Multi-portfolio Optimization : A Fairness-aware Target-oriented Model

We consider a multi-portfolio optimization problem where nonlinear market impact costs result in a strong dependency of one account's performance on the trading activities of other accounts. We develop a novel target-oriented model that jointly optimizes the rebalancing trades and split of market impact costs. The key advantages of our proposed model include the consideration of clients' targets on investment returns and the incorporation of distributional uncertainty. The former helps the fund manager circumvent the difficulty in identifying clients' utility functions or risk parameters, while the latter addresses a practical challenge that the probability distributions of risky asset returns cannot be fully observed. Specifically, to evaluate multiple portfolios' investment payoffs achieving their targets, we propose a new type of performance measure, called the fairness-aware multi-participant satisficing (FMS) criterion. This criterion can be extended to encompass the distributional uncertainty and has the salient feature of addressing the fairness issue with the collective satisfaction level as determined by the least satisfied participant. We find that, structurally, the FMS criterion has a dual connection with a set of risk measures. For multi-portfolio optimization, we consider the FMS criterion with conditional value-at-risk, a popular risk proxy in financial studies, being the underlying risk measure to further account for the magnitude of shortfalls against targets. The resulting problem, although non-convex, can be solved efficiently by solving an equivalent converging sequence of tractable subproblems. The numerical study shows that our approach outperforms utility-based models in achieving targets and is more robust in out-of-sample performance