Math is Money

An adventure story from my time as a substitute teacher where I taught how math can make you rich.

Aug 18, 2022 4 minutes

If a 15-year invested $6,000 today and earned 10% per year how much would it be worth at age 65?

The answer is $704,345! That’s more than 2 orders of magnitude! Compounding interest over time is insane, but you have to start early to unlock the real value.

Back in March 2019, I did my first substitute teaching gig for fun. It was high school algebra and the topic was solving systems of equations. For the first class, I followed the teacher’s written instructions. But the kids didn’t behave and played around on their Chromebooks. I spent my break between classes throwing together a quick Excel model and started the next class with a question:

“Who is interested in me showing you how math can make you rich?” It immediately got quiet and all the hands shot into the air.

I wrote the following system of equations and told them that solving them would make them rich. The students then inundated me with questions trying to understand what it meant.

\[ \begin{cases} Future Value= PresentValue(1+r)^n \\\ Annual Income At Retirement = FutureValue * 4\% \end{cases} \]

Let’s start with the 4% Rule: this conservative estimate suggests withdrawing 4% of your nest egg annually is unlikely to deplete your savings during your lifetime. So start by asking yourself what annual income would make you happy. One of the students said he’d need $250k, but I talked them into using a more reasonable $50k. Since it was an algebra class, they knew to divide our desired annual distribution by 4% to come up with our savings goal.

\( \frac{$50{,}000}{.04}=$1.25\: million \)

That was a big number and most considered it to be unrealistic to the students. But remember the answer to my first question above: $6k at 10% annual interest for 50 years is $704,345. We’re already over halfway there —56.3% to be exact. We calculate that by using the time value of money equation: \( FV=PV\: (1+r)^n \) Plugging in the numbers as follows: \( FV = 6{,}000 * (1+.1)^{50} = 704{,}345 \).

We can continue doing this for the next year by decreasing “n” by one. So for year two, we plug in: \( 6{,}000 * (1+.1)^{49} = 640{,}314\). Boom, we’re done! \( 704{,}345 + 640{,}314 = 1{,}344{,}659 \) which is greater than our $1.25 million target. In fact, with $1,344,659 in the bank, our 4% withdrawal jumps to $53,786 because of the second equation: \( 1{,}344{,}659 * 4\\\% = $53{,}786 \).

The amazing thing here is that this 15-year-old wouldn’t have to invest a single dollar more, and they would be fine. But once you discover the power of compounding, it’s hard to stop. How much would they have if they continued investing $6k per year until reaching age 65? Rather than do all the math here, I built a Google Sheet that lets you play around with the assumptions.1

But the answer is $7.68 million! Bonus points for the student that wanted $250k per year because 4% of $7.68 million is $307,200. It’s also noteworthy that investing $6k for 50 years is only $300k in total investments, and yet one can withdraw more than this every single year for the rest of their life!

Some may claim it’s unrealistic for a 15-year-old to save $6k. But so what? The math stays the same. Start with whatever is realistic. They may need to save for a few more years, but compound interest is still a powerful force. Also, is that a reasonable objection? I put together a quick thought experiment on the ‘Savings’ tab of the Google Sheet that shows how one might get to the $6k target.

Some may claim 10% is an unrealistic rate of return. I’ll remind you that my goal with this “lesson” was to inspire kids to appreciate math and begin saving for retirement which I think it achieved. The market return may be lower, or it may be higher; the fact that compound interest is supercharged by starting young is the valuable lesson.

If you’ve orbited the sun for a while already you might be thinking “This is great if I had known when I was 15, but I’m ‘x’ now, what should I do?”

Get started right now to take advantage of all the compounding you can. There’s a quote of which I’m a fan:

The best time to plant a tree was 20 years ago. The second best time is today.
-Chinese Proverb

I know math isn’t most people’s favorite subject, but do you think it would become more popular if we changed the name from Math 101 to Money 101?

  1. You can choose ‘File’ → ‘Make a copy’ if you have a Google account or else download an Excel version by selecting ‘File’ → ‘Download’. ↩︎

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