The VA version is in the Code of Federal Regulations, and it is narrower than it is usually described. Under 38 CFR 36.4303(k), no guaranty or insurance commitment will be issued on a loan financing a contract that (1) is for the purchase, construction, repair, alteration or improvement of a dwelling or farm residence, (2) is dated on or after June 4, 1969, (3) "[p]rovides for a purchase price or cost to the veteran in excess of the reasonable value established by the Secretary", and (4) "[w]as signed by the veteran prior to the veteran's receipt of notice of such reasonable value", unless the contract includes or is amended to include a provision reading substantially as follows:
"It is expressly agreed that, notwithstanding any other provisions of this contract, the purchaser shall not incur any penalty by forfeiture of earnest money or otherwise be obligated to complete the purchase of the property described herein, if the contract purchase price or cost exceeds the reasonable value of the property established by the Department of Veterans Affairs. The purchaser shall, however, have the privilege and option of proceeding with the consummation of this contract without regard to the amount of the reasonable value established by the Department of Veterans Affairs."
All four numbered conditions have to hold. The one people miss is the fourth: the requirement is aimed at a veteran who committed before knowing what VA thought the property was worth. "VA requires an escape clause in every contract" states the rule more broadly than the regulation does. Note also what the clause hands the buyer, which is an option in one direction only. It removes the obligation to complete and it removes the forfeiture, and it expressly preserves the privilege of going ahead anyway.
The FHA version has a regulatory root and a handbook text. The root is 24 CFR 203.15, which requires an insurance application to be accompanied by an agreement under which the seller or builder "will deliver to the purchaser of the property a written statement, in a form satisfactory to the Commissioner, setting forth the amount of the appraised value of the property as determined by the Commissioner." HUD implements that through the amendatory clause described in its Single Family Housing Policy Handbook 4000.1 at II.A.1.b.(i). In the handbook version last revised on 14 August 2019, the clause applies where the borrower did not receive form HUD-92800.5B, the Conditional Commitment Direct Endorsement Statement of Appraised Value, before signing the sales contract, and provides that the purchaser "shall not be obligated to complete the purchase of the property described herein or to incur any penalty by forfeiture of earnest money deposits or otherwise" unless given a written statement of an appraised value of not less than a stated amount, while keeping "the privilege and option of proceeding with consummation of the contract without regard to the amount of the appraised valuation." The same section lists sales that do not need it, including HUD real-estate-owned sales, the 203(k) program, sales where the seller is Fannie Mae, Freddie Mac, VA, USDA Rural Housing Services, another government agency, a mortgagee disposing of real-estate-owned assets or a seller at a foreclosure sale, and sales where the borrower will not be an owner-occupant. HUD revises the handbook section by section, and the footer of the consolidated PDF says so itself, directing the reader to "the online version of SF Handbook 4000.1 for specific sections' effective dates". So treat the wording above as the edition it came from, and check the current handbook before relying on it.
The blank in the FHA clause is not a threshold. HUD's own footnote to the clause directs mortgagees to insert the actual dollar amount of the sales price stated in the contract, and says that an increase in the sale price requires a revised amendatory clause. The number in the blank is therefore the price the parties agreed, not some minimum valuation the buyer negotiated.
On a conventional purchase the boundary with the financing condition matters more than the label. A lender's maximum loan is calculated against the lesser of price or value, so a low valuation shows up first as a loan that is smaller than the buyer planned. A buyer who waives a standalone appraisal condition but keeps a financing condition may still have an exit, and a buyer who waives both while promising to make up any shortfall in cash has none. Which clauses are in the contract, and how they interact, is a question to settle before the offer goes in rather than after the valuation lands.