Suppose you put money in the bank and it accrues interest at the rate of r per payment period. These terms need a little explanation. If the payment period is one month, and you started with $100 then the amount at the end of one month would equal 100
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In general, the amount you would have at the end of n months is 100
When a bank says they offer 6% compounded monthly, this means r, the rate per payment period equals .06∕12. Consider the problem of a rate of r per year and compounding the interest n times a year and letting n increase without bound. This is what is meant by compounding continuously. The interest rate per payment period is then r∕n and the number of payment periods after time t years is approximately tn. From the above the amount in the account after t years is
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Recall from Example 8.6.10 that limy→∞
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and so, taking the limit as n →∞, you get
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This shows how to compound interest continuously.
Example 8.6.11 Suppose you have $100 and you put it in a savings account which pays 6% compounded continuously. How much will you have at the end of 4 years?
From the above discussion, this would be 100e