is , and in that case the resulting wealth is equal to . The probability of losing is , and in that case the resulting wealth is equal to . Therefore, the expected geometric growth rate is: We want to find the maximum r of this curve (as a function of f), which involves finding the derivative of the equation. This is more easily accomplished by taking the logarithm of each side first. The resulting equation is: with denoting logarithmic wealth growth. To find the value of for which the growth rate is maximized, denoted as , we differentiate the above expression and set this equal to zero. This gives: Rearranging this equation to solve for the value of gives the Kelly criterion: Notice that this expression reduces to the simple gambling formula when , when a loss results in full loss of the wager. Bernoulli In a 1738 article, Daniel Bernoulli suggested that, when one has a choice of bets or investments, one should choose that with the highest geometric mean of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is different (Bernoulli wanted to resolve the St. Petersburg paradox). An English-language translation of the Bernoulli article was not published until 1954, but the work was well known among mathematicians and economists. Application to the stock market In mathematical finance, if security weights maximize the expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is growth optimal. Computations of growth optimal portfolios can suffer tremendous garbage in, garbage out problems. For example, the cases below take as given the expected return and covariance structure of assets, but these parameters are at best estimates or models that have significant uncertainty. If portfolio weights are largely a function of estimation errors, then Ex-post performance of a growth-optimal portfolio may differ fantastically from the ex-ante prediction. Parameter uncertainty and estimation errors are a large topic in portfolio theory. An approach to counteract the unknown risk is to invest less than the Kelly criterion (e.g., half). Criticism Although the Kelly strategy's promise of doing better than any other strategy in the long run seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints may override the desire for optimal growth rate. The conventional alternative is expected utility theory which says bets should be sized to maximize the expected