Kelly criterion

[HDB 2008-06-04] This text is lifted from some website and I can't remember where.

For those who are interested in the theory, here are the mathematical steps I used to generalize the above problem.

Let's define some variables:

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

Using the above infomation, 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)

Therefore, to maximize the geometric return M, we need to find F such that the Product Sum of (1+Wi*F)^Pi for all i is maximized. Unfortunately, there is no simple formula that can compute the Kelly Criterion for multiple possible outcomes. Fortunately, with the aid of computer, I constructed an optimization model that will find the Kelly Criterion for you.