Question

question 2

kelly and max are looking to buy a home together and have \\$60,000 saved up for a down payment. their budget allows monthly payments of \\$2,250 for 25 years, and the bank has offered them an annual interest rate of 5.7%, compounded monthly.

what is the most expensive home they can buy?

• \\$419,374.15
• \\$299,374.15
• \\$99,473.68
• \\$359,374.15

Explanation:

Identify the given parameters

We are given:

• Down payment: \(DP = \$60,000\)
• Monthly payment: \(PMT = \$2,250\)
• Time period: \(t = 25\) years
• Number of compounding periods per year: \(m = 12\)
• Total number of payments: \(n = m \times t = 12 \times 25 = 300\)
• Annual interest rate: \(r = 5.7\% = 0.057\)
• Monthly interest rate: \(i = \frac{r}{12} = \frac{0.057}{12} = 0.00475\)

Calculate the maximum loan amount

Using the Present Value of an Ordinary Annuity formula to find the maximum loan amount (\(PV\)) they can borrow:

$$ PV = PMT \times \frac{1 - (1 + i)^{-n}}{i} $$

Substitute the values:

$$ PV = 2250 \times \frac{1 - (1 + 0.00475)^{-300}}{0.00475} $$

Calculate the discount factor:

$$ (1.00475)^{-300} \approx 0.241331 $$

$$ 1 - 0.241331 = 0.758669 $$

$$ PV \approx 2250 \times \frac{0.758669}{0.00475} \approx 2250 \times 159.71979 \approx 359,369.53 $$

Using more precise values:

$$ PV = 2250 \times \frac{1 - (1.00475)^{-300}}{0.00475} \approx 359,374.15 $$

Calculate the maximum home price

Using the Mortgage Affordability Calculation concept, the total home price is the sum of the loan amount (\(PV\)) and the down payment (\(DP\)):

$$ \text{Home Price} = PV + DP $$

$$ \text{Home Price} = 359,374.15 + 60,000 = 419,374.15 $$

Answer:

• (A) $419,374.15 (Correct answer)
• (B) $299,374.15
• (C) $99,473.68
• (D) $359,374.15