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:...

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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





pis the probability of foreclosure





Bis the principal balance





Vis the sale price at foreclosure





Lis the legal fees incurred by foreclosure


Assume that the borrower’s probability of foreclosure is an increasing function of his/her ”balance-tovalue


ratio” (i.e. B/V):


p



p ==p (B/V)



p


0>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


Lincreases?



Answered Same DayDec 21, 2021

Answer To: A mortgage lender seeks to maximize the expected value of its portfolio. The portfolio, of course,...

David answered on Dec 21 2021
126 Votes
Answer:
Throughout, we’d assume that p is twice differentiable, countinuously w.r.t relevent domain
of B&V.
1.If p is increasing in B, benefit of reducing principal balances, B, is the decrease in foreclosure losses.
Foreclosure losses=B-(V-L)
We can re-write the entire equation as (1-p)B+p(V-L)= B –p(B-V+L)
Marginal benefit, is a decrease in foreclosure...
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