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Appendix C C-41<br />
The calculations corresponding to monthly payments #2 and #3 are<br />
now carried out in the same manner. In starting your work for monthly<br />
payment #2, remember that your unpaid balance now is P $14,791.60,<br />
rather than P $15,000. As a check on your work, after monthly payment<br />
#2 the loan balance should turn out to be $14,582.16.<br />
(b) There are formulas (that we won’t derive) from the mathematics of<br />
finance relating the loan amount, the monthly payments, and the interest<br />
rate. One such formula, which applies to our example (with monthly compounding<br />
and monthly payments) is<br />
1 (1 p) n<br />
L M<br />
(1)<br />
p<br />
where L is the loan amount, p is the periodic interest rate, and n is the total<br />
number of compoundings in the life of the loan (or, equivalently in our<br />
case, the length of the loan, in months). Use equation (1) to calculate what<br />
the monthly payments M should be for the example in part (a). Did the car<br />
dealer give you the correct figure?<br />
(c) Solve equation (1) for n in terms of the other variables. Then use the result<br />
to compute an answer for the following problem: You are thinking of taking<br />
out a bank loan of $315,848 toward the purchase of a beachfront condominium.<br />
In your notes, you’ve written down that the interest rate is 7 2%<br />
1<br />
per annum, compounded monthly, and that the monthly payments would<br />
be $953.90. You forgot, however, to write down the amount of time required<br />
to pay off the loan. Compute the required time.<br />
(d) You graduate, and the government reminds you that the time has come<br />
to begin repaying your student loan of L $5800. You have 4 years to<br />
repay this loan; the interest is compounded monthly; and the monthly<br />
payments are M $113.88. Find the periodic interest rate p and then the<br />
annual rate r 12p. Hint: Substitute the given numerical values for L<br />
and M into equation (1) and then convince yourself that you cannot solve<br />
for p using the algebraic techniques you learned in this chapter. Now use<br />
a graphing utility and the ZOOM feature to find p. It may take some work to<br />
find an appropriate viewing rectangle.
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