volatility) are constants. Taking expectations of the logarithm: Then the expected log return is For a portfolio made of an asset and a bond paying risk-free rate , with fraction invested in and in the bond, the expected one-period return is given by however people seem to deal with the expected log return for one-period instead in the context of Kelly: Solving we obtain is the fraction that maximizes the expected logarithmic return, and so, is the Kelly fraction. Thorp arrived at the same result but through a different derivation. Remember that is different from the asset log return . Confusing this is a common mistake made by websites and articles talking about the Kelly Criterion. For multiple assets, consider a market with correlated stocks with stochastic returns , and a riskless bond with return . An investor puts a fraction of their capital in and the rest is invested in the bond. Without loss of generality, assume that investor's starting capital is equal to 1. According to the Kelly criterion one should maximize Expanding this with a Taylor series around we obtain Thus we reduce the optimization problem to quadratic programming and the unconstrained solution is where and are the vector of means and the matrix of second mixed noncentral moments of the excess returns. There is also a numerical algorithm for the fractional Kelly strategies and for the optimal solution under no leverage and no short selling constraints. See also Risk of ruin Gambling and information theory Proebsting's paradox Merton's portfolio problem References External links Optimal decisions Gambling mathematics Information theory Wagering Articles containing proofs 1956 introductions Portfolio theories