Screening existing tenants

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Curious what the rule of thumb is on screening existing tenants. I currently have no background, credit checks or employment verification for 4 different inherited tenants. 

We're slowly repositioning the property from a D to a B, tenants we're used to really low rents and are scheduled for a second rent increase next month, everyone is month-to-month and have been consistent with payment for the past year. 

I'm almost through @Brandon Turner 's book on Managing Rental Properties and realizing I'm setting myself up for future problems if the tenants can't meet the 3x income minimum rental requirement. Which, I suspect at least two of them won't. 

So, do I hand everyone an application to renew tenancy and cut to the chase or wait it out? Pardon the semi-rhetorical question but do you issue a notice to terminate tenancy to a paying tenant on the assumption you're going to face problems down the road and swallow the cost of rehabbing a unit or two? or in other words, does the peace of mind in placing prescreened tenants into market-rate units outweigh to cost of an rehab and termination of tenancy. 

Thanks, 

Michael 

[John G.]

@Michael Wlosek if they have a history in your current property paying rent on time I would leave them alone. I was in the same boat as you. I fixed up units and raised rents. The bad ones just left and the others stayed and kept paying the raised rent rates on time...and have been there for 8 years. They do not meet 3x income standard but still pay the rent on time. I’m glad I never messed with them. I did walk them through the improvements I was doing and had done. IE. added laundry in basement, fixed deferred maintenance items, landscaping, exterior paint, etc. I showed them reasons for raised rents...they were happy to pay. Doesn’t always go that way.