Free and Open Source Software

Let's say, I create a bank with the caveat that all of my banking phone apps and webapps are FOSS (or if they depend on non-free components — banks probably do to communicate with each other —, then just OSS). Am I going to be behind the competition by doing this?

If the most secure crypto algorithms are the ones that are public, can we ensure the security of a bank's apps by publicizing it?

Are they not doing this because they secretly collect a lot of data (on top of your payment history because of the centralized nature of card payments) through these apps?

EDIT: Clarifying question: Is there a technical reason they don't publicize their code or is it just purely corporate greed and nothing else?