Enhancing an existing algorithm for small-cardinality constrained portfolio optimisation

Abstract

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.

Keywords:

• Cardinality constraint
• mean-variance portfolio optimisation
• portfolio selection

Disclosure statement

The authors report there are no competing interests to declare.

Data availability statement

Some of the datasets used in this study are publicly available at http://people.brunel.ac.uk/∼mastjjb/jeb/orlib/files/. The datasets we created are available upon request.

Notes

1 The code can be downloaded from https://github.com/MJCraven/SiftedQP.

2 Although this is not shown in Figure 1, an EF can dominate another EF even if parts of the curves overlap.

3 The datasets can be obtained from http://people.brunel.ac.uk/∼mastjjb/jeb/orlib/files/ and are titled port1.txt through to port5.txt.

4 Using log returns instead of simple returns for our calculations (which we also do for our own datasets, described in the next paragraph) means we are approximating the portfolio return rather than computing it exactly.

5 Graham and Craven (2021) state that they use an approximated third point that differs from what we have described. However, the code they provide uses the approximation we have outlined, and is much more reasonable than what is written in the paper (which we believe was written in error).

6 In some cases, a sub-EF may consist of only a single point because of the risks and returns associated with the assets in the portfolio. Such cases result in an error when running the algorithm and should be discarded. This can be done by adding a condition that the maximum and minimum return of a sub-EF must differ by more than some threshold.

7 This research was aided by support from Compute Ontario (www.computeontario.ca) and Compute Canada (www.computecanada.ca).

8 https://github.com/nphelps/Small_cardinality_portfolio_optimisation

9 These values are not continuously compounded. However, for values of this size the continuously compounded return is approximately the same.

10 https://github.com/nphelps/Small_cardinality_portfolio_optimisation

Additional information

Funding

This work was supported by the Canadian Financial Wellness Lab, which receives funding from the NSERC Alliance Grant Program (Grant Number: ALLRP 566997-21).