# Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing
## 6 Portfolio Optimization Implementation
### Theoretical Remarks
Figure 3: Annealing results with increased weight on return. 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 \(\pm\)9%. When increasing the value of \(\theta_{3}\) from 1 to 100, we get the results as seen in Figure 6.

Figure 4: Annealing results with increased weight on return and reduced weight on risk. 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 \(\pm 10\%\) too much (or too little) of the budget. Figure 5: Portfolio performance with optimization weights \(\theta_{1}=1000,\theta_{2}=1,\theta_{3}=1\) in the year 2020. Figure 6: Annealing results with increased weight on 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.