In investment we often come to problems like this:

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

Launching Product 1: 30% increase in stock price (ROI = 30%). Chance of happening: 20%.
Launching Product 2: 10% increase in stock price (ROI = 10%). Chance of happening: 15%.
Launching Product 3: 12% increase in stock price (ROI = 12%). Chance of happening: 25%.
Failure to launch: 15% decrease in stock price (ROI = -15%). Chance of happening: 40%

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, solving similar problems in gambling, cannot be directly used because it assumes binary outcome. That means it limits the outcomes to be either WIN or LOSE.

The problem above has four possible outcomes, and therefore the Kelly Formula is inadequate in solving it.

Another flaw of the Kelly Criterion is that it assumes 100% loss when the outcome is unfavorable. This works well in gambling (black-jack) or sports betting, as you lose all your wager when you lose. In the stock market, you almost never lose 100% of your investment in a single trade.

    Here is the model that will compute the Kelly Criterion for multiple outcomes.

Inputs:

Possible Outcome Name: This is the name of the outcome (ie. No Product Launched)

ROI: This is the return on investment if the potential outcome becomes real (ie. -15%)

Probability: This is how likely the outcome would become real (ie. 40%)

Note: All probabilities should add up to 100%

    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 formular 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.