Since the book "Fortune's Formula" came out, many investors use the Kelly Criterion to figure out how much they should invest. Most of these investors, though, haven't gone through the math behind the Kelly Criterion or read Ed Thorp's paper on how to use it in the stock market.
When used directly in stock trading, the Kelly Criterion is often wrong. The stock market is more complicated than most gambling games, and the assumptions that the criterion is based on do not hold.
Take the following problem as an example:
Company A is looking into three new products right now. We know that A might announce the launch of one of the new products at a convention that is coming up. We can also figure out how different events will affect the stock price:
If Product 1 is launched, A's stock price will go up by 30%. This could happen 20% of the time.
If Product 2 comes out, A's stock price will go up by 10%. This has a 15% chance of happening.
If Product 1 is launched, A's stock price will go up by 12%. This has a 25% chance of happening.
If A doesn't launch a new product, the price of its stock will go down by 15%. This is likely to happen 40% of the time.
You have $100 in your bankroll right now. How much of it would you invest in A's stock so that your money grows the most over time?
The Kelly Criterion can't help you solve this problem because it only considers two possible outcomes: FAVORABLE or UNFAVORABLE. It also assumes that if the result is bad, you will lose everything you put into it (the wager).
On the stock market, there are usually more than one way things could turn out, and you almost never lose all of your money in a single trade. So, the Kelly Criterion by itself is not enough to use in the stock market.
I've looked at the math behind how the Kelly Formula was made, and it can be used to find the answer to the problem above.
Let's define some variables:
F = the percentage of your money you put into A
W1 = 30% for the launch of Product 1.
W2 = Launching Product 2 Return on Investment = 10%
W3 = Return on Investment of Launching Product 3 = 12%
W4 = ROI for no new products = -15%
P1 = 20% Chance of Product 1 Coming Out
P2 = 15% Chance of Product 2 Coming Out
P3 = 25% Chance of Product 3 Coming Out
P4 = 40% Chance That No Product Will Be Released
B = First money in the bank
B' = Future Bankroll after N such investments
M = the average of N investments like this.
Using the information above, we can come up with:
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)
Finding the maximum Ln(M) lets us find the maximum 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)
In chapter 7 of his paper "THE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK MARKET," in which he talks about how to use the Kelly Criterion on the stock market, Ed Thorp wrote the above equation.
This optimization problem can't be solved in a clean way. Simulation, on the other hand, can be used to make a web app that finds the Kelly Percentage with the help of modern technology. For example, you can find one of these kinds of web apps at:
http://www.cisiova.com/betsizing.asp
The web application takes possible outcomes (ROI and probability) as inputs and figures out the Kelly Percentage and the maximised mean growth rate for you. Since the Kelly Criterion is just a special case of this maximisation problem, the web application works fine with simple Kelly problems like sports betting or gambling.
