## Wednesday, October 10, 2007

### The Kelly Formula make easy

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.
Let’s talk about this a little as I find it an interesting concept and it echoes somewhat on my perspective of life and principle towards investing – bet (read invest) when the odds are with you or otherwise do nothing. So I guess again I should try to be a copycat. But heck, it pays to be one.

What is Kelly Formula?

I first came across Kelly Formula in the book, Fortune’s Formula. The history of it is detailed in the book that also happens to be a favorite book of Charlie Munger. Despite all the odds, a minority of gamblers – unsure if I should really classify this group of minority as gamblers because of how they carry out their bets that in high likelihood they shouldn’t be for they know what they are doing - still able to make a lot of money. For example, in Hong Kong, people are wild over horse racing, yet somebody was actually making a lot of money despite the croupier’s take because that person was able to develop some form of an algorithms that taught him how much to bet each time given the odds. One of such algorithms is the Kelly Formula that seems quite plausible. The gist of it makes sense in terms of sizing your bet according to the odds.

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 simply figures out how much to bet if the odds are in your favor – in Vegas, in card games, in the stock market, in a coin flip, whatever. The equation of the Kelly Formula can be simplified to:

That is the simplified formula. The actual formula (for purists) is:

A Kelly Example

Let’s say you have \$1,000 in cash and someone offers you 2 to 1 on a coin flip. That is, they’ll pay you \$2 if it comes up heads or 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.

By inserting the numbers in the formula, it shows:

In this coin flip gamble, the Kelly Formula tells you that the maximum you should bet on any flip is 25% of your bankroll – \$250 in this case. Doing so will give you the maximum long-term growth with minimum downside.

The Kelly Formula Guarantees and its Weaknesses

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

You never want to overbet the Kelly Formula. That is, you never want to put more of your bankroll than the formula prescribes.

At any rate, investing is somewhat 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 more of the dollar-for-fifty-cent cases rather than its contrast), you’ll make good money. In short, the Kelly Formula helps maximize your return though it does nothing for volatility that is something to understand about on how to think about stock prices.

Kelly Formula Applied to Investing

There’s one thing to note with the Kelly Formula when applying it to investing in businesses when they are on hot-fire sale: It would prescribe you to put a large portion of your bankroll into one business. That may not be a bad event actually. As said earlier, bet big when odds are highly favorable.

When you wait patiently for dollars to sell for half of it, the odds of winning is so large that it overwhelms the odds of losing, and you will end up putting 85% or more of your available bankroll into one position.

Now, if you try to run the Kelly Formula on most value stocks, what the model will tell you is likely to be you ought to put a high percentage of your bankroll in such position.

Does this make sense? Hell, it does to me at least. Why? Because the odds of a loss are so low and the odds of winning are so high.

So, why should we care about the Kelly Formula?

For one, it helps us understand that it is alright to own just a few holdings when the odds are good – be it five or fifteen.

Don’t focus on calculating the Kelly Formula for your investments and then diversifying based on your mathematical models. Rather, spend the time and energy finding “no-brainers” – investments that would be in that 85%-of-bankroll or more range. Then, buy the heck out of them.