People always dunk on me for saying this outrageous statement, but if you put me in charge of a random Fortune 500 company, I give myself a 50% probability of the company performing better than if its real CEO was in charge, and a 50% probability of it performing worse. I really don't think these guys bring as much to the table as they think they do and as much as the general public seems to think they do. Business success does not come from paranormal insight and abilities.
I think you could randomly nominate CEOs from the set of the general population with, say, a minimum of a college degree, and the randomly nominated CEO's performance would not statistically differ from "qualified" CEOs.