A Quantum Computing-based System for Portfolio Optimization using Future Asset Values and Automatic Reduction of the Investment Universe

The present work aims to describe a quantum computing (QC, [1]) based system for solving the portfolio optimization problem (POP, [2]). Briefly explained, the POP intends to find the optimum asset allocation with the objective of i) maximizing the expected return and ii) minimizing the financial risk. More specifically, and following the Markowitz POP formulation, the financial risk is calculated based on the diversification of the built portfolio [3]. Following this philosophy, the system tends to distribute the whole budget into different and uncorrelated assets rather than investing large amounts of money into the highest expected, albeit correlated, returns.

Formally described, the problem to be solved counts with i) a group of $N$ assets $\mathcal{A}=\{a_{0},\dots,a_{i},\dots,a_{N-1}\}$; ii) a dataset $AD=\{ad_{0},\dots,ad_{i},\dots,ad_{N-1}\}$, in which $ad_{i}$ is a list of historical values $ad_{i,k}$ of an asset $a_{i}$, representing $k$ a specific day within the complete period of $K$ days, and iii) a total $bd$ budget.

Thus, the objective is to find the most promising assets in which to invest this budget, considering that i) all $bd$ must be invested and ii) the proportion of $bd$ that can be allocated to each asset is conditioned by the variable $p$. More concretely, and considering $w_{i}$ as the proportion of $bd$ invested on the asset $a_{i}$, this $w_{i}$ can be represented as the summation of any proportions $p_{i}$ in $P=\{p_{0}=bd,p_{1}=bd/2,...p_{p-1}=bd/2^{p-1},0\}$, Furthermore, it should be deemed that $w_{i}<=bd$. With this notation in mind, the goal is to find the $W=\{w_{0},w_{1},...,w_{N-1}\}$ that maximizes the total expected return while minimizing the financial risk.

The system described in this paper, coined Quantum Computing-based System for Portfolio Optimization with Future Asset values and automatic universe reduction (Q4FuturePOP), is a QC-based scheme for dealing with the $POP$ considering the following innovations:

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  Future projected values: most of the QC based techniques proposed in the literature solve the $POP$ using as input historical dataset values for $\mathcal{A}$ [4]. In other words, developed solvers select the appropriate $W$ considering the past values of the group of available assets. On the contrary, for Q4FuturePOP, the input dataset is composed of future predictions to try to account for realistic environments in the $POP$ formulation. Using this input, Q4FuturePOP builds the complete dataset used for formulating the $POP$ problem considering future projected values of $\mathcal{A}$. Thus, the $W$ chosen by the system is based on future predictions instead of historical values.

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  Automatic universe reduction: in order to calculate $W$ in a more efficient way, a search space reduction mechanism has been implemented in Q4FuturePOP. This mechanism works as follows: first, Q4FuturePOP takes as input the complete universe of $\mathcal{A}$. With this set of assets, the system conducts a number $E$ of preliminary executions which are used by Q4FuturePOP for detecting a subgroup of promising assets. After these preliminary executions, Q4FuturePOP builds an alternative $\mathcal{A}^{\prime}$, which is a subgroup of $\mathcal{A}$ ($\mathcal{A}^{\prime}\subseteq\mathcal{A}$). Using this newly generated $\mathcal{A}^{\prime}$, the problem is finally executed, and the obtained outcomes are returned to the user. Thanks to this procedure, the complexity of the problem to solve is automatically decreased, allowing the system to reach a higher level of accuracy.

The rest of the paper is structured as follows. Section 2 presents a brief overview of the background related to QC and POP. In Section 3, the inputs and outputs of Q4FuturePOP are described for the sake of understandability. After that, in Section 4, the whole system is described. Then, in Section 5, we discuss the preliminary performance of Q4FuturePOP. Section 6 finishes this work by outlining some of the planned future work.

The first POP-focused paper including real quantum experiments was published in 2015, in which the authors solved the problem using a prototype of the D-Wave’s quantum annealer [5]. With only 512 qubits at their disposal, the authors worked with a pool of 15 assets in which to invest. A second study focused on this topic was presented in 2017, exploring a specific investment case related to the Abu Dhabi Securities Exchange [6]. In the following years, some interesting theoretical papers appeared, exploring different formulations and their possible resolutions using quantum approaches. Examples of this scientific trend can be found in [7] and [8]. These papers, published in 2018 and 2019, respectively, theorize the implementation of two algorithms without actually testing them on real quantum devices.

Also in 2019, some advanced approaches were presented. In [9], for example, the authors employ a hybrid algorithm in which the quantum module is executed by the D-Wave 2000Q computer in order to improve the solutions found by a classical greedy algorithm. In the same year, the first paper focused on gate-based quantum computers was published, in which the authors present a Quantum Approximate Optimization Algorithm [10].

Particularly interesting are the papers [11] and [12], published by Chicago Quantum in 2020. In the first of these papers, the authors demonstrated how a hybrid solver can promisingly solve portfolios of up to 33 assets. In [12] a similar approach is proposed, in which the number of assets considered rises to 60 using another hybrid technique. Finally, in [13], the same authors managed to solve problems with a size of 134 assets, making use of the D-Wave Advantage System, composed of 5436 qubits.

Since 2021, the study of the POP through the quantum paradigm has experienced a significant increase. In [14], for example, a variant of the problem known as minimum holding time is solved by the D-Wave quantum annealer, considering a universe of 50 assets in which to invest. In [15], the authors present a problem in which different investment bands are deemed, allowing the fixing of a maximum permissible risk. Furthermore, a quantum-gate-based method was presented in [16], where a hybrid algorithm called NISQ-HHL is proposed.

The study presented in [17] is especially interesting for this paper. That work consists of a detailed analysis of the parameterization of the D-Wave’s annealer. To do so, the authors use POP as a benchmarking problem, presenting a formulation of the problem that allows an investment granularity adapted to the user’s needs. This study has proved to be interesting from a mathematical point of view since the formulation employed in this study is the one embraced in our work. Finally, it is worth highlighting the study presented in [18], where different quantum solvers are proposed based on both the quantum gate paradigm and the annealer. The authors of that work carried out different tests in a dynamic environment, solving problems with up to 52 assets.

As can be seen, the research conducted in recent years has been prolific. This section has attempted to briefly outline this vibrant activity. Being aware that the full state of the art is much broader, we refer interested readers to works as [19].

Now, let’s define some notation and terms in order to properly understand how Q4FuturePOP operates. From now on, we use the superscripts ^h for the historical data and ^f for the predicted future data. Furthermore, the daily return of an asset $a_{i}$ at day $k$ is $er^{h}_{k,i}=(ad^{h}_{k,i}-ad^{h}_{k-1,i})/ad^{h}_{k-1,i}$, resulting in $er^{h}\in\mathbb{R}^{K\times{N}}$ representing the daily returns of $\mathcal{A}$ in $AD^{h}$.

Thus, Q4FuturePOP receives as inputs i) $AD^{h}\in\mathbb{R}^{K\times{N}}$, which contains all the historical values of $\mathcal{A}$ assets during $K$ days, ii) a list $V=\{v_{0},v_{1},...,v_{N-1}\}$, in which $v_{i}$ is the initial value of $a_{i}$ at the time Q4FuturePOP is executed (in most cases, $v_{i}=ad^{h}_{K-1,i}$); and iii) $Er^{f}=\{Er^{f}_{0},Er^{f}_{1},...,Er^{f}_{N-1}\}\in\mathbb{R}^{N}$, which is the list of predicted expected returns. It should be highlighted that these $Er^{f}$ values are given by an expert from the Spanish company Welzia Management¹¹1https://wz.welzia.com/.

Once the input is received, the module named Predicted Dataset Generation (PDG) builds the complete dataset $AD^{f}\in\mathbb{R}^{K\times{N}}$, which is composed of all the projected future daily values of the whole $\mathcal{A}$ in the period that goes from the moment in which the system is executed and the following $K$ days. All values in the generated $AD^{f}$ must meet two requirements: i) $Cov(er^{f})=Cov(er^{h})$, meaning that the covariance of the input daily returns is the same as that of the daily returns generated by PDG; and ii) the expected return of the prices build must be the same as $Er^{f}$. How $AD^{f}$ is generated is described in Section 4.1.

As output, Q4FuturePOP returns i) the above-mentioned $W$, containing the list of investment weights $w_{i}$ given to each $a_{i}$; ii) the expected return of the chosen portfolio $er_{portfolio}$; iii) the risk associated with this portfolio $\sigma_{portfolio}$; and iv) the $SHARPE$ value. It should be considered that for the computation of outcomes ii, iii and iv, the Markowitz formulation of the $POP$ has been used as a base.

Q4FuturePOP consists of three interconnected modules: PDG, Assets Universe Reduction Module (AUR) and the Quantum Computing Solver Module (QCS). A schematic description of Q4FuturePOP is depicted in Figure 1, in which the relationship of the three modules is represented.

In a nutshell, PDG is devoted to generating the complete dataset of future predicted values of $\mathcal{A}$. The second module, AUR, is in charge of intelligently reducing the complete $\mathcal{A}$ into $\mathcal{A^{\prime}}$, while QCS is the module that solves the $POP$ and provides $W$ both for AUR or for the final user (as seen in Figure 1). In the following subsections, we describe PDG, AUR and QCS in detail.

At this moment, Q4FuturePOP is in a prototypical stage of development, waiting to be completely validated as a whole system. Anyway, each module has been checked separately. On the one hand, the PDG has been successfully tested using a pool of predicted values provided by Welzia Management. In any case, due to extension constraints and the fact that this paper is more focused on QC, we will deepen the validation of the PDG module in a future research paper.

On the other hand, AUR and QCS modules have been jointly checked using a dataset also provided by Welzia Management. This dataset contains the daily values of 53 different assets over 12 years (from 01/01/2010 to 13/12/2022). Welzia Management also provided us with a set of historical portfolios chosen by the company’s experts. Thus, using the dataset as input and the historical portfolios as baseline, six different use cases have been built for validation. Each of these instances consists of an excerpt of the complete dataset, with a depth ranging from 12 to 28 months. Therefore, for each use case, AUR+QCS modules of Q4FuturePOP have been run 6 independent times, and the results provided have been compared with the portfolios built by the experts. Also, it should be noted here that the quantum solver used for the conducted tests is the Advantadge_System6.2 of D-Wave, comprised of 5610 qubits and 40134 couplers spread over a Pegasus topology.

The results of this preliminary experimentation are depicted in Table 1, in which we represent the time frame that compose each dataset, the number of available assets, and the outcomes provided by both the experts and AUR+QCS. Each instance is coined as UCX_Y_Z, where $X$ is the ID of the dataset, $Y$ the time frame measured in months, and $Z$ the amount of assets deemed. Finally, it should be noted that both the employed datasets as well as the complete set of results obtained by AUR+QCS are available upon reasonable request.

Analyzing the results obtained by AUR+QCS modules, it should be noted that they have proved to be promising. These results have been approved by the experts from Welzia Management after several technical meetings. These meetings, and the fact of having the help of these experts, have been a really enriching point in the development of Q4FuturePOP. This is so since quantum-based solutions are usually analyzed from a purely academic point of view. That is, the solutions provided by the systems are usually analyzed based solely on their $SHARPE$ ratio. Although this ratio is a good indication of the quality of a solution, it does not represent the reality of the industry. High $SHARPE$ ratios can lead to triumphalist conclusions, which eventually clash with the reality of an industry with volatility that is difficult to perceive by a computer (whether quantum or classical). This is why an expert’s judgment when generating a portfolio is an absolutely necessary factor.

All in all, the solutions achieved by the proposed system have proven to be promising, offering better results than the experts in some cases. In any case, as has been described, the fact that they present $SHARPE$ ratios higher than the portfolios proposed by Welzia Management does not imply that in practice they are better than the latter. Even so, Welzia Management has valued very positively the results obtained by AUR+QCS, being aware of the value that a platform of these characteristics can have in its day to day operation, acting as an assistant in their decision making processes.

In this paper, a quantum-based approach for solving the well-known Portfolio Optimization Problem has been presented, coined as Q4FuturePOP. Two are the main innovations inherent to the system proposed: i) Q4FuturePOP is modeled for working with future prediction of assets instead of historical values; and ii) Q4FuturePOP includes an automatic universe reduction module, which is conceived to intelligently reduce the complexity of the problem. Along with the description of the system, we have also introduced a brief discussion about the preliminary performance of the different modules that compose the prototypical version of the tool.

Several research lines stem directly from the findings reported in this work. The first, and most obvious, is the validation of the complete system. Other future work includes the fine-tuning of the parameters that involve the generation of the POP’s QUBO. Also, other quantum-based solvers apart from the ones provided by D-Wave are planned to be tested.

This work was supported by the Spanish CDTI through Proyectos I+D Cervera 2021 Program (QOptimiza project, 095359). This work was also supported by the Basque Government through ELKARTEK program (BRTA-QUANTUM project, KK-2022/00041). The authors thank Miguel Uceda, Welzia’s Invesment Director, for his assistance and for providing the data employed for the tests conducted.