The descendants of oil tycoon John D. Rockefeller have them. So do Microsoft Corp. founder Bill Gates, Fidelity Investments Chief Executive Officer Abby Johnson and Hong Kong telecommunications tycoon Li Ka-Shing. They are family offices, the loosely regulated, privately owned companies that manage vast amounts of money for wealthy clans. They’re also among the most secretive firms in the world, and by all accounts they are expanding rapidly. Ask any expert on family offices to define them and t

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