## Saturday, December 16, 2006

### Compounding Isn't Intuitive

Not long ago, a colleague of mine asked me a bit about some financial stuff. I'm not really qualified to give out any advice, but I could always tell him what I thought about saving and investing for the long-term and how one should probably go about it. Anyway, in our discussion, we hit upon the subject of compounding, and at one point it was clear that while he understood how powerful it was, he clearly didn't give it enough credit. So, I bring to your attention the non-intuitive nature of compounding, especially at high rates of return.

Let me describe 3 scenarios, and without too much dwelling on the matter, tell me which of them you think is best for you. To simplify things a bit for this exercise, assume that you will pay all of the debt at the very end of the 25-year period, and not periodically. Also, for simplicity's sake, assume no external factors like tax, fees, etc.

Scenario A: You are given \$10,000 by an anonymous benefactor. You are allowed to put the money into a special risk-free investment that returns 6% a year for 25 years, compounded annually.

Scenario B: A rich guy lends you \$10,000 at the rate of 7.5%, compounding annually. After 25 years, you will have to pay him back the entire amount you owe him (\$10,000 + all the compounded interest at the 7.5% rate). With that borrowed money (\$10,000), you can invest it in a risk-free vehicle that returns 10%, again compounded annually.

Scenario C: Same as Scenario B, except that instead of \$10,000 at the rate of 7.5%, you get \$5,000 at the usurious rate of 16%. But, in this situation, you are allowed to invest risk-free at 17.5%. You still pay your debt at the very end of the 25 year period, and everything compounds annually.

So, of these 3 scenarios, which one is the best? Which one of these will net you the most at the end? This means you pay off all the debt in Scenario B and C (you have no debt to repay in Scenario A).

My colleague is not unintelligent; he's a very solid engineer. But, when he was given these three choices he chose poorly. It wasn't so much that he didn't make the best choice, but that he was surprised by the result, which convinced me that the math of compounding is clearly not intuitive. I suspect that the vast majority of people will also choose poorly, and it says a lot about how underappreciated the power of compounding is, especially when you are talking about relatively high rates of return.

Well, I guess this is where I give you the end results for each of the three scenarios presented above.

Scenario A: You were given \$10,000 and owe nothing. Free money, yay! So, right from the get-go, you're better off than the other two, since you've got \$10,000 net, and the others have \$0. After 25 years, your money grew to \$42,918 at the 6% rate. Not bad considering it was free money.

Scenario B: You borrowed \$10,000 at 7.5%. After 25 years, you owe the lender \$60,983. This is due to the 7.5% interest rate he was charging you. But, you were able to grow the \$10,000 you had borrowed at 10%. So, at the time you paid your debt, you had grown the initial money to a nice \$108,347. After paying off the debt, you are left with \$47,364.

Scenario C: You borrowed \$5,000 at a ridiculous 16% rate. After the 25 years is up, you owe a whopping \$204,371. But, you were blessed with a beautiful return rate of 17.5%. You grew the \$5,000 to a staggering \$281,784 after 25 years of annual compounding. After your debt repayment, you're much better off than the other two people with a tidy sum of \$77,413.

To be fair, I should note that after 10 years, the person in Scenario A is doing the best, followed by the one in Scenario B. The one in Scenario C is probably kicking himself at this point having only a net just over \$3,000 vs. the one in Scenario A with nearly \$18,000. It is in year 21 that the person from Scenario C overtakes them both and never looks back. Also, I should note that it is not until after the 24th year that Scenario B outpaces Scenario A.

So, the above scenarios are far from reality, because debt repayment doesn't work as above and also there's no risk-free investments yielding such high returns. But, the math is what it is. And, the fact that most people would not have chosen the best option says a lot about how little the public truly understands the powers of compounding.

Also, the above implies that if companies are able to achieve consistent high rates of return on investment, then it's not necessarily bad that they are taking on debt at fairly high rates. In fact, if they are able to keep their ROI at a high level over long periods of time, then taking on debt even at rates that aren't great would be hugely profitable.

The power of compounding interest is indeed great. However your post illustrates that using numbers that will never exist, not even plausible, without the consideration of risk.

So while readers can be awed once again by the power of compounding, there is no real tip here due to the ridiculous parameters passed into this scenario.

Unknown said...

I think the tip here is to realize that our intuition about investments may not always be correct, and we should run the math before making decisions. I know I tend to give too much weight to my intuitive feelings, especially when I'm pressed for time (or the math is complex). The example isn't a choice I'll ever have to make, but it did show me that I'd have chosen incorrectly, if I'd chosen intuitively.

Jon said...

When pre-paying a 6% mortgage rather than investing at 8%, most people think "Well I'm only losing 2%, that's not worth the risk." But as you've shown, compounding does not add linearly, so after 30 years you get:

1.08^30 - 1.06^30 > 1.02^30

In this case the difference is actually close to 1.05^30. The long-run behavior is more like:

1.08^n - 1.06^n = 1.08^n (roughly)

where n, granted, is a bit greater than 30