7.1 The coordinated max-median rule for portfolio selection  (Page 5/6)

Recent evaluations and results

Recent evaluations have been mostly focused on evaluating the following:

1. Repeatability of the procedure in terms of the variability associated with its possible tracks for each realization.
2. Determining any additional gain (if any) in terms of returns as a function of the number of portfolios evaluated ( J ) at any given year.
3. The existence of any indications between current year portfolios' medians and subsequent year (same portfolio) performance. Are there any associations and if so how weak or strong are these?
4. Investigating a stable and plausible stopping rule and assessing how beneficial it might be to run the random search until this condition is met.

Several experiments were set up to determine if it would be worthwhile to inspect more randomly sought portfolios on a yearly basis as part of the overall procedure. A job simulating a total of 104 tracks (each consisting of $J=25,000$ portfolios per year over a 43 year period, 1965 though 2008) was submitted to ADA and took approximately three days to complete. Several important observations can be made from the outcomes of these simulations (shown in Figures 3 and 4, below). We note that, here, we can exploit the independence regarding the portfolios evaluated to get 52 tracks of $J=50,000$ portfolios each by combining pairs of $J=25,000$ tracks and selecting the maximum of the pair (simply the maximum of a longer execution). Essentially this gives us information regarding what would have happened (in terms of the performance of the strategy should we have run it for twice as long). Analogously, tracks for $J=100,000$ and $J=200,000$ portfolios were constructed. Finally, some overall discussion of the results is given after the figures.

The total portfolio value was evaluated at the end of both years 2006 and 2008 and contrasted to both market performance (blue track) and the performance of a single track of a whopping $J=2,600,000$ portfolios considered yearly (green track). As expected the variability of the procedure compounds as a function of time, and by chance we might under-perform the market. However, more often than not the procedure out-performed the market and by a quite reasonable amount. The proportion of portfolio-tracks simulated that were over an equally-weighted alternative at the end of 2006 was over 80% (for the cases where $J=25,000$ and $J=50,000$ ) and over 90% (for the cases where we assessed more randomly sought portfolios, i.e. $J=100,000$ , and $J=200,000$ ). Also, there is weak evidence suggesting that, although running as many as $J=2,600,000$ portfolios might at times outperform the market, this approach is generally not consistently higher on-average than considering tracks consisting of less yearly-evaluated portfolios.

Another rather interesting observation is made through the scatter-grams produced (see Figure 5, below) assessing the correlation between current year portfolio median and (same portfolio) next year performance contrasted to the performance of the S&P 500 index. The number of portfolios evaluated for this purpose was $2,000,000$ and the those that are highlighted as producing the maximum of the medians represent ( $<0.1\%$ , i.e. $<2,000$ ). The main purpose of this effort was to assess any associations between the current year medians as a forward-looking measure of portfolio performance (as we intend to pick the maximum and by chance we can pick portfolios of performance similar to those in the top 0.999 percentile). As expected the associations are weak, though not extremely weak (correlations are $0.209$ for the first case and $-0.182$ for the second), however can be noticed and depend highly of the year evaluated.