**Introduction **

In my previous discussion on the similarities between gambling and investing, I reviewed J. L. Kelly´s Bell Labs paper, “A New Interpretation of Information Rate,” specifically focusing on how his conclusions can be linked to very simple and effective investing concepts.

I will now explore some of the impacts that paper had on the scientific community and how the related know-how was used to profit both in probability-based games and in the stock market.

**Ed Thorp**

One of the most active researchers in this area was Edward O. Thorp, who has been a math professor, writer, hedge fund manager and blackjack researcher.

Thorp said he was introduced to Kelly' paper by Claude Shannon at M.I.T. in 1960.

At the time, Thorp was looking for the optimal way to play Blackjack and he had already created a method called "card counting" to address it, but after reading Kelly's work, he built onto those conclusions and integrated them into his theory. He then became famous for systematically beating the "house" by playing Blackjack at several casinos. The related theory is explained in his 1962 book, "Beat the Dealer."

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Thorp also wrote a paper on the topic called “The Kelly Criterion in Blackjack, Sports Betting and the Stock Market,” which was published in 1997.

Let's see if the content of that paper can lead us to gain some more investing insights.

**The optimal fraction **

In the introduction, Thorp explains the relationship between gambling and investing:

"The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, i.e., how much to bet. In the stock market (more inclusively, the securities markets) the problem is similar but more complex. The gambler, who is now an 'investor,' looks for 'excess risk adjusted return.'”

In the paragraph "Coin Tossing", Thorp carries out a mathematical function study, which is intended to graphically clarify the different possibilities a gambler has with regard to the size of the optimal fraction to bet (Kelly fraction).

As we can see from the graph, the Kelly fraction (f*) is the “optimal” value that maximizes the expected value of the capital growth rate, or G(f). In the first part of this series, it was established that, for symmetrical bets, this value is equal to the difference between the win and loss probabilities.

By looking at the graph, we can also extrapolate some additional important aspects:

- Using a fraction which is different from Kelly's can lead to similar results, but suboptimal ones: Indeed, in this case, the capital will grow at a lower speed (due to the lower rate of growth). This holds if the used fraction is lower than a certain finite value, fc. In short, stick with the calculated probabilities (or, if you want, with your investment thesis).
- If f > fc, then growth is negative, which means that our capital will ultimately shrink and head toward 0. Just to simplify, this shows that we should never over-bet (or over-invest).

In real life, both gamblers and investors using the Kelly formula are usually not comfortable with the optimal fraction and reduce it a bit. This makes sense, not because there's a better fraction value, but because in most cases (and especially in investing) we're not able to precisely calculate the probabilities of success and failure. So we want a “probabilistic” margin of safety: it's better to grow our capital slower than (unknowingly) drifting toward over-betting and losing money.

**The Kelly criterion for asymmetrical bets**

As anticipated in the previous installment, we'll now pass from the hypothesis of a perfectly symmetrical bet to an asymmetrical one.

Let´s also stick with a binary outcomes scenario: this means that, as before, we only have two probabilities: one related to a favorable outcome (p) and a loss one (q = 1 – p).

Here's how Thorp introduces the asymmetrical bet case:

"The Kelly criterion can easily be extended to uneven payoff games. Suppose Player A wins b units for every unit wager. Further, suppose that on each trial the win probability p > 0 and pb − q > 0 so the game is advantageous to Player A."

The factor p*b – q is nothing more than the probabilistic outcome of our investment. So that number must be positive to convince us to invest, as we intend to rule out those situations in which we don't have an edge. We must thus have p*b – q = 0, or p*b > q.

This also means that, for this more general case, simply having a success probability greater than that of failure (p>q) is not enough to place a favorable bet as it is for the symmetrical bet case.

Now we have an additional variable, b, the win payoff (or simply, the odds), which contributes to the expected outcome.

Finally, here's what is commonly known as the Kelly formula:

f* = (p*b - q)/b

Where, p is the win probability, q is the loss probability and b is the win payoff (how much you win, if you win, in wager units).

Please observe that when the win payoff is b=1 (which means that if we win, we'll double our capital), the formula simply reduces itself to the one seen in the previous article.

In order to show how win-loss probabilities and different levels of payoffs combine together into the formula to generate the Kelly fraction optimal value, I created the following table:

As we can see, the formula gives out a positive number only for a subset of probabilities-payoffs couples. The negative numbers don't make sense even if they come from applying the formula. It depends on the fact that the expected outcome is also negative: this simply means that in those cases, we should not invest at all.

Another important observation is that win probability and payoff can compensate each other: e.g., in the case of a symmetrical bet (p=0.5), a payoff of 50% (b=0.5) is not enough to have a (probabilistic) positive outcome, but if we raise the payoff from 50% to 200% the Kelly formula suggests to invest 25% of our budget.

When both the win probability and the payoff are on the high side, the formula suggests to invest a substantial fraction of our budget.

On a side note (even if not strictly needed for our discussion), we could be interested in calculating the Kelly fraction for more than two outcomes, and consequently multiple probabilities (this is a generalization of our binary outcomes scenario). Unfortunately, there's no simple and linear formula that can be used in the case of more than two outcomes, but Thorp's paper can point investors in the right direction if they are curious about which approach to follow.

**Investment insights **

As we did in the first part of the series, let's now try to extract some investment lessons from the Kelly formula:

- The fraction of capital budget to invest does not only depend on the win-loss probabilities, but also on the payoffs associated to them.
- The probability (p) to be associated to the favorable (and consequently, to the negative) scenario can be calculated by means of a thoughtful investment analysis. In order for these probabilities to be more heavily skewed toward the favorable outcome, we need to be able to choose a good company. Such a company will preferably have a durable and solid moat, an honest and competent management team, a high return on capital and good growth prospects.
- The win payoff depends on how much we earn or lose in each scenario, which in turn depends on the difference between the company's intrinsic value and our purchase price. This is strictly related to the concept of margin of safety.

**Conclusion**

In a nutshell, in order to maximize the growth rate of our capital, we must find good companies selling for a price that allows for a good margin of safety.

I'm sure that this sounds now much more familiar than the cold formulas and numbers. The investment concepts that can be extrapolated from the Kelly formula reminded me of Joel Greenblatt (Trades, Portfolio)'s Magic Formula approach and its theoretical substrate.

In Greenblatt´s words:

"If you just stick to buying good companies (ones that have a high return on capital) and to buying those companies only at bargain prices (at prices that give you a high earnings yield), you can end up systematically buying many of the good companies that crazy Mr. Market has decided to literally give away."

My conclusion is that basic value investing principles make sense even if we look at them from different perspectives, because they are common-sense concepts.

Read more here:

- What Investing and Gambling Have in Common, Part 1
- Garrett Motion: A Margin of Safety and a Turbo-Catalyst
- Why Berkshire Hathaway's Future Is Safe

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### About the author:

*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.*

**Nicola Guida**

Thomas Macpherson- 1 month agoGreat stuff Nicola. I think for many investors the idea of applying the Kelly formula is beyond their means. But with a little effort and some focus, it can give investors some very interesting insights into their portfolio selection and oversight. Thanks again for a great article.. Best - Tom