In banks' furious scramble for wealthy customers' investments, size is sometimes overrated.

Banks of all sizes are competing to get more of their most affluent customers' asset management business, hoping to lock down a steady source of fee revenues. The market is dominated by the country's largest banks and investment specialists, from JPMorgan Chase (JPM) and Bank of America's Merrill Lynch (BAC) to UBS and Credit Suisse.

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