Deriving the Kelly Growth Rate at the Optimal Bet Fraction

Summary

The document asks how to simplify the expected logarithmic growth function at the Kelly-optimal fraction for a binary win-or-loss wager. It gives the proposed fraction, the growth expression evaluated at that fraction, and an attempted expansion that leaves two logarithmic terms whose sum is expected to equal the logarithm of two.

The post contains the algebraic setup but no answer or derivation. Its focus is a mathematical identity in the Kelly criterion rather than a trading implementation or empirical strategy. The result applies to the stated binary-outcome formulation; practical use of Kelly sizing also depends on correctly specifying outcome probabilities and payoffs, which the document does not discuss.

Key ideas

• The Kelly criterion selects a fraction of capital to maximize expected logarithmic growth.
• For a binary wager, the growth function combines the log returns from winning and losing outcomes.
• Substituting the optimal fraction leads to an algebraic simplification involving the win and loss probabilities.
• The document poses the derivation question but does not provide its solution or discuss practical estimation of probabilities.

Tags

# Kelly's maximum for G(f)


# Kelly's maximum for G(f)












In Thorpe's paper, Thorpe derives the Kelly criterion

$$f^* = p - q$$

and plugs this into the equation

$$G(f^*) = p \times \log(1+f^*) + q \times \log(1-f^*)$$

to get the following expression

$$G(f^*) = p \times \log (p) + q \times \log (q) + \log(2).$$

I am struggling to derive this result. I am left with

$$p \times \log(p) + q \times \log(q) + p \times \log\left(1 + (1-q)/p\right) + q \times \log \left(1 + (1-p)/q\right).$$ I expect the last two terms somehow equal $\log(2)$ but I have been scratching my head for hours trying to get there. Can anyone show me how please?

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