Why would a company pay a 3rd party to host payment application but then still take on all payment processing/refunds/etc as a company?

There are a number of transaction flow and responsibility models available for many company business processes. The variables in these models include the in-sourcing or outsourcing of various stages, the ownership of the required business systems, and the management of the systems themselves.

The first step in offering more insight for your current situation would be to know more about the specific go-to-market model in use. Is this an eCommerce scenario where you capture sales online, or a service scenario where you have lower volume higher dollar transactions handled manually by your team, or a retail scenario with payment options available in a physical location. Most likely it is one of the first two scenarios given your consideration of outsourcing the entire function.

The second step is to understand more about the specific flow contained in the go-to-market model. For an eCommerce scenario it could be (a) capture of the customer profile and credit card online; (b) real-time validation of credit card number against known issues database (initial fraud dectection); (c) consolidation of daily transactions into a file for your merchant banking institution and transfer of payment to them; (d) recording the consolidated transaction total in your ERP system (often at the merchant bank or credit card processor level); (e) acceptance (or rejection) of the transaction by the merchant banking institution; (f) feedback to the eCommerce / customer management system on the payment status; and (g) matching the cash clearance from the merchant bank against the ERP transaction. If you are going to outsource part or all of the business process you still need to understand the flow and how it fits with your other systems.

The third step would be the business evaluation of the situation given what you know. Business Process Outsourcing (BPO) can be a valuable business tool when leveraged correctly. What you’ll need to understand are the business goals and needs of your go-to-market model, the visibility / reporting needs for your company on the process as a whole, and the cost models for in-sourcing and out-sourcing the specific business process.