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economics, Mathematics

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A mortgage lender seeks to maximize the expected value of its portfolio. The portfolio, of course, is the
sum of all of the mortgages in it, so no generality is lost by examining the case of one loan:
E[port] = (1-p).B +p.(V-L)
where:
_ E[port] is the expected value of the portfolio
_ p is the probability of foreclosure
_ B is the principal balance
_ V is the sale price at foreclosure
_ L is the legal fees incurred by foreclosure
Assume that the borrower’s probability of foreclosure is an increasing function of his/her ”balance-to value ratio” (i.e. B/V):
P ? p (B/V) P’>0
In other words, borrowers who are deeper underwater are more likely to enter the foreclosure process. In such cases, reducing principal balances would reduce foreclosure-related losses (by reducing the probability of foreclosure). On the other hand, principal balance reductions are a direct loss for the lender.
1. Derive the marginal benefit of reducing principal balances.
2. Derive the marginal cost of reducing principal balances.
3. What is the necessary condition for maximizing E[port] with respect to the principal balance?
4. What is the sufficient condition for maximizing E[port]?
5. How does the marginal benefit curve shift in response to an increase in L?
6. How does the marginal cost curve shift in response to an increase in L?
7. How does the optimal principal balance change when L increases?

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