I will drop in my two cents here. First off, Tenants and current owner signs off on estoppel. You do not want a tenant say they paid 3 months ahead after you have closed. Also you need to know the deposit amount and repair request that have not been addressed. As far as the inherited tenants, I have never had a tenant 30% below market. It was usually within $100.00.
The day I close on the property, I go to the property and hand each tenant a letter or leave it on the door, where new rents will be mailed, that no cash will be accepted, it has to be in check or money order. I also inform them that at the end of their lease I will be offering them a new lease but the rents will increase, usually half of what they are below market on a six month lease. The next lease will be for a full year at full value.
I lose some tenants right up front, some say they will find another place. Usually, once they see the property is being cleaned up, deferred maintenance is being done, the good ones stay. I have one tenant that told me he was moving 10 years ago. His rent when I bought it was $200 per month, today he is at $575, but he recommends pretty regular and warns me of tenants that would be bad for the neighborhood. (the difference is, his sink doesn't leak, the old clay sewer pipe was replaced, so everything drains, he has insulated windows and new steel front door, and this year he got central heat and air. When the tax statements came in and my taxes went up again, i informed him that starting February when his lease was up I would have to raise his rent to $600. His statement was, "Raise it I don't care, I know I will be warm and dry and if I am not you will play Hell getting me that way." Guess what he is right.
Back to your point, I do not see how you let a 1200 rent go for 800, but you do need to be human about it, even if it is business. I would inform them that rents are going up and if they can afford it you would love them to stay, but give them the modified lease of raising the rent $100 every quarter until it is at market. If it needs some maintenance and they see you taking care of it, they will want to stay with you. But it gives them a way out.
I noticed above, someone recommended a month to month only, I personally hate these. My shortest lease is 6 months and that is to get someone up to market rents in newly acquired properties. I like 1 year leases and if they double the deposit, I will give them a 2 year lease, (for their protection against rent raises.)
Food for thought you can always get good tenants, that will stay when they move in at a little below market value, it is a headache to move for the tenant, so you can adjust your rent each year to get back to market in small increments. Very few people are willing to move for $10 to $25 more a month. One month vacancy, because you are at top dollar will cost you more than taking a little less rent for a qualified renter who is dependable and has staying power. Landlording is a people business. It is business but we are dealing with their daily lives, provide quality for the money and make sure they pay ontime and keep the property in good shape. Everyone comes out ahead.
I will get off rant and step off the soapbox.