Portfolio Optimization Problem Research Articles

Solving Constrained Mean-Variance Portfolio Optimization Problems Using Spiral Optimization Algorithm

Portfolio optimization is an activity for balancing return and risk. In this paper, we used mean-variance (M-V) portfolio models with buy-in threshold and cardinality constraints. This model can be formulated as a mixed integer nonlinear programming (MINLP) problem. To solve this constrained mean-variance portfolio optimization problem, we propose the use of a modified spiral optimization algorithm (SOA). Then, we use Bartholomew-Biggs and Kane’s data to validate our proposed algorithm. The results show that our proposed algorithm can be an efficient tool for solving this portfolio optimization problem.

Benchmarking the performance of portfolio optimization with QAOA

We present a detailed study of portfolio optimization using different versions of the quantum approximate optimization algorithm (QAOA). For a given list of assets, the portfolio optimization problem is formulated as quadratic binary optimization constrained on the number of assets contained in the portfolio. QAOA has been suggested as a possible candidate for solving this problem (and similar combinatorial optimization problems) more efficiently than classical computers in the case of a sufficiently large number of assets. However, the practical implementation of this algorithm requires a careful consideration of several technical issues, not all of which are discussed in the present literature. The present article intends to fill this gap and thereby provides the reader with a useful guide for applying QAOA to the portfolio optimization problem (and similar problems). In particular, we will discuss several possible choices of the variational form and of different classical algorithms for finding the corresponding optimized parameters. Viewing at the application of QAOA on error-prone NISQ hardware, we also analyse the influence of statistical sampling errors (due to a finite number of shots) and gate and readout errors (due to imperfect quantum hardware). Finally, we define a criterion for distinguishing between ‘easy’ and ‘hard’ instances of the portfolio optimization problem.

OPTIMIZATION OF PORTFOLIO USING FUZZY SELECTION

The problem of portfolio optimization concerns the allocation of the investor’s wealth between several security alternatives so that the maximum profit can be obtained. One of the methods used is Fuzzy Portfolio Selection to understand it better. This method separates the objective function of return and the objective function of risk to determine the limit of the membership function that will be used. The goal of this study is to understand the application of the Fuzzy Portfolio Selection method over shares that have been chosen on a portfolio optimization problem, understand return and risk, and understand the budget proportion of each claim. The subject of this study is the shares of 20 companies included in Bursa Efek Indonesia from 1 January 2021 until 1 January 2022. The result of this study shows that from 20 shares, there are 10 shares that is suitable in the forming of optimal portfolio, those are ADRO (0%), ANTM (43.3%), ASII (0%), BBCA (0%), BBRI (0%), BBTN (0%), BRPT (0%), BSDE (0%), ERAA (16%), and INCO (40.7%). The expected return from the portfolio is 0.0878895207 or 8.8% for the return and 0.0226022117 or 2.3% for the risk.

Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing

In this work, we introduce a new workflow to solve portfolio optimization problems on annealing platforms. We combine a classical preprocessing step with a modified unconstrained binary optimization (QUBO) model and evaluate it using simulated annealing (classical computer), digital annealing (Fujitsu’s Digital Annealing Unit), and quantum annealing (D-Wave Advantage). Starting from Markowitz’s theory on portfolio optimization, our classical preprocessing step finds the most promising assets within a set of possible assets to choose from. We then modify existing QUBO models for portfolio optimization, such that there are no limitations on the number of assets that can be invested in. Furthermore, our QUBO model enables an investor to also place an arbitrary amount of money into each asset. We apply this modified QUBO to the set of promising asset candidates we generated previously via classical preprocessing. A solution to our QUBO model contains information about what percentage of the whole available capital should be invested into which asset. For the evaluation, we have used publicly available real-world data sets of stocks of the New York Stock Exchange as well as common ETFs. Finally, we have compared the respective annealing results with randomly generated portfolios by using the return, variance, and diversification of the created portfolios as measures. The results show that our QUBO formulation is capable of creating well-diversified portfolios that respect certain criteria given by an investor, such as maximizing return, minimizing risk, or sticking to a certain budget.

Multi-objective interval portfolio optimization modeling and solving for margin trading

Margin trading is an important part of stock market. Compared with traditional channel business, it brings excess profits to both financial institutions and investors. As an important optimization problem in the field of finance, portfolio optimization with the purpose of spreading risks and improving returns is the essence of margin trading. In this paper, aiming at achieving the portfolio optimization for multi-period margin trading, we propose a multi-objective interval portfolio optimization method, including model construction and optimization. Based on real business rules, the process of model construction consists of three steps. First, an interval estimation method of securities’ returns and risks is designed according to market feedback to keep portfolios remaining robust under the market fluctuation. Second, a multi-objective problem of maximizing return and minimizing risk is constructed to seek portfolios with balanced risks and returns. Third, a rolling investment mechanism is formed based on the portfolio’s actual return of each period to maintain effective continuous investment. For the model optimization, a multi-objective interval optimization evolutionary algorithm with knee based decision-making is developed. It obtains knee solutions for different risk segments to meet the investment needs of investors with different risk preferences, so that their investment decisions do not need to rely on quantitative expression of their preferences. Furthermore, adapting to the interval model, an interval optimization method is developed to enhance the robustness of knee points. Experimental results on several benchmark problems and the real portfolio optimization problem demonstrate the superiority of the proposed model and algorithm.

On Polyhedral Coherent Risk Measures and Portfolio Optimization Problems

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.

Multi-period portfolio selection based on uncertainty theory with bankruptcy control and liquidity

Due to the impact of uncertain events, such as the 2008 financial crisis and the outburst of COVID-19 pandemic, the experts’ evaluations information is becoming increasingly important. This paper considers a multi-period portfolio optimization problem under uncertain circumstance, and the return rates of risky securities are regarded as uncertain variables, where the uncertainty theory is used to deal with experts’ evaluations. In light of the complexity of financial markets, we formulate an uncertain multi-period mean-entropy-variance model, where bankruptcy, liquidity, diversification and self-financing are considered as realistic constraints. Furthermore, the maximum return and the minimum risk are both achieved in a single-objective model through the normalization method. Then the equivalent deterministic forms of two secondary models for main model are provided. In addition, we develop a modified root system growth algorithm, which is more suitable for the proposed model. Finally, the effectiveness of the proposed model and designed algorithm is confirmed by numerical simulations.

An Algorithm for Business Management Based on Portfolio Optimization

An algorithm is derived for active business management. The key component of the algorithm is the definition and solution of an appropriate portfolio problem. For the last one, the disbursements for the business management are regarded as potential portfolio resources. For the portfolio definition, the increases or decreases of the disbursements are assumed to be the assets, whose weights are found as solutions from the portfolio optimization problem. These solutions recommend the reallocation of the resources between different disbursements, which increases the income of the business entity. The definition and solution of this portfolio problem are made sequentially in time, and the obtained solutions are applied as recommendations for future business management steps. An algorithm for business management, based on the sliding mode of repetitive definitions and solutions of portfolio problems with historical data of last disbursements, gives recommendations for the reallocation of resources for the next future period of management. The algorithm is numerically tested with real data on animal husbandry from Bulgaria. The empirical results demonstrate an advantage in increasing the husbandry income in comparison with the lack of such an active business policy. The algorithm can be implemented as a software solution in an appropriate programming system, supporting fintech service for active business management.

From deterministic to stochastic: an interpretable stochastic model-free reinforcement learning framework for portfolio optimization.

As a fundamental problem in algorithmic trading, portfolio optimization aims to maximize the cumulative return by continuously investing in various financial derivatives within a given time period. Recent years have witnessed the transformation from traditional machine learning trading algorithms to reinforcement learning algorithms due to their superior nature of sequential decision making. However, the exponential growth of the imperfect and noisy financial data that is supposedly leveraged by the deterministic strategy in reinforcement learning, makes it increasingly challenging for one to continuously obtain a profitable portfolio. Thus, in this work, we first reconstruct several deterministic and stochastic reinforcement algorithms as benchmarks. On this basis, we introduce a risk-aware reward function to balance the risk and return. Importantly, we propose a novel interpretable stochastic reinforcement learning framework which tailors a stochastic policy parameterized by Gaussian Mixtures and a distributional critic realized by quantiles for the problem of portfolio optimization. In our experiment, the proposed algorithm demonstrates its superior performance on U.S. market stocks with a 63.1% annual rate of return while at the same time reducing the market value max drawdown by 10% when back-testing during the stock market crash around March 2020.

Portfolio construction as linearly constrained separable optimization

Mean–variance portfolio optimization problems often involve separable nonconvex terms, including penalties on capital gains, integer share constraints, and minimum nonzero position and trade sizes. We propose a heuristic algorithm for such problems based on the alternating direction method of multipliers (ADMM). This method allows for solve times in tens to hundreds of milliseconds with around 1000 securities and 100 risk factors. We also obtain a bound on the achievable performance. Our heuristic and bound are both derived from similar results for other optimization problems with a separable objective and affine equality constraints. We discuss a concrete implementation in the case where the separable terms in the objective are piecewise quadratic, and we empirically demonstrate its effectiveness for tax-aware portfolio construction.

Neurodynamics-driven portfolio optimization with targeted performance criteria

This paper addresses portfolio selection with targeted performance criteria via neurodynamic optimization. Five portfolio optimization problems are formulated with a variable weight to maximize five risk-adjusted performance criteria in Markowitz’s mean–variance framework and reformulated as iteratively weighted convex optimization problems to facilitate subsequent problem-solving solution procedures. In addition, distributed portfolio optimization problems with separable performance criteria are also formulated. Three neurodynamic approaches are developed based on two globally convergent recurrent neural networks to solve the formulated and reformulated problems. Extensive experimental results on 13 datasets of world stock markets are elaborated to demonstrate the superior performance of the neurodynamic approaches against the baselines in terms of five given evaluation criteria and two investment returns.

A Generative Adversarial Network for Financial Advisor Recruitment in Smart Crowdsourcing Platforms

Financial portfolio management is a very time-consuming task as it requires the continuous surveying of the market volatility. Investors need to hire potential financial advisors to manage portfolios on their behalf. Efficient hiring of financial advisors not only facilitates their cooperation with investors but also guarantees optimized portfolio returns and hence, optimized benefits for the two entities. In this paper, we propose to tackle the portfolio optimization problem by efficiently matching financial advisors to investors. To this end, we model the problem as an automated crowdsourcing platform to organize the cooperation between the different actors based on their features. The recruitment of financial advisors is performed using a Generative Adversarial Network (GAN) that extrapolates the problem to an image processing task where financial advisors’ features are encapsulated in gray-scale images. Hence, the GAN is trained to generate, based on an investor profile given as an input, the ’ideal’ financial advisor profile. Afterwards, we measure the level of similarity between the generated ideal profiles and the existing profiles in the crowdsourcing database to perform a low complexity, many-to-many investor-to-financial advisor matching. In the simulations, intensive tests were performed to show the convergence and effectiveness of the proposed GAN-based solution. We have shown that the proposed method achieves more than 17% of the average expected return compared to baseline approaches.

A portfolio stock selection model based on expected utility, entropy and variance

In the context of investment decision-making, the selection of stocks is important for a successful construction of portfolios. In this paper the expected utility, entropy and variance (EU-EV) model is applied for stock selection, which can be used as preselection model for mean–variance portfolio optimization problems. Based on the EU-EV risk, stocks are ranked and the best ranked stocks with lower risk are selected in order to form subsets of stocks, which are then used to construct portfolios. The EU-EV model is applied to the PSI 20 index, to the Euro Stoxx 50 index and to the Nasdaq 100 index. Subsets of selected stocks are analysed and their portfolios’ efficiencies are compared with those of the portfolios obtained from the whole set of stocks using the mean–variance model. The results reveal that the EU-EV model is an adequate stock selection model for building up efficient portfolios with a lower number of stocks.

Distributionally robust portfolio optimization with second-order stochastic dominance based on wasserstein metric

In portfolio optimization, we may be dealing with misspecification of a known distribution, that stock returns follow it. The unknown true distribution is considered in terms of a Wasserstein-neighborhood of P to examine the tractable formulations of the portfolio selection problem. This study considers a distributionally robust portfolio optimization problem with an ambiguous stochastic dominance constraint by assuming the unknown distribution of asset returns. The objective is to maximize the worst-case expected return and subject to an ambiguous second-order stochastic dominance constraint. The expected return robustly stochastically dominates the benchmark in the second order over all possible distributions within an ambiguity set. It is also shown that the Wasserstein-moment ambiguity set-based distributionally robust portfolio optimization can be reduced to a semidefinite program and second-order conic programming. We use a cutting plane to solve our second-order stochastic dominance constraint portfolio optimization problem with ambiguity sets based on the Wasserstein metric. Then we decompose this class of distributionally robust portfolio optimization into semi-infinite programming and apply the cutting surface method to solve it. The captured optimization programs are applied to real-life data for more efficient comparison. The problems are examined in depth using the optimal solutions of the optimization programs based on the different setups.

Portfolio optimization using elliptic entropy and semi-entropy of coherent fuzzy numbers

This paper explores the elliptic entropy and the elliptic semi-entropy of a coherent fuzzy number and discusses several of their properties. We propose a methodology that incorporates the adaptive index, k, of the coherent fuzzy number representing an individual investor’s stock market assessment (pessimistic, optimistic, or neutral). We combine the adaptive index with the elliptic entropy and semi-entropy to obtain risk measures incorporating investor attitude. We use the proposed risk measures in a portfolio optimization problem that uses coherent fuzzy numbers to model the asset returns. The optimization models are solved using a genetic algorithm. We apply the proposed methodology to a large-scale case study involving 100 assets to demonstrate its effectiveness. The real-world performance of the proposed approach is illustrated using an out-of-sample analysis. Based on the results we obtained, we discuss the advantages and disadvantages of elliptic entropy and semi-entropy. Finally, we compare the proposed methodology with other approaches in the literature and demonstrate its superiority.

A novel dynamic reference point model for preference-based evolutionary multiobjective optimization

AbstractIn the field of preference-based evolutionary multiobjective optimization, optimization algorithms are required to search for the Pareto optimal solutions preferred by the decision-maker (DM). The reference point is a type of techniques that effectively describe the preferences of DM. So far, the reference point is either static or interactive with the evolutionary process. However, the existing reference point techniques do not cover all application scenarios. A novel case, i.e., the reference point changes over time due to the environment change, has not been considered. This paper focuses on the multiobjective optimization problems with dynamic preferences of the DM. First, we propose a change model of the reference point to simulate the change of the preference by the DM over time. Then, a dynamic preference-based multiobjective evolutionary algorithm framework with a clonal selection algorithm (ĝa-NSCSA) and a genetic algorithm (ĝa-NSGA-II) is designed to solve such kind of optimization problems. In addition, in terms of practical applications, the experiments on the portfolio optimization problems with the dynamic reference point model are tested. Experimental results on the benchmark problems and the practical applications show that ĝa-NSCSA exhibits better performance among the compared optimization algorithms.

Multi-period uncertain portfolio selection model with prospect utility function.

In this paper, we discuss a multi-period portfolio optimization problem based on uncertainty theory and prospect theory. We propose an uncertain multi-period portfolio selection model, in which the return utility and risk of investment are measured by prospect theory utility function and uncertain semivariance. More realistically, the influence of transaction costs and bankruptcy of investor are also considered. Moreover, to solve the portfolio model, this paper designs a new artificial bee colony algorithm by combining sine cosine search method. Finally, a numerical experiment is presented to demonstrate the proposed model and the effectiveness of the designed algorithm.

A novel dynamic reference point model for preference-based evolutionary multiobjective optimization

In the field of preference-based evolutionary multiobjective optimization, optimization algorithms are required to search for the Pareto optimal solutions preferred by the decision maker (DM). The reference point is a type of techniques that effectively describe the preferences of DM. So far, the reference point is either static or interactive with the evolutionary process. However, the existing reference point techniques do not cover all application scenarios. A novel case, i.e., the reference point changes over time due to the environment change, has not been considered. This paper focuses on the multiobjective optimization problems with dynamic preferences of the DM. First, we propose a change model of the reference point to simulate the change of the preference by the DM over time. Then, a dynamic preference-based multiobjective evolutionary algorithm framework with a clonal selection algorithm (ĝa-NSCSA) and a genetic algorithm (ĝa-NSGA-II) is designed to solve such kind of optimization problems. In addition, in terms of practical applications, the experiments on the portfolio optimization problems with the dynamic reference point model are tested. Experimental results on the benchmark problems and the practical applications show that ĝa-NSCSA exhibits better performance among the compared optimization algorithms.

Generalized-Hukuhara penalty method for optimization problem with interval-valued functions and its application in interval-valued portfolio optimization problems

In this study, a gH-penalty method is developed to obtain efficient solutions to constrained optimization problems with interval-valued functions. The algorithmic implementation of the proposed method is illustrated. In order to develop the gH-penalty method, an interval-valued penalty function is defined and the characterization of efficient solutions of a CIOP is done. As an application of the proposed method, a portfolio optimization problem with interval-valued return is solved.

Shadow Price Approximation for the Fractional Black Scholes Model

In this work, we used Tran Hung Thao’s approximation of fractional Brownian motion to approximate the shadow price of the fractional Black Scholes model. In the case to maximize expectation of the utility function in a portfolio optimization problem under transaction cost, the shadow price is approximated by a Markovian process and semimartingale.