Trading - Type I and Type II errors

According to curriculum:
"A return distribution of skilled managers that is highly distinct from the return distribution of unskilled managers:
-> high opportunity cost of not hiring the skilled managers" (NB: so, this would be a Type II)

“The smaller the difference in sample size and distribution mean and the wider the dispersion of the distributions, the smaller the expected cost of the Type I or Type II error.
More- efficient markets are likely to exhibit smaller differences in the distributions of skilled and unskilled managers-> lower opportunity cost of retaining unskilled manager and lower the cost of hiring an unskilled manager.

Isn´t this a contradiction? Shouldn´t the rule be:
Wide dispersion (high differences skilled vs unskilled) -> high cost of Type I or II
Narrow distribution (low differences skilled vs unskilled) -> low cost of Type I or II

Thanks!
C

Highly distinct here means the mean return of skilled managers is significantly different from the mean return of unskilled managers (back to hypothesis testing days) \to so it would suck big time not to hire the skilled manager (high opportunity cost).

The smaller the difference in sample size and distribution mean of the distributions (means the mean return of skilled managers and unskilled managers are not significantly different from each other (so does not make much of a difference who i hire or fire),

AND

…if the return distribution of the skilled managers and the return distribution of the unskilled managers had wide dispersions (high standard deviations ), this means both skilled managers and unskilled managers are not really consistent in terms of adding value (hence no big deal if i fired the good one and kept the bad one since they are equally inconsistent) \to small expected cost of Type I and Type II errors.

So when you read the question, you have to differentiate if they are referring to differences in mean returns or are they describing the dispersion of the distribution.

Wider dispersion & smaller differences in mean returns \to low opportunity cost of Type I and II errors.

Tight dispersion & larger differences in mean returns \to high opportunity cost of Type I and II errors

2 Likes

Yes :slight_smile: I understand, I was mixing things up.
Thanks for the nice explanation!

I still CANNOT believe CFAI did not take a decision yet…

Anytime now. Give it a few more weeks. :sunglasses:

Super Explanation!!

Please can you explain how “mean reversion” affects type I and II as well? Thanks