Kelly Criterion And The Stock Market

Posted by Nick Niesen on November 8th, 2010

Since the book "Fortune's Formula" is published, many investors are turning to the Kelly Criterion for determining the size of the investment. Unfortunately, most of these investors have not walked through the underlying mathematical derivation or read Ed Thorp's paper on how to apply the Kelly Criterion in the stock market.

There are many fallacies when using the Kelly Criterion directly in stock trading. Unlike most gambling games, the stock market is too complex and the underlying assumptions of the criterion do not hold.

For example, consider the following problem:

Company A is currently researching 3 different new products. In an upcoming convention, we know that A might announce the launch of one of the new products. We can also estimate the impact of different outcomes on the stock price:

30% increase in A's stock price if Product 1 is launched. There are 20% chance for this to happen.
10% increase in A's stock price if Product 2 is launched. There are 15% chance for this to happen.
12% increase in A's stock price if Product 1 is launched. There are 25% chance for this to happen.
15% decrease in A's stock price if no product is launched. There are 40% chance for this to happen.

Now you have $100 dollars in your bankroll, how much would you invest in A's stock so that your bankroll can have maximum growth in the long term?

The Kelly Criterion cannot help you solve this problem because it assumes only two possible outcome: FAVORABLE or UNFAVORABLE. It also assumes that if the outcome is unfavorable, you will lose 100% of what you invested (the wager).

In the stock market, you often have multiple outcome scenarios, and you almost never lose 100% of your investment in a single trade. Therefore, the Kelly Criterion alone is not directly applicable to the stock market.

I have looked through the mathematical derivation of the Kelly Formula, and it can be used to derive the solution for the above problem.

Let's define some variables:

F = % of your bankroll that you invest in A
W1 = ROI of Launching Product 1 = 30%
W2 = ROI of Launching Product 2 = 10%
W3 = ROI of Launching Product 3 = 12%
W4 = ROI of No Products Launching = -15%
P1 = Probability of Product 1 Launching = 20%
P2 = Probability of Product 2 Launching = 15%
P3 = Probability of Product 3 Launching = 25%
P4 = Probability of No Product Launching = 40%
B = Initial Bankroll
B' = Future Bankroll after N such investments
M = The Geometric Mean of N such investments

Using the above information, we can formulate:

B' = B * (1+W1*F)^(P1*N) * (1+W2*F)^(P2*N) * (1+W3*F)^(P3*N) * (1+W4*F)^(P4*N)

M^N = B'/B = (1+W1*F)^(P1*N) * (1+W2*F)^(P2*N) * (1+W3*F)^(P3*N) * (1+W4*F)^(P4*N)

M = [(1+W1*F)^(P1*N) * (1+W2*F)^(P2*N) * (1+W3*F)^(P3*N) * (1+W4*F)^(P4*N)]^(1/N)

M = (1+W1*F)^(P1) * (1+W2*F)^(P2) * (1+W3*F)^(P3) * (1+W4*F)^(P4)

We can find the maximum M by finding the maximum Ln(M):

Ln(M) = Ln[(1+W1*F)^(P1) * (1+W2*F)^(P2) * (1+W3*F)^(P3) * (1+W4*F)^(P4)]

Ln(M) = P1*Ln(1+W1*F) + P2*Ln(1+W2*F) + P3*Ln(1+W3*F) + P3*Ln(1+W3*F)

The above equation is what Ed Thorp stated in chapter 7 of his paper "THE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK MARKET", in which he discusses how to apply the Kelly Criterion in the stock market.

There is no clean solution to this optimization problem. However, with the aid of modern technology, a web application that finds the Kelly Percentage can be developed through simulation. For example, you can find such web application at:

http://www.cisiova.com/betsizing.asp

The web application takes possible outcomes (ROI and probability) as inputs and calculates the Kelly Percentage and the maximized mean growth rate for you. Since the Kelly Criterion is just a special case of this maximization problem, the web application works perfectly well with simple Kelly problems such as sports betting or gambling.