The Johnsons have accumulated a nest egg of $19,000 that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $1300/month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $1600. If local mortgage rates are 8.5%/year compounded monthly for a conventional 30-yr mortgage, what is the price range of houses they should consider?

Question

The Johnsons have accumulated a nest egg of $19,000 that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $1300/month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $1600. If local mortgage rates are 8.5%/year compounded monthly for a conventional 30-yr mortgage, what is the price range of houses they should consider?

Answer 1

R = the periodic payment

P = the amount borrowed
n = the number of payments and i = the periodic interest in decimal form.

The monthly payment, R, required to retire the debt of $P over a period of n = 30(12) = 360 months at the monthly interest rate of i = 8.5/(100(12)) = .0070833 derives from

R = Pi/[1-(1+i)^(-n)]

Solve for P.

Answer 2

To find the price range of houses the Johnsons should consider, we need to determine the range of mortgage loans they can afford based on their monthly payment constraints.

Let's break down the information given:

1. Down Payment: The Johnsons have a nest egg of $19,000 that they intend to use as a down payment toward the purchase of a new house.
2. Monthly Payment: They want to make monthly payments between a minimum of $1300 and a maximum of $1600. These payments will be used to take advantage of the tax deduction.
3. Mortgage Rate: The local mortgage rates are 8.5% per year compounded monthly for a conventional 30-year mortgage.

Now, let's calculate the price range of houses they should consider:

Step 1: Determine the loan amount:
The loan amount is the purchase price of the house minus the down payment.

Step 2: Calculate the monthly mortgage payment:
To calculate the monthly mortgage payment, we can use the formula for the present value of an ordinary annuity:

PV = PMT * [(1 - (1 + r)^(-n)) / r],

where PV is the loan amount, PMT is the monthly payment, r is the monthly interest rate (8.5% / 12), and n is the total number of monthly payments (30 years * 12 months).

Step 3: Determine the price range:
The price range will be the loan amount plus the down payment, since the loan amount is the difference between the purchase price and the down payment.

Let's now calculate the price range step by step:

Step 1: Loan Amount:
Loan Amount = Purchase Price - Down Payment
Since the purchase price is unknown, we need to find it based on the loan amount.

Step 2: Monthly Mortgage Payment:
Monthly interest rate = 8.5% / 12 = 0.7083% (converted to decimal form)
Total number of monthly payments = 30 years * 12 = 360 months

Using the formula mentioned above, we can calculate the monthly mortgage payment:

PMT = PV * r / (1 - (1 + r)^(-n))
PMT = PMT_lower_limit = $1300
r = 0.7083% (as calculated above)
n = 360

Initially, we will set an upper limit for the loan amount and iterate until we find a monthly payment within the specified range.

Upper Limit:

Let's set an upper limit for the loan amount by assuming a monthly payment of $1600, which is the maximum they can afford.

Step 2.1: Calculate the initial upper limit loan amount:
PMT_upper_limit = $1600

Substituting these values into the formula:
Loan Amount_upper_limit = PMT_upper_limit * (1 - (1 + r)^(-n)) / r

Step 3: Price Range:
Price Range = Loan Amount + Down Payment

Now, let's perform the calculations:

Loan Amount_upper_limit = $1600 * (1 - (1 + 0.7083%)^(-360)) / (0.7083%)
Loan Amount_upper_limit ≈ $234,809.23

Price Range_upper_limit = Loan Amount_upper_limit + $19,000
Price Range_upper_limit ≈ $253,809.23

Therefore, the upper limit of the price range for houses the Johnsons should consider is approximately $253,809.23.

Now, let's calculate the lower limit using the PMT_lower_limit of $1300:

Loan Amount_lower_limit = $1300 * (1 - (1 + 0.7083%)^(-360)) / (0.7083%)
Loan Amount_lower_limit ≈ $171,222.86

Price Range_lower_limit = Loan Amount_lower_limit + $19,000
Price Range_lower_limit ≈ $190,222.86

Therefore, the lower limit of the price range for houses the Johnsons should consider is approximately $190,222.86.

In conclusion, based on their monthly payment constraints and the given mortgage rate, the Johnsons should consider houses in the price range of approximately $190,222.86 to $253,809.23.