I don't think this is possible because first of all, one of bitcoin's features is that it allow transactions without risking your privacy and security. Loans and such, which you have mentioned, is not easy to process because it requires info (most of the time private ones) for them to be able to reach you if ever you decided to run from your obligations.
That is some valid point. The fact is there really would not be any necessity to have a bitcoin bank in the first place. Our wallet first and foremost is our bank, and even if we want to exchange to fiat, I am sure that is not even a problem anymore in most places except few. The lending part is not something that may really work out looking at all the modalities involved.
I am sure that there is some teams already thinking at this aspect for some time. Now nobody can predict a date for such things but a Bitcoin Bank is inevitable, i wonder what will be the politics of this bank considering that bitcoin is developed on decentralized technology.
I saw an ICO recently coming up with something like this although I did not get to take much look at the details or the whitepaper as the case may be, but may be in the long run, something like this can happen most especially having exchange to fiats, lending and some other banking stuffs integrated all in one. How they want to pull the lending aspect off is what I do not really know anyway but when it comes to exchanges in most localities, that is already existing.