A hybrid optimization and data-driven approach to understand the role of the risk-aversion profile parameter in portfolio optimization problems with shorting constraints

A mean-variance model is proposed for portfolio rebalancing optimization problems with transaction costs and minimum transaction lots. The portfolio optimization problems are modeled as a non-smooth nonlinear integer programming problem. A genetic algorithm based on real value genetic operators is designed to solve the proposed model. It is illustrated via a numerical example that the genetic algorithm can solve the portfolio rebalancing optimization problems efficiently.KeywordsGenetic AlgorithmTransaction CostPortfolio OptimizationGenetic OperatorPortfolio Selection ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Подано нову теорію портфельної оптимізації в умовах невизначеності, що ґрунтується на застосуванні теорії нечітких множин та ефективного методу прогнозування. Розглянуто пряму та двійкову задачі нечіткої портфельної оптимізації. У прямій задачі визначається структура оптимального портфеля цінних паперів, яка забезпечує максимум очікуваної дохідності при обмеженнях на ризик, у двійковій задачі – структура портфеля, який забезпечує мінімум ризику при обмеженнях на встановлений рівень критичної дохідності. Для визначення дохідності цінних паперів запропоновано метод прогнозування нечіткого методу врахування аргумента, що дозволить будувати нечіткі прогнозні моделі за експериментальними даними автоматично за незначної участі експерта. Описано експериментальні дослідження запропонованої теорії та проведено порівняння з класичною моделлю портфельної оптимізації.

The novel theory of investment portfolio optimization under uncertainty is presented based on fuzzy set theory and efficient forecasing methods. The direct problem of fuzzy portfolio optimization and dual problem are considered. In the direct problem structure of a portfolio is determined which provides the maximum profitableness at the given risk level. In dual problem the portfolio structure is determined which provides the minimum risk level at the set level of critical profitableness. The experimental investigations of the suggested theory were carried out and comparison with classical portfolio model was performed.

Portfolio optimization (selection) problem is an important and hard optimization problem that, with the addition of necessary realistic constraints, becomes computationally intractable. Nature-inspired metaheuristics are appropriate for solving such problems; however, literature review shows that there are very few applications of nature-inspired metaheuristics to portfolio optimization problem. This is especially true for swarm intelligence algorithms which represent the newer branch of nature-inspired algorithms. No application of any swarm intelligence metaheuristics to cardinality constrained mean-variance (CCMV) portfolio problem with entropy constraint was found in the literature. This paper introduces modified firefly algorithm (FA) for the CCMVportfolio model with entropy constraint. Firefly algorithm is one of the latest, very successful swarm intelligence algorithm; however, it exhibits some deficiencies when applied to constrained problems. To overcome lack of exploration power during early iterations, we modified the algorithm and tested it on standard portfolio benchmark data sets used in the literature. Our proposed modified firefly algorithm proved to be better than other state-of-the-art algorithms, while introduction of entropy diversity constraint further improved results.

The problem of portfolio optimization under uncertainty is considered. For its solution the application of fuzzy sets theory is suggested. Fuzzy portfolio optimization problem is stated; its model is provided and investigated, as well as the algorithm of its solution is presented in the article. The problem of multicriteria fuzzy portfolio optimization is also considered and investigated. This problem includes two main criteria – portfolio profitability and risk. A mathematical model of this problem is constructed, explored and the sufficient conditions for its convexity are obtained. For better estimation of stock profitability, Fuzzy Group Method of Data Handling (FGMDH) for stock price forecasting is suggested. The experimental investigations of the suggested approach are carried out and their results – optimal portfolios based on the projected stock prices are presented, and its efficiency is evaluated.

In this study, the portfolio optimization problem is explored, using a combination of classical and quantum computing techniques. The portfolio optimization problem with specific objectives or constraints is often a quadratic optimization problem, due to the quadratic nature of, for example, risk measures. Quantum computing is a promising solution for quadratic optimization problems, as it can leverage quantum annealing and quantum approximate optimization algorithms, which are expected to tackle these problems more efficiently. Quantum computing takes advantage of quantum phenomena like superposition and entanglement. In this paper, a specific problem is introduced, where a portfolio of loans need to be optimized for 2030, considering ‘Return on Capital’ and ‘Concentration Risk’ objectives, as well as a carbon footprint constraint. This paper introduces the formulation of the problem and how it can be optimized using quantum computing, using a reformulation of the problem as a quadratic unconstrained binary optimization (QUBO) problem. Two QUBO formulations are presented, each addressing different aspects of the problem. The QUBO formulation succeeded in finding solutions that met the emission constraint, although classical simulated annealing still outperformed quantum annealing in solving this QUBO, in terms of solutions close to the Pareto frontier. Overall, this paper provides insights into how quantum computing can address complex optimization problems in the financial sector. It also highlights the potential of quantum computing for providing more efficient and robust solutions for portfolio management.

Multiobjective optimization (MO) is the problem of simultaneously optimizing two or more conflicting objectives subject to certain constraints. Many real-world problems involve simultaneous optimization of several often conflicting objectives. The portfolio optimization problem belongs to this category of problems. According to Markowitz’s Mean - Variance model (MV) an investor attempts to maximize portfolio expected return for a given amount of portfolio risk or minimize portfolio risk for a given level of expected return. The portfolio optimization problem involves two conflicting objectives (i.e. expected return and portfolio risk) and thus belongs to the family of multiobjective problems. With the assistance of scalarization techniques a multiple objective problem can be converted into a single objective problem. However, the drawbacks to these conventional approaches lead to the development of alternative techniques that yield a set of Pareto optimal solutions rather than only a single solution. The problem becomes much more complicated when we incorporate to the portfolio model some real world constraints. These additional constraints made the portfolio optimization problem difficult to be solved with exact methods. In the last decade several metaheuristic optimization techniques have been developed to address the challenges imposed by complex multiobjective optimization problems. Due to the intrinsic multiobjective nature of the portfolio optimization problem, multiobjective approaches, particularly multiobjective evolutionary algorithms (MOEAs) are suitable in handling the difficulties imposed by this type of problems. Especially in the presence of multiple constraints the portfolio optimization problem becomes very complicate and efficient solution needs to be found.Furthermore, the existing multiobjective evolutionary algorithms (MOEAs) techniques cannot be used directly to solve the constrained portfolio optimization problem as a number of configuration issues related to the application of MOEAs for solving the constrained portfolio optimization problem must be addressed. The successful implementation of the constrained portfolio optimization problem by the MOEAs requires the development of novel algorithmic and technical approaches. In particular new multiobjective evolutionary approaches are needed to efficiently solve the constrained portfolio optimization problem. In this thesis we address these issues by examining a number of configuration issues related to the application of MOEAs for solving the constrained portfolio optimization problem. Furthermore we introduce a new multiobjective evolutionary algorithm (MOEA) that incorporates a novel representation scheme and specially designed genetic operators for the solution of the constrained portfolio optimization problem. These issues have been addressed in this thesis and a set of efficient solutions is found for each of the examined test problems. In this thesis we develop a methodological framework for conducting a comprehensive literature study based on the papers published in MOEAs for the Portfolio Management over a long time span across various disciplines. This framework is being used to gain an understanding of the current state of the MOEAs for the Portfolio Management research field. Based on the literature study, we identify potential areas of concern in regard to MOEAs for the Portfolio Management. Based on the examination of the state-of-the art we present the best practices from a technical and algorithmic point of view for dealing with the complexities of the constrained portfolio optimization problem. We introduce new genetic operators to enhance algorithms’ performance. We propose a novel representation scheme for the solution of the constrained portfolio optimization problem. Finally, we introduce a novel MOEA for the solution of the constrained portfolio optimization problem. The experimental results applied to the constrained portfolio optimization problem, indicate that the proposed approach generates solutions that lie on the true efficient frontier (TEF) for all of the examined cases for a fraction of time required by exact approaches.

Solving the Optimal Trading Trajectory Problem Using Simulated Bifurcation

The subject of the study in this paper is models and methods of optimization of the organization's project portfolio for the planning period, considering the effects of the previously made decisions. Project portfolio optimization is one of the responsible and complex tasks by company's top management solves. Based on the analysis of the known works in the field, the research purpose is described: to create a method that would allow solve the problem of multi-criteria project portfolio optimization for the planned period, considering the aftereffects of the previously made decisions. The research tasks are to enhance the method for solving the project portfolio optimization problem in terms of maximizing the difference between income and costs for all projects started during the planned period; to propose a method for solving the project portfolio optimization problem in terms of the social effects of projects that started during the planned period; create a method for solving the problem of project portfolio optimization for the planned period in a multi-criteria setting. There are the following results obtained in the paper. There is presents the mathematical model of the problem being solved, the problem objective functions include the difference between the receipt and expenditure of funds in time, the portfolio risks, and its implementation social effects. The mathematical model considers the provision of funds sufficiency for the implementation of projects in all periods, the required sequence of project implementation, and the mandatory inclusion of some projects in the portfolio for a given period. The problem under consideration belongs to the multi-criteria non-Markov dynamic discrete optimization problems. There is a proposed method for solving it in a multi-criteria formulation. The method is based on solving one criterion problem, and then a multi-criteria problem. The method is based on the minimax approach and implicit search. There has been developed solving method for the problem of enterprise project portfolio optimization for the planned period following the profit criterion. In contrast to the existing methods, this method considers the constraints on debt absence and the aftereffects of the previously made decisions. The method served as the basis for creating risk and social effect optimization methods. A method for enterprise project portfolio optimization of the planned period is provided, which, unlike previous, considers the criteria of profit, risks, and social effect, the constraints on debt absence, and the aftereffect of the previously made decisions. That makes it possible to improve the quality of the generated portfolio.

The portfolio optimization problem in which the variances of the return rates of assets are not identical is analyzed in this paper using the methodology of statistical mechanical informatics, specifically, replica analysis. We defined two characteristic quantities of an optimal portfolio, namely, minimal investment risk and investment concentration, in order to solve the portfolio optimization problem and analytically determined their asymptotical behaviors using replica analysis. Numerical experiments were also performed, and a comparison between the results of our simulation and those obtained via replica analysis validated our proposed method.

In Low Order-Value Optimization (LOVO) problems the sum of the r smallest values of a finite sequence of q functions is involved as the objective to be minimized or as a constraint. The latter case is considered in the present paper. Portfolio optimization problems with a constraint on the admissible Value at Risk (VaR) can be modeled in terms of a LOVO problem with constraints given by Low order-value functions. Different algorithms for practical solution of this problem will be presented. Using these techniques, portfolio optimization problems with transaction costs will be solved.

Introduction. The problem of decision-making under risk and uncertainty lies in the use of adequate criteria for assessing their optimality, in particular, in an adequate risk assessment. Various functions are known that are used as risk measures. For technical systems, the probability of an accident (failure) is used, in insurance – the probability of bankruptcy, in finance – Value-at-Risk, etc. At present, the concept of a coherent risk measure (CRM), in which its basic properties are postulated, is widely recognized. The paper considers CRMs and their subset, the polyhedral CRMs (PCMRs), which have attractive properties and contain a number of important risk measures. Such risk measures are well defined on complete information about the stochastic distributions of random variables. However, applications usually contain only partial such information from observational data. This only allows one to describe the stochastic distribution by an ambiguity set (AS). For such a case, robust PCMR constructions intended for risk assessment at AS are considered in the paper. The computation of such PCRM constructions in the form of linear programming problems (LP) is described. To demonstrate the use of the PCRM apparatus, the problems of portfolio optimization on reward-risk ratio are considered, where reward and risk are estimated by the average return and some PCRM respectively for known stochastic distributions, and by their robust constructions under uncertainty with AS. It is described how in both these cases the portfolio optimization problems are reduced to appropriate LP problems. The purpose of the paper is to describe the PCRM apparatus for assessing risks under uncertainty with AS and demonstrating the effectiveness of its application to linear problems on the example of portfolio optimization problems. Results. The use of the PCRM apparatus for the case of uncertainty with AS in the form of appropriate robust constructions and their application to portfolio optimization problems on reward-risk ratio is described. The conditions under which these portfolio problems are reduced to the corresponding LP tasks are formulated. Conclusions. The PCRM apparatus can be effectively applied to linear optimization problems under uncertainty with AS, which is demonstrated by the example of portfolio optimization problems. The reduction of portfolio problems to LP problems allows one to effectively solve them using standard methods. Keywords: coherent risk measure, polyhedral coherent risk measure, CVaR, ambiguity set, portfolio optimization, linear programming problem.

Solving norm constrained portfolio optimization via coordinate-wise descent algorithms

A Note on Sparse Minimum Variance Portfolios and Coordinate-Wise Descent Algorithms

Portfolio optimization models are usually based on several distribution characteristics, such as mean, variance or Conditional Value-at-Risk (CVaR). For instance, the mean-variance approach uses mean and covariance matrix of return of instruments of a portfolio. However this conventional approach ignores tails of return distribution, which may be quite important for the portfolio evaluation. This chapter considers the portfolio optimization problems with the Stochastic Dominance constraints. As a distribution-free decision rule, Stochastic Dominance takes into account the entire distribution of return rather than some specific characteristic, such as variance. We implemented efficient numerical algorithms for solving the optimization problems with the Second-Order Stochastic Dominance (SSD) constraints and found portfolios of stocks dominating Dow Jones and DAX indices. We also compared portfolio optimization with SSD constraints with the Minimum Variance and Mean-Variance portfolio optimization.