The descendants of oil tycoon John D. Rockefeller have them. So do Microsoft Corp. founder Bill Gates and hedge-fund manager Bill Ackman. 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 family-office expert how to define them and the likely response will be: “If you’ve seen one family office, you’ve seen o

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