Apple chief executive Tim Cook wakes up at 4 every morning, which is clearly the secret to his success . . . at getting up early.

It’s hard to say much more than that though, because, aside from the truly exceptional cases, it isn’t obvious how much of a difference chief executives really make in their companies. That hasn’t stopped us, of course, from lauding them as either visionary leaders who lead with their visions, or efficiency experts whose idiosyncrasies can teach us mere mortals how to win at life like they do at business. But the fact of the matter, as Dan Rasmussen and Haonan Li point out in Institutional Investor, is that there’s precious little evidence that most of them are actually improving their companies’ performances, as opposed to just occasionally getting lucky.

How can we say that? Well, the important thing to understand is that the business world has been taken over by two big ideas in the last 30 years. The first is that companies should solely focus on increasing their stock prices, and the second is that, to make chief executives care more about this, and attract the kind of top-tier talent that could presumably do a better job of it, businesses should pay them bigger performance-based bonuses in the form of stock options and grants. The result, as Rasmussen and Li report, has been an MBA arms race that has seen chief executive pay at the biggest firms soar, in inflation-adjusted terms, from an average of $1.8 million in the 1980s to $4.1 million in the 1990s, and then $9.2 million in the early 2000s.

So the question is what have companies gotten for these even more exorbitant pay packages? And the answer is: not much, other than having better-remunerated executives. Indeed, Rasmussen and Li found that supposedly elite chief executives, who either had an MBA or had worked at a top consulting firm or investment bank, weren’t statistically significantly more likely, if at all, to see their company’s stock outperform that of their peers or the broader market. And this was true for graduates of every business school, including Harvard, Stanford and Wharton.

On top of that, it turns out that past chief executive performance really isn’t indicative of future chief executive returns, either. The number of executives whose companies were in either the top half or top quarter of performers in one three-year period, who also were in that category over the next one, was almost exactly what you would have expected based on chance alone: 25 percent in the case of the former, and 7 percent for the latter.

All of which is to say that there’s no evidence that chief executives who have fancy résumés do a better job than chief executives who don’t, or that ones who have succeeded before will do so again. At least, that is, when it comes to how they say they should be judged: by how well their stocks do. If you were feeling particularly cynical, then, you might argue that we’ve set up a system where we allow people who’ve checked off all the right boxes — college, consulting and business school — to play a game of heads they win, tails shareholders lose. If they’re fortunate enough to be in charge when things are going well, then they’ll probably get paid a lot more than they deserve, but even if they aren’t, they’ll still walk away with more money than most people could dream of, all while the business suffers.

Now, this isn’t to say that chief executives never matter. Just that the Steve Jobses of the world are so few and far between that it doesn’t make sense to talk too much about them. Most chief executives, after all, aren’t changing the world, but rather changing how much of their company’s stock to buy back. They’re more proficient at thinking the way business schools want them to — and using the word “leverage” as a verb — than they are at, as you might put it, thinking different.

It doesn’t have to be this way. We don’t have to pay chief executives 312 times more than the average worker just to get the same type of results we’d expect from pure chance. We could probably get away with paying them, “only” say, 58 times more. That, as the left-leaning Economic Policy Institute points out, is how much we did back in 1989, in what was hardly a socialist dystopia. In fact, it was arguably even more capitalist not to pay them so much for delivering what, in a lot of cases, are fairly mediocre results.

They could even sleep in if they wanted, and get up at 5 a.m. instead.