macrocephalopod

But a stock could go from $5.20 to $4.80 and then back to $5.20, and instead of showing a flat geometric return and a positive arithmetic return, you just have an 8% loss in your data and not the 8.3% gain.

Yeah and a bunch of equally probable unpredictable things could also happen to all the stocks that go in and out of the index.

Its a one way problem isn't it? How often does a stock rise in price and then get dropped from the index for a stock that just fell in price?

Stocks that get removed tend to be those that have decreased in market cap. Stocks that get added tend to be those that have increased. I'm unclear why you think this is a form of performance bias in the index.

To bias the performance of the index you would need to believe that the stocks being added would have different future expected returns than the stocks being removed.

This isn't about the index its about the performance of the individual stock. If a stock falls below $5 and then comes back above it later, your data set does not include the gain the stock received. I the above example, your data says stock lost 8%. In real life it was flat.

That's a bias because there is no way for the opposite to occur. And this isn't about momentum or mean reversion, its just about randomness.

There’s no bias. The data doesn’t include moves from $4.80 to $5.20 but it also doesn’t include equally likely moves from $4.80 to $4.40. Adding or dropping stocks based on their past price moves can’t introduce a bias unless future price moves are correlated with past moves.