## 6 Evaluation
### Results
To illustrate this, assume we pick the top 10 stocks (\(k=10\)) and track them over the course of one month, _i.e._, 20 trading days (\(w=20\)). This is illustrated in Figure 14, using the highest ranked stocks selected by the snapshot aggregation technique. Figure 15 shows these same gains depicted in a histogram. The mean gain is a modest 1.7%. Since each day is an independent investment, we can show the evolution of the portfolio over time by compounding the cumulative gains. It is critical to realize that the \(CG_{d}\) only affects that fraction of the overall investment that was invested on that day, that is \(1/w\). Let us explain this fact with an example: we shall invest a fixed amount of money \(x\) in each of the top \(k\) stocks selected by one of our algorithms, and keep the funds invested for \(w\) trading days. At the end of the period, each stock has a rate of return \(\text{Gain}^{s}_{d}\), and the portfolio of \(k\) stocks has a cumulative gain \(CG_{d}=1/k\sum_{s}\text{Gain}^{s}_{d}\). That is, we invested \(x\times k\) on day \(d\) and recovered \(x\times k\times CG_{d}=x\times\sum_{s}\text{Gain}^{s}_{d}\) on day \(d+w\). Now, assume that we repeat this process every day, over a long period of time. Since we are investing a fixed amount per day, the total amount of money that needs to be available at the beginning of the process is Funds \(=x\times k\times w\). The cumulative gain \(CG_{d}\) on any given day only affects \(1/w\) of the total sum invested. A little algebra will show that:  
Figure 14: Gain realized by picking the top 10 tickers chosen from the snapshot aggregation algorithm and selling after 20 days. Figure 15: Histogram of the gains realized by picking the top 10 tickers chosen from the snapshot aggregation algorithm and selling after 20 days. \[\frac{1}{w}\times\text{Funds}\times CG_{d}=\text{Funds}\times\frac{1}{k}\sum_{s} \frac{\text{Price}_{d+w}^{s}-\text{Price}_{d}^{s}}{w\times\text{Price}_{d}^{s}}\]  
That is, the price difference is scaled by \(1/w\). To compute the evolution of the overall gain (\(OG\)) over time, we compound the cumulative gains as follows:  
\[OG=\prod_{d}\left(1+\frac{CG_{d}}{w}\right)\]  
The gain \(\text{Gain}_{\text{d}}^{s}\) and cumulative gain \(CG_{d}\) are both zero-based quantities; compounding these would converge to zero very quickly. Adding one to the quantity \(CG_{d}/w\) allows us to multiply them meaningfully. This is the same as saying that a 20% yield is equivalent to a factor of 1.2. Figure 16 shows our four algorithms side by side. We observe that Dynamis aggregation performs best. Snapshot aggregation and snapshot skyline basically performed the same; they are indistinguishable in Figure 16; but in fact, they are not identical. Finally, we compute the geometric average rate of return, or average yield \(Y\), as:  
\[Y=\sqrt[n]{\prod_{d=1}^{n}\left(1+\frac{CG_{d}}{w}\right)}-1=\sqrt[n]{OG}-1\]  
Comparing the average yield \(Y\) at the end of the evaluation period in Figure 17, we see that Dynamis performs better than snapshotting, with Dynamis aggregation being the clear winner. Figure 16: Overall gain (\(OG\)) with \(k=10\) and \(w=20\).