But that doesn't mean it lost half it's value. If I printed an extra 100 trillion dollars and launched it into space that wouldn't affect the value of the dollar.
And the same thing was true during the bailout. 99% of that money was never spent so it didn't have any affect on the economy.
My original point was that that money entered the system via large investment banks, whom use it to invest in publicly traded securities. Most of that money doesn't make it to circulation, it just inflates the price of those assets instead of all goods in the wider economy.
In the UK, at least, the Bank of England used QE money to buy government bonds. This meant that the bond-holders could exchange their illiquid bonds for cash, which they then needed to do something with.
The BoE then collected and destroyed government payments on the bonds, so the net amount of money doesn't change in the long-term.
One side-effect of this is that the cost of bonds went up -- demand had increased -- so the profitability of buying them went down, meaning the banks were naturally incentivised to do something else with that money.
It's not a panacea, but it is quite a neat lever to have when you want to increase the supply of money without actually spending any. Unfortunately it still looks like the banks are being given lots of free cash, but it's not actually free.
A lot of it was a backstop in case it was needed, but those reserves were not tapped.
Basically the Fed made lines of credit (with strings attached) open to banks to guarantee liquidity, increasing trust in markets. Banks didn't withdraw the money, and some that did (or were forced to take it to dissuade bank runs on the others) paid it back quickly to get out from under the strings.
Here's [1] a decent source on the finances of the bailout (which were actually loans).
