Dividing gains between a client and her agent

Abstract

A client(she) contracts with an agent(him), who has limited liability, as follows: she lends him one dollar at time 0 and he uses the money to trade in a security market. As return, he promises to give her a fixed amount \(e^{r_0T}\) at the final time T; in addition, if the real return rate of the strategy is larger than \(r_0\), she can also get a fixed proportion \((1-\alpha)\) of the “excess profit” and he will take the rest. Assume that the market is complete and the agent aims to maximize the risk-neutral value of his profit subject to some expected shortfall constraint. The reasonable benchmark return rate \(r_0\) and the proportion \(\alpha\) are explicitly worked out.