On Teaching Compound Interest and Discounted Cash Flows

“The most powerful force in the universe is compound interest”

– Albert Einstein[1]

“Tomorrow and tomorrow and tomorrow,

Creeps in this petty pace from day to day

To the last syllable of recorded time”

This Spring I began teaching Personal Finance to a group of bright college students, and we recently wrapped up a section on compound interest and discounted cash flows.

What I’m trying to get across to these undergraduates is that all of the key financial choices they will make in their lives – all of their future decisions about consumer debt, retirement, insurance, purchasing a home, tax preparation, and investing – will be much, much better decisions if they deeply understand compound interest and discounted cash flows.

What are these concepts for?

The compound interest formula tells these students, and any of us who use it, exactly how quickly, and to what ultimate size, money grows in the future.[2]

Discounted cash flows reverses the process, and tells us what the present value would be of any given cash flow or series of cash flows that occurs in the future.[3]

I’ve realized over the course of the last few weeks, however, that I’m trying to convince these students of the absolute centrality of an idea that 95% of them have never heard of before walking into my class.

Not only this, but also 95% of the people my students will meet in their life never have heard of compound interest and discounted cash flows, and therefore will not have the slightest idea how profoundly it affects their lives and their personal financial choices.

Picture me in front of the class jumping up and down and waving my arms wildly (metaphorically of course), trying to get them to believe me.

And yet, why should they believe me when I appear to be the first (and possibly insane) person to ever argue this case?

I’m afraid that after they leave my class, the Financial Infotainment Industrial Complex will never again reveal the importance of compound interest and discounted cash flows to personal finance decision-making.

**Why isn’t this taught as a requirement of Junior High School Math?**

I was a strong math student in junior high and high school.[4] I received a solid foundation in algebra, geometry, trigonometry, and calculus. Of these, algebra has frequently proved useful, but none of the others apply to my life or career.

Compound interest and discounted cash flows, however, dominated my professional life as a bond salesman and hedge fund investor, and I make use of insights from them in my personal financial life all the time.

And yet, nobody taught me compound interest or discounted cash flows in school. I’d be willing to bet that almost all of you reading this didn’t get taught these concepts in school. That knowledge had to wait until I started as a bond guy at Goldman. This, despite the fact that you only need junior high school level math – basically algebra and the concept of ‘X raised to the power of Y’ – to understand and use compound interest and discounted cash flows.

The fact that school taught, and I spent years learning, complex but ultimately very niche mathematical skills, combined with the fact that nobody taught the essential mathematical skills of personal finance (and Wall Street finance for that matter) really gets up my nose when I think about it.

More than gets up my nose, it puts me in a suspicious frame of mind.

Why would these essential skills not be taught to every junior high school student, and then re-taught to every high school student, and then elaborated on for every college student? Because that’s how important this stuff is. And how relatively unimportant trigonometry, geometry and calculus skills are for most citizens.

I’ve only come up with a couple of possible explanations, as I explain below, but please chime in with your own theories.

**1. Math teachers, as a group, do not understand the role of compound interest and discounted cash flows in personal finance.**

I fear this is true. I’ve become friends with a few of my high school math teachers as an adult and with one I’ve discussed the power of compound interest as a math concept and as a personal finance concept. Later in his career, long after I took his class, he taught compound interest as part of his lessons on mathematics skills known as ‘sequences and series.’

In these later days he emphasized to his students that if he had really understood compound interest – as a young man – as well as he does now, his working and his retirement years would look totally different. Could somebody please tell the Professional Math Teachers Association (or whoever is responsible for this stuff) that this is really *the key concept*, and I mean, for *everything*?

**2.** **The Financial Infotainment Industrial Complex wants to keep us down.** I’m afraid I’m coming around more and more to this explanation. Nothing else makes sense.

I mean, seriously folks, calculus: Not relevant (for most people.) Compound interest: relevant (for everyone.)

Coming up next: Part II - Compound interest and Wealth

Part III – Compound interest and Consumer Debt

Part IV – Discounted Cash Flows Formula

Part V – Discounted Cash Flows – another example, using annuities

Part VI – Conclusion, and why we need this math as a society

——Addendum by Michael to this post:

One of my high school math teachers (and my high school advisor!) responded to my post by pointing out that not only does he teach compound interest, but that its part of the math textbook he wrote. How about that? I can’t resist linking to his textbook on Amazon, as my way of atoning for casting aspersions on math teachers.

[1] Albert Einstein frequently gets credited with this wise statement. A quick interwebs search suggests Einstein didn’t necessarily say this, as the first mention in print is found circa 1983. But Einstein could have, and should have, because it’s true.

[2] To get started on your own learning journey on compound interest, I recommend beginning by watching a video here, with my favorite, Salman Khan. If you enjoy that, continue the process with videos on present value #1, present value #2, and present value #3

[3] A nice place to start on discounted cash flows is Salman Khan’s video on present value #4 (and discounted cash flow). Khan doesn’t go far enough on discounted cash flows, or as far as I’m going to go in this series of blog posts to follow, but he at least gets us started, which is more than I can say for almost anyone else available for free out there.

[4] I didn’t pursue math in college, beyond one statistics class required for my concentration, which was Social Studies. Shout out to the 0.0005% of readers (I chose an arbitrary but statistically insignificant number) who will recognize my major and salute me for it, rather than assume I spent my college years doing what the rest of you did in Social Studies in middle school – memorizing state mascots.

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Tags: albert einstein, compound interest, discounted cash flows, financial infotainment industrial complex, math teachers, personal finance math

I tried. I had a brief tenure as a high school math teacher in an inner city school, and I made sure to include compound interest calculations when they were learning exponents.

I like to think it hit home for some of them… but I’m not really sure.

DCF calculations would have been tougher (especially since I didn’t learn DCF until the job I had immediately after teaching) as the kids there didn’t know how to sum series. And in defense of calculus, that’s all integration is – summing over infinitely smaller slices.

Either way, I completely agree that these topics are really important. I’m on my own secret mission of normalizing “hard math” topics on our blog. (One of them is up today, actually – haha – and it links to the post I wrote about DCFs a few months back!)

We’ll teach ‘em yet! My own attempt at teaching DCF comes in parts 4 and 5 of this series. I’ll check out yours shortly!

I don’t understand the point about Calculus not being important. Does the author not understand that his compound interest formula’s were all created using convergence of series in calculus?

I’m being overly dramatic for the sake of a point. I agree with you of course calculus is important in the big picture. But I don’t think most people need to be able to do calculus for their life, nor do they need to derive ‘convergence of series’ using calculus in order to make use of compound interest and discounted cash flows. If time in a math class is limited for most students, then I’d focus on the stuff that can change lives (compound interest, discounted cashflows) rather than calculus

The issue there is to understand the crucial awareness of the compounding interest formula to kids as well as money management. Calculus is surely important for some of them who will reach universities for scientific studies and research. They are a huge minority though…

The compounding interest formula is based on time, something kids have way more than what we grown-ups have…

all one has to do in order to make sense or cents of the sum is to make sure one knows the integers being rendered. in other words…. re read what was said and you will get this: Not relevant (for most people.) Compound interest: relevant (for everyone.) – See more at: http://www.bankers-anonymous.com/blog/part-i-the-most-powerful-math-in-the-universe-goes-untaught/#sthash.JxdDDZ9D.dpuf

I’m a high school math teacher in an urban school district. This article raises some valid points, but from a classroom teaching perspective, it’s slightly misleading and it only tips the iceberg regarding what is really a nationwide, systemmatic issue, not something that can be solved at the classroom level.

Personal finance courses in both junior high and high school are great ideas. In fact, many math teachers I’ve worked with would strongly support and also be interested in teaching such courses. Sadly, the school board and district curriculum personnel have agreed the opposite – personal finance courses should not be offered. I won’t pretend to know a reason why. They just won’t develop such courses. I do know it’s said that we teach such things in our business courses. But those classes are not taught by math teachers or individuals with finance credentials and the evidence that many high school students graduate with poor personal finance skills demonstrates how little these classes do for our students regarding personal finance.

We also have to keep in mind that there are requirements at the state and district levels that require teachers to teach specific skills. That’s not a bad thing in principle, and in theory it’s possible for us to include other topics. But more often than not there are so many required skills per grade level that teachers barely have time to teach what they’re required to teach. By the time we consider differentiating work for absent, failing and struggling students, there’s just no time to incorporate those things into the traditional courses.

Then, of course, there’s college entrance requirements, including tests like the ACT and SAT. I’ve never looked at the SAT because in the midwest we use the ACT, but I know the ACT isn’t concerned with understanding compounding interest and discount cash flows. It’s more concerned with algebra skills, a few trig and geometry problems that incorporate algebraic skills, and the ability to interpret oversimplified word problems and solve them using traditional methods (graphing, solving equations, etc.) High schools want to send students to college and colleges aren’t looking for students with Personal Finance classes, they’re looking for traditional Algebra, Geometry, Trig, Calc, etc.

I apologize if this sounds like a rant, it’s really not. I just wanted to add clarity to the conversation from the education perspective. No one can argue that personal finance would be a waste of time. I completely agree with your assessed value of those skills. But I think we really need to take a step back and look at the way public education in America works before we start blaming teachers.

Tim – Thank you very much for taking the time to respond thoughtfully. I agree in a fundamental way that the last people we should be blaming are the teachers. My mom’s a teacher. My mother-in-law is a teacher. Some of my favorite people and best friends are teachers. And the last thing they need is some ex-Wall Street guy (me) blaming them for the fact that people are financially illiterate/financially innumerate. I really just mean to be a little challenging in terms of societal priorities, and I’m exaggerating to make a point. I actually don’t have a good explanation for the lack of financial training, but I’m hoping people (like you!) will point us in the right direction.

Totally agree with your comment. Teachers cannot be blamed for the lack of financial material in mathematics and other subjects. So you know, since you sound American, in my Canadian province, the school curricula do not really provide the awareness the author of the article promotes. In other words, the kids are not prepared financially.

It seems the teachers are always blamed for everything that goes wrong in schools and other educational institutions. Yet a teacher can always plant the seeds for the kids to be aware of his real future based on financials more than his/her actual knowledge of mathematics and other core subjects…

In 1995 I was lucky to have a very forward thinking math teacher in high school who convinced the math department to require all students to use a TI-82 calculator. They taught us some simple programming skills, and showed us how to plot graphs, which was helpful in some ways, a headache in others. Not being a strong math student, I don’t remember much from high school math. What I do remember, however, is Mr. Pettus explaining compound interest to us. This was re-enforced by a simple program he had us write. I remember clearly the theoretical $1000 invested every year, hitting RETURN again and again, and seeing not much change until year 20 or so, and then WOAH, now we’re talking serious money. Of course that was all based on a 7% CD…

I’m so glad to hear it! The ‘magic’ of compound interest is really the point, becuase its so powerful that it might, just maybe, affect behavior at an early enough age that people can take advantage of their biggest asset, time.

It’s all so much easier to ‘program’ now that so many students can access a powerful spreadsheet program on a laptop/ipad.

And yes, the move from 7% to 1% return on CDs is a setback, but the math still applies…

I just wanted to note that I did indeed learn compound interest in middle school and learned it well enough that the concept stuck as well as the algebra. I definitely agree that finance would serve students better than much of what is in mandated curricula.

That’s great! Not enough of us get it early enough. Thanks for your comment.

of all the math I learned in high school, the most important thing was the natural logarithm of 2. when you can figure out how fast things double, you are well on your way to understanding growth, debt, interest, amortization.

Love this post. I have often expressed the same opinion in a different way: learning time value of money was the most important thing I got from college as an accounting major. So much so, in fact, that my whole career has been based on it. I can’t tell you how many times people have asked me to calculate a payment on a loan amount or the present value of a cash flow. Most people can’t even wrap their heads around this concept, let alone do the simplest calculation with a financial calculator.

Great blog, keep up the good work.

Thanks Mike! Let’s help spread the word. This stuff needs to be taught in junior high and high school.

Students in Hong Kong have been learning the time value of money at the age of 16-17.

Which, among other reasons, helps explain why the 21st Century belongs to East Asia, not North America.

I graduated from a public HS in Minnesota in 2006 and remember doing personal finance and compound interest, in 7th grade econ. We also covered how to open and balance checking accounts; establish a budget based on pre-determained incom and family situation; and did mock job intervirws (i still remember blowimg mine because i blanked on calculating sales tax i could have done in my sleep because i was so nervous). I guess i didnt realize how fortunate i was to have that experience!

That’s great…I think 7th grade is the time to teach compound interest, and then repeat it every year thereafter!

So is there a book with sample problems that can take one from beginning financial math thru to a level of enlightenment? I see you have internet links. I am still a book learner as I am sure many are. Thanks for stirring up a valid argument. William

William – thanks for your question…my goal (I’ve been saying this for over a year to myself, so bear with me a little longer) is to post some explanatory videos in the short run, and then publish a book (the old-fashioned paper kind!) with clear how-to explanations on compound interest and discounted cash flows. So…I’m hoping to fill this gap myself in the medium run. In the short run, you could get a book like “All the Math you Need to Get Rich” by Robert Hershey, which I will review on this site shortly. The problem with this book, on this topic, is that he has his readers rely on Tables in the Appendix to do compound interest math, when in reality we should be teaching how to do this with Excel Spreadsheets, which are much more flexible, intuitive, and available to everyone. Because everyone at this point has a computer, but nobody walks around carrying Compound Interest tables in their back pocket.