## Taking It to the Limit A Note on Continuous Compounding

If you made a deposit in a savings account, how often could your money be compounded during the year? If you think about it, there isn't really any upper limit. We've seen that daily compounding, for example, isn't a problem. There is no reason to stop here, however. We could compound every hour or minute or second. How high would the EAR get in this case? Table 6.3 illustrates the EARs that result as 10 percent is compounded at shorter and shorter intervals. Notice that the EARs do keep getting larger, but the differences get very small.

As the numbers in Table 6.3 seem to suggest, there is an upper limit to the EAR. If we let q stand for the quoted rate, then, as the number of times the interest is compounded gets extremely large, the EAR approaches:

where e is the number 2.71828 (look for a key labeled "ex" on your calculator). For example, with our 10 percent rate, the highest possible EAR is:

In this case, we say that the money is continuously, or instantaneously, compounded. What is happening is that interest is being credited the instant it is earned, so the amount of interest grows continuously.

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