What is your breakup between CHA and market tenants?  

A nicer place with CHA, HDTV and streaming BluRay goes for $400/week. 

@Keith Rivera We manage a large portfolio of CHA tenants and a few in the suburbs in DHA and CCHA.  

So about 18 months ago, the pm told me that they trusted me enough to allow me to rent any new ones out. cha-ching!

Around $700, cha-ching.  

I'm sure you have a lot to choose from in Ra-cha-cha.

Typical term is 'property appraises for full purchase price", so 99% of the time, conventional or commercial, getting the loan approved covers the DSCR (self-sufficency) and the assumed FMV of the property.The diff to me has always been, the DSCR implies cash-flow and deposits to my account every month - - cha-ching :)

I always install it when I buy a new unit, so I can't say what kind of increase is warranted, but I will say that I estimate my houses would rent for $100-150 less per month if they did not have CHA (comparing them to the market where CHA is not installed). 

You can go to The CHA Housing Choice website and there is a PDF named CHA’s Rent Determination Standards which will give you a great guide to determining what amount CHA will allow based on the specifics of your property.As @Scott M. stated, consult with an attorney very carefully if you feel the tenant may not be able to cover the balance of the rent.  

@Michael Cha my advice would be to buy another property with 5% down instead of waiting to save 20%.