Division By Subtraction: The Costly Illusion Of Average Returns

Are you ready for the world’s shortest, easiest, and most important finance quiz? Here it is: your investment makes 35% and then loses 30%. What is your overall performance?

a) Better than a 5% gain

b) Between 0% and 5% gain

c) Around zero

d) Between 0% and 5% loss

e) Worse than a 5% loss

This is not a trick question. Trick questions are cleverly designed to deceive or confuse. This is basic arithmetic.

Yale professor Shane Frederick is a master of trick questions. He developed the Cognitive Reflection Test (CRT), a series of simple-sounding questions with nearly irresistible wrong answers. Here is the most famous example:

A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?

Now that is a trick question! The trick is the strong siren song of the wrong answer. The best way to answer these questions is to channel your inner Odysseus: tie yourself to the mast of your ship as you sail past the sirens. That way you can hear their beautiful song but you can’t dive to your death.

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You want to say the answer is ten cents. You need to say it. It’s almost hypnotic. A word problem with two small numbers like a buck and a buck and a dime and something like subtraction is involved? Easy! But the ball doesn’t cost ten cents.

I have run this test on hundreds of students, at the undergraduate and graduate level, in finance and analytics, at multiple institutions across various states. I’ve run it on senior executives at Fortune 500 firms across various countries. I’ve even run it on young children. But I don’t run this test the way it’s supposed to be run.

The way it’s supposed to be run is simply as a series of questions. You get some right, you get some wrong. Just three such questions seems to do a great job of distinguishing people who can reflect and control their impulses from those who can’t. It’s almost a cognitive marshmallow test, like the famous series of experiments at Stanford in the 1970s by Walter Mischel, where children who were able to resist eating a single marshmallow for a few minutes would then get a second marshmallow. That simple act of childhood patience seemed at the time to be linked with better lifetime outcomes.

When I run the CRT, I tell the participants, whether children, students, or executives, two important things. First, the scary news. Like the marshmallow test, the CRT may have long-term repercussions. It seems to correlate with your IQ, your future earnings, your career growth, your wealth, health, happiness, prosperity, satisfaction, and inner joy. Maybe even world peace. In other words, the test matters. That sufficiently boosts their stress levels, always a fun thing to do to executives, students, and children.

Second, I share the good news. You already know how to beat the test! It’s right there in the name. It’s the same as the kids with the marshmallows. It’s all about reflection. Can you pause? Can you wait to blurt out your answer? Then you’ll probably do just fine.

With this preamble, people on average get about two questions out of three correct. Without the preamble, that would be astonishing. In the original study, the average score was 1.2, and the highest scoring location was at MIT, where the average was 2.2. So we all feel a little surge of pride with our performance and disregard that it was partially hint-assisted.

Of course, when I ask them these questions, it’s not a scientific study aiming to document the base level of reflection or lack thereof. On the contrary, it’s to make an entirely different point about our ability to overcome our knee-jerk reactions simply by being aware of them.

Some groups get 2.5 questions correct out of the maximum possible of 3.0. But I’ve only ever seen one group consistently get all three correct. We recently launched a DBA program at Fairfield Dolan. A DBA is a terminal degree in business, sometimes synonymous with an Executive PhD. It’s a three-year program that culminates in an applied research dissertation. Our first two cohorts have been professionals with years or often decades of business experience, as well as a master's degree. These are people at the top of their field who also have an insatiable love of learning. I wanted them to bond together as a group, because they would be together as a cohort for three years, so I introduced one more little wrinkle when asking them these CRT questions: I let them come up with the answers as a team.

So what happens? It’s not that the loudest or most persuasive student wins. No one capitulates to an appeal to authority or succumbs to someone’s confidence. They are all too curious for the former and to experienced to be swayed by the latter. They all want to know. They don’t move on until each one of them is 100% convinced.

The group mulls everything over. Every proposed answer is double-checked: if that answer were correct, what would it imply? Ah, then the bat would cost too much: if the ball were $0.10, the bat would be $1.10, and the total would be $1.20, not $1.10. So that doesn’t work. What if it were a nickel instead of a dime? Aha!

Why is it that conversation can make everyone involved a genius? Is it the external reflection that substitutes for internal reflection? Something about exposing your ideas to open criticism, and being willing to hear other perspectives, seems to make everyone brilliant. When you’re alone and wrong, you may never know it.

That’s why this short and simple finance quiz is so important. You made 35% and then you lost 30%. Obviously you made about 5%.

But wait! Something stirs in your memory about compound returns. Pleased with yourself, you might nod sagely and say, “It’s a little less than 5%.” You sound measured and intelligent and just the right amount of humble. You’d be ready to receive your adulation and praise.

But you’d still be wrong. And who would be there to correct you?

If you start with $100, you can watch it grow by 35% to $135. Then, a loss of 30% means you end up with 70% of that $135. That’s only $94.50. Not only did you not make five percent, you didn’t make a little less than five percent. You didn’t even make! You have less than you started with.

The knee-jerk answer is that you made maybe a bit less than 5% but the reality is that you lost more than 5%.

That knee-jerk reaction is indeed a real jerk. It makes you think you’re doing well while you slowly go bankrupt.

The average return is often how we calculate alpha. We rarely include compounding. But as we’ve seen, compounding can reverse the sign.

Average returns don’t really exist. You can’t eat an average return. You can’t tell your spouse or your children or your investors or clients: “Good news and bad news. Our average return is nice and positive, but unfortunately we’re broke.”

Compound return is what matters. Wealth matters.

Let’s put it another way. If you lose 30%, how much do you need to make before you are back to break even? It’s far from symmetrical. It would essentially be the reciprocal of 70%, or a 43% return. If you lose 50% or half, you need to earn 100% or double. If you lose 80%, you need to earn 400%.

Imagine an investment opportunity that alternates between making 300% and losing 80% each year. Its average return is north of 100%! The marketing material writes itself. But you would actually lose about 20% of your wealth after each two-year period.

I ran the world’s shortest, easiest, and most important finance quiz recently on X.

About two-thirds of people got the correct answer. One in six erroneously thought the investment made money.

Trick questions are not all alike. If we make mistakes in the cognitive reflection test, or if we leap to similar conclusions in life, we might be okay. Occasionally we’ll be surprised by the price of a baseball. But if we forget about compound returns, we could end up bankrupt.

What’s the solution? There are two, one general and one specific. First, in general, talk it over with somebody, whether human or AI. Every large language model available today gets these questions right. On our own, we can get flummoxed and divided by simple subtraction. With a team, we can multiply our plusses. We can compound not only returns, but knowledge.

Second, specifically, remember the asymmetry. Instead of gaining 35% and losing 30%, what if you either made or lost 30%, randomly? You might think you’d break even on average. But 70% of 130% is only 91%. You’re losing 9% of your wealth every two periods. Where does it go? You lose more on your gains and don’t gain enough from your losses. It vanishes into the math of compound returns.

This is why it is so important to aim to control your downside risk. It’s not enough for your average gains to keep pace with your average losses. They have to more than compensate for them just for you to stay in place. If you lose 30%, you not only have to make 30% back, you have to make an additional 13%, just to break even.

Another way of thinking about is in terms of how much downside you can “afford” before you lose money. If you make 30%, you can now afford a 23% downside. Any more than that will slip you into the red. If you make 40%, you can afford a 29% downside. If you make 50%, you can only afford to lose 33%.

There are hundreds of behavioral anomalies, cognitive biases, and other mental illusions. It’s nice to take away a simple rule of thumb from each one. For the cognitive reflection test, one tip is to pause and reflect and double check your answer. For the marshmallow test, perhaps you can remind yourself that no matter when you eat your first marshmallow, it’s always going to be in the future, whether it’s a few seconds or a few minutes, so might as well wait and double your fun. For compound returns, we all have an intuition that compounding reduces the performance a bit, but we forget that it can also change the sign. Maybe one intuitive takeaway here is that the downside is not just a brake on profits, but a chasm and a void that takes more energy to climb out of than it does to fall in.