Pabrai Week: The Kelly Formula
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
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, 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.
Also Filed Under "How to Think About Investing"
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Allen
Oct 4th, 2007
47 comments
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Joe Ponzio
Oct 4th, 2007
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Allen
Oct 4th, 2007
47 comments
I'm aware this is a potentially dangerous practice, as one will tend to put a number to justify the conclusions of the analysis, but I was doing a valuation of a corporation that many of the investing "gurus" (including Warren Buffett) seem to be adding to their portfolios, but I can't see why they would believe the business was undervalued according to the financials.
The only thing I can tell is that the last year's free cash flow appears to be an anomaly in that it is much different than the previous 9 years' positive free cash flows (it's negative, and almost 13x worse than the positive FCF in 2005).
If you assume a more "reasonable" FCF in 2006, you get a valuation that is one and a half times the current market price.
Obvoiusly, you need sound reasoning to back up your feeling that last year's performance was an anomaly, but this seems highly subjective. What do you think? Should good investors substitute numbers in their valuations when they believe them to be exceptions or one-time occurences?
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Joe Ponzio
Oct 6th, 2007
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Everything we do is a guess of the future. We look at the past for consistency. Substitute to your heart's content. Just make sure that your numbers are reasonable and that your reasoning is sound.
Check out Do The Math In Your Head. It is all about false precision and guesstimating.
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Tim
Oct 7th, 2007
1 comment
I think Parbrai in his book uses it as a general illustration of the virtues of concentrating rather than saying it is mathematically correct.
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Robert Crawford
Oct 8th, 2007
24 comments
http://en.wikipedia.org/wiki/John_Larry_Kelly,_Jr
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Joe Ponzio
Oct 8th, 2007
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Kevin Maki
Oct 10th, 2007
1 comment
Of course, as other posters have also pointed out, one's expected value (or edge) can only be estimated and is not known precisely for a common stock investment. This is another reason to use a conservative fraction of the Kelly number for each purchase.
Fortune's Formula by William Poundstone is an excellent book, which I recommend to anyone interested in investing, trading or playing games of chance with a positive expectation.
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Joe Ponzio
Oct 10th, 2007
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Using a percentage (like 25% of bankroll) will ensure you never go broke because it is a percentage of total bankroll - not starting bankroll. If you invest $250 from a $1,000 bankroll, and you lose it all, you'll have $750 left. Your next bet would be 25% of $750 - or $187.50.
The smaller your bankroll gets, the less you bet on a dollar basis (even though the percentage remains constant). Eventually, if you get down to a $0.01 bankroll, you have to stop betting because you can't divide it any more.
Of course, that assumes you place one bet each time and wait for the result. If you bet 25% of your bankroll on 4 bets - all at the same time - you can lose it all. Otherwise, if you lost 25% at a time - on a $1,000 bankroll - you'd have to stop betting when you got down to $0.03 some 38 bets later.
Even still, as you have said, the Kelly Formula is a guide - not an exact science - in the world of investing. Then again, there is no exact science in investing.
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Max Olson
Oct 11th, 2007
2 comments
I think Tim was trying to say that in order for it to work in the stock market, one must use the REAL Kelly Formula, not the simplified Edge/Odds version. The Edge/Odds works with simple bets like a coin toss or horse race (as shown in Fortune's Formula). This is because it requires two things: 1) There are only two outcomes - "win" and "lose"; and 2) In the losing outcome, 100% of your bet is lost. So, you really couldn't use it for stock situations that have many possible outcomes with varying gains/losses.
Also - the Kelly Formula used for gambling assumes you are only placing one bet at a time, not a portfolio of diversified bets. By using the actual Kelly Criterion (maximizing the logarithm of the portfolio as a whole) you can see what fraction of the bankroll should go into many stocks with many outcomes. I think the reason the Edge/Odds formula is seen in Fortune's Formula and Pabrai's book (and this blog) is because it's much easier to understand and gets the very basic concept right.
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Chungst
Oct 21st, 2007
10 comments
Sorry, but that would fail the logic test. When you make a bet in your portfolio using the Kelly Criterion, how does the Kelly formula know if you are making a single bet itself or a single bet consisting of a "basket" of securities?
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Jeff
Oct 4th, 2008
12 comments
Pabrai also mentions as does Poundstone in "Fortune's Formula" that Edward Thorp published a paper "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." Given Thorp's investing record as mentioned in "Fortune's Formula" this should prove to be a interesting read for those looking to further dig into the subject. Thorp was a spectacularly successful investor with very low volatility of returns. I have not read this paper, but I might this afternoon.
Futhermore, applying the Kelly Criterion either using the website mentioned, Excel, or numerical methods (science phrase for pen, paper, and calculator), you gain some further incite as to why the current breakdown in the commercial paper markets is occurring. With such low potential returns and such high potential losses, any investor whose probability of default on their commerical paper deviates from 0% should sprint to the door. This is just what is happening. Fear is powerful.
Regarding portfolio construction and the Kelly Criterion. Theoretically, any investment decision that you make would have the construciton of a Kelly bet. I think that includes working from a top down or bottom up perspective on your investments. The decision outcomes generally always include a comparison of a risky investment to the certainty of cash. See past inflation here if you will. Buying a house? Benefits of home ownership versus certainty of cash. Allocating assets to bonds, stocks, or real estate versus cash. When you start constructing an equity portfolio, you are weighing probabilities of cash versus the expected outcomes for your equity positions. I'm pretty sure you could rollup your expectations from individual risky assets selections into to an bankroll optimization for your entire asset base. You would do this instead of using Markowitz's mean-varriance optimization. I think Poundstone talks about this.
Finally on overbetting the Kelly, there's a great graph in "Fortune's Formula" that shows the pitfall of overbetting the Kelly. The graph shows the geometric mean (what the Kelly maximizes) versus the proportional size of the Kelly bet. I don't have my copy of the book handy but here is what I remember.
The parabolic shaped graphed with the wide end pointing down. The geometric mean is increasing approaching a full Kelly bet where it is maximized. Past a full Kelly, the geometric mean starts decreasing. At twice the Kelly bet, the geometric mean is zero and beyond 2-Kelly it is negative.
Given the fuzziness of estimating probabilities (especially in investments) input into the Kelly Criterion, taking a full Kelly bet could actually be a 2-Kelly bet or more fairly innocently. Thus you would always want to take less then a full Kelly bet. This does not mean that you wouldn't use leverage when applying the Kelly Criterion.
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Francis CUIGNIEZ
Feb 20th, 2009
1 comment
Just curious, but how can you apply Kelly-bet system if you follow, say, all the S&P500 stocks with an average of 2 in-and-out's a day. Now I use 5.000 USD as fixed bet (paper trading .... for the moment)
Bye,
Francis.
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Your Name
Feb 9th, 2010