Compound interest confoundedly profound
Finally I can tell you why I started this maths blog series. I was supposed to be doing something for my financial future. I thought “hmm I better understand how mortgages work — what was the formula for compound interest again”? I vaguely remember doing these calculations when I was 15. But when I was 15 I had no money, no interest in taking out a $100,000 loan and no interest in calculating the interest of other people’s loans…
To make learning about mortgages more fun, or as a ridiculous procrastination exercise, I decided to derive the formula for compound interest without looking it up. I thought about what the difference is between simple and compound interest.
Simple interest is simple multiplication. Compound interest is calculating interest on the interest. The german word for compound interest is much cooler: Zinseszins Interesterest! The indonesian word for interest is also cool: bunga flower.
Compound interest: say you have a principal P and a yearly interest rate I. At time zero you have P. After one year you have P + P*I, the interest. After two years you have on top of the P + P*I, interest again, P*I, and interest on the interest, P*I*I, so you have all up P + 2P*I + P*I². After three years if you work it out you have P + 3P*I + 3P*I² + P*I³. See the pattern?
It’s that triangle. That triangle is used to determine binomial coefficients for expanding (x + y)ⁿ. If we factor out the P, we see that we are expanding the binomial expression (1 + I)ⁿ so the formula for compound interest is
where I is an interest rate like 0.05, and n is years. But compound interest is usually calculated per month or per day even. That’s fine, we just divide the (annual) interest rate by that unit, and multiply n by it, because we are now compounding more frequently but the rate is smaller. The textbook formula looks like this
where r is the annual interest rate, n is the frequency unit (eg. 12 if every month) and t is the number of years. I used I instead of r as the annual interest rate because I was freestylin’.
So, let’s calculate! I’m going to do something terrifying. I’m going to use a real example: the balance of my student loan. It’s terrifying because I never dealt with the fact that I have a student loan. I unwittingly signed up for this loan when I was 17 in order to go to university. It was always something mysterious in the background that I never had to worry about. Interest-free, they said. And I never dared to look at it because I had no idea how to handle it. I had no experience with money let alone loans. And here I am now some decades later, having done almost a degree in mathematics, breaking out into a cold sweat at the thought of doing this calculation. Thankfully Kylie had a song ready for this moment.
It’s in your hands now
To change your fortune
To shape your future
Be proud of yourself
Remember, things can only get better
At the end of 2012 (the last time I made a payment), the balance was $24,280. The average inflation rate in the 5 years since then is ~2%. If interest (technically it is “indexation”) is calculate yearly, how much will my loan be at the end of 2017?
Gulp, yup. It was about that last time I checked. Wow that’s the first time I’ve ever applied my own numbers to that equation. I feel better. But wait the government calculates the “indexation” twice a year, so it should be a little more
They made an extra $13 by calculating the interest twice a year instead of once a year. That’s nice of them, given that if they had’ve run it daily they could have cashed in a bit more
They made another $13! What if they started compounding every minute, every second, every microsecond? Would they be able to fund the shift to renewable energy with my student loan if they compounded as frequently as they can? Let’s chuck an arbitrary large number in there
Hmm, that only brought them an extra 20c. It’s definitely not worth running computer programs to calculate the interest at that frequency, whatever it is. Why isn’t the money increasing as we compound ever more frequently? It is increasing, just it increasing by increasingly smaller amounts.
Let’s take a closer look at the forces at work in that formula. The only numbers that are changing are the parts relating to the compounding frequency — let’s call that n and set all the other numbers to 1 so we can see more clearly how the expression is changing:
As n increases, we expect the expression to increase, because it is raised to the power n. But the number getting raised to the power n is getting smaller. If we calculate this for ever bigger numbers of n, the result will slightly bigger each time, but never bigger than 2.72
How can we prove that no matter how big n becomes, the result never goes beyond 2.7-something? What number is that exactly?
(1 +1/n)^n is a binomial expression, so we can use the binomial theorem and Pascal’s triangle to expand it. Whatever n is, we take the nth row of Pascal’s triangle to get the coefficients of the expanded binomial expression:
Substituting with the formulas for the (n choose k) terms we get:
As n approaches infinity, this series tends towards a limit, ie. it converges. It can be shown that
You can kinda see that the binomial expansion reduces to this series as n tends towards infinity. The proof is a bit more sophisticated[1]. You can see that this reciprocal factorial series converges because each term is less than the following geometric series, which converges.
This limit, is what we know as the number e.
I still haven’t paid off my student loan. But paying of a student loan was never a life goal for me. Understanding the number e on the other hand, was.
[1] This proof is really awesome. You can’t prove the limit term by term, because the terms themselves become infinite expressions as n tends towards infinity. For example, as n tends towards infinity, how would we even be able to express one of the terms in the middle of the sequence? It helped me to think of whether a program could type out all the terms. If it can’t, you can’t find the limit of the series by finding the limits of the individual terms.
[2] You can approximate e with the binomial expansion. It doesn’t converge as quickly as the reciprocal factorial series, which is quicker because it is like assuming big n already for the binomial expansion.
