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:

[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 (B/V)

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

6. How does the marginal cost curve shift in response to an increase in

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

118 Votes

Solution
Initial Working
As shown, E(port) = (1-p)*B + p*(V-L)
Given, p is an increasing function of (B/V)

p = f(B/V)
So E(port)
= [1-f(B/V)]*B + f(B/V)*(V-L)
= B-B*f(B/V) + V*f(B/V) – L*f(B/V)
= B + f(B/V)*(V-L-B)
1] Marginal benefit of reducing principal balances
Marginal benefit of reducing principal balances = dE/dB
= d(B + f(B/V)*(V-L-B)) / dB
= 1 +f’(B/V)*(V-L-B) + f(B/V)*(-1) * df(B/V)/dB = f’(B/V) +
= 1-f(B/V) + f’(B/V)*(V-L-B)
The above formula is the marginal benefit of reducing principal balances
If p=B/V (direct linear relationship)
Marginal benefit = 1-B/V+1/V*(V-L-B)
= 1- B/V + 1 –L/V – B/V
= 2-2B/V-L/V
If p=B2/V (2nd degree relationship)
Marginal benefit = 1-B2/V+2B/V*(V-L-B)
= 1- B2/V + 2B – 2BL/V – 2 B2/V
= 1 + 2B – 2BL/V - 3 B2/V
2] Marginal Cost of reducing principal balances
Marginal Cost of reducing principal balances = 0
How did this happen? It is because the Cost incurred by the lender, is not dependant on reducing
principal balances at all. What cost possibly a lender needs to incur is when the deal is getting done in...

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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:...

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

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