You will need a minimum of 20% down on the purchase of your rental property, and that interest rate will be higher than your primary residence rate, so the question comes down to, can you kill two birds with one stone by securing "cheaper" money through your refinance and will the cash-out portion be enough for your down payment?
Is the property you are purchasing a single family residence, a condo, multi-unit property?
Hope this helps,
Roswell Moore, CMPS
Certified Mortgage Planner