Using AS/400 systems would make them a computer service company. I actually work at a bank (developing apps) and we use a massive COBOL system for a lot of data. However, we also have a massive Hadoop cluster (much larger than the COBOL systems, and firewalls everywhere.

Banks might not be the most competent at web programming, but they higher a ridiculous number of awesome security folks. I would still put them behind Apple or Google in their given domains, but many banks have pretty robust technical systems for their domain.