A risk–reward framework for the competitive analysis of financial games

Competitive analysis is concerned with minimizing a relative measure of performance. When applied to financial trading strategies, competitive analysis leads to the development of strategies with minimum relative performance risk. This approach is too inflexible. Many investors are interested in managing their risk: they may be willing to increase their risk for some form of reward. They may also have some forecast of the future. In this paper we extend competitive analysis to provide a framework in which investors can develop optimal trading strategies based on their risk tolerance and forecast. We first define notions of risk and reward that are natural extensions of classical competitive analysis and then illustrate our ideas using the ski-rental problem. Finally, we analyze a financial game using the risk–reward framework, and, in particular, derive an optimal risk-tolerant algorithm.