The main problem with lower appraisals is that it can impact your ability to get a mortgage.
Banks don’t like loaning you more money than a house is worth because the risks are too high. If you fail to pay, and bank seizes the house, they can’t guarantee that they can sell it for enough to pay off the loan.
So if your down payment isn’t enough to account for the overage in sale price you may not qualify for the mortgage, will lose the bid, and will lose out on any deposit or fees.
For the sake of argument lets say you put $50,000 down on a house appraised at $500,000 (or 10% down) but the sale price ends up going to $550,000 because of a bidding war. Remove your down payment and the mortgage is for $500,000. The bank will treat this as a mortgage with zero down payment and can deny the loan.
If the house under appraises, then the buyer might need to come up with more cash in order for bank to fund the purchase.
Say they were buying a house for $500k, with $50k down and a bank loan for $450k. But appraisal comes back at $475k. Now, bank doesn’t want to lend $450k on a $475k appraisal, since that’s more than the 90% loan to value they’d agreed to (94.7%). So the buyer either needs to pay more as down payment to get loan down to a 90% LTV, or the seller needs to reduce their price.
When you are taking out a mortgage, that mortgage is secured by the house. If you stop paying it back, the bank gets to seize the house, sell it, and use the proceeds to pay themselves back.
As a condition of this agreement, the bank wants to know that if there were indeed to seize and sell the house, they would get back something like the amount they loaned out to you. The appraisal gives them an independent assurance that the house is worth at least as much as the mortgage.
If you’re trying to buy a house without a mortgage, none of this matters, though you may still want to know if the appraisal is well below a price you’re considering and the reason for the disconnect. If you do need a mortgage, you have two options in the event of a low appraisal. First, you can make up the difference in the down payment. The bank only cares about the money they loan to you, so if you reduce the size of the loan by paying more in cash, that’s fine. Second, you can back out of the sale. Depending on the details of the sale contract, this may leave you liable to, essentially, pay the seller some money for wasting their time (often about 1% of the asking price). You can add a standard clause to the contract that avoids this payment in the event that financing falls through, though all else equal, sellers will prefer to do business with a buyer who waives that clause.
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