Question: Why do companies on the stock exchange "buy back" their own shares sometimes? 

Answer: The fourth answer is the correct answer. A company can not declare profit on their own stocks, even if they buy back at 5 and sell it at $50, not $1 of it will be in income, just additional capital and higher owner's equity. The company itself has no good reason to declare a dividend or a buyback, it helps the share holders only. With less shares outstanding Earnings per share goes up, and that way the company price goes up. It's a way to increase the share price of the company despite no growth, or little growth. Most big companies who have slow growth, but high cash flow make up their slow growth by dividends and buy backs.
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