## Kelly Criterion for stock trading size

Posted by **Frugal** on April 16th, 2006

I’m sure some people know about Efficient Frontier, but I’m guessing that there are less investors that know about Kelly Criterion. So what is Kelly Criterion and who is Kelly? Kelly worked at AT&T, and published the original paper back in 1956. Its math is quite involved with communication and information theory, mostly dealing with probabilities. However, behind all the maths, there lies an astonishing result: by placing bet amounts according to Kelly Criterion (originally applied to horserace gambling), one can maximize the returns in the long term. Here is the betting formula which has been tailored to stock trading:

K% = ( (b+1) * p – 1) / b = ( b*p – (1-p) ) / b

Win probability (**p**): The probability that any given trade you make will return a positive amount.

Win/loss ratio (**b**) or odds: The total positive trade amounts divided by the total negative trade amounts.

If you think of b as the odds of b-to-1, payout of b when betting 1 unit of money, the numerator is simply the mean value of expected payout, or the so-called “edge”. Therefore, K% can be expressed as edge/odd. For obvious reason, you don’t want to bet in any game where the expected payout is 0 or negative.

If Kelly Criterion is so great, why is that this is not heard or used very often in the investing world. There are a couple of reasons that prevent it to be used practically:

- The volatility of strictly using Kelly Criterion is quite big. Despite that in the long term, probabilistically speaking your portfolio will have the maximum return possible, the ups and downs are too big to be digested by most people. Therefore, people talk about using “half Kelly” or half of the bet amount calculated from Kelly Criterion in attempt to reduce the portfolio volatility.
- To use Kelly Criterion, it requires knowing how good you trade stocks (in terms of
**p & b**). Obviously, if you don’t know exactly how much your “edge” is, the Kelly betting amount will probably be off from the correct amount. Estimating and knowing your edge will be a much harder task than calculating the Kelly betting amount.

Despite the mathematical correctness of Kelly Criterion, it is much harder to invest such in practice. Aren’t there anything that we can walk away from such a terrific investing formula? Indeed, there is. Here is what I personally learned after investing stocks for almost 10 years now. **The riskier the stock/or entry point is, the less amount that you should put in; the safer the stock/or entry point is, the more amount that you should put in.** This is exactly the spirit of Kelly Criterion that bet should be proportional to your edge or your supposed advantage. I have been burned by stupid bets so many times that I finally learned to carefully size each of my stock transaction. In fact, sizing of your transaction is equally important if not more than what stocks you pick. While most of the investment world talks about what to buy, much less attention is spent on how much one should buy. But for every transaction, it always consists of the following elements: what (stock) to buy/sell, when to buy/sell, and how much to buy/sell. For successful investing, all three elements must be carefully chosen. And Kelly Criterion helps you on deciding the last element: how much.

For more related articles, one can check out the article from Businessweek, and investopedia. Tom Weideman also has an excellent article using simple calculus for deriving Kelly Criterion with less math from information theory.

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April 17th, 2006 at 1:08 pm

Hi, Frugal,

Nice site and a thought provoking article.

I checked out Kelly’s original paper and agree Weideman’s derivation is a lot more intuitive. I was exposed to Kelly’s criterion as a blackjack player but have no background in communication theory otherwise.

The original formula refers to a sequence of bets. In blackjack, you can play more than one hand and calculate the covariance between them. In investing however, you’re simultaneously placing a large number of interdependent bets. In my opinion, it makes the Kelly criterion much less useful even as a conceptual tool. Just as the key to sucess in gambling is to find the right game. The key to successful investing is not to find the appropriate bet size but to find the appropriate bet.

Cheers, ML

April 17th, 2006 at 8:10 pm

[...] My 1st Million at 33 discusses a less well known (at least less publicly appreciated) investing calculation known as the Kelly Criterion. Using the formula, you can discern a mathematically correct amount you should be investing in a particular stock. This of course, depends very heavily on your ability to accurately determine the probabilities used by the equation. [...]

April 17th, 2006 at 9:06 pm

Hi, ML,

Thanks for commenting on my article. I agree that Kelly Criterion is hard to apply to investing. I try to capture the spirit of it. If I’m investing in an entire market index, I think of it as like a contrarian investing philosophy. If I’m investing in individual stocks, I size my bets according to the market cap of the stock in attempt to bet properly to control total volatility from this particular stock. But I think the principle is always the same, more the edge you have, the more you should invest; the bigger the odd is (or the less likely the winning event happens such as individual

small capstocks), the less you should invest. The most difficult thing is to have this edge (positive expected return). Obviously, in close to all gambling games or lotteries, this “edge” is either 0 or negative, which also means that one should not gamble or buy lottery ticket.