Mental Models

Law of Small Numbers and Investing

A random event, by definition, does not lend itself to explanation, but collection of random events do behave in highly regular fashion – Daniel Kahneman

Currently, I’m reading one of the gem of a book ‘Thinking Fast & Slow’ by noble prize winner Daniel Kahneman which is basically, as one of my friend say, an encyclopaedia in the field of psychology & behavioural finance. A lot of the contents in the CFA level 3 subject on behavioural finance has been taken up from the experiments about which Kahneman talks at length in this book. Do yourself a favour and read (and absorb) what he has to say on the subject.

(F)law of small numbers

In one of the chapters he goes at length to discuss what he calls ‘law of small numbers’

He cites a study of kidney cancer carried on in United States –

A study of the incidence of kidney cancer in the 3,141 counties of the United States reveals a remarkable pattern. The counties in which the incidence of kidney cancer is lowest are mostly rural, sparsely populated, and located in traditionally Republican states in the Midwest, the South, and the West. What do you make of this?

Your mind has been very active in the last few seconds.. You deliberately searched memory and formulated hypotheses. Some effort was involved.. You probably rejected the idea that Republican politics provide protection against kidney cancer. Very likely, you ended up focusing on the fact that the counties with low incidence of cancer are mostly rural. The witty statisticians Howard Wainer and Harris Zwerling, from whom I learned this example, commented, “It is both easy and tempting to infer that their low cancer rates are directly due to the clean living of the rural lifestyle—no air pollution, no water pollution, access to fresh food without additives.” This makes perfect sense.

Now consider the counties in which the incidence of kidney cancer is highest. These ailing counties tend to be mostly rural, sparsely populated, and located in traditionally Republican states in the Midwest, the South, and the West. Tongue-in-cheek, Wainer and Zwerling comment: “It is easy to infer that their high cancer rates might be directly due to the poverty of the rural lifestyle—no access to good medical care, a high-fat diet, and too much alcohol, too much tobacco.” Something is wrong, of course. The rural lifestyle cannot explain both very high and very low incidence of kidney cancer.

Interesting right? Before we answer this, let’s dive down to another, easier to understand case study –

Imagine a large urn filled with marbles. Half the marbles are red, half are white. Next, imagine a very patient person (or a robot) who blindly draws 4 marbles from the urn, records the number of red balls in the sample, throws the balls back into the urn, and then does it all again, many times. If you summarize the results, you will find that the outcome “2 red, 2 white” occurs (almost exactly) 6 times as often as the outcome “4 red” or “4 white.” This relationship is a mathematical fact. You can predict the outcome of repeated sampling from an urn just as confidently as you can predict what will happen if you hit an egg with a hammer. You cannot predict every detail of how the shell will shatter, but you can be sure of the general idea. There is a difference: the satisfying sense of causation that you experience when thinking of a hammer hitting an egg is altogether absent when you think about sampling.

A related statistical fact is relevant to the cancer example. From the same urn, two very patient marble counters take turns. Jack draws 4 marbles on each trial, Jill draws 7. They both record each time they observe a homogeneous sample—all white or all red. If they go on long enough, Jack will observe such extreme outcomes more often than Jill—by a factor of 8 (the expected percentages are 12.5% and 1.56%). Again, no hammer, no causation, but a mathematical fact: samples of 4 marbles yield extreme results more often than samples of 7 marbles do. Now imagine the population of the United States as marbles in a giant urn. Some marbles are marked KC, for kidney cancer. You draw samples of marbles and populate each county in turn. Rural samples are smaller than other samples. Just as in the game of Jack and Jill, extreme outcomes (very high and/or very low cancer rates) are most likely to be found in sparsely populated counties. This is all there is to the story.

We all have read about ‘law of large numbers’ somewhere or the other. It basically says that as the number of experiments (samples) increases, the actual ratio of outcomes will converge on the theoretical or expected ratio of outcomes. But it is the flip side i.e. the law of small numbers which gets lesser attention intuitively and unless that is understood, we have not truly grasp the former concept.

There are few things to internalise here:

  1. Large samples are more precise than small samples. (What constitutes large enough sample size is another discussion entirely. )
  2. Small samples yield extreme results more often than large sample does.
  3. We, as humans, are bad intuitive statisticians as the reasoning behind the finding of kidney cancer survey highlights. This has been the recurring theme across the topics covered in this book.

So what this particular concept has to do with investing?

 A lot I would say when it comes to evaluating businesses and making decisions.

Studying limited or recent history of a business:

In my experience, most of the participants in the market look at last 2-3 year operating history of a business. Then based on those numbers and adjusting for what company has to say they extrapolate and make their own estimates for the next year or two. This seems to be highly inadequate. Ideally, for analysing a business, we need to assess how it has performed over an entire business cycle which includes peaks and troughs. Depending upon the business, these cycles could take anywhere between 3-8 years orbiting across multitude of business conditions.

Even a five-year analysis of past numbers could be inadequate. Remember 2003-08 period? Everything was hunky-dory during this period and someone who thought those margins and growth rates could sustain made lot of bad bets in 2007-2008 period. This could have been avoided if one rather looked at numbers from say the year 2000. FY2000-03 was a painful time for the economy as a whole. So essentially, for most of the businesses out there, 2000-2008 would have covered substantial part of their entire business cycle.

Clearly, two years does not seem to be reasonable sample size.

Implications while evaluating smaller businesses:

Businesses which rely on one narrow / niche type of an activity accounting for bulk of their revenues and operating over smaller geographical area are bound to see higher level of extreme fluctuations in their business operations versus a bigger, more operationally & geographically diversified company. This partly explains why we see higher volatility in stock prices of small caps & midcaps over large caps leading to higher beta – something which ‘modern portfolio theory’ shuns.

This does not mean smaller businesses are bad. In fact, they happen to be an ideal hunting ground for spotting upon mis-priced securities from time to time (but definitely not all the time). Only thing is that we acknowledge the occurrence of such fluctuations (in business as well as market quotations) and prepare to take advantage of the same as when time favours and not run-like-hell when things turn bad temporarily.

As Buffett once said ‘look at market fluctuations as your friend rather than your enemy; profit from folly rather than participate in it.’

Adopting an incorrect time horizon in your investment decision making:

This is one of the worst things investor can do for himself. Thinking in terms of days, weeks or months is hazardous for your investing life. This, in essence, runs polar opposite to what compounding aims to achieve.

One is bound to see higher fluctuations in market price over shorter periods and hence frequent conversion from black to red and then, if he sticks along, black. As Michael Mauboussin notes in one of the chapters of his book ‘More than you know’, probability of making positive spread over one day is 50% while over one year it rises impressively to 72%. Over 10 years, it is about 100%. By trading frequently and focusing on daily price movement, we trade in a 100% probability event for a 50% one. How rational is this?

There is lot to learn from Kahneman and this piece focused on some of the commonly overlooked follies relating to smaller numbers and how it applies to investing. It is recommended to read this book and absorb what it aims to deliver. I aim to post couple of the articles, as when I get time, on some of these concepts and how they are relevant for us as investors.

Thanks for reading. Cheers!

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