"Sponsoring" means that the Member Bank has conducted an FI (financial investigation) and a very extensive due diligence process and is willing to allow the ISO/MSP to setup merchant accounts under its umbrella as the acquiring bank.
Ultimately, even when an ISO/MSP takes full risk on their merchant accounts, the bank is still held 100% liable at the end of the day if there ever were a circumstance where the ISO/MSP defaulted on covering chargeback losses or other liabilities.
Even beyond risk or loss liability, the Member Bank is also responsible for ensuring that its ISO/MSP's comply with all Visa and MasterCard operating rules & regulations, that various portfolio-wide and BIN-wide metrics for chargebacks are not exceeded, etc.
So it is a huge responsibility for a company to be an ISO/MSP with respect to their sponsoring bank and likewise, it is something that banks look at with a great deal of scrutiny due to the risks and processing volumes involved.
With respect to the term "BIN" - it stands for Bank Identification Number. Issuers and acquirers have their own BIN and it identifies them as the bank behind a given transaction.
When it comes to merchant processing, having access to your own BIN or BIN heirarchy is a significant advantage with respect to the merchant service and support capability it provides you with. (It is somewhat analgous to having your own Data Center if you are a Web hosting company.)
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