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- Ernie deposited $1,000 with annually compounded interest for 10 years, and wound up with an account worth
$4,000. If his interest rate had been twice as large, would he have had more than $8,000, less than $8,000, or
exactly $8,000? - In some situations it may be illegal for a lender to charge compound interest to a borrower. Suppose that you agree
to lend a business associate $10,000 for 5 years at 8% annually compounded interest. Later, though, you learn that it
is illegal to charge compound interest for this loan. You realize, though, that you can still end up with the same future
value using simple interest by using a different rate. What simple interest rate would you need to charge to wind up with
the same future value? - The Rule of 70 is a variation on the Rule of 72. It works in the same way, but uses 70 instead of 72. Use the Rule of
72 to approximate the interest rate needed to double your money in 4 years. Do the same thing using the Rule of 70.
Then, fi nd the future value of $1,000 in 4 years using each approximation.
3.2 Compounding Frequencies
The comparisons between simple and compound interest make it plain that the “interest on
interest” that compounding provides makes an enormous difference over time. Yet the dif-
ference only becomes significant after several compoundings have taken place.
In the examples we looked at in Section 3.1, there was no difference between simple
and compound interest at the end of the first year. Since no interest was credited to those
accounts until the end of the first year, there was no opportunity for interest on interest
until after that first crediting took place. In the second year, interest on interest began to
make a difference, and the difference became more pronounced the following year, as
interest on interest combined with interest on interest on interest. As the years went by
the cumulative effect of compounding on compounding accounted for the dramatic end
results we saw.
Of course, none of this can happen until interest is first credited to the account. Yet there
is no reason why we must wait an entire year for this to happen. Interest could be credited,
say, at the end of each month. Then, rather than waiting for an entire year for compounding
to begin, we would have to wait only one month for compound interest to start working its
magic. What’s more, over the course of each year compounding would take place a total
of 12 times, rather than just once. The advantage of this (to the lender at least) is obvious.
Putting the power of compounding to work sooner and more often can’t help but add up to
more interest and bigger account balances.
But how much more interest would this monthly compounding mean? If we invested
$5,000 at 8% compounded annually, when the first interest is credited at the end of the first
year we would have $5,400. Now suppose that we look at the same $5,000 at the same 8%,
but this time we compound the interest every month.
The interest earned for the first month would be:I PRT
I ($5,000.00)(0.08)(1/12)
I $33.33So at the end of the first month the account balance would be $5,033.33. Following the
same approach we find that the next month’s interest works out to $33.56—a bit more—
and continuing on through the rest of the year we get:3.2 Compounding Frequencies 101