Use the Kelly Criterion to determine position size

This is a classic trading post from my non-trading blog, GoodeValue.com.

The Kelly Criterion is a formula for choosing how large a bet to make on each trade/investment/gamble. It works for the stock market, though it was originally developed for gambling. The formula is simple: bet the proportion of your investment as defined by the ratio of expected return divided by maximum return. Expected return is what you expect in the long run.

So, the formula is: P_invest = E(r) / M(r)
where,
Proportion of portfolio to invest = P_invest
Expected return= E(r)
Maximum return = M(r)

Now, a couple of examples:

1. If you flip fair coin and win $1 if heads and lose $1 if tails, the expected return is $0 (.5 x $1 + .5 x -1). The maximum return is $1 (if heads). Therefore, the Kelly criterion suggests you bet no money ($0/$1). This makes sense, because you should not invest money where you expect to only break even.

2. You want to short Apple (AAPL) because you think there is an 80% chance the stock will go down in the next month. You think if that happens, the stock will go down 10%. You figure that there is a 20% chance that the stock will go up 5%. The expected return is 7% (.8 x 10% + .2 x -5%). The maximum gain is 10%. The Kelly formula suggests that you invest at most 70% (7/10) of your portfolio.

3. Same thing, shorting AAPL. You like the odds, so you increase your leverage by buying put options. You buy just out of the money options. Now, there is a 70% chance that your options expire worthless (-100% return) and a 30% chance that you make 300%. The expected return is +20% (.7 x -100 + .3 x 300). The maximum gain is 300%. The Kelly formula says that you should bet less than 1/15 (about 6.5%) of your portfolio (20/300).

One thing to consider is that the Kelly formula seeks only to maximize gains. If you wish to minimize portfolio variability as well, you should invest significantly less than the maximum allowed by the Kelly formula. Also, keep in mind that the formula is only as good as your guesses of probability. In order to minimize portfolio volatility and because it is very difficult to accurately estimate the expected return on a trade a priori, many traders stick to using a very small fixed percentage of their portfolio on each trade.

I recommend a Legg Mason article on the Kelly Criterion, or this paper by Edward Thorp (who used it to great effect).

Visit Cisiova’s website for their advanced online Kelly Criterion calculator, which allows you to enter a large number of possible outcomes.

If you liked this post you may want to check out William Poundstone’s book Fortune’s Formula.

Disclosure: I own no Apple stock, long or short. Unfortunately, I did once lose money shorting AAPL. My disclosure policy never loses me money.

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