# Probabilities, from Allen Gillespie

July 6, 2007 |

There is always much debate whether to equal weight or cap weight indices. If there are 30 securities (country ETFs or stocks) in a portfolio, given that they have similar though different return distributions, what is a good way to estimate how frequently one would expect a cap weighted portfolio to outperform an equal weighted portfolio?

## Scott Brooks writes:

It really comes down to what do you see doing better, the larger companies or the smaller companies (large or small in reference to that index/ETF that you are looking at).

If you expect the larger stocks in an index to do better, then go with the cap weighted. If you expect the smaller stocks to do better, then go with the equal weighted. For instance, RSP the SPEWI ETF has nicely outperformed the SPY SP cap weighted ETF for quite a few years now.

I would suggest a bootstrapping approach. Imagine the actual data arranged in a four column table:

Period Ticker  CapWgt  Return
1         GE        0.4     1.05%
1         IBM       0.2    -0.85%
1         …
1         XYZ       0.01   0.97%
2         …

From this table the cap weighted and equal weighted returns can be easily computed. Now generate artificial data by scrambling (i.e permuting) the entries in the return column while leaving the other columns unchanged; compute the cap weighted and equal weighted returns for the artificial table.

Repeat the process 10,000 times and see how the real-life returns stack up compared to the 10,000 artificially generated cases. Some details need to be filled in, but you get the general idea.

Alex is sending you on a snipe hunt. It is obvious by symmetry that the required probability is 50%.

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