A Question of Interest.

The compound interest formula is standard fare in Intermediate and Advanced Algebra courses. It is sometimes followed by a formula for the present value of an annuity, problems concerning accumulation of principal in $10,000 bank accounts, annuities that pay retirement benefits, and other problems which are far removed from the immediate concerns of most two-year college students. Of vital interest to these students is the matter of installment-buying. In particular, some of the questions most frequently heard are: "My car payments are part interest and part repayment of the loan. How much of each month's payment is interest?" . . . "How fast am I repaying the loan?" ..... "How much of the loan remains to be repaid after 18 months?" The conventional formulas that are used to answer such questions have been developed around the concepts of annuities and sinking funds. It is, however, completely unsatisfactory to explain to a student that in order to answer his questions he needs to consult an annuity table because, in paying his car loan, he is in effect "purchasing an annuity." In fact, the preceding questions can be attacked directly, and answered with mathematics accessible to students who are completing a second course in Algebra. Let Ak (k = 0, 1, 2, . . . , n) be the unpaid balance on a loan after the kth payment. The loan of AO dollars is made at an interest rate of r% per. month compounded monthly on the unpaid balance. The loan is to be repaid in n fixed monthly payments of M dollars each. M, although constant, is, for each month, the sum of the interest payment (decreasing) and repayment of principal (increasing), and does not include carrying or service charges. If Pk (k = 1, 2, 3, ... , n) represents the amount of principal repaid in the kth monthly payment, then M = rAO + Pi = rAl + P2 = rA2 + P3= rAk-I + Pk, (1) and Ak may be found recursively if Pk is known. Ak = Ak-I Pk. (2) From (1) and (2) we notice that Pi = M rA0, P2= M rA1 = M r(A0 P1) = P1(1 + r),