Wrong. While the Fed is a consortium of privately owned banks, it is chartered by Congress, from which it gets its authority as the sole producer/manipulator of currency and credit. The President appoints the board of governors, who have control over the system.
Basically, the Federal Reserve is a collusion between government and the banking industry, in which government is allowed to borrow far beyond its means to ever repay and at rates of interest much lower than a free market would demand; in return for this service the banking industry is rewarded with the power to produce as much currency and credit as it likes, guaranteeing that it can't go broke, regardless of how stupidly it loans and invests its money (bank bailouts anyone?).