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How do share buybacks work?
The short version of JB King's excellent answer is that the company will typically buy back shares from the open market at market price. Sometimes, it will specifically target larger stakeholders, even controlling interests, who are making noise that they want to divest; if such an investor were to just dump their stock on the open market, neither the investor nor the company would be very happy with the resulting price collapse. In those cases, the company may offer an incentive price above market rates. In recent times, the investor looking to divest has often been the U.S. Government, who received stock in return for bailouts, and (with notable exceptions) turned a modest profit on many of them. Not enough to break even on the entire bailout, but the Government didn't just throw $700 billion in taxpayer money down a hole as conservative pundits would have you believe. In the '80s, a specific type of buy-back was made famous, called the "leveraged buyout". Basically, the company took out a huge loan against itself, and used that money to buy up all the company's publicly-traded shares, essentially becoming a private company. This became a popular tool among private equity groups, for better and worse.