Using MOEAs To Outperform Stock Benchmarks In The Presence of Typical Investment Constraints

Portfolio managers are typically constrained by turnover limits, minimum and maximum stock positions, cardinality, a target market capitalization and sometimes the need to hew to a style (such as growth or value). In addition, portfolio managers often use multifactor stock models to choose stocks based upon their respective fundamental data. We use multiobjective evolutionary algorithms (MOEAs) to satisfy the above real-world constraints. The portfolios generated consistently outperform typical performance benchmarks and have statistically significant asset selection. In finance, a portfolio is a collection of assets held by an institution or a private individual. The portfolio selection problem seeks the optimal way to distribute a given monetary budget on a set of available assets. The problem usually has two criteria: expected return to be maximized and risk to be minimized. Classical mean-variance portfolio selection aims at simultaneously maximizing the expected return of the portfolio and minimizing portfolio risk. In the case of linear equality and inequality constraints, the problem can be solved efficiently by quadratic programming, i.e., variants of Markowitz‟s critical line algorithm. What complicates this simple 1 Andrew Clark, Chief Index Strategist, Thomson Reuters Indices & Lipper, [email protected] 2 Jeff Kenyon, Lead Software Engineer, Thomson Reuters Indices, [email protected] 2 statement of portfolio construction are the typical real-world constraints that are by definition non-convex, e.g. cardinality constraints which limits the number of assets in a portfolio and minimum and maximum buy-in thresholds. In what follows, we use multi-objective evolutionary algorithms (MOEAs) 3 as an active set algorithm optimized for portfolio selection. The MOEAs generate the set of all feasible portfolios (those portfolios meeting the constraints), calculates the efficient frontier for each and also their respective Sharpe ratio. The portfolio with the best Sharpe ratio becomes the portfolio used for the next time period. We chose MOEAs to solve a non convex optimization problem because there are certain outstanding problems in terms of their use: 1) In the literature MOEAs have not been used to solve multiperiod financial problems (or multi-period problems in general), 2) The number and types of constraints in a real world financial portfolio problem exceeds what has been done with MOEAs so far and 3) It is not known if MOEA stock selection is statistically significant. We answer all of these questions with a yes thereby advancing the understanding and use of MOEAs. This especially true when it comes to solving a moderately difficult, multi-period real world problem such as those encountered in finance. A multi-objective optimization problem (MOP) differs from a single objective optimization problem because it contains several objectives that require optimization. When optimizing a single objective problem, the best single design solution is the goal. But for multiobjective problem with several (possibly conflicting) objectives, there is usually no single optimal solution. Because of this the decision maker is required to select a solution from a finite set of possible solutions by making compromises. A suitable solution should provide acceptable 3 See Coello et al Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer, 2002, for a good introduction to MOEAs. 3 performance over all objectives. The main motivation for using evolutionary algorithms (EAs) to solve multi-objective optimization problems is that EAs can deal simultaneously with a set of possible solutions which allows us to find several members of what is called the Pareto optimal set 4 in a single run of the algorithm. This differs from deterministic mathematical programming techniques where a series of separate runs is required. Additionally EAs are less susceptible to the shape or continuity of the Pareto front, e.g., they can easily deal with discontinuous and concave Pareto fronts. Discontinuity and concavity problems are known obstacles for deterministic mathematical programming. Adapting any stochastic optimization algorithm (such as an EA) so it can perform a multiobjective optimization requires a change to the method of archiving possible solutions. Any solution on the Pareto front can be identified formally by the fact that it is not dominated by any other possible solution. A solution X is said to be dominated by solution Y if Y is at least as good on all counts (constraints) and better on at least one constraint. Stated mathematically: ( ) ( ) 1, i i f Y f X i M and ( ) ( ) i i f Y f X for some i As several possible solutions can be generated, an archive of the non-dominated (Pareto optimal) solutions needs to be maintained. A possible archiving scheme is: All feasible solutions (Pareto optimal vectors) generated are candidates for archiving 4 A solution or a set is considered Pareto optimal if there exists no feasible solution which would decrease some constraint without causing a simultaneous increase in at least one other constraint. 4 If a candidate solution dominates any existing members of the archive, the dominated solutions are removed If the new solution is dominated by any existing member of the archive, the new solution is not archived If the new solution neither dominates nor is dominated by any members of the archive, the new solution is added to the archive Using such a scheme, as the search progresses, the archive will converge to the true trade-off surface between constraints. As to the EA part of MOEA, a generic EA assumes a discrete search space H and a function : f H  where H is a subset of the Euclidean space (in a multiobjective problem H is a subset of the Euclidean space M where M is the number of constraints). The general problem is to find