House mortgage
You have just purchased a house and have obtained a 30-year, $200,000 mortgage with an interest rate of 10 percent.
Required:
a. what is your annual payment?
b. Assuming you bought the house on Jan. 1st, what is the principle balance after one year?
c. After four years, mortgage rates drop to 8 percent for 30-year fixed-rate mortgages. You still have the old 10 percent mortgage you signed four years ago and you plan to live in the house for another five years. The total cost to refinance the mortgage is $3,000 including legal fees, closing costs and points. The rate o a five-year CD is 6 percent. Should you refinance your mortgage or invest the $3,000 in a CD? The 6 percent CD rate is your opportunity cost of capital.
a. The present value of a mortgage equals the period payment times the annuity factor
? =
Payment =
Payment =
b. After one year: Principal =
=
After ten years: Principal =
=
c. The remaining principal after year 4 is
Principal =
=
If the house is remortgaged at 8 percent, the payments are
? =
Payment =
Payment =
The difference between the two payments is ? The present
value of the incremental difference over five years is
PV of Savings =
=
a. To calculate the annual payment for a mortgage, you can use the formula for the present value of an annuity:
Payment = [P * (r * (1 + r)^n)] / [(1 + r)^n - 1]
Where:
P = Principal amount (in this case, $200,000)
r = Interest rate per period (in this case, 10% or 0.1)
n = Number of periods (in this case, 30 years)
Plugging in the values, the formula becomes:
Payment = [200,000 * (0.1 * (1 + 0.1)^30)] / [(1 + 0.1)^30 - 1]
Using a calculator or spreadsheet software, you can solve this equation to find the annual payment.
b. After one year, to find the remaining principal balance, you need to know the monthly payment amount. You can calculate the monthly payment using the same formula as above, but with the number of periods as 12 (for monthly payments over one year):
Monthly Payment = [P * (r * (1 + r)^n)] / [(1 + r)^n - 1]
Then, multiply the monthly payment by 12 to get the annual payment.
After calculating the annual payment, subtract it from the original principal amount to find the remaining principal balance after one year.
c. To determine whether to refinance the mortgage or invest the $3,000 in a CD, you need to compare the present value of the savings from refinancing with the opportunity cost of capital, which is given as the CD rate of 6%.
First, calculate the new monthly payment for the 8% mortgage using the same formula as in part a, but with the interest rate of 8% and the remaining number of periods (26 years instead of 30).
Next, find the difference between the original annual payment and the new annual payment. Multiply this difference by the number of years (5) to find the total savings over five years.
To calculate the present value of these savings, you need to discount the future cash flows using the opportunity cost of capital. You can use the following formula:
PV of Savings = Total Savings / (1 + r)^t
Where:
Total Savings = Difference in annual payments * Number of years (calculated in the previous step)
r = Opportunity cost of capital (6% or 0.06)
t = Number of years (5)
Using the values in the formula, solve for the present value of the savings. If the present value of the savings is greater than the $3,000 investment in the CD, it would be more beneficial to refinance the mortgage.