Why IID Stock Returns Do Not Guarantee Equal Betas

Summary

The document addresses whether two stocks with independent and identically distributed returns must have the same beta. Its answer is no: beta depends on a stock’s relationship to the market benchmark, so identical marginal return distributions do not determine equal market covariances. The response uses perfectly negatively correlated stocks as an illustration, noting that their betas can have opposite signs when beta is nonzero.

It also asks whether sample size matters and replies that the number of observations does not matter provided the statistics are significant. That statement is terse and should not be read as saying sample size is irrelevant in practice: sample size affects estimation uncertainty and the ability to establish significance. The discussion does not specify the market benchmark, estimation procedure, or assumptions behind the example, and offers no empirical evidence or confidence analysis.

Key ideas

• Identical return distributions alone do not imply identical market betas.
• Beta reflects a stock’s relationship with a market benchmark.
• Perfect negative correlation can produce betas with opposite signs when beta is nonzero.
• The response conditions its sample-size claim on statistical significance but does not discuss estimation uncertainty.

Tags

# Beta and the Assumption of IID Returns


# Beta and the Assumption of IID Returns












For two stocks that are independent and identically distributed (iid), do they have the same beta? Does the number of data points matter?

## Answer by Aksakal almost surely binary (score 0, accepted)

https://quant.stackexchange.com/a/10421

> For two stocks that are independent and identically distributed (iid), do they have the same beta?

No. If you have two stocks perfectly negatively correlated, then ones' beta would be positive, and the other's - negative, as long as the beta is not zero.

> Does the number of data points matter?

No, as long as your statistics are significant.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.