Considering even a used card will set you back around $150, it would take nearly 100 days to ROI. This of course does not include the other components needed to build a rig and the countless other parts people often overlook, riser cables, racks, fans for cooling, adequate power, etc.
Excellent writeup. But, I think it is necessary to address the above point.
When you buy a GPU, it doesn't turn into a brick. It is still worth something. Consider a 40% hardware depreciation year-over-year (i.e. computer hardware loses value at 40% per year). During this 3 month ROI, you will lose approximately 10% of the value of the GPU, leaving you with $135. At the end of your mining period, you can sell your GPU and recover some costs associated with its purchase.