Big wealth-management firms that have expanded into Asia are running into a bit of a snag: More and more of Asia’s millionaires are taking it upon themselves to manage their wealth. A June report from Boston Consulting Group found managers at banks in Asia have full discretion over 4% of clients’ assets—down from 7% in 2006. According to annual reports, HSBC Holdings Plc. earned $2.10 for every $100 of assets it managed in the Asia-Pacific region in 2011—a 25% decline since 2007. Easaw Thomas, an anesthesiologist in Asia, keeps less than 10% of his money with wealth managers and doesn’t give bankers full discretion. “They are essentially like secretaries who help facilitate a trade that you may want to make,” Thomas said.