1. Introduction
To this day, a multitude of highly relevant problems are thought to be intractable. This means that there are currently no known exact algorithms that can solve every instance of those problems in a worst-case polynomial time span. Due to the lack of exact algorithms, stochastic and heuristic algorithms inspired by nature, such as simulated annealing (SA), genetic algorithms (GA), swarm intelligence methods, and many more, have been developed. Those heuristic methods may not always find an optimal solution; however, in some cases they may yield a sufficiently good answer in a significantly shorter time span compared to known exact algorithms.
Based on several successful developments in the past decade, another tool to solve certain currently hard-to-solve problems has become available: quantum computing. Quantum computing itself, as a theoretical concept of performing calculations, has been known for decades. However, it was not clear whether quantum computers could eventually be built or not. Since the first machines potentially able to harness quantum phenomena to perform calculations became available in the last decade, intensive research efforts have been directed to understand these new computational possibilities. In theory, quantum computers promise to solve currently intractable problems, such as the factoring of numbers (see Shor’s algorithm [
1]), in polynomial time. However, currently available quantum computers suffer from noise, missing error correction, low numbers of qubits, and more, and are thus far from reaching the theoretically possible speedups.
Current research efforts can be split into two main classes. The first class is concerned with the investigation of possible near-term advantages using currently available quantum and classical hardware. The second class is concerned with the creation of quantum algorithms that may exhibit speedups over current classically available algorithms, but require so-called fault-tolerant quantum hardware.
Commercially available quantum computers can also be split into two main classes: quantum annealers and quantum-gate computers. The former, manufactured for example by D-Wave Systems, are based upon the adiabatic theorem to perform calculations. Many of the (decision) problems currently thought to be intractable have an optimization version that can easily be mapped onto the native input format of quantum annealers, namely the Ising model (or the equivalent QUBO model). As of now, these models have become a quasi-standard for examining NP-complete and NP-hard problems in the realm of quantum computing. There is a plethora of known transformations for NP-complete/-hard problems to the QUBO model (see e.g., [
2]), as well as guidelines on how to formulate QUBO models in general (see e.g., [
3]). The performance of various QUBO formulations for certain problems has been subject to intensive investigations in the past couple of years (see e.g., [
4,
5,
6,
7,
8,
9,
10,
11]) and is still an ongoing effort.
In this paper, we focus on the applicability of annealing techniques to the NP-hard problem of portfolio optimization [
5], a well-known topic for investment funds and individual investors.
The pioneering work by Harry Markowitz [
12] in 1952 can be considered the foundation of portfolio optimization. The goal here is to distribute an investor’s capital on assets such that a given objective, such as maximizing the return or minimizing the risk, is optimized. Portfolio optimization is an actively researched question in scientific domains such as combinatorial optimization [
13], operations research [
14], data science [
15], and the application of quantum computing [
16]. Not only the academic community is focusing on this topic, but it is also a core business of banks and financial advisers and the combination of portfolio optimization with quantum techniques is addressed by numerous companies and consultants [
17,
18].
Current approaches of solving portfolio optimization with annealing techniques (see [
5,
11,
19,
20,
21]) exhibit, amongst others, the following limitations:
These approaches also try to solve portfolio optimization as a whole, which limits the possible problem sizes significantly, as current quantum hardware is not advanced enough yet to tackle bigger problems. Thus, we present a new workflow combining a classical preprocessing step with a modified QUBO model for portfolio optimization that is able to solve significantly larger optimization problems. In our modified QUBO model, we enable an investor to place an arbitrary amount of money into an arbitrary number of assets, which was not possible previously. The solutions to our QUBO model tell investors how to split their capital amongst available strategies in order to reach their goals (i.e., achieve a certain return and minimize risk). We evaluate our QUBO model via quantum annealing (QA), simulated annealing (SA), and digital annealing (DA).
This paper is organized as follows. First, we provide the reader with some preliminaries: in particular, we recall the foundations of portfolio optimization (
Section 2), quantitative investing (
Section 3), the QUBO model, and simulated and quantum annealing (
Section 4). In
Section 5 we give an overview of portfolio optimization in the field of quantum annealing. In
Section 6 we present our approach to portfolio optimization followed by an evaluation in
Section 7. The conclusion together with some discussion is given in
Section 8.
2. Portfolio Optimization
Modern portfolio theory is about the optimal selection of assets for a given level of risk to achieve the maximal return [
12]. This theory was first presented in 1952 by the pioneer in this field, Harry Markowitz [
12]. His theory shows that investors should diversify their portfolios to achieve a maximum return under a given level of risk. The investors decide for themselves how to calculate the expected return and what return indicators are important. Suppose there are
l assets. The expected return of a portfolio is given by
Variables
${X}_{i}$ with
$0\le {X}_{i}\le 1$ denote the portion of the asset
i in the whole portfolio and
${\mu}_{i}$ is the expected return of asset
i. Of course
$\sum _{i=1}^{l}}{X}_{i}=1$ holds. The expected risk of a portfolio is given by
Here, ${\sigma}_{ij}$ denotes the covariance between assets i and j.
The investor has the possibility to select different combinations of
$(E,V)$. It is assumed that all combinations are distributed according to
Figure 1. The
x-axis represents the expected return (the more right the better) and the
y-axis represents the expected risk (the lower the better):
The dashed/solid line in
Figure 1 denotes the convex hull around a set of portfolios. That is, all envisioned portfolios lie inside that area. Every dot represents a single portfolio. The blue dot marks the portfolio where the expected return is the highest, while the portfolio marked with a green dot has got the lowest risk. As a trade-off for the lower risk, the portfolio represented by the green dot also yields lower expected returns than the portfolio represented by the blue dot. The solid part of the line in
Figure 1 denotes the area where optimal portfolios may lie, which is also called the
efficient frontier. Portfolios within the efficient frontier are considered optimal because they exhibit the best balance between maximizing the return
E while also minimizing the risk
V. The red dot, for example, is not optimal, because the green dot has a lower expected risk with the same expected return. Similarly, the yellow dot has a lower expected return compared to the green dot with the same expected risk.
To find the best investment portfolios, the problem needs to be formulated as an optimization problem:
This is the constrained quadratic form of the portfolio theory of Markowitz. Out of
l assets, the ones with the lowest covariance
${\sigma}_{ij}$ and highest return
${\mu}_{i}$ shall be selected. The weighted percent values
${X}_{i}$ describe how the different assets build up the solution portfolio [
5].
3. Quantitative Investment Strategies
Quantitative trading can be defined as the systematic implementation of trading strategies that humans create through rigorous research. In this context, systematic is defined as a disciplined, methodical, and automated approach [
22] (p. 15).
The quantitative investment process starts by testing a possible strategy. Therefore, one has to start by gathering the necessary historical data, which might contain several flaws. For example, the data might be not adjusted to stock splits or dividends, there might be missing values, reported values might not be correct, and so on. If this is the case, one has to include a data cleaning step before one proceeds [
22] (pp. 135–136). After establishing the data set for the potential strategy, the strategy is fitted to the historical data. There are additional pitfalls that need to be considered. For example, a historical database of stock prices might not include stocks that have “disappeared” due to bankruptcies, delistings, mergers, or acquisitions, and thus suffer from a so-called survivorship bias, because only “survivors” of those often unpleasant events remain in the database [
23] (p. 33). Another example of a possible bias is the so-called data-snooping bias. This bias can be thought of as a kind of overfitting of the parameters to given historical data. It occurs when one tries to optimize the parameters of a strategy such that it performs exceptionally well on historical data. It is very likely that such an optimization can be performed [
23] (p. 35). The look-ahead bias is about situations where information is used that was only available after the investment has been made. This information was not available when an asset was bought or sold [
23] (p. 58). After the strategy shows acceptable performance on historical data, the strategy is tested on an out-of-sample data set, usually taken from the near past. Finally, the out-of-sample test (so-called backtest) is evaluated, and the strategy will either be used for investments or discarded [
23] (p. 29).
In our work, the price series of assets are the foundation of a trend strategy and a mean reversion strategy. Trend strategies (also called momentum strategies) are based on the theory that sometimes markets move long enough in one direction such that the trend can be identified and exploited. Mean reversion strategies assume, however, that there exists a center of a price series (e.g., the mean of a price) and that the price including its fluctuation always returns to its center [
22] (pp. 39–45). The price series must be stationary for a mean reversion strategy. Stationarity can mathematically be tested by the augmented Dickey–Fuller (ADF) test [
24] (p. 49). Price changes in a price series are described as
The ADF test finds out if
$\lambda =0$. If the hypothesis
$\lambda =0$ can be rejected, then the next price change
$\Delta y\left(t\right)$ depends on
$y(t-1)$. Therefore, the series is not a random walk. The test statistics is the regression coefficient
$\lambda $ divided by the standard error of the regression
$\lambda /SE\left(\lambda \right)$. Since mean reversion is expected,
$\lambda /SE\left(\lambda \right)$ must be negative and lower than the critical
z-
$score$ of the hypothesis. Variable
k denotes price lags of the price series,
$\beta t$ is the constant drift of the price series, and
$\alpha $ is the auto-regressive parameter. Error term
${\u03f5}_{t}$ is independent and identically distributed with a mean of 0. Intuitively formulated, being a stationary price series means that the price moves away from its initial value slower than a geometric random walk. The speed of this moving can mathematically be calculated by the variance
in which
z is the log price
$(z=log(y\left)\right)$,
$\tau $ an arbitrary point in time, and
$\langle \dots \rangle $ the average over all
ts. For a geometric random walk the following is known:
Notation ∼ indicates that the relation for large
$\tau $ becomes equality and for small
$\tau $ the relation deviates from a constant line. If the price series is mean reverting or in a trend, then the formula is the following:
Here,
H is the Hurst exponent with a value of
$H=0.5$ for a geometric random walk,
$H<0.5$ for a stationary price series, and
$H>0.5$ for a trend. The trend or stationarity is stronger the larger the difference of
H from
$0.5$ [
24] (p. 51).
Since price series are finite sets, the statistical significance of
H must be determined. The hypothesis test for that is the variance ratio test [
24] (p. 52), and it tests if
The null hypothesis is that the price series is a random walk. If the test equals 1, then a random walk can be rejected; if the test equals 0, then the series may be a random walk. The
p-value of the test gives the probability that the null hypothesis can be accepted [
24] (p. 52).
Variable
$\lambda $ in the ADF test formula (Equation (
4)) is a measure for the time a price series needs to return to its center, also called half-life. For that, the price series must be transformed into its differential form. For this, the Ornstein–Uhlenbeck formula can be used:
where
$d\u03f5$ represents Gaussian noise. If
$\lambda $ is positive, then the price series is not stationary, while a
$\lambda $ close to 0 means that it will take a long time until the series returns to its center. Additionally, the measure of half-life provides a guideline for how the period of the moving average and the moving standard deviation for strategies shall be selected. The period is optimal if it is a small portion of
$\lambda $ [
24] (p. 53).
Most price series are not stationary, however, stationary portfolios can be created out of several distinct price series. This works via the cointegration of price series. Cointegration means that if a linear combination of non-stationary price series can be found, then they are cointegrated with each other. A common combination consists of two price series. Here, one price series is invested in long (an investor bought and owns the asset) and the other is invested with a factor (hedge ratio) in short (an investor lent and sold an asset and returns it later). That is called the pairs-trading strategy. To test the stationarity of the cointegration, the augmented Dickey–Fuller (ADF) test can be used. First, the optimal hedge ratio is calculated by linear regression
where
S is the alleged stationary price series,
h the hedge ratio, and
${V}_{0},{V}_{1}$ are the two distinct series. Using
S, a portfolio will be created. On top of the price series of the portfolio, the ADF test is executed to show stationarity. Not all pairs of series are suitable for a pairs-trading strategy. The mathematical techniques presented in this section are crucial for the preprocessing step in our portfolio optimization step. By the use of backtesting strategies on different assets and discarding strategies on assets where the backtest results are unsatisfactory, the search set for the portfolio optimization can be reduced. However, the complexity of the optimization problem still remains.
5. Related Work
There are several methods for portfolio optimization. The largest influence comes from Markowitz’s work [
12] for portfolio selection. Probability and optimization theory is combined to find a portfolio with minimized risk and maximized return [
34]. In the following paragraphs, existing portfolio optimization approaches using quantum methods are presented.
The portfolio optimization problem has been solved using two different quantum computing methods. The first method is the quantum linear systems algorithm. The algorithm creates a risk-return curve in which the portfolio with the lowest risk for a given return can be found [
35]. The second method is quantum annealing. In the following, we will present existing quantum annealing approaches.
Elsokkary et al. [
11] selected the expected price value of a stock as the return. For the risk measure, the variance of the stock price was used. Additionally, a budget for the portfolio can be chosen. The corresponding QUBO model consists of three parts.
The first part is for the expected return with
$E\left({R}_{i}\right),{a}_{i}\in 0,1$ depending if a stock
i is selected for the portfolio or not. The second part represents the risk with
$Cov({R}_{i},{R}_{j})$, and if
$i=j$ then it is the variance of the stock price. The third part constrains the model to a given budget
B.
${A}_{i}$ is the price of the stock
i. The positive weights
${\theta}_{1},{\theta}_{2},{\theta}_{3}$ describe how important each term in the model is. In the application of that approach, data from the Abu Dhabi securities exchange has been used, and the algorithm was executed on the D-Wave Quantum Annealer. The thetas were all set to 13 to have equal importance for the three terms. Given a budget of USD 100, their algorithm constructed a portfolio worth USD
$121.176$. Therefore, the thetas were adjusted to
${\theta}_{1}=0.8$,
${\theta}_{2}=0$,
${\theta}_{3}=0.2$, and the constructed portfolio was worth USD
$119.007$, where no optimization for risk was made [
11]. Their approach lacks the ability to have more than one stock for each security in the portfolio. Additionally, the authors did not mention how the expected return was calculated. Venturelli et al. [
5] used a similar approach for portfolio optimization via quantum annealing. The authors used investment funds instead of stocks as assets, the annual return of each fund as the expected return, and the annual standard deviation of the logarithmic return as the risk. Using the two factors of return and risk, the Sharpe ratio was calculated for each fund in order to put each fund into one of the 12 classes for the “attractiveness” of the asset. The QUBO model has been defined as follows:
The attractiveness class of each fund is represented by
${a}_{i}$, while
${q}_{i}\in 0,1$ indicates whether fund
i is in the portfolio or not.
${b}_{i}j$ is the covariance of the corresponding returns as a measure to construct a diversified portfolio:
Additionally, the approach has an extension to choose the amount of funds in the portfolio out of
N funds. The amount of funds that shall be in the portfolio can be selected using
M.
P is a penalty that sets the global minimum of the function to the wanted amount such that
$\sum _{i=1}^{N}}{q}_{i}=M$. The approach was executed via quantum annealing and reverse quantum annealing. The results show that reverse annealing is 100 times faster than quantum annealing [
5]. Their approach lacks the ability to tell the theoretical investor how much of the capital shall be invested into which fund. Their work did not show any portfolios constructed by the quantum annealer, instead, they just showed if the problem has been solved and in what time. Additionally, the expected return of each fund was computed naively by means of historical returns.
The approach by Phillipson and Bhatia [
19] is similar to Venturelli et al. [
5] and Elsokkary et al. [
11].
N assets to invest in are available out of the Nikkei225 and S&P500. The expected return of asset
i is
${\mu}_{i}$ calculated over a quarterly 5-year data period. The risk is modeled by a risk matrix
${x}^{T}\sum x$, where the diagonal is the variance of each assets and the other cells represent the covariance of the assets.
n determines how many assets shall be contained in the resulting portfolio. The return of the result portfolio shall be higher than
${R}^{*}$ and the approach searches for the portfolio out of
n assets that achieves the target return with minimal risk. The corresponding QUBO formulation is as follows:
The lambdas weigh each term for its importance. The results in that approach only show how fast quantum annealing is compared to other optimization algorithms such as simulated annealing [
19].
To conclude, existing approaches lack the ability to tell a theoretical investor how much of the capital needs to be allocated to which asset to have the optimal portfolio. Additionally, taking the mean of historical returns is a naive estimate of asset returns.
7. Evaluation
We evaluate our new QUBO formulation for portfolio optimization by assuming a hypothetical investor who splits their money according to the suggestion of the QUBO optimization process and observing the development of their portfolio. We show that it is possible to add additional risk minimization, if needed, by adjusting the parameters of our model. The same is true for the maximization of the expected return.
7.1. Dataset
Our data set consists of stocks of the New York Stock Exchange (NYSE) as well as common exchange-traded funds (ETFs). The data set contains the daily adjusted closing prices of those stocks and respective ETFs over a period of 5 years, starting on 31 December 2014 and ending on 31 December 2019. The daily adjusted closing prices of each stock and respective ETF over the 5-year period form a price series. Since the data were not complete (e.g., daily closing prices were missing for several days for certain stocks or ETFs), we decided to only keep price series that contain more than 900 daily prices. All the data were acquired via Yahoo Finance.
7.2. Evaluation Framework
The experiments are carried out on D-Wave System’s Quantum Annealer (Advantage System 4.1), on Fujitsu’s Digital Annealing Unit (DAU Version 2), and on a classical computer using simulated annealing. We use default parameters for all algorithms on their respective hardware systems. We use UQO [
38] as a platform to handle the execution of our experiments. Since D-Wave’s Advantage System 4.1 only provides about 5000 qubits, arranged in the Pegasus graph, we were only able to use 18 strategies. We chose 8 pairs-trading strategies on ETF pairs and 10 trend-following strategies on different NYSE stocks as our 18 strategies. Each of the 18 strategies were selected by the best Sharpe ratio of the backtest results. To compare the annealing results to a random process, we created randomized portfolios, each of them consisting of an arbitrary amount of strategies that were assigned an arbitrary fraction of the capital.
7.3. Diversification Measure
In the following figures to be seen, the coloring indicates the diversification of the portfolios. We used the following equation to determine the diversification of a portfolio:
In this equation, P denotes a portfolio, the price series of the portfolio are denoted by ${s}_{i}$, ${s}_{j}$, $1\le i,j\le 18$, and $0\le {w}_{{s}_{i}},{w}_{{s}_{j}}\le 1$ are the fractions of the capital that were invested into the strategies ${s}_{i}$, ${s}_{j}$ in this specific portfolio P. In the denominator of the first part of the equation, we added the term ${10}^{-10}$ to the entropy of the portfolio to prevent division by zero, as it might be possible that ${w}_{{s}_{i}}=1$, which implies that all other weights ${w}_{{s}_{j}}\ne {w}_{{s}_{i}}$ must be zero, and thus the value of the denominator would be $ln\left(1\right)=0$. Intuitively, the first part of the equation measures how similar the price series within the portfolio are, while the second part of the equation adds a reward for portfolios with a larger number of selected strategies. We normalize the result of the above diversification measure to the closed interval $[-1,1]$. The best value for diversification is 0, while $-1$ and 1 denote the worst possible diversification. Here, 1 (dark-red coloring) means that all price series of the strategies in the portfolio behave exactly the same, while $-1$ (dark-blue coloring) means that (groups of) time series behave exactly opposite, i.e., if the price of one series increases then the price of the other series decreases by the same amount. In our case, no portfolio received a normalized diversification result below 0, as none of the strategies within the created portfolios had negative correlation.
7.4. Results
We now present the results of our experiments. For all our experiments, we created 1000 random portfolios as well as $10,000$ samples for each of the annealing strategies simulated annealing, quantum annealing, and digital annealing. In our visualizations, we plot all the random portfolios, while for the portfolios generated with any of the annealing strategies we only plot the 10 best results. We start off by evaluating the created portfolios when using equal values for ${\theta}_{1}$ (weight for return) and ${\theta}_{2}$ (weight for risk).
Please note that in
Figure 2 all portfolios generated by simulated annealing (marked as diamonds) and digital annealing (marked as stats) are almost stacked on top of each other, while portfolios generated by quantum annealing (crosses) reside in the same area as portfolios generated by simulated annealing and digital annealing, but exhibit an observable distribution. Nevertheless, the diversification, denoted by the coloring of the respective shapes, is roughly the same.
If we now increase
${\theta}_{1}$ to 400, which means we reward the creation of portfolios that yield higher returns, we obtain the results visualized in
Figure 3. Note that there is nothing special about the value of 400, it is just an arbitrary value that emphasizes the creation of portfolios that yield higher returns.
The results show that, apart from a few outliers, the
y-values of the annealing portfolios of
Figure 2 and of
Figure 3 are approximately equal. However, when comparing the results in
Figure 2 to
Figure 3, we can observe that the portfolios of the latter experiment possess a higher
x-value. As the
y-value indicates risk and the
x-value indicates return, the conclusion is that while maintaining an equal amount of risk, the portfolios generated in
Figure 3 yield a higher return.
In the next experiment, we increase
${\theta}_{1}$ to 1000 and decrease
${\theta}_{2}$, which is the indicator for risk, to 1. This leads to portfolios that yield higher returns than previous experiments but also exhibit a lot more risk. The results of this experiment are visualized in
Figure 4.
In this experiment, we observe that the resulting portfolios created by simulated annealing (diamonds) and digital annealing (stars) increase their Sharpe ratio as well as their risk significantly. One can also see that the color shifts from a greenish coloring in
Figure 3 to a yellow color in
Figure 4. These portfolios are less diversified than portfolios in the experiment of
Figure 3. This is the intended effect of increasing the risk and the return. The portfolios generated by quantum annealing (crosses), however, only exhibit a slight trend towards increasing their Sharpe ratio. A significant change in risk can not be observed. One explanation for this might be that the decrease of
${\theta}_{2}$ from 100 to 1 is not significant enough for quantum annealing to mitigate influences such as noise or the additional scaling of parameters performed by the quantum annealer itself.
We now use the best portfolios generated by each of the annealing approaches in the previous experiment (${\theta}_{1}=400,{\theta}_{2}=100,{\theta}_{3}=1$) to investigate the potential returns we would have received if we invested in these portfolios in the year 2020.
TPOR is the abbreviation of the ETF
Direxion Daily Transportation Bull 3X Shares while DUSL is the abbreviation of the ETF
Direxion Daily Industrials Bull 3X Shares. TPOR-DUSL in
Figure 5 is a portfolio that only contains the pairs trading strategy with ETFs TPOR and DUSL. The pairs-trading strategy is long in the asset TPOR and short in the asset DUSL. Long in TPOR means that 75% of the investor’s capital is allocated to TPOR. Short in DUSL means that a number of stocks equal to 25% of the investor’s capital was borrowed from a bank, with the intention of selling it back to the bank at a later point in time. As can be seen in
Figure 5, this portfolio lost approximately 60% of its value in 2020. The portfolios generated by the different annealing strategies have allocated between 23% to 25% of the investor’s total capital to TPOR-DUSL. Hence, these portfolios underperform as well. The slight differences between the performances of the portfolios generated by simulated annealing, quantum annealing, and digital annealing can be explained by slightly better (or worse) diversification and slightly more (or less) allocation to TPOR-DUSL and other strategies in the respective portfolios.
In the above experiments, we observed that the created portfolios used between 80 and 120% of the available capital. Thus, sometimes we observed overspending and sometimes underspending. Although this is not a big problem in our approach, as we can just keep the proportions and scale the whole portfolio such that exactly 100% of the capital is used, we still can enforce this constraint by increasing the parameter
${\theta}_{3}$. To investigate the effect of increasing
${\theta}_{3}$ in our QUBO model, we examined the results of
Figure 3 with respect to the capital spent in the created portfolios. In the experiments visualized in
Figure 3, we observed that the portfolios created by simulated annealing and digital annealing already used exactly 100% of the capital, while the portfolios created by quantum annealing were off by approximately ±9%. When increasing the value of
${\theta}_{3}$ from 1 to 100, we get the results as seen in
Figure 6.
An interesting result when comparing
Figure 6 with the previous experiments of
Figure 3 is that when increasing
${\theta}_{3}$, portfolios with a lower Sharpe ratio seemed to be created. For the portfolios created by simulated annealing and digital annealing, we observed that all the created portfolios used
$99.2$% of the whole capital. The increase of
${\theta}_{3}$ did not show any significant impact on portfolios created by quantum annealing. Here, we still observed that the portfolios used ±10% too much (or too little) of the budget.
8. Conclusions
In this paper, we have introduced a new workflow to solve portfolio optimization problems. We have started with a classical preprocessing step in which we used classical backtesting for all pairs-trading and trend-following strategies. We applied the strategies to historical real-world data within the time span of 2015–2019. Based upon the results of this step, we chose the most promising strategy/asset combinations as an input to our modified QUBO formulation for portfolio optimization. Our modified QUBO model enables an investor to place an arbitrary amount into arbitrary many assets. The solutions to our QUBO formulation create portfolios in which a certain fraction of the overall budget is dedicated to certain strategies, while also respecting the amount of risk that should be taken and the magnitude of return that should be achieved. We evaluated our QUBO formulation via simulated annealing, digital annealing, and quantum annealing. The simulated annealing results show that our approach works as intended, meaning that portfolios were generated that respect the given preferences of the investor (expressed by the chosen values of ${\theta}_{1},{\theta}_{2},{\theta}_{3}$) while also creating diversified portfolios. The results of quantum annealing were also promising, yet not as good as the simulated annealing and digital annealing results. The probable cause for this is a combination of inherent noise, missing error correction, scaling of the parameters, and others. We note, however, that we only used default parameters for all of our evaluations. Hence, for future work, finding and using more suitable parameter configurations or adding techniques such as reverse annealing or certain types of low polynomial running-time post-processing might significantly increase solution quality in all cases. The combination of classical algorithms and reverse annealing might also provide benefits over purely classical or purely quantum methods.