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Nicola Guida
Nicola Guida
Articles (19)  | Author's Website |

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

May 21, 2020


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 gambler’s 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

Let´s 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, let´s 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.


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 don’t. It’s 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.

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About the author:

Nicola Guida
I'm a Software Engineer with a big passion for Value Investing. I love looking for undervalued companies both to feed my investment pipeline and to write articles in order to share my investment thoughts.

Visit Nicola Guida's Website

Rating: 5.0/5 (9 votes)



Thomas Macpherson
Thomas Macpherson premium member - 2 weeks ago

Great article Nicola. I wrote abut the Kelly formula several years ago and how investors can use it to help make better allocation decisions. Great stuff. Thanks! - Tom

Nicola Guida
Nicola Guida premium member - 2 weeks ago

Hi Tom, thank you! I´m glad you liked it. :) I am about to read your article as well (I think I missed it because of my random reading habits..).

Suveer premium member - 1 week ago

Nicola Thanks for your article - I had read about Kelly fromula in a book earlier but had not gotten into the details - instinctively i do allocate a higher amount to my best ideas - now that you have tickeld my curosity - would love to dive in deeper and make a formula for a more disciplined approach in bet size based on probablities - thanks

Nicola Guida
Nicola Guida premium member - 1 week ago

Hello Suveer, thank you for your comment. I´m happy to know that the article raised your motivation to improve your investing process. Please note that, even if prof. Kelly´s conclusions are mathematically precise, indeed he demonstrated that there´s no better way to grow your capital once you know the probabilities of success and failure, figuring out that probabilities is the most difficult thing (that´s why the concept of margin of safety is so important). It´s similar to what happens with the DCF model: in theory that is the best way to calculate a company´s intrinsic value but the numbers you use as inputs are very difficult to predict. BRs, Nicola

Suveer premium member - 1 week ago

Indeed - my take has been that all the stocks that you want to own tend to be popular and have virtually no margin of safety ( guru focus calculation ) - I feel that the game then depends upon relative valuation etc - it's like buying art - if you want to own a Picasso or Rembrandt then be prepared to pay top $ - of course look at all the other parameter of quality / profitability / efficient use of capital etc ... the risk you run is that it will market perform and not outperform !
who said investing is easy !!

Nicola Guida
Nicola Guida premium member - 1 week ago

Suveer, I would be careful about relative valuations: e.g. when approaching a bubble you could conclude that a company is cheap even if it´s actually quite expensive if you look at its asset and/or cash flows. The high flyers have usually (but not always) a reason to fly so high, which is high growth and strong moats. So it´s up to you to give that growth a value (and how much) or being more conservative. But my advice is that to link the estimated value of a company to its cash flows, simply because that are the ones that will be distributed (either in the form of dividends or capital appreciation or both). We don´t know how much a Picasso will be worth 10 or 20 years for now (e.g. the same holds for gold), because the value in this case cannot be measured objectively, it is actually "in the eye of the beholder".

Suveer premium member - 1 week ago

Agree with everything you say !
who said inventing is easy - thanks for your input and comments - best of luck

Rschick13 - 1 week ago    Report SPAM

Good article! I'd almost equate value investing to gambling with an edge. If you spent time researching a team and had inside information that their best player was hurt so you bet against them, you are gambling but with a higher probability of winning than the normal house edge. When you buy a stock with great fundamentals you are not guaranteed a good result but you have information where the odds are in favor of you making money.

Nicola Guida
Nicola Guida premium member - 1 week ago

Hello Rschick13, thank your for your comment. I agree with you: having an edge is what triggers a real investor as opposed to pure greed. Sometimes, when irrationality prevails, you don't even need to have a special edge: you just have to do your homework and stay the course. In the next article I will show how success/failure probabilities and payoffs combine to determine the amount of invested capital. Stay tuned ;)

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