### Trust The Code

I’m teaching a “Mathematical Analysis for Business” course at a Community College. Mostly it’s a quick survey of (the more-or-less standard) Precalculus topics (other than Trig). Way too quick, indeed: ideas taking weeks for similarly-abled students to understand in Precalc are alloted mere days in this course … the old “mile-wide, inch-deep” problem. But that’s another rant. Today I want to talk about what I’ve learned from this course: Annuities and Amortizations and suchlike Business Math (rightly so-called). But really about teaching: I hope to throw some light on some phenomena that could just as well be illustrated with any number of other topics. Here we go.

I posted some calculator code recently for evaluating $a_{n\rceil r} := {{1-(1+r)^{-n}}\over r}$ . This quantity is used in calculating, for example, the “Present Value” of an annuity. Specifically (in our text, whose notations I’ll [mostly] follow just in case any current student of mine should happen to read this post), this is given by $A = Ra_{n\rceil r}$, where $R$ is the periodic payment, $n$ is the number of periodic payments, and $r$ is the periodic rate (unfortunately … there’s some room for improvement here since the text also sometimes uses $r$ for the nominal rate [AKA the APR]). Thus, for example, if I were to have been frightened by a cabal of unscrupulous bankers and lawyers into agreeing to pay \$30 every month for the next 33 years (what is in fact the case), and if my bank will pay me 4% (annual) interest on a savings account, compounded monthly (which might be about right; I don’t know and don’t much care), and if I had a big pot of money (unlikely at best), then I could calculate the “Present Value” of my future payments: $Ra_{n\rceil r} = \30a_{(33\cdot 12)\rceil(.04/12)} = \6590.50$ and cover the debt by depositing this amount once-for-all.

The text derives the formula for $a_{n\rceil r}$ (pronounced “a-angle-n at r”, by the way)—as well they should … unfortunately, they use “Sigma notation” to do it, and this population of students aren’t prepared to actually read the derivation with understanding (nor, as I’ve already hinted, is there anything like time enough to prepare ’em to read it now). Luckily, the whole thing can be done without Sigmas, so I did that (and, with any luck, at least one student got something out of it since it took an entire lecture).

All of the substantial mathematics for these sections of the text is done once the a-angle-n formula is derived; everything else follows from this derivation by very simple algebra (which, yes, does of course count as substantial somewhere … but not here: all of these students knew how to solve linear equations before they ever saw my face or cracked this book). So I say: let’s make this as clear as we know how. For example, since finding $R$ when $A$, $r$, and $n$ are known is obviously accomplished by $R = {A\over{a_{n\rceil r}}}$, let’s put the stress right there instead of, for heck sake, expanding the “angle” symbol to get $R = A[{r\over{1-(1+r)^{-n}}}]$.

Now, of course, this represents the same number … but it might not be obviously the same. Anyway, it seems to miss the whole point of having given $a_{n\rceil r}$ a name in the first place! It’s one of the main tricks of the algebraic method: anything you’re going to need to keep referring to again and again in performing calculations should be given a name: “ $x$ stands for the Unknown Quantity” in what seems likely to be the best-known example. And on and on it goes. It’s easy (for students like mine) to understand that if payments are made at the beginning of the pay period, rather than at the end (so-called “annuities due” as opposed to “ordinary annuities”), then the first payment earns no interest and there are $n-1$ more payments after that, so one has $A_d = R + Ra_{(n-1)\rceil r}$ … so why obscure this simple fact by displaying $A_d = R + R[{{1- (1+r)^{-(n-1)}}\over r}]$ instead (as if this were some whole new idea)?

It gets worse. The final exam is written by the course co-ordinators; well and good. They’ve also prepared a review packet; better still. Trouble is, there’s a “Formula Sheet” with a whole bunch of the messy formulas and no mention at all of $a_{n\rceil r}$! It begins to appear that somebody’s going out of their way to prevent us from doing this right.

1. David

Hi! Interesting article. I am also teaching math at a community college, but I haven’t taught business math. I like your suggestion of minimizing the number of formulas by focusing on a_n|r. I would like to describe how I would derive the formula.

If you invest 1/r at an interest rate of r, then the investment will generate a payment of 1 each period. Therefore, a perpetuity that pays 1 each period has a present value of 1/r.

Purchasing an annuity that pays out over n periods is equivalent to buying a perpetuity today, and selling it back n periods later. Thus the present value of the annuity is (1/r) – (v^n)*(1/r). In this expression, v = 1/(1+r) is the discount factor, i.e. the present value of receiving a unit payment, one period in the future.

Therefore a_n|r = (1-v^n)/r = (1-(1+r)^(-n))/r.

2. vlorbik

this is terrific (and sure oughta be in the book!).

i just showed ’em that, just as
x^2 – 1 = (x-1)(x + 1)
and
x^3 – 1 = (x-1)(x^2 + x + 1)
(“familiar” formulas from a previous class–
the scarequotes are there because most
103 students learn the difference-of-squares
well enough but there’s lots of puntage
on the difference-of-cubes),
we also have (as an in-class exercise)
x^4 – 1 = (x-1)(x^3+x^2+x+1)
and (back to lecture, alas) indeed that
x^n – 1
*always* factors as (x-1) times
the sum of the first n powers of x
[starting at 0: 1 + x + x^2 + … + x^(n-1)].

(one scribbles a couple lines on the board here
[x times “each” power on one line &
-1 times each power on the next,
lined up for cancellation; the scarequotes *here*
are because of the “…” notation …]).
dividing by x-1 on both sides gives a “formula”
for the sum-of-powers-of-x; at this point
i’m back to the textbook for the rest.
(they do okay in making the connection
from sums-of-powers to annuities.)

one could quicken this development by simply
performing a “long division” on (x^n – 1)/(x-1).
in fact, i did this first but the glazed eyeballs
convinced me that maybe i’d do well to make
the connection to differences of squares or cubes
more explicit …

3. vlorbik

new “formula does not parse” spotted today goddammit.

1. 1 45th Carnival of Math

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• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)