Every investor hopes for the ideal tenant mix - one is home during the day to let in contractors, two are willing to shovel the sidewalk and salt the steps and all three pay the rent for the coming month no later than the first (preferably by automatic deposit to the investor's rent account)!
Good leases and credit checks coupled with strong tenant management can move most tenants on the continuum towards that ideal status (then the investor fears that their perfect tenant will become a homeowner, leaving a vacancy). A realtor will be a tremendous resource in upgrading the standard lease to protect the investor.
Some investors see existing tenants as a bonus, keeping cashflow on the property strong right from the day of closing. Other investors would rather have a month or two of vacancy to "refresh" the units and bring in tenants who will pay the going rate for rent.
Almost every investor agrees that a previous lease signed for below-market rent depresses the value of the property to them! (Hint for multi-family sellers: If you are planning in selling within the next 3 years, talk to your realtor today about improvements you should make to the building-which will be funded by bringing the monthly rent from your longstanding tenants up to market rates!!!)
My wildest low-rent situation was two 1,300 sf units on a quiet street within one block of a park which were rented out for $160 per month on the 2nd floor and $150 per month on the 3rd floor (the third floor tenant received a discount for cutting the tiny lawn). Market rent at the time was $850. These tenants both left 30 days after closing and BOUGHT their own properties!