Enhancing an existing algorithm for small-cardinality constrained portfolio optimisation

The efficient frontier (EF) allows an investor to (in theory) maximise their return for a given level of risk. Portfolios on the EF may contain many assets, making their management difficult and possibly expensive. An investor may wish to impose an upper bound on the number of assets in their portfolio, leading to so-called cardinality constrained efficient frontiers (CCEFs). Recently, a new algorithm was developed to find CCEFs for small cardinalities. Relative to other algorithms for this problem, this algorithm is very intuitive, and its authors demonstrated that it performs at nearly the state-of-the-art. However, we have found that the algorithm seems to struggle in certain situations, particularly when faced with both bonds and equities. While preserving its intuitiveness, we modified the algorithm to improve its CCEFs. This improvement comes with longer runtimes, but we think many practitioners will prefer the algorithm with modifications. Some practitioners may prefer other algorithms, due to the runtimes or because some points on our CCEFs still fall short of optimality. However, in addition to its intuitiveness, our modified algorithms (and the original version) find low-risk points on the CCEF that a state-of-the-art algorithm does not.