Did the alpha significantly decay between periods?

Question

A strategy's alpha is estimated in two periods. In the early period it is $6.0$ bp with standard error $2.0$ bp ( $z \approx 3.0$ , significant). In the later period it is $2.0$ bp with standard error $1.5$ bp ( $z \approx 1.33$ , not significant). A portfolio manager concludes the alpha has "significantly decayed."

Test whether the change in alpha between periods is statistically significant, and explain why the two separate p-values do not establish decay.

Accepted Answer

The error. Each period's p-value tests that period's alpha against zero. "Significant early, not significant late" does not test whether the alpha changed. Decay is a claim about the difference between periods, so you must test that difference directly. Test the difference. The estimated decay is 6.0 - 2.0 = 4.0 bp. Treating the two period estimates as independent: SE{	ext{diff}} = \sqrt{2.0^2 + 1.5^2} = \sqrt{4.0 + 2.25} = \sqrt{6.25} = 2.5. z = \frac{4.0}{2.5} = 1.6, \qquad p = 2\,[1 - \Phi(1.6)] \approx 0.11. The conclusion. At p \approx 0.11, the decay is not statistically significant. T…

Did the alpha significantly decay between periods?

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