Home Buyers (and Sellers) can be excused for thinking that the purpose of an appraisal — typically, done at the behest of the Buyer’s lender — is to determine the home’s fair market value.
Not exactly.
Rather, the purpose of the appraisal — at least one done in the context of a home sale — is to address this question: “If the Buyer defaults 
on their mortgage, and the lender subsequently forecloses on the home, is the value of the collateral (i.e., the home) sufficient to make the lender whole?”
Or, in plain English: if the lender gets stuck with the home, can they sell it for at least as much as the mortgage?
In practice, once the appraiser can answer that question affirmatively . . . they’re done.
And since they know the sales price, that’s invariably the price the appraiser enters as the home’s estimated value (there’s no extra credit for overshooting).
How High (or Low) is the Bar?
Of course, the other key variable that goes into an appraisal is the size of the mortgage.
The larger the mortgage, the higher the loan-to-value ratio; the lower, the reverse.
From the lender’s perspective, the lower the ratio, the better (and safer the mortgage).
Loan-to-Value Illustrated
For illustrative purposes, imagine a home under contract for $1 million, and two different Buyers.
Buyer #1 puts down 50%, while Buyer #2 only puts down 5%.
In each case, the home’s fair market value is the same — that is, it’s not affected by the Buyer’s financing.
However, the risk of the home not appraising is virtually nil with a $500k mortgage, while it’s substantially higher with a $950k mortgage.
