Central-planned Portfolio Selection, Pareto Frontier, and Pareto Improvement

In delegated portfolio management, we formulate a central-planned portfolio selection problem by multi-objective programming (utilities of the investor and the manager) to study the Pareto optimal portfolio and find Pareto improvement. First, we solve out two cases of the closed-form Pareto optimal portfolio based on non-smooth and non-concave utility optimization. One case has a special terminal outcome that the manager suffers a loss and the investor loses nothing, resulting that the optimal portfolio has a novel two-peak-three-valley pattern. We originally divide the optimal portfolio into three terms (Merton term, Aggressive term and Conservative term) to explain the pattern and conduct asymptotic analysis to illustrate economic insights. Second, we establish the collection of Pareto points of a single contract and prove that it is a strictly decreasing and strictly concave frontier. Third, we use Pareto frontiers to compare different contracts, showing that among first-loss contracts with long evaluation time, the investor benefits from the one with a smaller incentive rate and a smaller managerial ownership proportion. In addition, when the evaluation time is short, we discover a way of Pareto improvement by simultaneously adding the investor's utility into the manager's investment objective and increasing the manager's incentive rate.