What Investing and Gambling Have in Common, Part 1

Professor Kelly's paper offers the chance to look at some investment concepts from a different perspective

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Nicola Guida
May 21, 2020
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Introduction

A few days ago, I rediscovered a very interesting paper written by J. L. Kelly, who in the fifties was an associate of professor Claude Shannon (the father of Information Theory) at the famous Bell Labs (later AT&T Bell Labs).

The paper is titled A New Interpretation of Information Rate and was released on March 21, 1956. Let's take a look at the introduction:

"If the input symbols to a communication channel represent the outcomes of a chance event on which bets are available at odds consistent with their probabilities (i.e., fair odds), a gambler can use the knowledge given him by the received symbols to cause his money to grow exponentially. The maximum exponential rate of growth of the gamblers capital is equal to the rate of transmission of information over the channel."

While the intention of Professor Kelly was not that of delving into the basics of gambling (indeed he only wanted to find an alternative scenario for which the transmission rate was significant even in absence of any coding on the transmission channel), I think that the conclusions of this study can be used to (mathematically) confirm some very important investing concepts.

What gambling and investing have in common

Before looking at the paper content, I'll explain the meaning of the title I used for this article.

As Value Investors, we can be annoyed (and sometimes even offended) by whoever tries to associate our investing endeavors to gambling. A gambler (or a speculator) is in our mind someone who intends to earn money trying to guess the outcome of a probabilistic event, and the guess is not an educated one but rather one based on hope or other arcane reasons. Investing on the basis of technical analysis, or betting at a race track, are some examples of such a speculative attitude.

I think that investors and gamblers do have something in common, and that is the act of guessing or trying to imagine the outcome of a chance event or series of events.

We can name it as we prefer, but every investor must deal with uncertainty and therefore with the need of guessing. The difference only lies in the methodology used to produce that guess.

A Value Investor simply tries to minimize the probability of capital loss and to maximize that of capital appreciation. Our biggest effort is that of increasing the probability of success by replacing (as much as possible) the uncertainty with an educated guess.

Game results sent on a private channel

Lets now go back to Kelly's paper. Here's how he introduces the problem:

"Let us consider a communication channel which is used to transmit the results of a chance situation before those results become common knowledge, so that a gambler may still place bets at the original odds."

Let's also make the hypothesis of a symmetric bet: that means that in case of loss, we lose the whole capital we bet, and if we win, we'll double our money.

In the trivial case of a perfect transmission channel, we could simply bet all our capital and fully reinvest it all the time, so that after N bets our initial capital would be multiplied by two to the power of N. For example, if the initial capital is $10 and we place N=10 bets, our final capital would be $10 * 2^10 = $10240.

We can also notice that in this case, as there's no way we can lose the bet, the speed of growth of our capital is equal to the transmission rate, that is to the number of game results we receive within a certain amount of time (e.g. once per week).

The paper then continues with a more interesting scenario:

"Consider the case now of a noisy binary channel, where each transmitted symbol has probability, p, of error and q of correct transmission. Now the gambler could still bet his entire capital each time, and, in fact, this would maximize the expected value of his capital (...) This would be little comfort, however, since when N was large he would be probably be broke and, in fact, would be broke with probability one if he continued indefinitely."

If betting the entire capital each time is a sure recipe for failure, we must necessarily be more conservative and invest only a part of it. The main question the scientist poses is: which is the fraction of our budget we should reinvest each time in order to maximize the exponential rate of growth?

Kelly mathematically demonstrated that, in order to maximize the exponential rate of growth (that is, the compounding power of our investment outcomes), the fraction of the budget we should invest is equal to L = q p, with L being the above mentioned fraction and q and p being the probabilities of a positive and a negative outcome, respectively.

For example, we could have q=0.6 (60% chance of positive results) and p=0.4 (40% chance of negative results), so L=0.2, which means that, in the case of a 60/40 distribution (a positive outcome being here more probable than a negative one), we should invest 20% of our budget if we want our capital to grow as fast as possible. If the distribution is more favorable, lets say 80/20, we should invest 60% of it.

How is Kelly's thesis linked to the investment concepts we're already familiar with? Here are a few considerations directly derived from his simple formula:

  1. Never invest all your money on a single idea (unless you are absolutely certain about it, but that's usually not the case with investing).
  2. Allocate your capital accordingly with the estimated probabilities of success and failure. Hope, random guesses and speculation in general are not wise approaches and do not produce good results in the long run.
  3. The fraction of our budget to invest depends on the difference between the probabilities of success and failure: the bigger the difference, the greater the amount to invest. This means that we should have the courage of allocating greater amounts of capital to our best ideas.

Conclusion

Kelly's paper practically demonstrates that we can take advantage of a specific situation (in our case, an investment scenario) if we have an edge. In investing, having an edge is always the result of a sound investment thesis.

Charlie Munger (Trades, Portfolio) once said:

"The wise ones bet heavily when the world offers them that opportunity. They bet big when they have the odds. And the rest of the time, they dont. Its just that simple."

In the next article on this topic, I will extend the discussion to the case of asymmetric bets, describing what is commonly known as the Kelly Formula (which is a generalization of the simple one described in this article), how to practically use it and how this and other studies have been used in the past to profit off of both asymmetric outcomes and gain/loss probabilities.

Read more here:

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