Vol 3, Issue 16. Quarter 3 – 2026.
Nerd alert. Nerd alert!!!!! If you don’t want to hear any nerdspeak today, move on to another post. If you are still reading, here it goes.
About 2500 years ago a Greek philosopher named Zeno asked a number of questions for which he saw no obvious solution even though common sense told him that the answer was sure to exist. Such a question is called a paradox. There remains one form of one of his most famous questions that is particularly useful to you as you manage your money. It goes like this.
Consider two points along a road. Let’s call them A and B. There is some finite distance (D) between A and B. I am going to refer to this distance as D = 1 unit. In order to travel from A toward B, one has to first get half way. This puts you at a new position of d = ½ – let’s call this spot A2. To go from A2 to B, you first have to travel half way between those two points. This puts you at a new position d = ½ + ¼ – let’s call this spot A3. To go from A3 to B you first have to cover half of the remaining distance. Let’s call d = ½ + ¼ + 1/8 this new spot A4. Similarly, we can define A5, A6, A7, A8, and so on. Getting from any point Ai to Ai+1takes an amount of time that is finite, but strictly positive – meaning more than 0. Since we ultimately end up with an infinite sum of times (or distances) that are each greater than 0, it seems to be impossible to complete the trip in a finite amount of time, even though we know full well, that we can get from A to B. In other words, we have an infinite sum, and it is not clear how you prove that this sum will be a value less than infinity. Since the distance between A and B is 1 unit, Zeno is saying that the total distance is D = ½ + ¼ + 1/8 + 1/16 + 1/32 . . . and so on. How do we explain why this adds up to one, since we know that must be true?
Here is a clever approach. Multiply each term in the list by ½. The new set of terms win this sum become (½ * ½) = ¼ , (1/2 * ¼) = 1/8, (1/2 * 1//8), 1/16, (1/2 * 1/16) = 1/32 and so on. What we have done here is to define ½ * D. When you compare this list to the original list you see that we have simply subtracted ½ from the original equation, because that first term (1/2) is no longer part of the list. Notice that we accomplish the same thing if we subtract 1/2 from the total. Again this simply drops the first term from the list. This also gives the sum of ¼ + 1/8 + 1/16 + 1/32 and so on. Clearly, this is D – ½. Since both lists are now identical, we know that these two sums must be equal to each other. Thus, we get ½ * D = D – ½. Multiply both sides by 2 and get D = 2D – 1. Add 1 to both sides and subtract D and you get D = 1, which we knew all along had to be the result in the first place.
The non-nerds in the group are not thinking, “Cute trick Chester, but who gives a damn?”
Why, thank you for asking. I can rewrite what we did above in a much simpler form which is: ½ / (1 – ½) = 1. But here is the truly useful part. I can do a similar trick dividing the distance into thirds, fourths, fifths, etc. In fact, this works for any fraction of the original distance less than 1 (let’s call this number /delta.) What this shows is that the sum of \delta^n as n grows from 1 to infinity is simply \frac(\delta)(1-\delta). Isn’t that wonderful?
The non-nerds are now thinking: “Apparently, you missed my earlier question, Chester. Cute trick but who gives a damn?”
You do. Let me explain why. Think about \delta as a discount factor. By this I mean something like \delta = 1/(1+r) . If r is the discount or interest rate, then \delta is just the present value of a $1 bill, delivered 1 period from now. \delta^2 is the present value of a $1 bill delivered 2 periods from now, etc.
Here is where this understanding becomes valuable. Ask yourself, what is the present value of $1 per year that goes on forever if the discount rate is r? Well, the answer is – – -you guessed it \frac(\delta)(1-\delta). In fact, I can make this even simpler and say that the value of $1 per period forever is simply 1/r. This is called the present value of a perpetuity.
The non-nerds are now saying: “Still not sure why I should care!!!”
You care because this is the basis for the calculations of the value or price of many important things like mortgage payments, and car loan payments. Of course, they don’t literally go on forever, but it sure will feel like they do. This is also the foundation for the calculations of the prices of things like annuities or other long-term contracts. For example, an annuity that pays you $1000 per month if your interest rate is 6% per year, which is about r = 0.5% per month is going to be worth a bit less than $1000/0.005 = $200,000. Since you are not going to live forever, you now know that if someone offers to sell you an annuity that cost $200,000 and pays less than $1000 per month, it’s a pretty bad deal.
Obviously, I am not giving you all of the bells and whistles today. But here is the point. If you can simply hold onto this very simple idea, you will have a very good way to estimate the fair price of lots of things that you do care about like annuities, pensions, and mortgages. Stay with us in future posts as we give you a little more at a time so that you end up with enough of a mathematical education to be able to smell out of crummy deal when you see one. If you accomplish that, I promise you will have a much happier life.
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