chex system may not only be used for previous negative banking relationships, but it also provides a negative response for identifying information that may erroneously supplied by the customer or input improperly by the employee. It also, if you subscribe, performs an OFAC verification. Since time deposits may be considered high risk, this type of verification may enhance your AML procedures and CIP. Therefore, IMHO, I would run chex system verification for time deposits.
Regarding existing accounts, this may depend, if they are currently in good standing and verification previously performed there may not be a benefit, simply cost. However, if your bank reports deposit accounts to chex systems, I have, believe it or not, had customers who had previous c/o accounts try to open another account at the same bank. If your system does not reflect c/o deposit accounts, chex system review of customers with previous relationships may assist with recovery.
Allowing PO box only for an address will create non-compliance with the Patriot Act and BSA. Customer may supply PO Box as mailing, however, I would always obtain and verify a physical address.
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An error is not a mistake until you refuse to correct it