Contracting with a Mean-Volatility Controlled Outcome

We study how Holmström–Milgrom Continuous-time contracting models mean-volatility control Second best, risk-based continuous time mean-volatility controlmodel can be extended by allowing not only the mean, but also the volatility of the outcome to be controlled. The extension utilizes the representation of each admissible contract, in transforming the agent’s problem in a weak formulation with the mean-volatility controlled outcome into an equivalent problem in a strong formulation with a volatility controlled outcome. We present a general Hamiltonian for the principal’s problem and then consider special cases with Markovian mean-volatility controlled outcomes. The results are applied to agency problems where the agent can improve the outcome either by private effort or by project selection which may or may not be observable to the principal. Holding other things constant, our project selection models indicate that the higher the PPS, the more conservative the agent becomes in both private and public decisions of his project selection. We discuss numerical examples which suggest that if the manager privately makes project selection decisions for the firm which currently operates under a high-risk environment, then strikingly, the PPS can appear to increase with the overall volatility of the firm, if choices of risky projects are prompted by changes in the overall quality of project opportunities.