Brands like Ferrari, Prada and Hermès have built lucrative businesses by focusing squarely on rich customers. First Republic Bank is trying to do the same.

Over three decades, the bank has catapulted from a tiny thrift to a lender of choice for some deep-pocketed clients, including Facebook Inc. chief Mark Zuckerberg. The bank draws in clients with excellent terms on mortgages and perks like branch parties and private museum tours.

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