Compound interest and repeated payments
A Series About Series, pt. I
Now that I have your rapt attention:
Let’s say you have a ten-year-old. You open for them a bank account with a yearly interest rate of 6% (yeah, right) that compounds monthly. Every month you put ten dollars into that account. How much do they have when they turn 18? Easy. They’ll have a bit more than 1,244 dollars.
Wait, easy? Sure. I just plugged the appropriate numbers into
where P is the principal (ten dollars), i is the effective interest rate (6% divided by twelve), and n is the number of compoundings (eight times twelve). My Google Chrome address bar did the rest of the work. Again, easy.
This is a great illustration of the power of mathematics. I can take a process as at-first-pass complicated as putting $10 into a bank account for ninety-six months, where once you deposit one amount all preceding deposits grow a little bit, and condense it into a single formula that isn’t all that bad.
It’s also a great illustration of the beauty of mathematics, because this formula had to come from somewhere. Let’s find out where. It may be surprise you that the core idea at work here is the most fundamental notion in mathematics: counting things.
On your child’s tenth birthday, you put ten dollars into the bank. Let’s let Sm keep track of amount at the m-th deposit, so S0 = 10. So far so good. Next month, we put in ten more dollars, but our original deposit has grown half a percent, or by a factor of 1.005. So S1 = 10 + 10(1.005). Then next month, we add ten dollars, our second ten dollars grew by a factor of 1.005, and our original ten dollars grew by a second factor of 1.005. Great! We can see S2 = 10 + 10(1.005) + 10(1.005)^2. It isn’t hard to see that by the n-th compounding, our sum looks like this:
This is called a geometric series, but to tell you that story, I have to tell you what a series is. A series is just addition. I take a bunch of terms, add them together, and call it a series. 1 + 2 is a series satisfying, among other things, the rule
The strange symbol on the left, the capital Greek letter “sigma” (S for “sum”) tells us we’re adding. Below it is the starting point for the index, which in this case is k (letters between i and k are generally used for indices) and above it is where k stops. To its right is the rule that the terms follow. In this case, we just add the indices together, starting at 1 and ending at 2. That’s 1 + 2. That’s a series.
A geometric series is a series whose terms differ by a common ratio. In other words, if I divide any term by the one immediately before it, I will always get the same value, which we call r in the following symbolism:
Match this with our example. There, a = 10, r = 1.005, and n = 96. The index k starts at 0 since you don’t compound your last ten dollars. (You actually deposit ten dollars ninety-seven times; your child’s birthday is the 0-th deposit.)
This is all well and good, but adding together ninety-seven terms is not as easy as working out something like the formula given at the beginning of this article. Or is it? We can show that the formula of an n-term geometric series is actually
If you think of the geometric series instead as P(1+i)^n—which ours is—then it isn’t hard to see the formula above is exactly the same as the one I used at the beginning of the chapter. Whoa! What happened?
Extend our series out to n terms. Now we use the general ar^n form, but with some simple replacement it will work for our repeated payment example.
Notice that if we multiply by r, most of the terms are the same. Why don’t we subtract these two series and see what happens?
Sure enough, every term in the middle gets killed off, and we’re left with
which we can divide by (1-r) to get
which is the formula we saw earlier.
But it gets better! If we take -1 < r < 1, we can add together an infinite number of terms and still use this formula. If r is small like we just supposed, multiplying it by itself over and over will make a smaller and smaller number. If we could do it an infinite number of times, it would be zero. So for an infinite number of terms, the geometric series formula is
An infinite number of things added together resolves to just simple division. That’s pretty incredible.
The general public has a love affair with mathematical facts which, while true, aren’t particularly loaded with information. Here’s one. The number 0.999…, which is the number whose decimal expansion is zero, the decimal, and then an infinite sequence of nines, is equal to 1. As an exercise, why don’t you use geometric series to show that this is true? We’ll talk about the solution next time, when we use series—remember, just addition!—to express more complicated functions like sines or logarithms.
