Cardinality-Constrained Portfolios: Optimization Approach & Algorithm

Every portfolio can be partitioned into multiple asset groups defined by asset classes, sectors, styles, and other features. A cardinality-constrained portfolio caps the number of stocks to be traded within each of these groups. These limitations arise from real-world scenarios faced by fund managers who seek to satisfy certain investment mandates or achieve their asset allocation objectives.

As an example, suppose you want to construct a portfolio by investing in stocks across m sectors you favor. And in each sector you select up to k stocks and each sector should not constitute more than q% of your portfolio. Moreover, you don’t know which and how many stocks should be included yet. You’ll also need to determine the portfolio weights based on your risk-return tradeoff. Now, imagine you can do all that automatically by running an algorithm.

In this paper, we develop a new approach to solve cardinality-constrained portfolio optimization problems with different constraints and objectives. In particular, our approach extends both Markowitz and conditional value at risk (CVaR) optimization models with cardinality constraints. A continuous relaxation method is proposed for the NP-hard objective, which allows for very efficient algorithms with standard convergence guarantees for nonconvex problems.

For smaller cases, where brute force search is feasible to compute the globally optimal cardinality-constrained portfolio, the new approach finds the best portfolio for the cardinality-constrained Markowitz model and a very good local minimum for the cardinality-constrained CVaR model.

As the total number of assets grows, brute-force exhaustive search quickly becomes prohibitively expensive. For instance, choosing 10 assets out of 30 requires solving more than 30 million optimization problems over the subsets. Our algorithm can solve problems of this scale on an average of 20 runs. We find feasible portfolios that are nearly as efficient as their non-cardinality constrained counterparts.

We are given a total of n candidate assets and a certain selection criterion f(w). The portfolio weights satisfy the simplex constraint...