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More often than not, we observed a positive correlation for the years inspected (the strongest correlations are those shown in the figures below). It turns out that for certain years (those with a negative correlation), we ought to utilize the “min-median” as a selection criterion. However, this cannot be known ex-ante , and the best we can do is utilize a measure that more often than not, produces above-average results. Here again, we can appreciate how these conflicting effects would average-out with time in a favorable direction, reiterating the fact that a strategy such as this one, if considered, should be evaluated over the long-haul.
Lastly, several evaluations were performed comparing the various max-medians of the portfolios simulated as a function of the number of portfolios run (i.e. J ) and compared to the single-stock max-median (See Figure 6 below), which could, at least heuristically, serve as an upper bound. This resulted (empirically) to be somewhat unstable as there is no guarantee that any thresholds set in terms of percentage to the bound could be attained in any reasonable computing time, mainly due to the fact the after a reasonable amount of simulations (namely $J=500,000$ and up to $J=2,000,000$ ) the percentages of this single-stock max-median attained depended considerably on the year inspected, making a generalization impossible. The most recent evaluations were performed with stopping after 5 ticks past $J=10,000$ simulations, which seems stable, however based on aforementioned results it seems to not provide any incremental benefit when contrasted to, for instance a hard-coded constant J stopping rule.
Several items are open at this point that might be worthwhile investigating in future research. Amongst them are the following (to mention a few):
In this module, we have presented the details of a modified version of the existing Max-Median Rule allowing for the joint selection of securities within this long-term investment strategy. This modified rule, namely the Coordinated Max-Median Rule , essentially bases the median selection criterion on the joint portfolio performance, rather than on single-stock individual performances. We saw that these modifications came with a cost of increased combinatorial complexity and that due to the impossibility of evaluating all potentially-investible portfolios, a parallelized computational approach had to be considered to assess a satisfactory number of portfolios on a yearly basis for potential investment. The algorithm's implementation was discussed, and several conclusions were drawn, the most significant being that our modified algorithm, much more often than not, seems to out-perform the market (in terms of the S&P 500 Index) when a disciplined investor adheres to it for a reasonable amount of time. The data suggest that one of the contributing factors for this on-average higher performance, at least in part, are the correlations between current year portfolio medians and next year portfolio performance, which seem both weak and not always positive. We noted that, more often than not, these correlations tend to be positive, an effect that seemingly averages out in a positive direction over the long-haul. We have also evaluated the performance of the described procedure on real-world S&P 500 data consisting of 43 years, and several potential future improvements, such as further work regarding a more robust stopping rule and the assessment of the procedure reproducibility with other indexes and or markets, were discussed.
Special thanks are given to Drs. James Thompson and Scott Baggett, as well as to Drs. Linda Driskill and Tracy Volz, for their overall help and coaching throughout this summer research project. In particular special thanks are given to both the NSF and VIGRE for making this research a reality.
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