Diversifying Investments and Maximizing Sharpe Ratio: A Novel Quadratic Unconstrained Binary Optimization Formulation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper is devoted to study of maximizing Sharpe Ratio using a novel QUBO formulation and computer modelling methods based on quantum annealing approaches. First of all, authors describe an overview of optimization in economics. After that, they formulate the problem of optimization and describe quantum annealing. Then they speak about quantum computers, Sharp Ratio maximization in quantum computers and diversification of investments over multiple sectors. Then they pass to a QUBO formulation of their problems. It is interesting that the quadratic formulation is used. Then they describe the features of the problem of diversified portfolio optimization and then pass to the computer modelling, giving some important examples. After that, they pass to the conclusion.

The paper is quite interesting and applicable for the journal. However, it is necessary to take into account some remarks:

1. The methods of minimization are well-known, and the authors describe well-known mathematical facts. What is the unique of their work? It should be stated exactly.

2. Such methods have been used in another branches of science, especially in physics and industry. They should be described in the paper.

3. The reference list was made carelessly, so it is difficult to find corresponding papers in databases. For example, important reference [10] does not even contain a year of publication.

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

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Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

The authors' work falls within the portfolio selection problem. The authors propose using quantum optimization techniques to maximize the Sharpe ratio of the portfolio, with an additional objective added, specifically the need to diversify between economic sectors.

In my opinion, the work presents several problems. Some of them have relatively simple solutions. However, others are structural.

1. Although the work is well-written, the financial terminology used is not orthodox: "portfolio" is preferable to "basket," and when the authors refer to "financial services," someone might think of payment methods or internet banking. "Financial Economy" would be better.
2. Some acronyms, such as QUBO, should be expanded the first time they appear in the abstract and introduction.
3. Some concepts and claims are not supported by the literature. For example:
• The economic sectors in which stock indices are divided can be more or less comprehensive. Thus, they are always somewhat arbitrary. Where does the author get this? (lines 79 and onwards)
• Despite the simplicity of this approach, its NP-hardness guarantees the possibility of expressing a vast class of problems. Which problems? (Line 115)
• The authors indicate that "Portfolio Optimization is a family of combinatorial optimization problems where the objective is to find an optimal allocation of weights for different assets." Where do the authors get this definition? Portfolio simply involves finding the optimal portfolio based on a certain criterion, and it has nothing to do with "combinatorial optimization," although it can be used. It may consist, for example, of minimizing the price to invest in a set of bonds to match cash flows on a specific date. It is related to financial economics and not optimization theory. The Markowitz model defines w1, w2, etc., as variables in [0,1], and therefore, it is not "combinatorial optimization," although it may be necessary to pose a discrete problem if we seek integer solutions (for example, the number of assets to acquire from each class).
• "For our dataset, we find that µmin = 0.00245" - a deeper explanation must be provided.
4. A key aspect is the fact that the authors introduce the need for assets to be diversified between sectors. However, as it is introduced, the need for diversification seems like an artificial problem proposed by the authors, since initially, the relationship between assets is given by the correlation matrix. This aspect should be better motivated, as it is the main focus of the work.
5. The assumptions underlying the model should be established: borrowing is not possible, and neither are short positions in stocks. These are common assumptions in portfolio selection models, but they need to be explicitly stated.
6. As the work is developed, it is difficult to see what the mathematical program to be solved finally is and what the auxiliary one solved with combinatorial optimization is. Although the authors number the equations, there is no reference to the ones they originate from in the generation of new equations. For example, there should be a more integrated view of the relationship between the basic problem (5) and onwards with (15) and onwards, how the diversification constraints between sectors are integrated into the Sharpe ratio problem, or how the auxiliary combinatorial program is formulated. The authors present concepts of quantum optimization and portfolio selection in a disjointed manner, making it difficult to see how they integrate.
7. The problem seems convex if we work with proportions defined in [0,1]. Authors must better motivate the need to discretize decision variables. It would make sense if the problem consisted of determining the number of shares to buy of each type, as they must be integers. However, as it is presented (the decision variable being proportions), "discretizing" the decision variable doesn't seem justified.
8. It would be beneficial to provide a more detailed description of the dataset used for the analysis, including the number of stocks in each sector, average returns, standard deviations, etc.
9. The numerical application simply analyzes the sensitivity of their methodology to changes in the parameters for implementing QUBO, but it remains self-contained. The proposed method should be compared with alternative methods: does it yield "better optima," is it computationally more efficient? Note that as the authors present the programs (5) and onwards and (15) and onwards, the decision variables are continuous.
10. Related to 9, there should be a discussion on what exactly this work contributes to the literature. The authors did not motivate why extending the Sharpe ratio problem to additional diversification conditions (between sectors) is important, nor do they compare their quantum method with alternative resolution strategies.
11. Some references are incomplete. For example, in reference [10], the journal is missing.

Comments on the Quality of English Language

Some financial concepts terms must be revised.

Author Response

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Reviewer 4 Report

Comments and Suggestions for Authors

The authors make use of quadratic unconstrained binary optimization (QUBO) to maximize a Sharpe ratio in a model for diversifying investments.

  The problem is interesting and it appears to be well posed. But some points need to be clarified before publication:   1) In eq(1), and some other equations after it, there is an "arg". It is not clear what it represents in such an expression.      2) One of the main steps for the new formulation proposed by the authors is a different encoding of the variables defining the QUBO setup in eq.(8). In eq.(9) the authors propose to replace the 1-bit variable by a rescaled p-bit variable. One of the hallmarks of the QUBO formulation is the fact that for a 1-bit variable x, one has x^2 = x. This allows for the fact that many polynomial expressions can be mapped in the quadratic form of the QUBO. Is there any reduction in the range of applicability by considering the encoding proposed by the authors?    I believe the paper deserves to be published, but these issues should be addressed first.

Author Response

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Reviewer 5 Report

Comments and Suggestions for Authors

In this manuscript ‘Diversifying Investments and Maximizing Sharpe Ratio: a novel QUBO formulation’, the authors investigate classical-quantum hybrid methods to solve The Portfolio Optimization task and multi-objective optimization problem. In details, they optimiz the Sharpe Ratio and a diversification term by a new QUBO formulation and then solve the problem via quantum annealing devices or Hybrid Computing.

The authors also discuss the behavior of the hybrid computing approaches, and find the results show the trade-off between the observed values of the portfolio’s Sharpe Ratio and diversification.

 

The optimization problem is important and widely applied in different areas. I believe that their calculations are correct in principle. Therefore, I approve their manuscript to be accept after rewriting the abstract which should be compacted in one paragraph and focus on their own work.

 

Author Response

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Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I think that the paper has been improved. So, to my mind it can be published

Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

Even though the paper has improved, there are still some aspects that are critical and need to be enhanced.

1. Regarding the response to query 4, it's not very convincing. It's an intuition from the authors that doesn't seem to be supported by the literature. In fact, there are portfolios that specialize in specific sectors (for example, real estate) (https://hedgelists.com/hedge-fund-lists-by-strategy/real-estate-hedge-fund-list-p-47/).

2. The basic problem of Sharpe ratio optimization can become convex. Regarding the issue of "diversification" among sectors (which the authors have not shown to be a real concern in portfolio selection), linear constraints could be used, so the programming problem would not lose convexity. For j=1,2,...,M sectors,

Sum of wi values belonging to sector j <= maximum allowed proportion in sector j

3. As I mentioned in the previous evaluation, there is no discussion about what the work contributes to the literature. The problem itself doesn't seem to be connected to financial literature (sector diversification), nor does it seem to require abandoning the convexity of the original problem. We also don't know how the proposed optimization methodology improves upon more common ones. Therefore, it's difficult to determine what this work contributes to financial literature and/or literature related to computational methods.

Minor issues:

1. When referring to equations by their number, it should be enclosed in parentheses, for example, (8).

2. Sometimes capital letters seem improperly used.

Comments on the Quality of English Language

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Author Response

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Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

 

The authors either did not want to or were unable to address the most critical issues. In fact, in the new draft, no new text is perceived, either highlighted or in a different color. (a) The authors do not seem to address a real financial problem, as they are unable to reference financial literature where the need to limit values associated with the same economic sector is motivated. Certainly, the literature on portfolio selection is extensive. (b) The authors still do not develop a discussion that references their approach and results to alternative approaches or connects their approach/results with those reported by the portfolio selection literature. (c) As minor errors, typos/format errors still persist (e.g., "4. QUBO FORMULATION... must not be written in capital letters").

Comments on the Quality of English Language

No work to improve the paper has been done. So, I must reject the paper. 

Author Response

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