Question

Suppose that you plan on purchasing a home and you are offered the following financing options. (1) Annual interest rate of 6%, with additional upfront fees of $20,000 (These fees are not included in the $300,000 you will be financing, they will be paid upfront in cash). (2) Annual interest rate of 7% with additional upfront fees of $2,000 (also will not be financed, will be paid upfront in cash). You will make monthly payments for 30 years and you will have a loan of $300,000 in either case. Assume your opportunity cost of capital is 10% annually. What is the monthly payment for each loan? What is the PV of the savings by taking option (1)? Which option should you take assuming you will keep the house for the entire 30 years?

Answer 1

Step 1
Calculation of monthly payment Option 1 Loan ($300000 loan @ 6% for 30 years)
We can use the present value of annuity formula to calculate the monthly loan payment.
Present value of annuity = P x {[1 - (1+r)^-n]/r}
Present value of annuity = loan amount = 300000
P = monthly loan payment = ?
r = interest rate per month = 6%/12 = 0.005
n = number of months loan payment = 30 years x 12 = 360
300000 = P x {[1 - (1+0.005)^-360]/0.005}
300000 P x 166.7916
P = 1798.65
Monthly loan payment of Option 1 loan = $1798.65
Step 2
Calculation of monthly payment Option 2 Loan ($300000 loan @ 7% for 30 years)
We can use the present value of annuity formula to calculate the monthly loan payment.
Present value of annuity = P x {[1 - (1+r)^-n]/r}
Present value of annuity = loan amount = 300000
P = monthly loan payment = ?
r = interest rate per month = 7%/12 = 0.005833
n = number of months loan payment = 30 years x 12 = 360
300000 = P x {[1 - (1+0.005833)^-360]/0.005833}
300000 P x 150.3076
P = 1995.91
Monthly loan payment of Option 2 loan = $1995.91
Step 3
Calculation of PV of the savings by taking option (1)
Savings per month if we choose option 1 = $1995.91 - $1798.65 = $197.26
We can use the present value of annuity formula to calculate the PV of monthly savings
Present value of annuity = P x {[1 - (1+r)^-n]/r}
Present value of annuity = present value of monthly savings = ?
P = monthly savings = $197.26
r = interest rate per month i.e. opportunity cost of capital = 10%/12 = 0.00833
n = number of months = 360
Present value of annuity = 197.26 x {[1 - (1+0.00833)^-360]/0.00833}
Present value of annuity = 197.26 x 113.95
Present value of annuity = 22477.47
PV of monthly savings = $22,477.47
PV of the savings by taking option (1) = PV of monthly savings - additional upfront fees of Option 1
PV of the savings by taking option (1) = $22,477.47 - $18,000
PV of the savings by taking option (1) = $4,477.47
You should chosse option 1.