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Applied Sciences

Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing

Jonas B. Lang, Sebastian Zielinski, Sebastian Feld

Published Dec 1, 2022• 31 citations
Publisher

Abstract

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.

DOI
10.3390/app122312288

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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.
Recent advances in quantum hardware offer new approaches to solve various optimization problems that can be computationally expensive when classical algorithms are employed. We propose a hybrid quantum-classical algorithm to solve a dynamic asset allocation problem where a target return and a target risk metric (expected shortfall) are specified. We propose an iterative algorithm that treats the target return as a constraint in a Markowitz portfolio optimization model, and dynamically adjusts the target return to satisfy the targeted expected shortfall. The Markowitz optimization is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. The use of the expected shortfall risk metric enables the modeling of extreme market events. We compare the results from D-Wave's 2000Q and Advantage quantum annealers using real-world financial data. Both quantum annealers are able to generate portfolios with more than 80% of the return of the classical optimal solutions, while satisfying the expected shortfall. We observe that experiments on assets with higher correlations tend to perform better, which may help to design practical quantum applications in the near term.
This paper investigates and compares quantum and classical investment strategies for portfolio construction under realistic trading and allocation constraints. The study considers a multi-period portfolio re-balancing setting in which asset weights are subject to lower and upper bounds, class-level exposure restrictions, and turnover limitations. On the quantum side, the portfolio selection problem is reformulated as a quadratic unconstrained binary optimization (QUBO) model through a finite binary encoding of asset weights, and subsequently embedded into a constrained quadratic model (CQM) solved by a hybrid quantum-classical optimization workflow. On the classical side, benchmark allocation strategies are introduced to provide an economically meaningful reference for performance assessment. The proposed framework jointly evaluates portfolio efficiency from two complementary perspectives: business performance, measured through return, volatility, and a measure of the trade-off between them, the Sharpe ratio; and computational performance, assessed through solver structure, constraint handling, and hardware execution characteristics. The resulting comparison provides a rigorous basis for understanding the practical role of quantum annealing in constrained portfolio optimization, while clarifying both its modeling advantages and its current implementation limitations relative to established classical approaches.
In this note, we describe an experiment on portfolio optimization using the Quadratic Unconstrained Binary Optimization (QUBO) formulation. The dataset we use is taken from a real-world problem for which a classical solution is currently deployed and used in production. In this work, carried out in a collaboration between the Raiffeisen Bank International (RBI) and Reply, we derive a QUBO formulation, which we solve using various methods: two D-Wave hybrid solvers, that combine the employment of a quantum annealer together with classical methods, and a purely classical algorithm. Particular focus is given to the implementation of the constraint that requires the resulting portfolio’s variance to be below a specified threshold, whose representation in an Ising model is not straightforward. We find satisfactory results, consistent with the global optimum obtained by the exact classical strategy. However, since the tuning of QUBO parameters is crucial for the optimization, we investigate a hybrid method that allows for automatic tuning.
One of the problems frequently mentioned as a candidate for quantum advantage is that of selecting a portfolio of financial assets to maximize returns while minimizing risk. In this paper we formulate several real-world constraints for use in a Quantum Annealer (QA), extending the scenarios in which the algorithm can be implemented. Specifically, we show how to add fundamental analysis to the portfolio optimization problem, adding in asset-specific and global constraints based on chosen balance sheet metrics. We also expand on previous work in improving the constraint to enforce investment bands in sectors and limiting the number of assets to invest in, creating a robust and flexible solution amenable to QA. Importantly, we analyze the current state-of-the-art algorithms for solving such a problem using D-Wave's Quantum Processor and compare the quality of the solutions obtained to commercially-available optimization software. We explore a variety of traditional and new constraints that make the problem computationally harder to solve and show that even with these additional constraints, classical algorithms outperform current hybrid solutions in the static portfolio optimization model.
Portfolio optimization (PO) is extensively employed in financial services to assist in achieving investment objectives. By providing an optimal asset allocation, PO effectively balances the risk and returns associated with investments. However, it is important to note that as the number of involved assets and constraints increases, the portfolio optimization problem can become increasingly difficult to solve, falling into the category of NP-hard problems. In such scenarios, classical algorithms, such as the Monte Carlo method, exhibit limitations in addressing this challenge when the number of stocks in the portfolio grows. Quantum annealing algorithm holds promise for solving complex portfolio optimization problems in the NISQ era. Many studies have demonstrated the advantages of various quantum annealing algorithm variations over the standard quantum annealing approach. In this work, we conduct a numerical investigation of randomly generated unconstrained single-period discrete mean-variance portfolio optimization instances. We explore the application of a variety of unconventional quantum annealing algorithms, employing both forward annealing and reverse annealing schedules. By comparing the time-to-solution(TTS) and success probabilities of diverse approaches, we show that certain methods exhibit advantages in enhancing the success probability when utilizing conventional forward annealing schedules. Furthermore, we find that the implementation of reverse annealing schedules can significantly improve the performance of select unconventional quantum annealing algorithms.
Recently, several researchers proposed portfolio optimization as a potential use case for quantum optimization. However, the literature is lacking an extensive benchmark quantifying the potential of quantum computers for portfolio optimization. In this work, we contribute to closing this gap. We provide a computational study, comparing quantum approaches against state-of-the-art classical methods on a meaningful, real-world instance set. In particular, we compare quantum annealing and the quantum approximate optimization algorithm against classical mixed-integer programming, simulated annealing, steepest descent local search, tabu search and a problem-tailored heuristic. We consider a variance-minimizing variant of portfolio optimization which we show to be more difficult to solve for classical optimizers than return-maximizing or multi-objective formulations. Our benchmark data set comprises 270 instances with up to 1,000 assets from actual stock data. Due to hardware limitation, quantum methods could only be tested for instances with at most 30 assets. The results show that all instances can be solved to proven optimality by mixed-integer programming in the order of seconds. Moreover, the problem-tailored heuristic consistently outperforms quantum approaches in terms of solution quality for fixed runtime. Thus, we conclude that there is only very limited room for a potential quantum advantage for the considered variant of portfolio optimization.