Question

A private university is considering how it should best go about recovering the outstanding tuition balances owed by students who leave the school. There are currently 100 students who left the school this quarter who owe an average of $6000 each to the university. The school has several options for how it can go about recouping the amount that these former students owe.

First, it can immediately sell these debts to a third-party collections agency for 50 cents on the dollar. Once the university sells these debts to a third party, it will make no financial difference to the university whether the students pay or not.

Alternatively, the school can spend the next quarter trying to get their former students to pay them back themselves. For every former student who they pursue in their first quarters after ending their enrollment, the school has a 40% chance to recover the full amount owed to them and a 60% chance to recover nothing. These probabilities are not affected by the amounts that the former students owe.

If, at the end of the first quarter post-enrollment, a former student still has not paid the university back what he or she owes, the university still has options. It can still sell the debt to a third-party collections agency, but now they will only receive 20 cents on the dollar from the agency. They can also continue to pursue the former student on their own. If they choose to continue collections themselves, the school now has a 5% chance to recover the full amount, a 20% chance to recover half of what they are owed, and a 75% chance to recover nothing. Again, the amount that a former student owes does not affect their probabilities of repayment. At the end of two quarters in collections, the university will automatically sell all unrecovered debts to a third-party collections agency for 5 cents on the dollar.

Question 1: If the university pursues the optimal financial strategy for the recovery of its debts from former students, what amount can they expect to recover from the students who left the school with outstanding debts this quarter?

Question 2.: A student who just left the university with $5,000 in unpaid tuition walks into the Bursar’s Office (like the CFO of a university) and offers to pay $2,800 in exchange for having all of his debt cleared. True or false: the bursar should take him up on the offer. Why or why not?

Question 3: The university’s Head of Collections believes that she is able to separate their former students into two equally-sized categories: those that are more likely to repay their debts and those that are less likely to repay. These two categories have no correlation with respect to the amounts owed.

The Head of Collections believes that, if pursued through internal collections, those former students who she identifies as more likely to repay have a 55% chance of repaying their full amounts owed in the first quarter after leaving the university, whereas those that she deems less likely to repay have only a 25% chance of repaying in the first quarter. She does not believe that either group’s likelihood of repayment after the first quarter will be different from each other or from the estimates provided in the original question prompt.

If the Head of Collections is correct, and the university pursues the optimal financial strategy for the recovery of its debts from former students after splitting them into these two groups, what amount can they expect to recover from the students who left the school with outstanding debts this quarter?