# Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing
## 5 Related Work  
There are several methods for portfolio optimization. The largest influence comes from [PERSON]'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. [PERSON] 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. \[ \begin{split}& Minimize:\theta_{1}\sum_{i}[-\alpha_{i}E(R_{i})]\\ &+\theta_{2}\sum_{i,j}\alpha_{i}\alpha_{j}Cov(R_{i},R_{j})\\ &+\theta_{3}(\sum_{i}\alpha_{i}A_{i}-B)^{2}\end{split}  \tag{15}\]  
The first part is for the expected return with \(E(R_{i}),a_{i}\in 0,1\) 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. [PERSON] 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:  
\[O(q)=\sum_{i=1}^{N}a_{i}q_{i}+\sum_{i=1}^{N}\sum_{j=i+1}^{N}b_{ij}q_{i}q_{j} \tag{16}\]  
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. \(b_{ij}\) is the covariance of the corresponding returns as a measure to construct a diversified portfolio:  
\[O_{penalty}(q)=P(M-\sum_{i=1}^{N}q_{i})^{2} \tag{17}\]  
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 \(\sum\limits_{i=1}^{N}q_{i}=M\). 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.