What the appraisal does
Your lender lends against the lower of the price or the appraised value. The appraisal protects both of you from overpaying relative to the market. It is ordered by the lender, performed by an independent licensed appraiser, and typically costs $550-$900.
When it comes in low
Suppose you offer $500,000 and the appraisal returns $485,000. The lender bases the loan on $485,000, leaving a $15,000 gap. With an appraisal contingency you have options: renegotiate the price, split the difference, pay the gap in cash, or walk away with your earnest money.
Without the contingency, you must cover the gap or forfeit your deposit. That is the risk you accept when you waive it to compete.
Challenging a low appraisal
A reconsideration of value can work when the appraiser missed a truly comparable recent sale. Your agent supplies the comps; your loan officer submits the request. Success is not guaranteed, but we have seen five-figure corrections when the evidence was strong.
Our advice
Keep the contingency in all but the most extreme bidding situations - and if you do waive it, cap your exposure: an "appraisal gap coverage up to $X" clause competes almost as well as a full waiver with a fraction of the risk.