Portfolio Delegation with Limited Liability

We consider the portfolio delegation problem in a world with complete contingent claim markets. A principal hires an agent to manage a portfolio. When the agent has limited liability (that is, there is a lower bound on the compensation contract), she may have an incentive to take on excessive risk. With complete markets, the precise nature of the risk the agent may take on is a large short position in the state with lowest probability, and a long position in every other state. We impose an incentive constraint that prevents the agent from taking on risk in this form. We show that the optimal contract requires that the compensation function be bounded above, and that this prevents excessive risk taking. The size of the bound controls the degree of risk taken on by the agent. The upper bound alone is sufficient to prevent the deviation mentioned. Our main result is that, with limited liability and a large number of states, incentive compatibility alone restricts the feasible contract to be either a flat one or one with exactly two compensation levels (equal to the lower and upper bounds on compensation). Even a small positive slope to the compensation function over other regions of realized wealth will lead to the agent deviating in the prescribed manner. We then compare the outcome induced by the optimal contract to that induced by a Value at Risk compensation scheme. Value at Risk is a popular risk management tool currently used in the portfolio delegation context. An appropriately defined Value at Risk scheme can be effective in controlling excessive risk taking. A Value at Risk constraint is equivalent to a short-sale constraint in this case. However, it does not necessarily achieve the second best outcome. Our model highlights a potential downside to financial innovation. While it may lead to superior gains from risk sharing in an exchange economy, in our context, it lets the agent gamble on a finer set of states. This intensifies the agency problem.