How to Calculate the Kelly Formula

In my previous article dealing with the Kelly formula, I attempted to convince you that the Kelly formula was the most important formula in investing. Simply put, I believe using this formula will significantly improve your investing performance over time. This article will delve into using actual numbers and estimates to help determine how much of your portfolio you should allocate to an investment idea.

The Kelly formula simply and elegantly states that an investor should calculate edge divided by odds to determine how much to invest in a security. In order to turn edge/odds into a number, we have to know a few key pieces of information: the probability of winning, the probability of losing, and the payoff. In the stock market, none of these can be precisely quantified, but if you don't know enough to estimate them with a reasonable degree of confidence, it's time to move onto the next investment idea. An oft-repeated saying is: If you're at the poker table and you don't know who the sucker is, it's you. If you've decided to buy a stock and you can't articulate why the sellers are fools (with a lower-case f), then consider the fact that whoever is selling to you might have a better reason why you're the fool.

The Kelly formula (edge/odds), in expanded form, is: (P*W-L)/P. In this formula, P is the payoff, W is the probability of winning, and L is the probability of losing. Basically, the formula states that for any given stock, you should invest the probability of winning times the payoff minus the probability of losing, divided by the payoff. Let's break this down a bit further.

The probability of winning and the probability of losing are self-explanatory. For a coin flip, the probabilities of both winning and losing are .5, or 50%, and the sum of both probabilities must equal 1 (assuming no ties), since you must either win or lose. Thus, any stock that has a high probability of tanking, such as a cash-strapped biotech company with only a prayer of getting a new drug to market, not only has a high probability of losing, but a low probability of winning as well. On the other hand, if you can invest in solid companies at low prices trading near liquidation value, then you've put yourself in a situation where it's hard to lose money. After all, it's hard to lose money if you're paying $100 million for a company that has $70 million of net cash in the bank and decent financial results, and if you're not losing money, then the alternatives are pretty good.

The payoff is how much money you'll make or lose for every dollar you invest. For example, for an even-money coin flip, if you wager a dollar, you'll either win or lose a dollar, so the payoff is $1. If you think a stock can triple, then you're getting $2 in profit for every $1 you invest, so the payoff is $2.

Microsoft (Nasdaq: MSFT  ) and General Electric's (NYSE: GE  ) market capitalizations are $285 billion and $363 billion, respectively. In order for their stocks to quadruple, their market caps would have to surpass $1 trillion, an incredibly unlikely scenario in the foreseeable future. The market caps of Yahoo! (Nasdaq: YHOO  ) , eBay (Nasdaq: EBAY  ) , and Amazon (Nasdaq: AMZN  ) (which, combined, we'll call YEA) are $36 billion, $46 billion, and $16 billion, respectively. Add those up (total $98 billion) and you still don't get to Google's (Nasdaq: GOOG  ) $145 billion market cap. Is Google really worth 50% more than YEA -- in other words, is it worth more than Yahoo!, eBay, and Amazon combined? That's debatable, and it's difficult to say whether Google is overvalued or the companies in YEA are undervalued. However, it would behoove investors to know what other people are paying for similar companies. Consider this: If Google doubled, it'd be worth more than Microsoft. If you're buying a house for $1 million, and the guy down the street is buying a similar-looking house for $500,000, you might want to do a little more research.

Once we've estimated the probability of winning and losing and the payoff, all we have to do now is some simple arithmetic to estimate how much of our portfolio we should invest in a company. Once again, the Kelly formula is the payoff times the probability of winning minus the probability of losing divided by payoff, which is edge/odds. So to run through a simple scenario: Suppose I offer you a coin flip that pays even money. How much of your money should you bet on this? The probability of winning is .5, the probability of losing is .5, and the payoff (because this is even money) is 1 -- for every $1 you bet, you make or lose $1. Thus, the formula is (1*.5-.5)/1, which equals 0. You don't have an edge, because the coin flip is 50/50, and you don't have an advantage in payoffs, so you shouldn't invest.

Suppose instead that I offered you a coin flip, except this time I offered you 2-to-1 odds. Now the formula becomes (2*.5-.5)/2, which equals .25. You should invest 25% of your money. Now let's say I offer you a coin flip, except I'll give you $1 if you lose, and $1 if you win. The formula becomes (1*1 - 0)/1, which equals 1, or 100%. The probability of losing is 0, and since you can't lose, you might as well put all your money in.

Although any formula is only as good as the estimates and data plugged into it, this formula forces investors to think in terms of payoffs and probabilities when investing in a company. It also prevents investors from investing in low-payoff, high-risk companies -- which is the definition of most "hot" stocks, where the easy money has already been made and the risk that the stock will tank is high -- and instead guides investors toward low-priced stocks where most of the risk has been taken out and potential payoffs are high. For example, USG (NYSE: USG  ) is an out-of-favor stock whose stock has cratered to $48 from around $110. However, after doing quite a bit of research on this company, it becomes clear that an investment in USG would have a high probability of winning, as well as a high payoff, which is probably why Berkshire Hathaway (NYSE: BRK-A  ) (NYSE: BRK-B  ) has recently been scooping up shares. More on this later.

For related articles:

Microsoft and Berkshire Hathaway areInside Valuepicks. Take a30-day free trial of the service and see which other stocks Philip Durell and his team have chosen as good companies selling at bargain prices. Yahoo!, Amazon, and eBay areStock Advisorselections.

Fool contributor Emil Lee is an analyst and a disciple of value investing. He doesn't own any of the stocks mentioned in this article and appreciates your comments, concerns, and complaints. The Motley Fool has a disclosure policy.


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  • Report this Comment On February 09, 2011, at 2:00 AM, MadhurKotharay wrote:

    Kelly's formula gives the best size of your bet, if you were to SEQUENTIALLY (SERIALLY) bet on the same setup, such as in a casino. In stock market, you bet in PARALLEL on multiple scrips. In such times, Kelly's formula cannot tell you the best size of your bet.

    After all, Kelly's formula tells you what the best bet size should be to get the most returns after 'N' rounds. And it is to account for the drawdown (losses) that might occur in the process.

    In stock market, the ratio Edge / Odds is irrelevant. The Edge is the only thing that matter. Go back to the derivation of Kelly's formula.

    Kelly's formula can help a trader who has to take huge number of punts one after the other based on a trading system but cannot help a fundamental investor.

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