If you are referring to a "branch" vs. a separately chartered subsidiary, then the FDIC's FIL 46-98, "Guidance on the Use of Trade Names" is where you need to look. This is an interagency statement that provides some specific steps you should consider. The following in an excerpt:
"While there are no federal laws or regulations that specifically require that all branches of an insured depository institution operate under a single name, the Agencies are concerned that if customers believe they are dealing with two different institutions, they may inadvertently exceed FDIC insurance limits by depositing excess amounts in different branches of the same. The Agencies believe it is important that customers understand the scope of FDIC insurance in these circumstances. Accordingly, an insured depository institution that intends to use a different name for a branch or other facility should take reasonable steps to ensure that customers will not become confused and believe that its facilities are separate institutions or that deposits in the different facilities are separately insured."
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Patricia Cashman,CRCM
Cashman Compliance Solutions, LLC
This is NOT legal advice