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Why would a bank take a lower all cash offer versus a higher offer via conventional lending?
Often the counter-party has obligations with respect to timelines as well -- if your buying a house, the seller probably is too, and may have a time-sensitive obligation to close on the deal. I'm that scenario, carrying the second mortgage may be enough to make that deal fall through or result in some other negative impact. Note that "pre-approval" means very little, banks can and do pass on deals, even if the buyer has a good payment history. That's especially true when the economy is not so hot -- bankers in 2011 are worried about not losing money... In 2006, they were worried about not making enough!