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Joe Ponzio

Pabrai Week: The Kelly Formula

October 09, 2007

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.


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:

Pabrai's Kelly Formula

The actual formula (for purists) is:

The Kelly Formula

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.

The Kelly Formula on a 2:1 coin flip

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.

Originally published at Joe Ponzio's F Wall Street.

About the author:

Joe Ponzio
Charlie Tian, Ph.D. - Founder of GuruFocus. You can now order his book Invest Like a Guru on Amazon.

Rating: 2.7/5 (31 votes)


David Pinsen
David Pinsen - 10 years ago    Report SPAM
I read and liked The Dhando Investor, and found the discussion of the Kelly Formula interesting, but Pabrai doesn't really follow it himself. His rule of thumb seems to be to put approximately 10% of his assets in each investment.
JJINVEST - 10 years ago    Report SPAM
I liked the discussion on Kelly formula. The spirit is not to follow it by heart but to understand and allow yourself the priviledge to concentrate in 10 holdings or less.
Buffetteer17 premium member - 10 years ago
Many investors find that betting the Kelly fraction has more potential downside than they would like. The probability that your bankroll will be reduced to a fraction x of the starting value at some point is x. For example, there is a 90% probability that you will be down to 90% of your original stake, a 50% probability that you will be down to 50%, etc. If you persist, you still maximize your growth rate compared to any other strategy, but many are not comfortable with a high probability of losing half their bankroll at some time during the process.

For investors, there is the "half Kelly" bet, which is simply half the bet size that the Kelly formula produces. The "half Kelly" bet has two notable advantages over the "full Kelly."

1) Your expected growth rate of your bankroll is 3/4 that of the "full Kelly" bet.

2) Your expected temporary downside follows a cube law. The probability that your stake will be x% of your original bankroll at some point is x3. So your chance of being down to 90% is 0.93, or 73%, your chance of being down to 50% is 0.5 cubed, or 12.5%.

Smaller fractions of the "full Kelly" become even safer, with the degree of safety from temporary losses growing much faster than the expected portfolio growth rate decreases.

Be extremely careful not to bet more than a "full Kelly." Not only does your chance of temporary loss increase greatly, but your expected return also decreases. In fact, if you bet too much more than a "full Kelly," your expected growth rate passes through zero and becomes negative. Take for example the "win 2 lose 1" example in the original post, where the optimal bet is 25% of your bankroll. The expected growth rate of your portfolio is about 6.1% per turn. If you instead bet 50% of your bankroll, your expect growth rate becomes 0, and if you bet 75% of your bankroll, your expected growth rate changes to -21% per turn. Yes, it is quite possible to turn a favorable bet into an unfavorable one by overconcentration. Since one generally has to guess at the probabilities and returns in real life, it is better to stick to "half Kelly" or smaller bets. This still does not precude very concentrated portfolios, where the chance of each individual investment to work out is very high.

source: "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market," Ed Thorp, 1997, paper presented at The 10th International Conference on Garmbling and Risk Taking.

Ccyork - 10 years ago    Report SPAM
i ain't no scientist or nuthin', but you folks should know by now how i like to poke holes in other people's logic an such

first off, how do you calculate any of those probability numbers you stick into the kelly formula? you're just guessin' at best

now supposin' you really could get those numbers accurate and everything, here's another problem: businesses just ain't static like coin flips and all. your odds of winnin' are a-changin' with every passing month. that messes the kelly all up

but iffin' you could get accurate probabilities and businesses were static, it would seem like placin' that 95% kelly bet would makes lots 'o sense. but here's why it just ain't practical: kelly works better the more iterations you do. or in smaller words, the more bets that you place, the more chances you have to win

a man can flip a coin 1,000 times in 'bout seven minutes. but you place that 95% kelly bet on a stock, and i guarantees you that crazy mr. market is gonna make you wait three-and-half years to see how your coin flip turns out. you ain't gonna make much headway that-a-ways

like i said, i ain't no scientist, but when in comes to applyin' it to investing, that kelly formula thing just don't make much sense to a simple feller like me
Billytickets - 10 years ago    Report SPAM
well put buffetteer. Being a person who plays head to head poker an average of 10 + hours a week and makes "hundreds" of capital allocation decisions outside the poker table buying memorbalia and tickets ( 23 buys and 11 sells in the last 6 days) each week I love the"THEORY" of the Kelly Criterion and its formula

its"weakness" with me is EVERY PERSON THINKS WHEN THEY BUY a STOCK the ODDS are in their FAVOR.

Why else would they buy?

EACH person needs to develop a "FORMULA of their own" which has passed the test of time.

WEB's formula has worked for 50 years now and some of our readers have earned 7 figures or more in their career allocating capital.

Writing my book made me a better investor because i had to"quantify" things that I used to just estimate( Like the WEB story that you dont need to know exactly"how tall" someone is to know that they are really tall)

My "filters" and formula were "tuned up" after studying WEB's purchases over a 30 year time frame .

Projecting the growth of earnings in a business over the next 5 years is MUCH EASIER for Hershey or JNJ that are more than 100 years old than it is for GOOGLE.

That statement DOES NOT mean GOOGle is not a GREAT INVESTMENT or a GREAT business .

It means that we don't know how it would fare during unemployment rates of 22% or Fed Funds rates of 16% like we know about JNJ or Hershey

Buffett defines risk as 1) misjudging the future earnings of the business and 2) time frame you are investing ( obviously in a short term like a week your investment will NOT necessarily be correlated with the earnings of the business ,over a 15 year period an 8% increase in EPS compounded over that time will more than likely be reflected in the stock price during some time in the 15 year span.

EVERY PERSON NEEDS to find THEIR OWN FORMULAS and FILTERS to pick stocks and make their capital allocation decisions

I worked backwards "studying" WEB's purchases and devloping a set of rules.IRONICALLY many now are taking my formulas and filters and "twaeking them" like i have done to WEB's and like EACH "great chef' adding their own PERSONAL TWIST.

I enjoy listening to everyone's feedback and "testing" the ideas I like best to see if they can ( or should be) added to my arsenal.

I like Mungers idea best . when you think the odds favor you BET BIG when they dont.DON"T bet.

Its really that simple folks.peace

Vooch - 10 years ago    Report SPAM
You calculate your own system's accuracy (eg. 80%)

0.80 times 2 = 1.60

1.60 subtract 1 = 60% bet size.

We've learned from Buffett Partnership Letters to not put more than 40% into the best idea.

Kelly Formula is only one aspect of bet sizing. Decision Trees, Expected Values, and Margin of Safety also play roles.

- Vooch

Billytickets - 10 years ago    Report SPAM
As usual great post vooch. CCYork my sentiments EXACTLY

I would really like to talk to WEB now and find out what he believes about the 40% rule that he talked about 40 years ago

In a company like AXP or KO where their is basically one product I believe it has alot of validity

But in companies like BRK KFT P&G and the OLD Altria (before the spin off) and JNJ where the "income streams" come from MULTIPLE "unrelated" brand names I am curious to what he would say in 2007

Many "multimillionaires" who just bought BRK have fared well and it is hard to argue that BRK itself is NOT as diversified as a portfolio of 10 stocks

Take this scenario. A comapny which has over 70 brand names OVER 100 million and 12 overa billion is selling at 8 times eranings and paying you a dividend yield of over 10% when a HELOC is charging you 5% and sells a product that cost a penny for 3 dollars to 'addicts" and the FOOD business and beer business itself is WORTH more than the purchase price meaning you get domestic and INTERNATIONAL tobacco for free

WEb put 40% of his partnership into AXP duringsalad Oil as you well know which arguably didnt have the diversification div yield or valuable brand names that MO had when the div yield was over 10% and PE was 8

This is a great question to ask at the annual meeting.peace

Kfall - 10 years ago    Report SPAM

I love this site, and the intellect it harbors. I think it's a privilege that this info is basically free. Being a novice, I'd like to know, what 3 aspects of a stock do you focus on, and what figures would need to appear, to make it a no-brainer. I realize alot of you do vast amounts of research, just answer the question as presented. What makes a stock jump out at you?
Musto - 10 years ago    Report SPAM

to make your life even easier I'll reduce your 3 aspects into 2.

1) If you had the money, would you be excited to own the company in whole?

if so,

2) Is the current price cheap enough to make the purchase right now?

If the answers are yes to both of those questions, then buy the stock and enjoy the ride.

Everything else is just details.

Hameed - 10 years ago    Report SPAM
Great reply musto, however your answers, especially the second one will make a ton of sense to someone who has figured out how to value companies. However a novice will have no idea how to figure out this "2) Is the current price cheap enough to make the purchase right now?"
Billytickets - 10 years ago    Report SPAM
My book is full of my filters and formulas which decide WHAT and WHEN I will buy something. There is almost half the book dedicated to that. Not trying to advertise but my point is answering the question is at least 10 or 20 pages for me personally maybe others have a different view.peace
Musto - 10 years ago    Report SPAM
hameed Wrote:


> to figure out this "2) Is the current price cheap

> enough to make the purchase right now?"

This is what the masters mean when they talk about the "Margin of Safety".

In theory this could be complicated, such as figuring out the future growth potential

and comparing it to the current earnings yield, but in practise the price should be screaming cheap so that you don't need to resort to tools such as DCF etc.

The original question was asking for shortcuts for security analysis. In real life

there is no shortcut to insight, and depending on your skills, a lot of that comes with much practise,

and learning from people wiser than ourselves.

Billytickets - 10 years ago    Report SPAM
yes musto's last post was excellent as usual. we are ALL shaped by our past experiences liek it or not ,keep soaking up the "free info".peace
Armeetofo - 10 years ago    Report SPAM
musto Wrote:


> kfall,

> to make your life even easier I'll reduce your 3

> aspects into 2.


> 1) If you had the money, would you be excited to

> own the company in whole?

> if so,

> 2) Is the current price cheap enough to make the

> purchase right now?


> If the answers are yes to both of those questions,

> then buy the stock and enjoy the ride.


> Everything else is just details.


i would like to boil musto's 2 aspects down to 1 sentence:

buy the great business at the right (dirt cheap) price.

is that simple?!

Armeetofo - 10 years ago    Report SPAM
"free info".

due to free, i don't think people take it seriously and benefit from it.

Cimacircle - 10 years ago    Report SPAM
billytickets --

to what book do you refer?
Robinsonfunds - 10 years ago    Report SPAM
the kelly forumula will never be as accurate when dealing with outcomes such as business performance. The inherant flaw with the kelly formula is the probabilities that one assigns a percentage. Coin flips and other semi-absolute events are one thing, but a businesses future is so intricate and complex as it has many, many more factors that shape the outcome, that properly guaging and predicting the the outcomes is very hard.

I suggest to be more conservative with your kelly percentages, err on the side of caution with the outcome predictions. the 1/2 kelly is one conservative measure, and another is lowering the prediction percentage of each possible outcome to skew more towards the "worst-case" scenario of the busineses future.

I am by no means saying the kelly formula is a bad or flawed idea. The kelly formula is a great tool for capital allocation, but I would suggest being EXTRA conservative on the "rosey" future scenarios, this will lower the percentage of bankroll in any 1 idea naturally.

Billytickets - 10 years ago    Report SPAM
cimacircle : it is the book consume Consume and Consume More which gives a dupilcable formula for security analysis and shows you where ,How to get the money to invest if need be why what and when to buy astock. read these reviews if intersted http://www.gurufocus.com/forum/read.php?2,9805 and email me at [email protected] with any questions.

Robinsofunds: I agree with you . However companies with long track records are "easier" ( but not easy) to project their future earnings.When i made my huge bet on MO in 1999 and 2000 I fely KFt and Miller was worth what i paid for the whole company getting tobacco for free.My specific filters and Formulas" eliminate all but a handful of companies. There is NO way to calculate the exact probablity of a stocks increase like u can do with blackjack poker or dice. great posts everyone.peace

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