Tuesday, May 22, 2007

Puff the Magic Mortgage Part II

We will continue our investigation of mortgages. In parts (a), (b) and (c) below, you will analyze the effect of accelerating repayment by paying off the loan in one year with 2 equal payments at 6-month intervals, instead of one payment each year for 2 years, [NOTE: Some lenders do not allow this without a prepayment penalty, but we'll assume Stan the Mortgage Man is sorry for the error of his ways and wouldn't charge this.]

In (d) we will analyze data tables corresponding to the same loan of $100 but there will now be several payments over the course of one year (you will need to determine how many).

(a) From your knowledge of compound interest you know that if payments are made semiannually (in 6 month intervals), the interest rate is divided by 2, the number of interest periods; thus the rate would be 5% on each of these payments. Using an analysis (algebraically) similar to part of the earlier activity, show that each of these equal payments would be $53.78.

(b) How much is saved in total by repaying the debt in one year by this method, compared to one payment a year for 2 years? Explain why this happens. [By the way, if your parents do make mortgage payments, ask them if they are making two payments a month and, if so, why?]

(c) What could you do to reduce the total payment even more, assuming that the debt is paid off in one year?

(d) Study the 4 tables below. The data in Y1, Y2, Y3, Y4 all relate to the loan of P = $100 at 10% annual rate of interest. The loan is repaid at the end of one year but is paid in several payments. We will not tell you what the meaning of each of the columns (functions) are. That's part of the challenge! Your job is to interpret the data and respond to the following questions:

(i) How many interest periods (payments) are there? How do you know? Be careful here!
(ii) Which column (function) corresponds to each monthly mortgage payment. Give reasons.
(iii) Which column corresponds to the amount of debt remaining after each payment? Give reasons.
(iv) Which column corresponds to the amount paid toward the principal (P = $100) at each payment? Give reasons.
(v) Which column corresponds to the amount of interest paid at each payment? Give reasons.
(v) The total dollar amount of which column should be exactly $100? Explain why.
(vi) How much is the first interest payment? How much is the first payment toward principal?
(vii) Explain the meaning of the zero value in Y4.
(viii) Which function (column) is best modeled by an exponential function of the form
f(x) = a ⋅ bx-1? Determine the values of a and b and their relationship to the loan.

Hint: Consider a simpler example. Suppose the first few terms in a sequence or list are 3,6,12,24,... This is known as a geometric sequence because, starting with the 2nd term, the ratio of each term to the preceding term is constant: 6/3 = 12/6 = 24/12 = 2. The function that describes this sequence is 3 ⋅ 2x-1, for x ≥ 1. Thus, every geometric sequence can be modeled by an exponential function. Use this approach to answer this part.

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