The Kelly Criterion: A Framework for Making Favorable Bets in Investing
May the odds be ever in your favor.
“Move only when you have an advantage. It’s very basic. You have to understand the odds and have the discipline to bet only when the odds are in your favor.”
- Charlie Munger
Ultimately, investing is about betting on the future price of a stock. As the late Charlie Munger alludes to, it is our job as investors to make favorable bets, even though they don’t always lead to favorable outcomes.
In this post, I’ll introduce you to the Kelly criterion, a framework for determining if a bet is favorable. I hope you’ll find it helpful.
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The Kelly Criterion
The Kelly criterion is a bet sizing formula that maximizes long-expected growth rate. The fraction of your wealth to bet—known as the Kelly fraction—is
f = p - (1 - p)/b where
p is the probability of winning, and
b is the odds of the bet, e.g., b=1 in a double-or-nothing bet.
Now, the Kelly criterion as described above is only applicable to binary bets where the bettor either wins or loses a fixed percentage of their bet. Still, I believe it helps provides useful insights into how to think about investing. After all, investing is nothing but a bet on future returns!
A Few Examples
First, consider a sure bet with double-or-nothing odds, i.e., p = 1 and b = 1. In this case, f = 1 - (1-1)/1 = 1. In other words, you should go all in on a sure bet, as you might have expected.
Next, consider a double-or-nothing bet with a 50% of winning, i.e., p = 0.5 and b = 1. In this case the Kelly fraction f = 0.5 - (1-0.5)/1 = 0, i.e., you should bet nothing and walk away.
The takeaway? When taking a low(er) probability bet, you should demand better odds. For instance, if instead odds b = 2, then f = 0.5 - (1-0.5)/2 = 0.25. In other words, you should be willing to bet up to 25% of your net worth on this bet.
Now consider a high odds bet with b = 9, i.e. you invest one dollar for the chance of getting ten in return. In this case, you should be willing to take the bet if there’s a greater than 10% chance of getting ten in return.
Good investments in start-ups or turn-arounds tend to be based on the potential of a high return, i.e., high odds, whereas investments in incumbents tend to focus on high predictability, i.e., high win rates.
The Kelly criterion helps us understand why many successful investing strategies can co-exist. Good investments in start-ups or turn-arounds tend to be based on the potential of a high return, i.e., high odds, whereas investments in incumbents tend to focus on high predictability, i.e., high win rates.
Favorable Bets ≠ Sure Bets
As the examples show, favorable bets do not make sure bets. For instance, when offered the bet to make triple your money (b=2) with a 50% chance of winning (p=0.5), you should be willing to wager 25% of your wealth on it if you want to maximize your growth-rate. However, statistically, you will still lose half the time. This has some very important implications for how we view the investment process, namely that the quality of an investment decision cannot be judged solely by its outcome.
The quality of an investment decision cannot be judged solely by its outcome.
Nassim Taleb illustrates this perfectly with a simple thought experiment published in The New Yorker in April 2002.
“Suppose that there were ten thousand investment managers out there — which is not an outlandish number — and that every year, entirely by chance, half of them made money and half of them lost money. And suppose that every year the losers were tossed out, and the game replayed with those who remained. At the end of five years, there would be three hundred and thirteen people who had made money in every one of those years, and after ten years there would be nine people who had made money every single year in a row — all out of pure luck.”
It is possible to make a good investment decision but have an unfavorable outcome. Similarly, it is possible to make bad investment decisions but have a favorable outcome. As investors, all we can do is focus on the quality of decision making process to ensure, that, on average, we are making favorable bets.
Try It Out
The next time you consider an investment, try to ask yourself the following questions.
How sure are you of making money on the investment over, say, 3, 5, or 10 years?
How much money do you expect to make?
What is an appropriate bet size?
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