That is what drives up the pay of CEOs. If you want someone who will be top-notch in running organizations as huge and complex as Fannie Mae or Freddie Mac, there is no point offering $5 million a year if similar enterprises elsewhere are paying $20 million for people with the kind of ability required.
Who is going to take a $15 million pay cut to go run these enterprises, in addition to having to put up with politicians?
The money that can be saved by limiting CEO pay is chump change compared to the money that can be lost because you cannot attract top-notch talent.
Congress itself is a classic example of what can happen when penny-wise policies restrict the caliber of people who can be attracted.
No top-level doctor, lawyer, economist, engineer or CEO can become a member of Congress without taking a big pay cut, perhaps costing that person's family millions of dollars over a lifetime.
On the other hand, if you paid every member of Congress a million dollars a year, it would cost less than the cost of even a small government boondoggle, much less a whole agency.
It is not that the turkeys in Congress today deserve a raise. They don't even deserve their current pay. But that is the very reason for attracting different people. Cheap politicians are actually very expensive and the same principle applies to CEOs.