but continuously. Many people (erroneously) believe that if they could have their savings compound continuously, then they would have an infinite amount of money. However, taking the formula above, in the limit that the 365 goes to infinity (continuous compounding), the formula results in this one:

\$10,000 x exp(.05) = \$10,512.71

…or about 4 cents more than compounded daily. Note: This formula uses the e^x key on a scientific or financial calculator.

In reality, no bank offers continuous compounding. But if it did, you would probably realize that it was certainly an advertising gimmick because even continuous compounding doesn't make much of a difference.

DC