Latticework of Mental Models: Denominator Blindness

Imagine you are engrossed in a very interesting book and your concentration is broken by a call from a friend (let’s call him Hobbes). It’s rather unusual to receive Hobbes’ call at this time during the day so, expecting to hear something urgent, you pick up the phone and ask him –

“What’s up, buddy? Is everything alright?”

“Dude, the stock that I bought last month has reported a decline in their quarterly profits by 50 crores! Is that a bad news? Should I sell it?” The panic in his voice is clearly evident.

What would you tell him?

As they say, the most useful way of answering a question is to ask another question in response. And in this case, you must ask Hobbes, “50 crores compared to what?”

“What do you mean ‘compared to what’? Isn’t 50 crores a huge number in itself?” You friend is little confused by your response.

The right answer is that the figure 50 crores needs to be considered in full context. The company’s profit declined by 50 crores but what’s the net worth of the company? Is the net worth comparable to 50 crores or is it something in the order of 10,000 crores? Asked in another way, was the profit decline almost 90 percent? Or was it less than 5 percent of the total profits, i.e. part of the daily noise?

Now consider these news headlines:

• ABC Fund Loses Rs. 100 crores;
• XYZ Corporation Lays off 100 people;
• Share Market Drops by 250 points.

Based on the headline alone all you see is a big scary number. Whether the number is meaningful or not depends on the total base on which that number stands. What makes these numbers meaningful is the denominator on which they sit.

Denominator blindness is an all too common thinking error. It is the failure to put big numbers into proper context. Put simply, information that provides only a numerator is of limited use.

Denominator blindness is part of a bigger mental model called Innumeracy, an inability to deal comfortably with the fundamental notion of numbers. Call it a lack of numerical perspective. Innumeracy is an important mental model which sits at the intersection of Psychology and Mathematics.

But before we dig deeper into this shortcoming of human mind, let me remind you that Innumeracy isn’t something to be ashamed of.

Homo Sapiens, i.e you, me and 7 billion other people on planet earth, carries a very heavy evolutionary baggage of 2.5 million years. For a large part of this human history, we have lived as hunter-gatherers and what helped us survive in the hostile African Savannah was the ability to be good with places, images, shapes, sounds, smell etc.

Numbers and text are fairly new invention and evolution is yet to catch up on this new development. Which means dealing with numbers doesn’t come naturally to us and it won’t for next few thousand years unless science finds a way to rev up the evolutionary engine. So we have to put conscious efforts to develop a feel for numbers.

The Jellybean Syndrome

I guess the most interesting part of being an experimental psychologist is to run harmless experiments on unsuspecting fellow humans.

One such team of psychologists took two bowls, a small one, and a larger one, and filled each bowl with jellybeans. The small one contained 10 jellybeans of which 9 were white and 1 was red. The large bowl held 100 jellybeans including 95 white ones and 5 red ones. People taking part in the experiment (usually referred to as subjects by these clever experimenters) could earn one dollar if they were able to pick a red jellybean out of either of the two containers. The subjects were informed beforehand the ratio of white/red jellybeans in each bowl. People were blindfolded and the bowls were shaken well so that the draw would be random, however, the participants were given the freedom to choose the bowl they wanted to pick from.

If you had to earn a dollar in this experiment, which bowl would you pick? The smaller one, where 10 percent of the jellybeans were red? Or the bigger bowl, where only 5 percent of the jellybeans were red colored? In spite of being aware of the low probability of red in the bigger bowl, most people in the experiment preferred to pick from the bigger bowl (where the odds of success were clearly, or at least mathematically, low) because they “felt” the bigger container offered more ways to win.

If your friend Hobbes was also in the race to win a dollar in one such experiment he would probably argue, “Dude, the bigger bowl has 5 red jellybeans which is a much big number as compared to the single red jellybean in the small bowl.”

Psychologists have dubbed this folly as The Jellybean Syndrome, which is a friendlier name to remember than the obtuse looking jargon – Denominator Blindness.

What Lies Beneath?

Any number can essentially be represented in this form –

Numerator / Denominator

The figure on the top of the division line is the numerator, and it tells you the absolute change that took place in any measured data; the number beneath is the denominator, and it indicates the total size of the dataset.

In the examples above, people (including your friend Hobbes) were only seeing numerators, not denominators. A company firing 100 employees does sound like a disturbing news but is that necessarily a sign of distress in the business? What if that company is a software giant like Infosys with 200,000 employees?

What about a 250 points drop in the market index? The question is which index are we talking about? If we are talking about Nifty, where the base is around 8,500, then it surely is a cause for concern. But if it’s Sensex then 250 points correction is barely 1 percent fluctuation. This example may seem very basic since most investors understand that market corrections are usually measured in percentages but I brought it up to illustrate the role of the denominator in determining the usefulness of any data point.

Risk Blindness

Denominator blindness gives rise to another related cognitive error called base rate neglect.

Every time there is a news of a plane crash, which are increasingly far and fewer these days, I shudder at the thought of being involved in such mishap and silently take a vow to never climb on an airplane again in my life. And I am not alone in fearing the possibility of dying in a plane crash. But this irrational fear is largely an outcome of our insensitivity to base rates.

We imagine the numerator—the tragic story we saw on the news—and forget the denominator. The denominator here is the millions of flight passengers who safely reach their destination every year. But the numerator, vividity of the event and the hype created by media around such disasters, is too compelling to ignore.

The base rate of surviving an air travel is so high that a person is more likely to die on his way to the airport than in a plane crash. But 200 people dying in an air crash is a worldwide news where a million people dying in road accidents is just a statistics which news channels find neither interesting nor worth sensationalizing.

As a tangent to our discussion on base rate, my prescription for a longer life is to live on an airplane. But on a second thought, you could still die early because of other adverse effects of flying which scientists are slowly discovering.

Daniel Gardner, in his book The Science Of Fear, writes…

To get a basic sense of the risk, we have to divide the numerator by the denominator. So being blind to the denominator means we are blind to the real risk. An editorial in The Times of London is a case in point. The newspaper had found that the number of Britons murdered by strangers had “increased by a third in eight years.” That meant, it noted in the fourth paragraph, that the total had increased from 99 to 130. Most people would find that at least a little scary. Certainly the editorial writers did. But what the editorial did not say is that there are roughly 60 million Britons, and so the chance of being murdered by a stranger rose from 99 in 60 million to 130 in 60 million. Do the math and the risk is revealed to have risen from an almost invisible 0.0001 percent to an almost invisible 0.00015 percent. (163)

In Investing

When it comes to investing, the impact of every investment you make can be expressed in following way –

Amount Of You Gain or Loss / Total Amount Of Your Wealth

Find out how much was the change in your portfolio value during any single trading day – that would become the amount of gain/loss in above equation and insert the value of your total net worth in the denominator. Do this for few days in a row and capture the values of those fractions. Now give a long hard thought to those fraction values. Assuming that you have a fairly diversified portfolio (say 10-15 stocks), you’d quickly realize that the numerator fluctuates constantly, and sometimes widely, but the denominator varies much more gradually over time.

The leap in the numerator values is usually very exciting but it’s the denominator that matter because that’s where the money is. After all, the sum total of your wealth is a much more important number than the amount by which it rose or fell on any given day. Still, many investors fixate on the numbers that change the most, overlooking the much larger amounts of money that are at stake overall.

If you don’t invest in equities directly and find mutual funds safer, here’s a caveat from the famous investment journalist Jason Zweig.  In his book Your Money And Your Brain, Zweig writes…

The fees and expenses charged by mutual funds are a small number, —typically less than 2% a year—, while performance can be a big number, sometimes surpassing 20% a year. And the expense figures barely fluctuate at all, while the performance numbers are forever flashing up and down. No wonder individual investors consistently say that they consider past performance to be much more important than current expenses when they pick a fund.

Decades of rigorous research have proven that the single most critical factor in the future performance of a mutual fund is that small, relatively static number: its fees and expenses. Hot performance comes and goes, but expenses never go away. The flashier factors like performance and reputation have almost no power to predict a fund’s return— but they are more vivid and changeable than the fund’s expenses, so they hijack our attention.

Similarly, transactional cost (brokerage) and taxes remain constant and silently eat away long term returns while the denominator blind investor is merrily focused on the numerator i.e. the money he made on frequent individual trades.

Conclusion

Whether it’s maths, investing or life, anytime someone throws a big number at you, make sure you understand the context and look for the denominator. Asking “compared to what?” can save a lot of trouble.

Denominator blindness is a cognitive bias which comes hardwired in our mental machinery. While we cannot exchange our brains with other people nor can we upgrade it at a hardware shop, we can certainly minimize mistakes that our biases cause by just taking notice of them.

I’ll wrap up the discussion by reminding you the significance of mental models. While mental models bolster thinking they are not crutches. They aren’t meant to provide you ready made answers. The last thing you want these mental models to do for you is to save you from thinking hard. These models should be treated as thinking tools to nudge you to discover even more useful questions. Questions that reveal penetrating insights on the problem being addressed.

In every field of human endeavour, one consistent source of innovation has been the transfer of ideas from one field to another. The practice of developing and refining your own Latticework of Mental Models adheres to the philosophy of multidisciplinary learning and constructing a view of the world which is increasingly useful.

Take care and keep learning.