Pabrai Week: The Kelly Formula

October 4, 2007 by Joe Ponzio

Some 50 years ago, John Larry Kelly came up with a formula to determine how much you should bet on a gamble or investment to optimize your bankroll. Now known as the Kelly Formula, the equation determines the optimal percentage of your cash to bet on a favorable bet.

Mohnish Pabrai talks about it. Pabrai applies it to some of Buffett’s past purchases. I guess we should take a look at it too. Heck, in this game, it pays to be a copycat.

What Is The Kelly Formula?

If you haven’t picked up a copy of Pabrai’s The Dhandho Investor, do it now. Pabrai explains it well. In essence, the Kelly Formula is a mathematical formula that is used to maximize the long-term growth rate of a series of repeated bets that have a positive expected value.

Huh?

The Kelly Formula basically figures out how much to bet if the odds are in your favor-in Vegas, in the stock market, in a coin flip…whatever. Pabrai simplifies the equation to:

The actual formula (for purists) is:

A Kelly Example

Let’s say you have $1,000 in cash and someone offers you 2-1 on a coin flip. That is, they’ll pay you $2 if it comes up heads; you’ll lose $1 if it comes up tails. The Kelly Formula will tell you how much you should bet on the coin flip to earn the maximum amount of money.

In this above coin flip, the Kelly Formula tells you that the maximum you should bet on any flip is 25% of your bankroll. Doing so will give you the maximum long-term growth with minimum downside.

The Kelly Guarantees (and Weaknesses)

Don’t fool yourself. There is no “perfect” system to avoid all loses. All we can do is minimize losses, maximize gains, and optimize bankrolls. The Kelly Formula insures that you’ll never lose everything; still, it doesn’t guarantee that you won’t lose sometimes.

You never want to overbet the Kelly Formula. That is, you never want to put more of your bankroll than the Formula suggests. In a moment, you’ll see how Pabrai puts that to work.

At any rate, investing is just like a coin flip offering favorable odds. On any given flip of the coin, you can lose money. Still, over the long term, if the odds are in your favor (as they are when you buy dollars for fifty cents), you’ll make money-good money. In short, the Kelly Formula helps maximize your return (though it does nothing for volatility, so you need to know how to think about stock prices).

Kelly Formula Applied To Investing

There is one major flaw with the Kelly Formula when applying it to investing in businesses when they are on sale: It would force you to put too much of your bankroll into one company.

When you patiently wait for dollars to sell for half off, you are waiting until the odds of winning are so large, and the odds of losing are so small, that you would end up putting 85% or more of your bankroll into one position.

In this video from Morningstar (link changed or removed by Morningstar), Pabrai asserts that the percentage would be even larger-upwards of 95%. (The Kelly Formula discussion starts 18 minutes into the video.)

Now, if you actually run the Kelly Formula on most value stocks…what the model will tell you is that you ought to put 97% of capital or 95% of capital, 95% of your bankroll into that bet.

How does that make sense?

…the odds of a loss are so low and the odds of a gain are so high.

So, Why Should We Care About The Kelly Formula?

What does the Kelly Formula do for us if we aren’t going to follow it by putting 95% of our bankroll into one company? For one, it helps us understand that it is okay to own just a few holdings-be it five or fifteen. (Pabrai started with ten but is possibly bumping it up to fifteen as his capital base swells)

Don’t focus on calculating the Kelly Formula for your investments and diversifying based on mathematical models. Rather, spend the time and energy finding “no-brainers”-investments that would be in that 95%-of-capital range. Then, buy the heck out of them.

A Note From Joe Ponzio

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