Marc Lasry, chairman and chief executive of New York–based hedge fund firm Avenue Capital Management, has specialized for years in distressed-debt investing, a topic with which he grew intimate in a previous life as a bankruptcy lawyer. He has become even more steeped in it lately, as Avenue in recent months has bought — without government support — some of the so-called legacy assets owned by troubled banks. Lasry, 49, co-founded Avenue in 1995 with his sister Sonia Gardner and has quadrupled assets under management in the past four years, to about $16 billion. A donor to the Democratic Party and a student of politics, he supports...