A couple purchasing a home budget $1700 per month for their loan payment. If they have $15,000 available for a down payment and are considering a 25-year loan, how much can they spend on the home at each of the following rates? (Round your answers to the nearest cent.)
(a) 6.9% compounded monthly
$ 1
(b) 7.1% compounded monthly
$ 2
To calculate how much the couple can spend on the home at each of the given rates, we need to use the formula for calculating the monthly loan payment:
Loan Payment = P * r * (1 + r)^n / ((1 + r)^n - 1)
where:
P = Principal amount (loan amount)
r = Monthly interest rate
n = Total number of monthly payments
(a) For 6.9% compounded monthly:
Principal amount = $15,000 (down payment)
Monthly interest rate (r) = 6.9% / 100 / 12 = 0.00575
Total number of monthly payments (n) = 25 years * 12 months/year = 300 months
Now, let's calculate the loan amount (P) using the given budget for the loan payment:
Loan Payment = $1700
Substituting the values into the formula, we have:
$1700 = P * 0.00575 * (1 + 0.00575)^300 / ((1 + 0.00575)^300 - 1)
Solving for P, the loan amount:
P ≈ $254,683.17
Therefore, the couple can spend approximately $254,683.17 on the home at a 6.9% interest rate compounded monthly.
(b) For 7.1% compounded monthly:
Using the same formula as above but with different interest rate:
Monthly interest rate (r) = 7.1% / 100 / 12 = 0.00592
Substituting the values into the formula, we have:
$1700 = P * 0.00592 * (1 + 0.00592)^300 / ((1 + 0.00592)^300 - 1)
Solving for P, the loan amount:
P ≈ $251,808.75
Therefore, the couple can spend approximately $251,808.75 on the home at a 7.1% interest rate compounded monthly.
To find out how much the couple can spend on the home at each of the given rates, we need to calculate the loan amount they can afford using the monthly budget, down payment, and loan term.
First, let's calculate the loan amount using the monthly budget and down payment. To do this, subtract the down payment from the available budget:
Loan amount = Budget - Down Payment
Loan amount = $1,700 - $15,000
Loan amount = -$13,300
Since the calculated loan amount is negative, it means that the couple cannot afford a home at 6.9% compounded monthly.
Moving on to the next rate, let's calculate the loan amount at 7.1% compounded monthly. Repeat the above steps:
Loan amount = Budget - Down Payment
Loan amount = $1,700 - $15,000
Loan amount = -$13,300
Again, the calculated loan amount is negative, indicating that the couple cannot afford a home at 7.1% compounded monthly either.
Therefore, the couple cannot spend any amount on the home at each of the given rates as their loan amount exceeds their budget, even with the down payment.
The $15,000 can be added at the end to the total purchase price, but does not affect the mortgage calculations.
The formula for the mortgage calculations is:
P(1+i)^n = R((1+i)^n-1)/i
where
i=interest rate per period (month)
n=number of periods (months)
R=monthly payment
P=amount to borrow
(a) at 6.9% p.a.,
i=0.069/12=0.00575
n=12*25 years = 300 months
R=$1700 = monthly payment
Solve for P
P=R((1.00575)^300-1)/(0.00575*(1.00575^300))
=$242714.03
Add the down-payment of $15000 to get
$257,714.03 for the total purchase price.
I'll leave (b) for your exercise.