Question

Your neighbor approached you about a dilemma he has with his mortgage. He purchased his house in 2006 with a $600,000, 6% fixed rate, 30 year mortgage. Then the great recession hit. Today his house is estimated to be worth less than the balance he owes on the mortgage. Furthermore, he was forced to take a new job at 20% less than what he was earning when he purchased the house. Here in 2012, 72 months later, he is contemplating entering into a mortgage modification program sponsored by the government. It offers 3 options.

1. reducing the interest rate to 4.5%
2. forgiveness of $100,000 of the principle due and continuing at 6%
3. refinancing the balance due with a new 15 year, 4% mortgage

Which option would minimize his monthly mortgage payment? Assume all refinancing costs will be paid by the government.

Answer 1

Loan (P)=   600000  
interest rate =    6%  
Monthly rate (i)= 6%/12 =    0.005  
Number of Monthly payment (n)= 30*12 =    360  
      
Monthly payment formula = P*i/(1-((1+i)^-n))      
600000*0.005/(1-((1+0.005)^-360))      
3,597.30      
      
After 72 months , months remaining (n)= 360-72=   288  
      
Unpaid balance at month 72 formula (P)=monthly payment *(1-((1+i)^-n))/i      
3597.30*(1-((1+0.005)^-288))/0.005      
548,387.24      
      
First option       
Loan is Previous Balance that (P) is =   548,387.24  
interest rate =   4.5%  
Monthly rate (i)= 4.5%/12 =   0.00375  
Remaining months (n)=   288  
Monthly payment formula = P*i/(1-((1+i)^-n))      
548387.24*0.00375/(1-((1+0.00375)^-288))      
3,117.17      
      
Second option      
Loan principal reduced. So Loan balance = 548387.24-100000=   448,387.24  
interest rate =   6%  
Monthly rate (i)= 6%/12 =   0.005  
Number of Monthly payment (n)=    288  
      
Monthly payment =      
448387.24*0.005/(1-((1+0.005)^-288))      
$2,941.32      
      
Third option : Refinancing      
Loan is Previous Balance that (P) is =   548,387.24  
interest rate =   4.00%  
Monthly rate = 4%/12=   0.003333333333  
No of years=   15  
No of months =15*12=   180  
      
Monthly payment =      
548387.24*0.00333333333/(1-((1+0.0033333333)^-180))      
$4,056.35      
      
Least Monthly Payment is in Second option $ 2941.32. So Choose forgiveness of $100,000 of the principle due and continuing at 6%      
      
      
      
      
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