It is said to be that, in a QE programme, the Fed reduces long-term interest rates by purchasing long-term securities. How does this work? Why does buying longer maturity securities from banks reduce long-term interest rates?
"Long-term securities" (i.e. bonds) work like this: You're a big company wanting to expand. You write a bunch of IOU notes that say "I'll pay you back $100 in 10 years". Then you sell these notes by auction.
The amount that people bid defines an interest rate. If you manage to sell them for $90 each, you get $90 and have to come up with an extra $10 in the next 10 years, or $1 per year of interest. If you sell them for $80, you have to come up with $2 per year of interest. If you sell them for $99, it's only $0.10 per year. From that, you can calculate an interest rate.
This also applies to the secondary market. If it's 5 years later (halfway through the term) and I buy the bond for $95, I am getting $1 interest per year for the next 5 years. If I buy it for $99, I am only getting $0.20 interest per year for the next 5 years.
(Of course, I don't actually get any of that money until I sell the bond, or it matures.)
If the Fed is buying up bonds, they are paying an above-market price to do so (otherwise people with bonds would sell them on the market instead of selling them to the Fed). They are also raising prices in the whole bond market as there's less supply of bonds for people who want bonds for some reason. Higher prices mean lower interest rates, as explained above.
Bonds sold by banks and private parties have a liquidity premium because these bonds can be liquidated for cash in the secondary market.
The amount of liquidity depends upon the amount of cash available in the system relative to the amount of bonds in the market. If there are more bonds and less cash available to buy them, then there will be less liquidity and the liquidity premium will be higher.
If there is more cash and fewer bonds, the liquidity premium will be lower. This is the effect of Quantitative Easing because central bank buys bonds and releases cash, causing the liquidity premium to drop, resulting in a decrease in medium term interest rates.