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

(1 +r)

= 100 + 100r. In this the second term is the interest and the first is called the
principal. Now you have 100

(1+ r)

in the bank. This becomes the new principal. How
much will you have at the end of the second month? By analogy to what was just done it
would equal

2
100(1+ r)+ 100(1+ r)r = 100(1+ r) .

In general, the amount you would have at the end of n months is 100

(1 +r)

^{n}.

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

^{y} = e^{x}. The expression in 5.23 can be
written as

[( ) ]
P 1+ r- n t
n

and so, taking the limit as n →∞, you get

P ert = A.

This shows how to compound interest continuously.

Example 5.16.1Suppose you have $100 and you put it in a savings account whichpays 6% compounded continuously. How much will you have at the end of 4 years?

From the above discussion, this would be 100e^{}

(.06)

4 = 127.12. Thus, in 4 years, you
would gain interest of about $27.