Saturday, May 19, 2007

Taking the Magic out of Mortgages Part I: Exponential Functions and Geometric Sequences to the Rescue

[Note: For an exceptionally clear and definitive exposition of all things financial, the best resource I have found is MoneyChimp. There are interactive calculators to thoroughly understand the concepts in this post and much much more. More importantly, for math nerds like me, the formulas are explained and, in some cases, derived. The mathematics is accurate and the analysis is excellent. Enjoy it!]

In Algebra 2 and Precalculus (or whatever it may be entitled in your local schools), students often do compound interest problems. Typically, the author of the text and/or the instructor will derive the formula for what your original investment of P dollars will be worth in t years, if interest is compounded n times per year for t years at an annual rate given by r (a decimal for this discussion):
Compound Interest Formula: A(t) = P(1+r/n)nt.

This is a nice practical application of exponential functions, exponential growth in particular. A similar, but more sophisticated, concept applies to annuities and amortization of a mortgage (paying off debt over time in n equal payments). In both an annuity and a mortgage, the original amount of money (whether it's the amount invested or the debt you owe) generally decreases over time. In an annuity, you receive a fixed amount at the end of each period, whereas, in a mortgage, you pay a fixed amount. In an annuity, your original investment is earning (accruing) interest (it may be possible to 'live off' the interest and not touch the principal), while you are receiving periodic equal payments that are deducted from your account. A central concept in both annuities and mortgages is that that interest is applied before receiving an annuity payment or before making a mortgage payment.

The following is the first part of an activity introducing students to the mysteries of mortgage calculations. The fact that the formulas for monthly payments or the decreasing amount of debt seem very intricate lead many to believe that this topic is too sophisticated for most secondary math students. Just give them the formulas, mention that it is related to exponential functions and let them plug it all into their graphing calculators. We know most adults, other than those in the business of lending, punch the numbers into the computer and read the results. Before calculators, bankers would look it up in those mortgage tables on some well-worn-out card. This activity may demystify a little bit of this. Students need good algebra skills, knowledge of exponential properties and functions in particular, a basic knowledge of compound interest and background in geometric sequences and series (later on). I am well-aware that sensitivity is needed here for students whose parents do not own a home, however, all students can benefit from these ideas since these principles apply to far more than a monthly mortgage payment.

You can find many excellent web resources for mortgage calculations. You can also find the actual formulas for all of this either in your text or in other sources. Most of us would probably use the built-in applications typically found on a graphing calculator or more likely use those free mortgage calculators all over the web. In this activity you will take an active role in the process of borrowing and lending and see what lies behind those sophisticated formulas.

In actual practice, mortgages can range from less than a hundred thousand into the millions of dollars. Therefore, these loans are typically repaid over 5, 10, 15, 20, 25 or 30 years to make the monthly payments more manageable. In this activity you will be borrowing a small amount and considering an oversimplified form of repayment, leading up to more general considerations.

You will be borrowing $100 from a reputable lender, Stan, The Mortgage Man.
Stan is charging the going rate at the moment, which is 10% compounded annually.

(a) If you repay the loan in one year, explain why your single payment would be $110.
(b) If you agree to repay the loan at the end of two years, in one single payment, explain why that single payment would be $121?

Discussion: Parts (a) and (b) should remind you of the compound interest formula you've learned:
One year re-payment: 100(1+0.1)1
Two year re-payment: 100(1+0.1)2

Surely, increasing the payment schedule to TWO payments over two years or one year, cannot be that much more difficult? Let's find out...

(c) This time you will make two equal payments over two years. Stan gives you the repayment schedule: Two equal payments of $60. He explains it as follows: The loan (your debt) of $100 is divided into two equal payments of $50 each. The interest charges are $10 (10% of the amount you borrowed) on each payment. Explain the mistake that Stan is making (or is he trying to take advantage of an unsuspecting borrower who didn't pay attention in algebra?). We're not asking you to correct Stan's error here - just explain why his calculation is either wrong or unfair.

(d) Now that you figured out that the two equal annual payments should not be $60, we will tell you what the actual payments would be according to mortgage formulas:
Each annual payment is $57.62 (rounded to the nearest penny).
Show why these two payments correctly repay the loan of $100 and the interest that is due on each payment. Show method clearly. Use calculator as needed.

(e) Do you think you could figure out an algebraic way to determine those equal payments of $57.62? You are about to...
Let A represent the equal annual payments you will make.
At the end of the first year, before you make your payment, you owe $100(1.1) = $110.00. Now the fun begins:
(i) Represent, in terms of A, the amount of debt (loan + interest) you will owe AFTER you make your first payment.
(ii) Represent, in terms of A, the interest you will be charged by the end of year 2 BEFORE you make your final payment.
(iii) Represent, in terms of A, your debt, AFTER making your final payment.
(iv) What should be the numerical value of your debt AFTER making your final payment? Now, write an equation and solve for A. You should come up with $57.62!

(f) Are you up to the challenge of solving for the general formula for A given an original loan of $P at an annual interest rate of i (expressed as a decimal)? Of course, you are! For now, you only need to do this for TWO payments, just as we did in (e). Of course, your formula for A should be in terms of P and i.

More to follow...


Totally_clueless said...

Very nice, systematic way of explaining the concept. I like to take a couple of shortcuts, but this is highly accessible.

For general n, you can probably induce the kids to use induction (sorry :-)).

BTW, Dave, did you take a look at the "proof" that pi=2. Any insights/comments?


Dave Marain said...

thanks, tc!
i worked hard on this and i'm not done!
The graphing calculator part is coming very soon. It will require considerable analysis of data and lead students to consider an exponential model for P_k, the part of the kth payment that goes toward the principal. I'm trying to decide if I want students to develop the general case using a geometric series to represent the debt after k payments.

I have not had a moment to look at your pi = 2 proof - sorry! I will...

Dave Marain said...

I finally got a moment to think about your devilishly brilliant
pi = 2 paradox. It may be well known but I believe it's too rich to be relegated to a footnote or a comment in this blog! Either you or I should develop it into an activity for students, even before calculus!
Anyway, here's my take. Paradoxes like this one or Zeno's Paradox or other apparent contradictions or fallacies usually require a deeper understanding of concepts than students normally take the time to develop. But these questions invite many students into the realm of mathematics. The 0.99999... = 1 argument is not a fallacy but it's a paradox that 'blows the mind'.

Here's my attempt to explain the apparent contradiction in the semicircle problem. Although at each stage, it appears that each small semicircle 'approaches' its diameter, the visual effect of this is intentionally misleading (but insidiously clever).

In fact, at the kth stage, there are 2^k semicircles each of length pi/(2^k) and 2^k diameters (segments) each of length 2/(2^k). Now here's where the the trap is set. Although the differences between each small semicircle and each small segment are approaching zero, the SUM of those differences is not! Since there are 2^k differences comprising this sum, the real sum = (2^k)((pi-2)/(2^k)
= pi-2. Thus, although it appears visually that the semicircles are approaching their diameters, the SUM of the semicircles will always be pi-2 more than the sum of the diameters, which is constantly 2!

I'm not sure which famous mathematician said it, but I've never forgotten the following definition of integral calculus:
The study of more and more of less and less!
This absolutely applies here, since, even though the individual semicircles appear to be approaching a segment geometrically, the fact is their lengths are approaching zero, just as the lengths of the segments are. BUT the sum of the lengths of the semicircles are NOT approaching 0 or 2 - the SUM is constantly pi.
Did I mess up?

Totally_clueless said...

Hi Dave,

First of all, I have to point out that the pi=2 paradox is not an original concoction of mine, but something I picked up from someone in college.

You have the same idea as I have on pi=2. The limit of the sums of the semicircle diameters, even though it visually appears to approach 2, is pi.

Another way I like to look at it is that the local radius of curvature for a line is infinity at each point, but it is approaching zero for the series of semicircles, so the series of semicircles is not converging to the line.

This may be a good illustration for the calculus students when discussing limits.


Totally_clueless said...

It is also interesting to note that the area between the circles and the diameter goes to zero, but the length of the arc stays constant.


Dave Marain said...

Isn't it amazing how profound questions stimulate the brain and lead to profound thinking?

Anyway, I posed the semicircle question to my calc class this morning and I told them they could earn up to 3 bonus points by explaining the fallacy - the more rigorous, the more points!' I haven't had a chance yet to analyze the results - I only received 3 responses, but I did hear some interesting dialogue (they were supposed to work independently but I shut my mouth!). I'll let you know how that turned out.

Anyway, your comment about curvature opened my eyes. Several key ideas here. First, although the DIFFERENCE between the length of each semicircle and its corresponding diameter is approaching zero, the RATIO is constantly pi/2!! Thus, if we were to 'zoom in' on the each tiny semicircle, we would see (on a microscopic level) that the length of the semcircle is always about 1.57 times the length of each small segment (diameter)!

The deception in your problem is that we are led to assume that because the total area between the semicircles and the line approaches zero, the same must be true for the TOTAL lengths of the semicircles!
By the way, if memory serves, if the radii are approaching zero, doesn't that mean that the curvatures of the semicircles approach infinity, since the curvature is the reciprocal of the radii?

Ok, now here's an extension that might fool some:
If the sum of the lengths of the semicircles is not approaching the length of the diameter, why then do we argue that the perimeter of an inscribed polygon of n sides approaches the 'curved' circumference of a circle. How are these two problems essentially different (yes they are!)... I know you'll see through this one!