Consider a prediction market, in which participants can trade shares (binary options) at the current market price pm. Each share is worth $1 if the event occurs, and nothing otherwise. What fraction of your wealth w should you risk if you believe the probability of the event is p? Buying is favorable if p > pm, in which case risking your entire wealth will maximize your expected profit with respect to your belief. However, that's extraordinarily risky: A single stroke of bad luck loses everything. On the other hand, risking a small fixed amount cannot take advantage of compounding growth.