*5.3. Robustness Check*

As mentioned above, we motivated the target time of 120 min based both on the available literature and the results of our event study; see Section 3. Since data snooping is a major problem in many financial applications, this subsection examines the sensitivity of our strategies to deviations from their parameter value. In Table 5, we vary the target time in two directions and report the annualized returns before and after transaction costs for BHS, FTS, GVS, RVS, and JDS.

First of all, we see that our results were robust in the face of parameter variations and always led to statements similar to those in Section 5.1. As expected, the results of a target time of 120 were identical to those of Table 3. Furthermore, the annualized returns for each strategy converged as the relative change decreased with increasing target time. The naive S&P 500 buy-and-hold strategy (BHS) always led to an annualized return of 1.81 percent, which is not surprising, since this approach is completely independent of the target time (Section 4). Furthermore, the performance of FTS increased slightly with ascending target time, e.g., the annualized return after transaction costs was −9.37 percent if we closed the trade at 9:50 and −8.36 percent if we closed it at 13:10. The same statement applies to GVS (−9.70 percent vs. −4.28 percent). Due to their mean-reverting component, RVS and JDS showed a slightly declining performance. For each target time, JDS remained the best variant with annualized returns between 49.65 percent and 62.61 percent, after transaction costs. Obviously, we were not on an optimum, but we found robust trading results, regardless of fluctuations in our parameter setting.

**Figure 4.** Development of an investment of 1 USD after transaction costs for FTS, GVS, RVS, and JDS (first column) compared to the S&P 500 buy-and-hold-strategy (BHS) (second column). The time period from January 1998–December 2015 is divided into three sub-periods (March 1998/December 2006, January 2007/December 2009, January 2010/December 2015).

**Table 5.** Yearly returns for BHS, FTS, GVS, RVS, and JDS for a varying target time from January 1998–December 2015.


Motivated by the findings in Section 3, Table 6 examines the annualized returns for a target time of 5, 35, 65, and 95 min. Most interestingly, annual returns were substantially lower for a target time of 5 min for FTS, GVS, RVS, and JDS because high market turmoil during the opening minutes reduced the results. For a target time of 35, 65, and 95 min, increasing market efficiency during the first minutes of each trading hour did not affect yearly returns before and after transaction costs; our strategies seem to be robust against this effect.

**Table 6.** Yearly returns for BHS, FTS, GVS, RVS, and JDS for a target time of 5, 35, 65, and 95 min from January 1998–December 2015.


Next, we take a closer look at our S&P 500 buy-and-hold strategy (BHS). The S&P 500 index was purchased in January 1998 and was held for the entire sample period. Of course, BHS is only a baseline approach for betting on the market. Therefore, we followed Endres and Stübinger (2019b) and developed a more realistic benchmark: The S&P 500 strategy buys the index at 9:30 and reverses it after 120 min. We observed an annualized return of 1.03% compared to 1.81% for BHS (see also Table 4). This insufficient performance is not surprising, as it is a baseline approach without modeling.

Finally, this manuscript supposed a high-turnover strategy of an institutional trader on high-frequency prices. Motivated by the literature, our back-testing framework assumed transaction costs of five basis points per share per half-turn, resulting in 20 basis points per round-trip per pair. However, other traders may be less aggressive in implementing this strategy. Therefore, we analyzed the breakeven point of the statistical arbitrage strategy since investors are exposed to different market conditions. We found that the breakeven point of JDS was between 35 basis points and 40 basis points. Concluding, this strategy generated promising results, even for investors that are exposed to different market conditions and thus higher transaction costs.
