A surety is a legally binding promise or guarantee given by a person or other entity (a guarantor) by which they commit to repay a debt on behalf of another entity if that entity fails to meet their debt obligations.
A person with an irregular income wants to mortgage a property but does not meet affordability requirements. They convince a guarantor (a parent or friend, for example) with good creditworthiness to provide a surety to the bank which makes the guarantor liable to repay the mortgage if the person with poor creditworthiness defaults on the loan.
The bank provides the loan because the guarantor acting as co-borrower meets affordability requirements.
As long as the person with poor creditworthiness meets their mortgage payments, the mortgage does not cost the guarantor anything. If, on the other hand, the home buyer is unable to make an amortization or interest payment on time, the payment will be charged to the guarantor.
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