The general rule is that where a person borrows money through a loan or guarantees a loan for a company or another person they are liable for the full amount of the loan. When a loan is secured by collateral which a lender can repossess and sell, if the sales price is less than the loan amount still owed, then the lender does usually have the right to file suit against the borrower or guarantor for the full amount of the loan borrowed, unless the lender violates state repossession laws, fraudulently or improperly sells the collateral so that they failed to obtain as much as possible for it (under some state laws), or otherwise acts illegally. Collateral taken in repayment is not valued at the "book value" of the collateral but at what the market will pay the lender for the value, meaning that if a remaining loan balance is $12,000 and is secured by a car with a blue book value of $10,000 , which at auction only sells for $6,000, then the lender can ask the borrower to pay them the $6,000 remaining on the loan after they receive payment from the sale (not the $2,000 left if the car had sold for its "blue book" or appraised value).
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