13.6
6.Your good friend Bono has noticed that interest rates are currently low. He is considering refinancing his Irish Castle and he wants your advice. If you can save him some money, he might get you some tickets to U2’s show in Seattle. He currently has 20 years left (240 payments) on his 30 year mortgage. The rate on his existing mortgage is 9%, but current rates for 20 year mortgages are 6%. Bono’s original mortgage was for $1,500,000. Assume all rates are APRs, compounded monthly and use the above information to answer the following questions.
a. What are Bono’s current monthly payments? (the payments on the existing 30 year mortgage (taken out 10 years ago at 9% for $1,500,000) (5 pts)
PV=$1,500,000 r=.09/12, n=360, P3=?
\( \Large P3 = \frac{PV} { \left [ \frac{1}{r} - \frac{1}{r \left (1+r \right )^{n}} \right ] } = \frac{1{,}500{,}000} { \left [ \frac{1}{\frac{.09}{12}} - \frac{1}{\frac{.09}{12} \left (1+\frac{.09}{12} \right )^{360}} \right ] } = $12{,}069.34 \)
b. If Bono wants to pay off the existing mortgage today, how much cash would he need? Assume that he made his 120th monthly payment today. HINT: What is the PV of the remaining payments on the old mortgage at the old interest rate? If you didn’t get (a), assume his payments are $8000/month. (5 pts)
P3=$12069.34, r=.09/12, n=240, PV=?
\( \Large PV = P3 \left [ \frac{1}{r} - \frac{1}{r \left(1+r \right )^{n} } \right ] = 12069.34 \left [ \frac{1}{\frac{.09}{12}} - \frac{1}{\frac{.09}{12} \left(1+\frac{.09}{12} \right )^{240} } \right ] = 1{,}341{,}446.24 \).
c. If Bono decides to refinance at current rates with a 20 year mortgage for the remaining balance on his old mortgage , will his payments be higher or lower? You do not need to do any calculations for this, just tell me which direction the payments will go and why you think that. (4 pts)
His payments will be lower. His payments are a combination of interest and principal repayment. While he will still have the $1,341,446 principal to repay, his interest will be less due to the lower interest rates charged for loaning money.
d. Let’s say he keeps his old mortgage. Under the current interest rate, should the present value of all of his future payments be more or less than the payoff to his loan (from part b)? Why? Again, no calculations are necessary. (4 pts)
Under the current interest rate (6%), the present value of the stream of 240 payments of $12,069.34 will be more than the payoff to his loan ($1,314,446). The intuition is the same as for bond prices and interest rates. Because interest rates are lower, you would have to put more in the bank today to produce the same stream of future cash flows. Note that since he can eliminate the loan by paying it off and the value of the loan (the PV of his future payments) is higher than the payoff, it is a good idea to pay it off and refinance. This confirms part (c), where we found that he could lower his payments by refinancing.