You question contains two parts: foreclosure and right of redemption. To understand right of redemption laws, it helps to understand the types of foreclosures and how the foreclosure procedure works.
Let us start with foreclosure basics. There are two types of foreclosure: judicial and non-judicial foreclosure. Depending your state, you may have a "right of redemption". Also, there are options to avoid foreclosure such as deed in lieu of foreclosure or short sale.
The right of redemption
Right of redemption laws are complicated and vary from state to state, thus, making a definitive statement on the subject that applies to all US jurisdictions all but impossible. In general, the right of redemption is the ability for the former owner of a foreclosed property to reclaim it after the foreclosure. How long a homeowner has this right is set by each state. Some states disallow redemption.
The right of redemption comes into play when a homeowner loses their home to foreclosure, then immediately after receives a windfall (i.e., wins the lottery, or receives a large inheritance) and then wants to return to their home. The redemption price may include foreclosure fees, legal fees, and other fees incurred by the lender -- in a lump sum.
When a new buyer purchases a foreclosed property in a right-of-redemption state, it is customary for the new buyer to approach the foreclosed owner and buy the redemption rights for a nominal amount.
Occasionally, a former homeowner will receive the necessary funds to redeem the property. It is highly unlikely that the owner will be able to find a lender willing to lend the money required to redeem the property because of the foreclosure and proof of solvency. Therefore, for the vast majority of foreclosures, redemption is an infeasible option and rare occurrence due to the high cost. If you contemplate purchasing a foreclosed property, or you were foreclosed on a property, won the lottery and want your old house, consult with an attorney in your state to discuss your rights and options for redemption.
Buying your house on the courthouse steps
You may be asking about the romantic notion of buying your old house in an auction on the courthouse steps. In theory this is possible because when a bank auctions foreclosed property someone eventually buys the house, so it might as well be the old homeowner.
This is possible if 1) the bank decides to auction the property, and 2) the homeowner finds financing.
There are risks to relying on this strategy. The bank may not auction the property. The bank may decide to list the property as a "REO" (Real Estate Owned) in the multiple-listing service, and sell the property through a real estate agent. Or, if the bank auctions the property, it may do so with a high reserve price.
It is highly unlikely that the owner will be able to find another bank willing to write a mortgage for a borrower who recently foreclosed on the same property. The borrower would need to show a significant change in their financial picture to warrant the risk of a sequel mortgage that would result in a different outcome.
Based on the information you provided, it appears you live in Arkansas but the situation you are asking about may not have taken place in Arkansas. Arkansas redemption law is complicated, and the right of redemption varies based on the type of foreclosure the creditor uses. I suggest the owner consult with an attorney specializing in property law in the state where the property is owned for a discussion of the rights that person has in their state.
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