What is the formula or procedure creditors and debt collectors use to figure out how much money they will settle a debt for?

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This answer is best explained by the process in which the collector assumes the debt in the first place.
Let's say you owe the bank $1000.00 because they charged you fees you didn't think were valid and simply didn't pay them and shut down your checking account. You are not the only person who has done this to the bank. They have thousands of dollars in uncollected fees from customers all over the world. Let's say the bank is owed $100,000 by a bunch of people who had the same problem.
Since the bank specializes in offering financial services, such as checking accounts, mortgages, cds, etc., they will no longer try to collect these debts after a few months. Another company that specializes in debt collecting will buy all of the bank's debt. We will call them Sharky's.
Sharky's and the bank both know that they will not be able to collect 100% of the money the bank is owed. So the bank sells the debt to Sharky's for a discount. The more "risky" the debt. The cheaper it sells for. Sharky's is a particularly suave company and they buy the debt for thirty cents on the dollar. Sharky's pays the bank $30,000.00 for the right to collect $100,000.00
Now Sharky's has the right, by law, to collect the full $100,000.00 originally owed to the bank. The bought the debt just like you or I would buy a car. Your $1,000.00 and your name, address, social security number, etc., is also included in the purchase. Sharky's then proceeds to call you and send you letters, notifying you that they now own the debt and you are expected to pay in full the amount owed.
Since they bought the debt for $30,000.00, anything over approximately 40% will be a profit because they have expenses. So you can settle with Sharky's for that amount if you haggle with them enough and you look like you have no financial prospects.
Short answer
There are some companies who only buy very "risky" debt. This is debt that will most likely not be collected. They will pay PENNIES on the dollar for the debt. Then they will call you an harass you to no end trying to collect the money at all costs. Many times this harassment strategy works. You can often settle with these companies for half or less of the original amount owed. Possibly even 20% of the amount owed.
If you do decide to negotiate with one of these companies, get EVERYTHING in writing. Better yet, do not do any business with them over the phone. They will most likely lie, cheat, and therefore steal the money that you send them. Tell them to quit calling you in a stern manner. Then write them, with a certified letter to confirm reciept, and tell them to stop calling you and write - if they would like to communicate. If they continue calling, THEY ARE BREAKING THE LAW. Report them to your state's consumer protection division. Every state has one.
More input
The short answer is perfect, just a little add-on.

Do not stop communicating with collection companies at all.
Make sure you communicate with them in written.
Written communication has 2 major benefits.

1. You have evidence of everything.
2. They CANNOT harass threat or abuse you on papers. :)

Fair Debt Collection Practices Act regulates collection companies. Any violation of law, would help you settle for lesser. Please download this act from www.ftc.gov.

Report abuse of law to your Attorney General, Consumer division and hire a local attorney (most of them would agree to take your case for free, they wont charge you anything, they will charge the collection company.) But make sure you have adequate evidences against them.
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