The Kelly Criterion was published in a paper in 1956 discussing optimal race track betting. In race track betting you either win or lose so the outcome is binomial. In the markets there are many possible amounts you can win or lose with varying probabilities. So strictly speaking you have to modify Kelly's formula to cover the full range of outcomes.
I wrote a book titled Optimal Portfolio Modeling published by Wiley Trading. One chapter of the book describes how to modify the Kelly formula to achieve an optimized value for the markets. However that only means that it is the optimal value for one single trade repeated over and over. It is not the best criterion to use in managing a portfolio.
The reason is that if the formula called for an optimal fraction of 20% to be invested in each position and you decide to have 10 positions then you are going to be investing 200% of your net worth. That would probably be way too much in most circumstances. Also the modified Kelly formula only takes into account optimizing expected return - it says nothing about reducing risk.
The other factor not being considered in a portfolio situation is that most stocks are correlated with the market. By investing in 10 positions one is not necessarily getting 10 independent positions because of the correlation. So unless one further modifies the formula to take into account the correlations one will tend to over trade and suffer more risk than is justified. In the later chapters of my book there is a detailed discussion of ways to adapt the formulas for risk, return and correlation.
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