Fairness and Optimality in Trading

This thesis proposes a novel approach to address the issues of efficiency and fairness when multiple portfolios are rebalanced simultaneously. A fund manager who rebalances multiple portfolios needs to not only optimize the total efficiency, i.e., maximize net risk-adjusted return, but also guarantee that trading costs are fairly split among the clients. The existing approaches in the literature, namely the Social Welfare and the Competitive Equilibrium schemes, do not compromise efficiency and fairness effectively. To this end, we suggest an approach that utilizes popular and well-accepted resource allocation ideas from the field of communications and economics, such as Max-Min fairness, Proportional fairness and a-fairness. We incorporate in our formulation a quadratic model of market impact cost to reflect the cumulative effect of trade pooling. Total trading costs are split fairly among accounts using the so-called pro rata scheme. We solve the resulting multi-objective optimization problem by adopting the Max-Min fairness, Proportional fairness and a-fairness schemes. Under these schemes, the resulting optimization problems have non-convex objectives and non-convex constraints, which are NP-hard in general. We solve these problems using a local search method based on linearization techniques. The efficiency of this approach is discussed when we compare it with a deterministic global optimization method on small size optimization problems that have similar structure to the aforementioned problems. We present computational results for a small data set (2 funds, 73 assets) and a large set (6 funds, 73 assets). These results suggest that the solution obtained from our model provides a better compromise between efficiency and fairness than existing approaches. An important implication of our work is that given a level of fairness that we want to maintain, we can always find Pareto-efficient trade sets. Thesis Supervisor: Dimitris Bertsimas Title: Boeing Professor of Operations Research