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Many buybacks are funded with debt issuance. That increases a company's liabilities, and since the proceeds are used to buy stock that is then immediately canceled, it does nothing for the company's assets. The result is that under generally accepted accounting principles, the company's liabilities can end up exceeding its assets. Technically, that's the definition of insolvency.
The other side: The companies with the largest negative shareholders' equity also tend to own extremely valuable brands. If you included a reasonable brand value on the asset side of Starbucks' balance sheet, then it would no longer look insolvent. The same goes for most of the other companies on this list.
One curiosity: The buyback-happy company at the top of this list, Philip Morris International, was spun out of Altria in 2008, around the same time that Kraft was also spun out. But Kraft, having merged with Heinz, now finds itself at the other end of the league table, with $65.4 billion in shareholders' equity.
Trivia: Which company has the most shareholders' equity? That would be Berkshire Hathaway, on $375.6 billion.