It is possible to lie with numbers, or at least mislead. We’ve seen this before when talking about how many hours you have available to work in a given week.
It’s not the numbers that are doing the lying, of course; it’s the interpretations we make with them. When Einstein added his famous cosmological constant to his field equations, it wasn’t because the data fit the equation better that way. Instead, it was because he believed in a static universe, and changed the equation to allow for this belief.
(Recent research has actually validated the existence some form of cosmological constant, leading to the term “cosmological irony”, denoted by the Greek letter “iota”. But I digress.)
Another potent way of lying with numbers is when talking about interest rates and, especially, compound interest. And this confusion could cost you.
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Half-wrong
I was at a financial seminar recently where we were talking about investing for retirement. In order to stress that time is more important than money, a statistic was brought out that said that $100 invested every month in a product that returned 12% interest would return over a million dollars over the course of a working lifetime.
(I’ve talked about this sort of thing before.)
Run the numbers on a calculator for $100 a month for 40 years at 12% interest and the figure I get is $1,030,970.88. So this is correct so far.
But as I’ve pointed out before, you’re probably not going to make 12%. Not because you’re not smart enough or good-looking enough (you’re both, I assure you), but because consistent 12% returns just don’t happen for most people. Periodic higher returns are possible, but usually they involve market timing (or buying loaded funds), and they aren’t consistent over the long run.
So touting 12% is misleading and unfortunate, but fine, whatever.
But I’ve more than once I’ve heard this type of statistic accompanied by a rhetorical question like this:
“What if I’m half-wrong?”
Interestingly, I’ve never heard this question answered, and I know why. So I’ll answer it.
Four-fifths wrong
The psychology of asking a question that is never answered is brilliant: it allows a person to make their own logical leap without needing to be tied down to whatever that logical leap implies.
So what did you think when I mentioned being half-wrong about making a million dollars? How much money did you think of?
If you thought a half-million dollars, you’d be forgiven. Unfortunately, you’d also be totally wrong.
And that’s why this type of insinuation bothers me.
Let’s deconstruct. What does it mean to half-wrong in this case? The only reasonable way to read this for me is to think “half-wrong about my return”. But our input to the calculation was the rate of return, not the amount of return.
So what is half-wrong of 12%? That’s 6% to me.
So what happens if you invest $100 a month for 40 years making the half-wrong interest rate of 6%?
You would only make $196,857.18. That’s less than one-fifth of the return.
So being half-wrong means that you’re four-fifths wrong. Whoops.
Exponentially wrong
Now, it’s easy to understand while this trick works. The mind understands linear changes, because our everyday experience uses linear measurements. We know what half a mile is relative to a mile. We know what half a pound is relative to pound. Even if we don’t know exactly, we have an intuitive sense of these things.
But if something doesn’t grow linearly, we don’t grasp it at all.
There’s a story that circulates around (I know it from the educational TV show Square One) where a kid asks his parents for a special kind of allowance. Instead of a weekly stipend of a few dollars, he asks to be paid each day, starting with one penny, but with the amount doubling each day. The parent foolishly agrees to this, believing the amount to be small, but it turns out that 30 days later, that measly penny will have turned into 10 million dollars.
Don’t believe me? Well, what’s 2 to the 30th power? 1,073,741,824. In pennies.
But the mind doesn’t have an intuitive sense of exponential growth, so we don’t know this.
And so too with rates of return.
Right about being wrong
Incidentally, what would your interest rate have to be in order to make half the return?
I ran the numbers, and found that you’d have to make 9.56%. That’s not half-wrong, that’s 20% wrong. And being 20% wrong means you’d be 50% wrong about your returns.
Now, let’s be honest: how much do you really think you’re going to make over the long term when investing in the market? All the research I’ve done points to earning between 5% and 8% annually, depending on what one takes into account.
So on this, we can at least we can take heart in one way: the person who asks “what if I’m half-wrong?“, is.
