Looks like your question got cutoff at the end. As I suspected, since your home is valued somewhere North of $350,000, a $15K difference to your asking price is still within the margin of error (5%). Meaning, it's all too easy at this price level for a $15K difference to be very hard to pin down. Another appraiser could easily be $15K higher and the lending community might ask a few questions as to why the differences, but they could actually accept either appraisal as reasonable. Note, itâ€™s also easy for another appraiser to be $15K less!
However, please understand that we're all still presuming the buyer is getting financing, especially since you said they want help with $11K in closing costs.
But here's the real rub, if what you're saying is the buyer offered you $360K, and they have an appraisal showing a value of $360K, but also wants $11K in closing costs, then they are really offering you $349K for a house that their own appraisal shows is worth $360K. The appraiser is required by USPAP to show value net of any seller paid closing costs, so the appraiser has already accounted for similar sales in your area and netted out the closing costs.
Contrary to prior answers, if the foreclosure in the area was a "distressed" sale, and "distressed" foreclosures are not common in your neighborhood (usually within a mile or less depending on the type of property), then the appraiser needs to defend why he is using it. Appraisals are supposed to be done for "like" properties.
If you're talking about potentially having to cough up $9,000, go spend $350 on your own appraisal. And once done, ask the appraiser to help educate you on his conclusions. It might not hurt to provide your appraiser with the same kind of "help" the buyer's appraiser may have gotten from the buyer's agent or the buyer. Some will comment that the appraiser for the buyer shouldn't have gotten any information from the buying side, but that would be a perfect world.
Real question is: you have a buyer interested enough to have made an offer on your home and you're only $15,000 different between offer and estimated value. If you can't afford to take money out of your pocket, but you can afford to put the house back on the market and wait for another buyer, then do it. Otherwise, consider either negotiating with the buyer to cough up some more of the closing costs, or increase their down payment to make the ratios work.