The Kelly Criterion is a mathematical formula used to determine the optimal size of a series of bets. It maximizes the expected logarithm of wealth, balancing risk and reward to prevent total loss (a.k.a minimization of risk of ruin) over time.
Credit: Grant Yang and Rohan Ramkumar, two high school friends of mine who never cease to inspire me.
\[ \text{Let } f^* = \frac{bp - q}{b} \quad \text{where } b = \text{odds},\ p = \text{probability of winning},\ q = 1 - p \] \[ \text{This maximizes expected log wealth: } \mathbb{E}[\log(W)] = p \log(1 + f \cdot b) + (1 - p) \log(1 - f) \] \[ \text{Set derivative to 0: } \frac{p b}{1 + f b} - \frac{1 - p}{1 - f} = 0 \Rightarrow f^* = \frac{bp - q}{b} \]
This is the standard derivation of the Kelly Criterion, which maximizes the expected logarithmic growth of wealth across repeated bets. In Information Theory taught by Dr. Anu Aiyer, Grant and Rohan used the letter W for doubling rate, and S(x) to mean wealth.
You can access the original write-up authored by Grant Yang and Rohan Ramkumar here: grant-rohan-kelly-criterion.tex
I talk about this in my Instagram Reel, where I poke fun at the Cockney dialect and explain what the Kelly Criterion is.