Summary
This study addresses the problem of Sharpe ratio maximization under a cardinality constraint, referred to in the paper as an m-sparse constraint. Adding a cardinality constraint typically makes optimization problems NP-hard. Existing studies usually approach this problem using heuristic methods or relaxations. The former can be time-consuming and do not guarantee optimality, while the latter struggles to control cardinality accurately. This research proposes transforming the m-sparse fractional optimization problem into an equivalent m-sparse quadratic programming problem, which ensures convergence to a locally optimal solution and allows for more precise control of cardinality. However, as will be discussed in the question section, the study essentially transforms the problem into a mean-variance optimization form. This raises the question of why not simply perform m-sparse mean-variance optimization. Since Sharpe ratio maximization is just one of many Pareto optimal points in mean-variance optimization, addressing the more general problem could be more advantageous. Also, as this paper deals with practical issue (i.e., cardinality constraint), the authors should discuss more about practical implementation of the proposed method (e.g., computational time, adding more practical constraints).
Strengths
- The proposed method appears to be theoretically sound and effectively controls cardinality.
- The approach has been tested on various datasets, and the experimental results indicate strong performance.
- The writing structure is well-organized and the content is presented in a clear and readable manner.
Weaknesses
- Ultimately, this research focuses on a portfolio optimization problem with a cardinality constraint driven by practical needs, making practical implementation the most critical aspect. However, there is a lack of discussion or analysis on this aspect.
- Practical implementation would require consideration of various additional constraints (e.g., lower or upper bound of portfolio weights, turnover constraint). It would be beneficial to demonstrate whether the proposed model can accommodate a range of additional constraints.
- There is a need for a comparative analysis of computation time to evaluate the efficiency of the proposed method.
- The following paper addresses the exact same problem of cardinality constrained Sharpe ratio maximization. While their approach, based on relaxation, has the advantage of transforming the problem into a convex optimization issue, it fails to control cardinality exactly.
[1] Kim, M. J., Lee, Y., Kim, J. H., & Kim, W. C. (2016). Sparse tangent portfolio selection via semi-definite relaxation. Operations Research Letters, 44(4), 540-543.
Questions
The authors emphasize several times in bold that directly maximizing the Sharpe ratio is a key point of this study. However, as shown in equations (3.4) and (3.7), the problem is eventually transformed into a mean-variance optimization form, with Theorem 1 demonstrating how this can be converted back to a solution for the Sharpe ratio maximization problem. This raises the question of why the study is framed as a Sharpe ratio maximization problem. Why not simply present it as m-sparse mean-variance optimization, which is more general? After all, Sharpe ratio maximization is just one of many Pareto optimal points in mean-variance optimization. Is this framing due to overlapping aspects with previous research? Further clarification on this point would be helpful.
Limitations
The authors mention the inability to directly apply fractional optimization as a limitation. However, I believe that more emphasis should be placed on practical implementation. Since this study defines and solves a problem based on practical needs, more attention should be given to practical implementation rather than purely mathematical formulation. Ensuring that the proposed method can be effectively implemented in real-world scenarios is crucial for its overall utility.