Mathematics 126 -- Calculus for Social Science 2

Discussion -- Mathematics of Finance and Integration by Parts

October 3-5, 2001

Working in a Group

Since this is the first of the discussion class of the semester, a few words about this way of working are probably in order. In the discussion meetings of this class, we will be aiming for collaborative learning -- that is, for an integrated group effort in analyzing and attacking the discussion questions. The ideal is for everyone in each of the groups to be fully involved in the process. The idea is that, by actively participating in the class through talking about the ideas yourself in your own words, you can come to a better first understanding of what is going on than if you simply listen to someone else (even me!) talk about it.

However, to get the most out of this kind of work, some of you may have to adjust some of your preconceptions. In particular:

This is not a competition. You and your fellow group members are working as a team, and the goal is to have everyone understand what the group does fully.
At different times, it is inevitable that different people within the group will have a more complete grasp of what you are working on and others will have a less complete grasp. Dealing with this a group setting is excellent preparation for real work in a team; it also offers opportunities for significant educational experiences:
- If you feel totally "clueless" at some point and everyone else seems to be "getting it," your job will be to ask questions and even pester your fellow group members until the point has been explained to your full satisfaction. (Don't forget, the others may be jumping to unwarranted conclusions, and your questions may save the group from pursuing an erroneous train of thought!)
- On the other hand, when you think you do see something, you need to be willing to explain it patiently to others. (Don't forget, the absolutely best way to make sure you really understand something is to try to explain it to someone else. If you are skipping over an important point in your thinking, it can become very apparent when you set out to convey your ideas to a team member.)

In short, everyone has something to contribute, and everyone will contribute in different ways at different times.

Background

Last week we discussed computing the future and present values of income streams. We have also seen the method of integration by parts, which can help us compute antiderivatives of functions that cannot be found by a substitution alone. In this discussion, we will practice on some integration by parts examples, then see how that technique can help us analyze the future value of an income stream where the rate of income generation is changing over time.

Discussion Questions

A) To practice integration by parts, work through problems 3,7,11 from Section 7.1.
B) Note: The ideas in this question are also used in problem 19 from section 6.7 on this week's problem set. A reverse annuity mortgage (RAM) is a financial product aimed at senior citizens who are living on a fixed income, but who have substantial equity in the form of real estate that they own outright (often a family home on which they had a conventional mortgage loan, now completely repaid). An RAM works like this: in exchange for anticipated income from the future sale of the real estate, the lending party (usually a bank) makes regular monthly payments to the borrowers (the homeowners). At the end of the term of the loan (or on the death of the borrowers), the property is sold and the proceeds are used to repay the lender. Usually, the borrowers determine a monthly payment P that meets their requirements for extra income. There is some prevailing interest rate r that must be factored into the calculations. The term of the RAM is some number T of years. The questions are: what is the equivalent total dollar amount A for a loan of this type, and how much will you owe the bank at the end of the term of the loan? These questions are usually answered by making A be the present value of the income stream consisting of the payments to the borrower:

A = (12 P/ r)(1 - e^-rT)
Explain why this is reasonable, in one or more paragraphs made up of coherent, grammatical English sentences.
C) Old Stuffed Shirt Clothiers (OSSC) is an established haberdashery business whose clientele is composed almost entirely of over-50 business executives.
1. Its rate of income generation is projected to be decreasing over the next ten years as its customers retire and need fewer dress clothes: R(t) = 200000 - 20000t dollars per year in the tth year. What is the future value of the income stream from revenues of the business over 10 years, assuming that the firm does not make any ``image'' or product line adjustments and assuming a prevailing 10% interest rate on investments?
2. You have just been hired by OSSC. Seeing the ``handwriting on the wall,'' you propose to the owner that a complete ``image makeover'' is necessary if the business is to survive in the long run. Aggressively changing the marketing focus to aim at a younger, ``hipper'' clientele will drive away the ``old stuffed shirts'' in the short run and it will take some time to build a new customer base because of the established reputation of OSSC. So under your plan, you project that the rate of income will look like R(t) = 40000 - 10000t + 5000t² over the next 10 years (decreasing income at first, but then increasing as a new clientele is developed). Which plan: ``business as usual'' as in 1, or yours yields the income stream with the higher future value? (Again assume a prevailing 10% interest rate on investments.)

Assignment

One write-up per group of solutions for these problems, due Friday, October 12.