Numeracy is the ability to intuitively understand numbers, their magnitude, their relationships, and how they are affected by operations. Unfortunately, innumeracy is far more widespread and socially accepted than illiteracy. To avoid poor decisions in money and investments it’s supremely important that one understands the language of numbers and develops a sense of probability.
In the 2.5 million years of human evolutionary history, homo sapiens never had to deal with numbers until about 12,000 years ago when complex societies began to appear. In the wake of the Agricultural Revolution, a completely new type of information became vital – numbers.
A critical step was made sometime before the ninth century AD, when a new partial script was invented, one that could store and process mathematical data with unprecedented efficiency. This partial script was composed of ten signs, representing the numbers from 0 to 9 – the Arabic numerals. When several other signs were later added to the Arab numerals (such as the signs for addition, subtraction and multiplication), the basis of modern mathematical notation came into being.
Unfortunately, human brain never got a chance to adapt to storing and processing numbers. Yet the development of modern society is firmly standing on the foundation of mathematical principles. In fact, until last few centuries, the common man didn’t have to deal with much mathematics. Anything beyond simple addition or subtraction was reserved for scientists and scholars who indulged in advance concepts for their own amusement.
Today, you can’t escape dealing with numbers. They’re everywhere. Trying to avoid them will invariably lead to poor choices and bad decisions.
For most people, only a casual mention of the word mathematics conjures up images of sine functions, logarithms and probability distribution. And that becomes a major barrier for people to overcome a condition called innumeracy.
Most of us are very familiar with the idea of Illiteracy. It’s the inability to read and write spoken words. So extending the logic one would guess that inability to read or write numbers would be called as innumeracy. But innumeracy is more than that. Just because you have studied mathematics in school doesn’t make you numerate. Innumeracy is the inability to deal comfortably with the fundamental notions of number and chance.
So, what makes one numerate? One aspect of numeracy is having a number sense.
Once, the mathematician G.H. Hardy was visiting his student in the hospital. To make small talk, Hardy remarked that 1729, the number of the taxi which had brought him, was rather a dull number. His student replied immediately, “No, Mr. Hardy! No! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.” The student was none other than genius Indian mathematician Ramanujan.
Ramanujan had an extraordinary number sense. He was a super-numerate. He was the Shakespeare of mathematics. Ramanujan’s observation about 1729 may be a remarkable insight but it doesn’t really matter that much when it comes to making better decisions in daily life. To practice rationality, one doesn’t need super-numerate kind of abilities.
Being numerate is less about numbers themselves and more about how those numbers help us make a sense of the world around us. In fact, when it comes to money, for most people, a number beyond few crores is pretty much incomprehensible. As the zeros at the end go beyond 5-6 digits, for a majority of the so called educated people, the intuitiveness to wrap their head around those large numbers degrades rapidly.
If you were to write a Rs ten crores cheque, you won’t have much trouble until your pen moves to the box where you must write that figure in numbers. And it’s not just about the digits. What if I were to ask you how much is 10 crores in terms of weight and volume? In other words, if you were carrying that much money in a suitcase, how big and how heavy would it be? Assuming Rs 2,000 bills, it would all weigh about 50 kg. I’ll leave the volume calculation to you.
How about Rs 478,000 crore? In case you’re wondering, that’s the market cap of Tata Consultancy Services, the company with the highest market capitalization in Indian stock market.
So why should an investor give a hoot about all this?
On Monday, an investor buys a stock that has fallen by 90 percent from its peak. He reasons that how much more can it fall? Assuming that the same stock falls further to 95 percent from its peak on following Friday, how much has that investor lost who had bought on Monday? It’s not 5 percent. He has lost 50 percent of his capital.
A discount apparel brand advertises – Buy 1 shirt at 50 percent off. Buy two at an additional 40 percent off. “Wow! A 90 percent discount on buying two shirts!” an innumerate would squeal. Not so fast buddy. A 50 percent off followed by an additional 40 percent discount leaves you with a final discount of 70 percent, not 90 percent. You will pay 3 times more than you thought.
Now you know why, as an investor and as a consumer, you should be bothered about these innumeracy issues.
As we have seen, because of evolutionary reasons, the human mind is not wired to deal with complex numbers and percentages. However, our minds are quite good in relating to real things that we see around us. So, one trick to make sense of numbers is to put a relevant context to the numbers. What do I mean by context? Here’s an example.
Warren Buffett, in his 2011 letter to investors, wrote –
You could take all the gold that’s ever been mined, and it would fill a cube 67 feet in each direction. For what it’s worth at current gold prices, you could buy — not some — all of the farmland in the United States. Plus, you could buy 10 Exxon Mobils (XOM), plus have $1 trillion of walking-around money. Or you could have a big cube of metal. Which would you take? Which is going to produce more value?
Instead of using a number (possibly a very large figure in tonnes) to describe the total amount of gold present in the world, Buffett used other smaller figures (number of Exxon Mobils and 67 feet high gold cube) that the mind can comprehend. This is Buffett’s style of simplifying complex looking ideas by thinking about them in a different way.
So that’s one way to deal innumeracy. Take large numbers and convert them into smaller denominations made up of numbers/things that we can understand.
Another hack, particularly useful for investors, is to rely on some heuristics like the rule of 72. The rule of 72 is a quick and reliable way to estimate the time it would take to double your money, given the compounding rate of return. How much time does it take to double your money with 12 percent CAGR? Divide 72 by 12 and you get 6 years.
Innumeracy is not limited to inability to deal with large numbers. Here are few more characteristics of an innumerate mind –
Confuses precision with accuracy – A precise (but very inaccurate) estimate of pi is 8.5278403. A more accurate (but less precise) estimate is 3. Foregoing accuracy for precision invariably leads to incorrect conclusions. However, people still side with precision because precision breeds confidence. It makes numerical estimates look more authoritative than they really are. For example, target prices recommended by research analyst reports is a popular way to exploit investor psychology. But any attempt to put a precise number on the value of a business, which is constantly changing and controlled by a host of other non-numeric factors, is an exercise in futility. The best one could do is to estimate a range for the intrinsic value of a company. The love for precision is what makes Microsoft Excel the most loved tool in the investor community. Don’t forget Charlie Munger’s words – “It’s better to be approximately right than precisely wrong.”
Confuses correlation with causality – Stand-up comedian George Carlin explained it best when he concluded, “Death is caused by swallowing small amounts of saliva over a long period of time.” Assume we detect a high correlation between money and happiness, writes Peter Bevelin in his book Seeking Wisdom, “But that doesn’t tell us if money causes happiness, if happiness causes money, or if some third factor causes them both.”
It’s not uncommon for amateur investors to find patterns in stock prices and other macroeconomics factors. These patterns could either be mere coincidences or there could be a correlation between them. But assuming a cause and effect relationship can lead to serious errors in making investment decisions. This logical fallacy is also known as – post hoc, ergo propter hoc (after it, therefore because of it).
Fixate on insignificant but exotic risks while ignoring more significant but mundane risks – Every time there’s a plane crash or a terrorist attack, people start worrying about their own risk of dying in similar events. Statistics say that one is more likely to die on his way to the airport than in a plane crash. Similarly, you’re more likely to die from a heart disease than in a terrorist attack.
An investor is more likely to lose money by investing in an IPO than investing in Index funds. A new investor is more likely to incur serious losses in day trading and derivatives than losing capital in mutual funds. For that matter, a perfectly safe looking option like fixed deposit is the riskiest investment instrument over long term because it earns less than the rate of inflation.
See meaningless patterns in random occurrences – Pattern recognition is the essence of neural intelligence, writes John Allen Paulos in his book Innumeracy. “Organisms survive by recognizing and responding to different patterns of sensory perception of their environments. They learn to distinguish predators from food, suitable from unsuitable habitat, males of their species from females, etc.; otherwise they die. Most patterns have some meaning, but some perceived patterns do not; they are purely specious, appearing by random chance.” Finding patterns where none exist is a typical case of getting fooled by randomness. The world we live in is more random than what we would like to think. For example, Pareidolia is a phenomenon which causes people to interpret random images, or patterns of light and shadow, as faces. About two decades back, NASA’s Viking 1 spacecraft was circling the planet Mars, snapping photos of possible landing sites, when it spotted the shadowy likeness of a human face. An enormous head nearly two miles from end to end seemed to be staring back at the cameras. When the images were made public, many people were convinced that this was a gigantic monument of some sort created by humans or human-like aliens who once lived on Mars. NASA re-photographed it from different angles and it was confirmed that the face was a pareidoliac image. But some people, to this day, remain convinced that it was a face.
Miscalculating probabilities – Once you’re out of school, chances are you haven’t had a chance to think about probability much. But no matter how hard you try, the probability won’t leave you. A wise man once said – we’re deterministic beings thrown into a probabilistic world. Imagine the weather forecast predicts that the probability of rains is 50 percent on Saturday and 50 percent on Sunday. Someone with a superficial knowledge of probability will conclude that there’s a 100 percent chance of rain during the weekend. Probabilities don’t add up that way. The probability of rain over the weekend is only 75 percent. Another interesting probability puzzle that stumps most people is the birthday problem. What’s the probability that in a group of 23 people, two people share the same birthday? It’s 50 percent. If it seems unreasonably high, you’re forgetting that we’re not talking about two people sharing a specific day as a birthday. Any day in 365 days would do. Birthday problem highlights that many events (e.g finding out that you and the stranger sitting next to you on a flight were born on the same day), which look like unusual coincidences, have a pretty high likelihood of transpiring. Monty Hall problem is another similar puzzle which will help you understand the basic idea behind probability.
Number sense is basically an intuitive understanding of numbers, their magnitude, their relationships, and how they are affected by operations. Although no one, except few gifted people like Ramanujan, is born with number sense it can be developed. And develop you must. If you want to be an above average investor, understanding the language of numbers and developing a sense of probability is supremely important.
For an illiterate, it’s almost impossible to survive in the modern world. But innumeracy is taken for granted. Unfortunately, innumeracy is far more widespread and socially accepted than illiteracy. Which means the most common tool for fooling the investors is numbers. Obfuscating the truth, misleading and misdirecting is much easier with numbers than words.
In today’s world, ideas like probability are not just for mathematicians anymore. One who doesn’t have the elementary knowledge of numbers and probability is pretty much like the proverbial one-legged man in a butt kicking contest.[/show_to] [hide_from accesslevel=’almanack’]
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