How does the Federal Reserve increase its liquidity?

Question

through raising the discount rate?

Answer 1

The Fed doesn't have liquidity banks and financial/money markets do. If the Fed buys bonds it "creates" the money to purchase them out of thin air increasing the money supply (creating liquidity). It does not actually buy the bonds but intervenes in the overnight market and buys repurchase agreements (Fed funds) which banks use to balance books everyday.
http://en.wikipedia.org/wiki/Repurchase_agreement

Answer 2

Let's look at this from two standpoints:

1) Liquidity for maintaining it's operations - The fed funds its operations from interest earned on Treasury Bills it owns as collateral for currency issued. The Fed gets enough interest in a month to pay for it's operation for the year. At the end of the year, the Fed returns the excess interest to the Treasury. Since 1913, it has returned about 95% of the interest earned back to the Treasury.

2) Liquidity from increasing the nations money - Here the Federal reserve has virtually unlimited liquidity. It can create money out-of-thin-air to purchase however many T-Bills are required to meet targets.

So, in general, the Fed is awash in cash and money-creation power, and rarely worries about it's liquidity.

The discount rate does not affect the Fed's liquidity. However, adjusting the discount rate may influence how much the member banks may choose to keep in excess reserves.