No, that's incorrect. The Federal Reserve System is an unusual hybrid unlike NASA or any other government agency I can think of. NASA doesn't have Boeing or Raytheon as "member agencies" or as owners of regional space administrations that take their marching orders from a Board of Space Governors appointed by the president.
As others who have replied, that's not quite correct. However, substantially all the net income of the Federal Reserve is remitted to the Treasury Department.
Some of it is remitted to the "owning" banks, as a kind of rebate for the token capital they pay. But "ownership" here is not the same as beneficial ownership.
In other words, the Federal Reserve is designed to be not directly linked to the Executive Branch. But it is also designed not to be a profit-making private enterprise.
Actually, I'm pretty sure the Fed is a privately-owned company, with preferred shareholders and everything. That's not to say the government isn't hugely influential on what it does, but that it's a completely separate entity.
Of course, I could be completely wrong. I'd actually be interested if someone had proof for either argument.
Seriously, if it involves the U.S. federal government, information about it will be easy to find. You don't need to guess or assume, just do a few simple searches.
edit: Well, not all of it, but the structure of the Fed isn't exactly a national secret.
That's a regional Federal Reserve Bank, not "the Fed". Also, you leave out that those share holders get at most 6% annual return for their investment, since it's capped by law.
