450 Chapter 10 Consumer Mathematics
Example 10.3.1 Bill is shopping for new dining room furniture. A furniture store
offers a dining room set he likes for $1,600. The store offers to arrange fi nancing for
this purchase for $200 down, and 9 monthly payments on an installment plan with a
10% carrying charge. What would the monthly payment be?The amount being fi nanced is $1,600 $200 $1,400. The 10% carrying charge amounts
to (10%)($1,400) $140. Adding this to the amount borrowed gives us $1,400 $140
$1,540 to be paid on the plan. (We could equivalently have multiplied $1,400 by 1.10). To
compute the payment, we divide to get $1,540/9 $171.11 monthly.It is easy to misunderstand the 10% carrying charge used in this example, and think of it
as an interest rate. It is not! For one thing, interest rates are understood to be rates per year,
while this rate applied in full even though the term was only 9 months. Also, this 10% rate
applied to the entire $1,400 initial balance, even though payments are being made that
should be reducing the balance over time. With a little thought, it should be apparent that
the interest rate for this loan is actually significantly higher than 10%.
This can be even more easily misunderstood with so-called simple interest loans. This
sort of loan is a type of installment plan where the carrying charge is expressed as a simple
interest rate. However, the interest is calculated by applying the simple interest rate to the
entire original principal of the loan, even though the payments are working to pay off the loan
during its term. Interest calculated in this way is sometimes called add-on interest.Example 10.3.2 To nya bought an electronic piano for $1,850. She fi nanced this
purchase with a $50 down payment and a 2-year simple interest loan at an 8% rate
and monthly payments. Find her monthly payment amount.Subtracting her down payment leaves $1,850 $50 $1,800 to be fi nanced. Her carrying
charge is:
I PRT
I $1,800(0.08)(2)
I $288
We add this on to arrive at a total of $1,800 $288 $2,088. Dividing this by the
24 monthly payments, we arrive at $2,088/24 $87.00 per month.Again, the rate quoted here is easily misunderstood. The actual interest rate that Tonya is
paying would actually be higher than 8%. In this case, the rate quoted is per year, just as
we would expect an interest rate to be, but it applies to the entire initial principal for the
whole 2 years. The fact that payments are being made on the debt is completely ignored in
the interest calculation.The Rule of 78 (Optional)
Back in Chapter 4 when we studied loans whose payments were calculated using annuities,
we found that we could use an amortization table to find the breakdown between principal
and interest for each payment. We also could use either an amortization table or the present
value of the remaining payments to determine the remaining balance owed at any point in
time. This was all based on a certain way of looking at the interest calculation. For each
monthly payment, interest was calculated on the balance, the payment was used to pay this
interest, and the remainder of the payment went to reduce the balance. For the next month,
the same process applied, using the new, lower balance, and so on until the balance was
reduced to zero.
That method does not entirely seem to fit the situation we are describing here. While
it still seems reasonable that there must be some way of splitting the payments between
principal and interest or finding the amount owed midway through the loan’s term, the
amortization process is exactly reflective of how the loan works. One traditional way of
handling these sorts of questions is the Rule of 78 (not to be confused with the Rule of 72,
despite the similarity of the names). In fact, fixed-term installment loans are sometimes
referred to as Rule of 78 loans.