If an appraisal is ordered for a home by a bank, and you then switch banks, the new bank will have no idea that the original bank had an appraisal done on the home. If they order a new appraisal and it comes in higher then good for you. If it comes in lower then you can show them the original appraisal and maybe some the information can be used to bring up the value for the second appraisal.
If your bank ordered a second appraisal because they were questioning the value of the home or because of guidelines, then they will use the lower of the two appraisals.
If your bank ordered an appraisal and it came in low, they can choose to order and use the second appraisal if it came in higher and they feel it is a better appraisal overall.
An appraisal does expire in six months, but it doesn't "attach" itself to the home.
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Hope this was helpful.