# Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing
## 6 Portfolio Optimization Implementation
### Theoretical Remarks
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. ### 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. Figure 2: Annealing results with equal weights for return and risk. 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.