>Say a company is worth a billion dollars and has issued a million shares. Each of those shares is worth $1,000 (because $1,000 times one million is $1 billion). The company uses its profits to buy half of those shares, so now the remaining 500,000 shares are worth $2,000 ($2,000 times 500,000 is $1 billion).

Why would the company still be worth $1B after the buybacks? The company just spent $500M on buybacks. So the company has $500M fewer dollars in the bank. I assume the company would now be worth $500M less. So the company would be worth only $1B - $500M = $500M. So with 500k shares in existence, the stock price would be unchanged from the beginning: $1k.