Vanna has just financed the purchase of a home for $200 000. She agreed to repay the loan by making equal monthly blended payments of $3000 each at 4%/a, compounded monthly
How much sooner would the loan be paid if she made a 15% down payment?
How much would Vanna have saved if she had obtained a loan 3%/a, compounded monthly
To determine how much sooner the loan would be paid if Vanna made a 15% down payment, we need to calculate the new loan amount and the new monthly blended payments.
Step 1: Calculate the down payment amount.
Down payment = 15% of the purchase price = 0.15 * $200,000 = $30,000
Step 2: Calculate the new loan amount.
New loan amount = Purchase price - Down payment = $200,000 - $30,000 = $170,000
Step 3: Calculate the new monthly blended payments.
First, we need to calculate the monthly interest rate.
Monthly interest rate = (Annual interest rate / 100) / 12 = (4 / 100) / 12 = 0.00333
To calculate the monthly blended payments, we can use the formula for monthly mortgage payments, which is:
P = (r * PV) / (1 - (1 + r)^(-n))
Where:
P = Monthly blended payment
r = Monthly interest rate
PV = Loan amount
n = Total number of payments
We know the monthly blended payment (P = $3,000), the loan amount (PV = $170,000), and the monthly interest rate (r = 0.00333). We need to find the total number of payments (n).
So, to find n, let's rearrange the formula and solve for n:
n = - log(1 - ((r * PV) / P)) / log(1 + r)
Calculating n using the above formula:
n = - log(1 - ((0.00333 * $170,000) / $3,000)) / log(1 + 0.00333)
n ≈ 117.52
Since the number of payments will be fractional, we will round it up to the nearest whole number (118) because most loan terms are defined in whole months.
Therefore, the loan will be paid off in approximately 118 months (9 years and 10 months) with the 15% down payment.
To determine how much Vanna would have saved if she obtained a loan at 3% per annum, compounded monthly, we can calculate the new monthly blended payments using the formula and subtract it from the original monthly blended payments.
Step 1: Calculate the new monthly interest rate.
New monthly interest rate = (3 / 100) / 12 = 0.0025
Step 2: Calculate the new monthly blended payments using the new loan amount ($200,000) and the new interest rate (0.0025) with the same formula mentioned earlier.
Calculating new monthly blended payments:
n = - log(1 - ((0.0025 * $200,000) / $3,000)) / log(1 + 0.0025)
n ≈ 119.47
Again, we round up to the nearest whole number (120) to get 120 months (10 years) as the new loan term.
Finally, we can calculate Vanna's savings by multiplying the difference in the number of payments (120 - 118 = 2) by the original monthly blended payment amount ($3,000).
Vanna's savings = 2 * $3,000 = $6,000
Therefore, Vanna would have saved $6,000 if she had obtained a loan at 3% per annum, compounded monthly, instead of 4% per annum, compounded monthly.