The Mind-Blowing Math of Money

Actual student quote from last week's class: (after we completed the section on financial applications of algebra)

It's sad that it took until now, when I'm an adult college student, that I had a math class like this one that taught me about stuff I really need: how to make good decisions with my money.

It's true:  we need to do a better job connecting K-12 mathematics with the real-life "math of money" that people actually USE.

I'll present to you the retirement scenario I gave to my students at the beginning of class & give you the chance to do your own math and make your decision, then I'll give you the shocking results.  Are you ready? Ok, here goes:

You are 21 years old and will make monthly deposits to an account paying 10% annual interest compounded monthly.  Which is your better option?

Option 1:

Pay yourself \$200 a month for 5 years and then leave the balance in the bank until age 65.

Option 2:

Wait until you are 40 years old and then put \$200 a month in the account until age 65.

The question, of course, is, "Which option should you choose if you want a higher balance at age 65?"

Take a few minutes (or a few hours, whatever you need) to do your own calculations and figure it out.  I'll wait...

Before we actually work these problems, I want to point out to you as I did to my students that a common symptom of innumeracy (lack of complete mathematical literacy or numerical fluency) is that your commonsense judgment is often wrong.  We refer to cases like this as being "counterintuitive"; meaning it goes counter to what we originally think (e.g. your intuition is wrong).

Intuitively, many people think the right answer must be option 2; after all, you've invested WAY more money into the retirement account than with option 1. Here, look, I'll show you:

\$200 x 12 months x 5 years = \$12,000 invested in Option 1

\$200 x 12 months x 25 years = \$60,000 invested in Option 2

Ok, ready for the surprising math? (Stop & figure out your own answers before reading if you want to be able to check yourself.)

First, we need a couple of formulas:  the "annuity" and the "future value" formulas to be specific.  Here they are:

It is important to note what each variable stands for (a fact often overlooked by  my students) as well as the unit (e.g. time is in years, not days or months).

It is also critical to understand when to use which formula.  In plain English (the kind I like best), you would use the annuity when you are making regular payments or investments that do not change over the course of time; you would use the future value formula if you have one "lump sum" amount to invest (all at once) and then it just sits there accruing interest for the time given.

Now, let's do some calculating.  For our problem, the rate is given at 10%; so, for all our calculations, r = 0.1.  Also, n is given in that it told us we are making monthly payments and the interest is compounded monthly; so, for all our calculations, n = 12.  Everything else, variable-wise, will, uh, vary depending upon which option we are looking at and at which point in time.

Because most of my students thought Option 2 was the best one, I'll start with it (which is what we did in class as well).  So, we know r and n (from the above paragraph), now let's figure out the rest.  We'll be making regular monthly payments (that won't change for the entire period of time) -- that means we'll use the ordinary annuity formula.  We are trying to figure out how much we'll have so we're going to solve for A (that's the unknown variable; a.k.a "the answer" we're looking for).  We know m, the amount of the monthly payment: \$200.  We also know t, the amount of time in years, 25. (Did you catch that? You waited until you were 40 years old and then you made payments until you're 65 so that means you're paying for 25 years:  65 - 40 = 25) So, we're all set!  Just plug everything in and calculate!

Wow!  \$265,366.68!  That's a pretty good return on a \$60,000 investment!

Right about now in class my students who'd picked Option 2 as "the answer" were feeling pretty pleased with themselves.  I asked them if they were more confident of their choice now than they were before and they all answered in the affirmative.

So, of course it was time to burst their bubble. *evil grin*

Let's think about Option 1.  Like before, we already know r and n.  Now, t is 5 (because we are only investing for 5 years) and the monthly payment is still \$200 so m = 200.  The first 5 years, this problem is just like what we did for Option 2:  an ordinary annuity.  (Why?  Because, like before, the amount we are investing is the same for the entire 5-year period).  So, we do the exact same problem as before, but with t = 5.  I will leave this to you and just say that, after 5 years, we will have a total of \$15,487.41.  Hmmm, not that great since \$12,000 of that was what we put in.  But, wait!  We're not done!  At this point in solving the problem, we are only 26 years old!  That money (the \$15,487.41) will now "sit" in the account and just grow with interest compounded monthly.  That means we NOW need to use the future value formula to figure out how much will be in the account when we are 65 years old.

We know r and n from before. Since we are 26 and will leave it in the account until age 65, t = 39.  The "lump sum", P, is \$15,487.41. As before, plug everything in an calculate!

OMG!  It's almost THREE TIMES MORE!!!

THAT, my dear readers, is the Mind-Blowing Math of Money!

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The Solver Blog

Author:  Dr. Diana S. Perdue

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