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Why does an unweighted synthetic index have better returns than the S&P 500?

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As part of a statistical study of stock returns I noticed that using unweighted windowed gains has a much larger return than the S&P 500. The data are all from Yahoo and standard data massaging techniques are used to remove bad data from the Yahoo data. As required, the results are essentially the same independent of the window length (from 1 to 400 trading days) except for the expected reduced volatility. The unweighted data is selected using the criteria that once a stock closes below $5 it is dropped going forward, and a 20 day exponential trailing moving average of dollar volume is above some discriminator (in this case $1e6 and $1e8). All of these data are computed using the Adjusted Close and the result for each point is the mean of all returns (e.g. on Day 1 you bought $1 of every stock and on Day 2 you sold every stock).

Depending on the discriminator and the date range there are between 20 and 3000 stocks in each window (here just the 1 day window is shown). The gain is then cumsum(log(a[2:length(a)]/a[1:(length(a)-1])) for a window of length 1 (as shown).

Obviously an index constructed this way is artificial but it in some ways represents are more accurate representation of the overall stock market as a holder of individual stocks approaches the market. I suspect many people who hold individual stocks (as I always have) keep their portfolios roughly balanced by dollar holdings not market cap.

I noticed some time back that over 30+ years in the market I appeared to have better returns than the S&P 500. Of course not nearly what these data show since 1970 (S&P 500 total return ca. 25x (7.1%/yr), index return 400x (13.6%/yr), both nominal not real).

Anyway what most impressed me with the unweighted index approach is how linear it is in log space. This does support an efficient market hypothesis view, the gains from rebalancing every (window length) days appear substantially more log linear than the S&P 500.

The questions are 1.) Is there an inherent bias in taking the windowed gains? 2.) If the result is mostly valid what does it say about market cap weighting?

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Posted by John Daschbach
Posted on 05/19/2017 2:09 PM
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maybe compared to all shares the s&p500 shares have already grown a lot to qualify for the s&p500 and so don’t grow as much once in the S&p500.

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Posted by Dinero
Answered on 05/24/2017 5:50 AM
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    I don’t trust backtests that don’t have out of sample testing. I’d like to see these results in real-time with taxes and fees included. I suspect they won’t be nearly as robust.

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    Cullen Roche Posted by Cullen Roche
    Answered on 05/25/2017 1:16 AM
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      The implicit out of sample testing is showing the two series with different weighted trailing average daily dollar volume. A pure out of sample statistical test would be, at each time step, choose at random only some fixed fraction of stocks which meet the criteria, and repeat this millions of times. This gives you statistics at each point in time (mean, median, most probable, quantiles, quantile skew, quantile kurtosis, ….)

      I didn’t include taxes and fees as the model idea was to create an index which was maintainable with a simple algorithm and which was possible to implement by an entity like an ETF (each day you recompute the next n-period index allocation and buy-sell to rebalance, pretty much what an S&P 500 ETF does)

      I think it can be shown on a statistical basis the only way a stock market can exist in perpetuity is if the mean of unweighted randomly selected index (based on whatever criteria is operationally possible and reasonable) exceeds over very long periods the market weighted index. Otherwise the market disappears and eventually becomes a single stock.

      Given that the single stock limit has to hold as t -> infinity if mean(weighted) > mean(unweighted) I find the results consistent with what we should expect, even if they are not accurate.

      I will note that the post 2010 results show only a 17% relative advantage for the unweighted index. This is not surprising as the increase in algorithmic trading should bring the two measures closer together.

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      Posted by John Daschbach
      Answered on 05/25/2017 3:34 PM
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