The INFORMS OR & Analytics Student Competition is an annual competition organized by the Institute for Operations Research and the Management Sciences (INFORMS) to encourage undergraduate and master’s-level students to work on real industry problems using advanced analytical tools. This special issue includes papers based on entries to the 2018 INFORMS competition, sponsored by Principal Investments.
The challenge posed by Principal Investments required students to develop advanced algorithms to address a complex portfolio optimization problem. Portfolio optimization refers to the process of finding the optimal proportion of each asset in an investor’s portfolio, given a particular investment objective. The two main objectives typically used are to maximize return for a given level of risk or, equivalently, to minimize risk for a given level of expected return. The traditional Markowitz mean–variance optimization framework offers a way to allocate stocks by considering a trade-off between risk and expected return. Further diversification can be achieved by avoiding concentration in countries, company sizes, sectors, etc. Institutional investors are often also required to take into consideration additional constraints such as limits on the number of unique assets in the portfolio or how many assets outside a specified benchmark can or must be added. Such constraints are designed to control the costs that occur when an asset is bought, sold, or held, while creating a portfolio that is concentrated enough to outperform the market.
Principal’s portfolio optimization problem formulation had the following characteristics:
(1) Active weights, defined as the difference between the asset weight in the portfolio and the asset weight in a benchmark; (2) A Markowitz objective function balancing risk (tracking error) and excess return; (3) Long-only (nonnegative weights); (4) No leverage; that is, the entire portfolio had to be invested; (5) Limits on the active weight per asset to within 5% of the benchmark weight; (6) Limits on sector active weights to within 10% of the benchmark; (7) Limits on the market cap quintile active weight to within 10% of the benchmark; (8) Limits on the portfolio beta (β) calculated in terms of active weights: −0.1 ≤ β ≤ 0.1; (9) Cardinality of the portfolio, with a target range on the number of stocks between 50 and 70; and (10) Upper and lower bounds on the portfolio tracking error.
Solving the problem to optimality is difficult given its nonconvexity, size, and combinatorial nature. The collection of papers in this special issue represents a number of approaches—some more successful than others—that utilized numerous heuristics and analytical tools to solve the problem. We have chosen to include this range of tools here to report both on techniques that work and on others that do not quite work, hoping to inspire further research on the algorithms outlined by the papers in the issue. These articles show that to the extent that both performance and ease of use of a mathematical tool are important for use in practice, several different mathematical techniques and heuristics can be effective in solving practical problems to provide high-quality solutions. Furthermore, many of these methods formalize managerial intuition or rules of thumb used in industry, which leads to easier adoption. At the same time, from an academic perspective, these techniques are effective in solving the problem to different extents. It will be of academic interest and value to understand what specific form or structure makes a problem amenable to specific heuristics, with the ultimate goal of developing a mapping between problem characteristics and successful solutions.
The articles in this issue are of two types: three full-length articles that describe theory-driven approaches whose portfolio allocations outperformed the benchmark and four technical notes that describe heuristics that either did not outperform the benchmark or performed well but could benefit from further theoretical development and computational studies.
The concepts of learning and local search are central in the three full-length articles. “GAN-MP Hybrid Heuristic Algorithm for Nonconvex Portfolio Optimization Problem,” by Yerin Kim, Daemook Kang, Mingoo Jeon, and Chungmok Lee, combines generative adversarial networks (GANs) with mathematical programming to tackle the cardinality constraint in the problem posed by Principal. The authors use GAN to generate the initial set of assets, optimize the portfolio, and conduct local search to improve on the current set of assets. In “A Three-Phase Approach to an Enhanced Index-Tracking Problem with Real-Life Constraints,” Oliver Strub, Stefano Brandinu, Dennis Lerch, Jürgen Schaller, and Norbert Trautmann break up the portfolio optimization process into three stages: preprocessing, optimization, and learning. They determine feasible portfolios that are as close as possible to proven optimality for given values of learning parameters associated with penalties on some constraints through the preprocessing and optimization phases and then learn better values for the penalty parameters over time. In “Variable Neighborhood Search Heuristic for Nonconvex Portfolio Optimization,” Andrijana Bačević, Nemanja Vilimonović, Igor Dabić, Jakov Petrović, Darko Damnjanović, and Dušan Džamić suggest an efficient local search procedure for finding optimal solutions that relies on search space analysis and “shaking” or perturbing the solution.
The four technical notes represent applications of interesting methodologies from the literature. Laurenz Roebers and co-authors (“Using Column Generation to Solve Extensions to the Markowitz Model”) suggest using column generation to handle the cardinality constraint. In contrast, Tao Jiang and co-authors (“An Inexact l2-Norm Penalty Method for Cardinality-Constrained Portfolio Optimization”) control the cardinality of the portfolio using an l2-norm penalty term on the portfolio weights. Juan Díaz and co-authors (“Index Fund Optimization Using a Hybrid Model: Genetic Algorithm and Mixed-Integer Nonlinear Programming”) use genetic algorithms for preliminary selection of the desired number of stocks in the portfolio. Enis Kayiş and co-authors (“An Extension to the Classical Mean–Variance Portfolio Optimization Model”) explore a desirability criterion to select a subset of stocks with an element of randomization to avoid getting stuck in local optima.
We thank the many people without whom this issue would not have been possible. Joe Byrum, Chief Data Scientist at Principal, was a tireless advocate for the INFORMS competition and this special issue, and was incredibly generous with his time. Professor John Birge of the University of Chicago, chair of the judging committee of the competition, advised on the parameters of the special issue. Terry Cryan of INFORMS helped with the initial outreach. Several members of the organizing and judging committees also served as referees for this issue: Jim Bander, Hande Benson, David Bergman, Janne Kettunen, and Irv Lustig. Our sincerest thanks also go to the many other anonymous reviewers, including authors who submitted papers for the special issue and agreed to help with the review process. Last but not least, we thank Professor Sarah Ryan, Editor-in-Chief of The Engineering Economist, for her guidance and responsiveness.