Question

A couple believes they can afford a monthly house payment of $1,900 and would like a 30-year fixed-rate loan. Currently, the rate for such a loan is 4.2% compounded monthly. To the nearest dollar, how much can the couple afford to borrow?

Answer 1

Using the present value of an annuity formula, a couple who can afford a monthly payment of $1,900 can borrow approximately $353,573 for a 30-year fixed-rate loan at an interest rate of 4.2% compounded monthly.

Step-by-step explanation:

Calculating the Maximum Affordable Loan

To calculate the maximum amount that a couple can afford to borrow with a monthly payment of $1,900 for a 30-year fixed-rate loan at an interest rate of 4.2% compounded monthly, we use the formula for the present value of an annuity:

PV = PMT [{1 - (1 + i)^-n} / i]

Where:

• PV is the present value, or the amount that can be borrowed.
• PMT is the monthly payment amount, which is $1,900 in this case.
• i is the monthly interest rate, which is the annual rate divided by 12. For 4.2%, i = 0.042/12.
• n is the total number of payments, which is 360 for a 30-year loan (30 years * 12 months per year).

Applying these values to the formula, we find that the maximum amount the couple can borrow is:

PV = $1,900 * [{1 - (1 + 0.042/12)^-360} / (0.042/12)]

After calculating, the present value (or the loan amount) comes out to approximately $353,573 (rounded to the nearest dollar).

Thus, the couple can afford to borrow up to $353,573 with their desired monthly payment on a 30-year fixed-rate loan with the given interest rate.

Answer 2

the couple can afford to borrow approximately $388,534 for their house with the given conditions.

To calculate the amount the couple can afford to borrow for a 30-year fixed-rate loan at 4.2% interest rate compounded monthly with a monthly house payment of $1,900, we use the formula for the present value of an annuity. Here's the step-by-step calculation:

1. Convert Annual Interest Rate to Monthly:

- Annual interest rate = 4.2% or 0.042 (as a decimal)

- Monthly interest rate = 0.042 / 12 = 0.0035 (or 0.35%)

2. Convert Loan Term to Months:

- Loan term = 30 years

- Loan term in months = 30 years * 12 months/year = 360 months

3. Use the Present Value of Annuity Formula:

- The formula is:
`\( P = PMT * \left[ (1 - (1 + r)^(-n))/(r) \right] \)`

- Where
`\( P \)` is the loan amount,
`\( PMT \)` is the monthly payment,
`\( r \)` is the monthly interest rate, and
`\( n \)` is the total number of payments (months)

4. Plug in the Values:

-
`\( PMT = \$1,900 \)`

-
`\( r = 0.0035 \)`

-
`\( n = 360 \)`

-
`\( P = 1900 * \left[ (1 - (1 + 0.0035)^(-360))/(0.0035) \right] \)`

5. Calculate and Round to the Nearest Dollar:

-
`\( P \approx \$388,534 \)` (rounded to the nearest dollar)

Therefore, the couple can afford to borrow approximately $388,534 for their house with the given conditions.