A question for Kim or Victor: Since IWM has more stocks than SPY, does it follow that daily returns on IWM are closer to the Normal Distribution than SPY? - A Reader

Victor Niederhoffer replies:

It does, as a consequence of the Central Limit Theorem .

Kim Zussman replies:

Let's look at it empirically. Here is the "Anderson - Darling" test for normality of daily SPY returns, 2000-present (SP500).

Next is the same test for IWM (Russell 2000 ETF), 2000-present.

Rocky Humbert writes: 

Vic, I'm not sure that the central limit theorem is the right paradigm. An unknown is whether the covariance within the two groups is sufficiently different to offset the CLT. I have never tested this. And testing is tricky because you need to use compounded total returns with dividends reinvested. The index and stock prices produce misleading results because dividends are greater for big caps.

Intuitively, I believe that most of the perceived differences can be explained by 2 things:

1) the dividends…which is really just a duration effect and 2) the reality that companies leave the R2k only when they are incredibly successful or when they die. Stocks only leave the SP500 when they die. They never leave the SP500 and go to the R2k when they are successful. So over time, the perceived differences are a micro sampling of a survivor bias between the 2 indices. Not sure how to test this theory…

What we do know is the implied volatility of r2k is almost always higher than the implied volatility of the SPX. I think this could be an analogue to the fact that out of the money puts are more expensive than out of the money calls. Put another way, if you are long SPX and short r2k in equal dollar amounts, you will usually make money during violent and persistent market downdrafts. I think this is proof that the distributions are different.

Victor Niederhoffer writes: 

Those are good points you make about areas that I should have considered in estimating the departures and distributions of comparative performances. It is also amazing to me that the statistical tests, especially the Kolmogorov Smirnov, show such departures. I am a great believer that the risk premium on untried and small stocks is much bigger and that they should perform better and that buying two handfuls of them will have a limiting distribution that converges to a return a percentage or two above the 8 % you get from the average NYSE stock. I must go back and check my premises. It reminds me of how I told the people in my family to buy the riskiest vanguard over the counter fund, and they tell me that they are always getting notes in the mail that the funds I recommended are being sued by their holders as the worst performing funds in history due to all sorts of wrongs of a practical and theoretical nature. I mean this response in a humble and appreciative way although it is sometimes hard to communicate that by email in the face of all the errors that are elicited.

Ralph Vince writes: 

Like everything else in this realm, it depends on the unit of time used in analysis. If you use annual data, things play much more nicely to Normal. The shorter the time unit used, the less so.


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