Generally, "private banking" can be broadly defined as providing personalized financial services to wealthy clients. However, in a specific BSA/AML context the term is limited to an account:
* for a non U.S. person which
* requires a minimum deposit of $1MM.
That does not sound like the folks you are talking about. If yours is a community bank, a response to a pre-examination questionnaire that indicates your bank is involved in "private banking" will probably cause your EIC to swallow his or her tongue.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.