Refinance: cash out vs buying equity

[Paul Kubin]

I was just daydreaming about refinancing (aren't we all?) and I wondered if there was a "buy equity" instead of "cash out" option during a refinance.  If not, I suppose you could cash out and then use the cash to pay down your principal by that amount. 

But when might it make sense to do that?  

Assume: 

- long term, single-family rental 

- market rate for rent with modest cash flow

- Refinancing makes $ sense (such as rates have dropped 2% or more since you mortgaged the property with 20% down)