Audrey Sanborn has just arranged to purchase a $550,000 vacation home in the Bahamas with a 20 percent down payment. The mortgage has a 6.1 percent stated annual interest rate, compounded monthly, and calls for equal monthly payments over the next 30 years. Her first payment will be due one month from now. However, the mortgage has an eight-year balloon payment, meaning that the balance of the loan must be paid off at the end of Year 8. There were no other transaction costs or finance charges. How much will Audrey’s balloon payment be in eight years?
So in effect, we want to find the outstanding balance of the mortgage after 8 years.
original mortgage = .8(550,000) = 440,000
i = .061/12 = .00508333...
P(1 - 1.0050833..^-360)/.00508333.. = 440000
p(165.01792..) = 440000
p = 2666.38
After 8 years, outstanding balance
= balloon payment
= 440000(1.0050833..)^96 - 2666.38(1.0050833..^96 - 1)/.0050833..
I got $ 386,994.15
check my arithmetic
Well, if Audrey's balloon payment is in the Bahamas, I hope it's not filled with tropical fish and palm trees!
But in all seriousness, let's calculate the balloon payment.
To do this, we first need to find out the remaining balance on the mortgage after 8 years.
To calculate the monthly payment, we can use the formula for the present value of an annuity:
PV = PMT * (1 - (1+r)^(-n))/r
Where:
PV = present value (loan amount)
PMT = monthly payment
r = monthly interest rate
n = total number of payments
First, let's calculate the monthly interest rate:
r = (6.1% / 100) / 12 = 0.00508
Next, let's find the number of payments remaining after 8 years:
n = (30 years - 8 years) * 12 months = 22 * 12 = 264
Now, let's calculate the loan amount:
PV = Loan amount - Down payment = $550,000 - 20% * $550,000 = $440,000
Now, we can calculate the monthly payment:
$440,000 = PMT * (1 - (1+0.00508)^(-264))/0.00508
Solving this equation, we find that PMT ≈ $3,090.17
Finally, let's calculate the balloon payment at the end of Year 8:
To find the remaining balance after 8 years, we can calculate the future value of the mortgage starting from Year 9:
FV = PMT * ((1+r)^n - 1)/r
Where:
FV = future value (remaining balance after 8 years)
r = monthly interest rate
n = number of remaining payments
Here, n = (22 - 8) * 12 = 14 * 12 = 168 months
FV = PMT * ((1+0.00508)^168 - 1)/0.00508
Calculating this, we find that FV ≈ $258,495.05
So, Audrey's balloon payment in eight years is approximately $258,495.05.
I hope that helps! And remember, don't pop the balloon payment too early, Audrey!
To find Audrey's balloon payment in eight years, we need to calculate the remaining balance of the loan at that time. Here are the steps to do that:
Step 1: Calculate the loan amount based on the down payment:
Loan amount = Total cost of vacation home - Down payment
Loan amount = $550,000 - (20% of $550,000)
Loan amount = $550,000 - $110,000
Loan amount = $440,000
Step 2: Calculate the monthly interest rate:
Monthly interest rate = Stated annual interest rate / 12
Monthly interest rate = 6.1% / 12
Monthly interest rate = 0.00508
Step 3: Calculate the number of monthly payments over 30 years:
Number of monthly payments = Number of years * 12
Number of monthly payments = 30 * 12
Number of monthly payments = 360
Step 4: Calculate the monthly payment using the loan amount, interest rate, and number of payments:
Monthly payment = Loan amount * (Monthly interest rate / (1 - (1 + Monthly interest rate)^(-Number of monthly payments)))
Monthly payment = $440,000 * (0.00508 / (1 - (1 + 0.00508)^(-360)))
Monthly payment ≈ $2,604.71
Step 5: After making monthly payments for 8 years, calculate the remaining balance of the loan:
Remaining balance = Loan amount * (1 + Monthly interest rate)^Number of monthly payments - Monthly payment * ((1 + Monthly interest rate)^Number of monthly payments - 1) / Monthly interest rate)
Remaining balance = $440,000 * (1 + 0.00508)^96 - $2,604.71 * ((1 + 0.00508)^96 - 1) / 0.00508
Remaining balance ≈ $390,159.96
Therefore, Audrey's balloon payment in eight years will be approximately $390,159.96.
To calculate Audrey's balloon payment, we need to determine the outstanding balance of the mortgage at the end of Year 8.
Step 1: Calculate the initial loan amount
The down payment for the vacation home is 20% of $550,000, which is $110,000. Therefore, Audrey's initial loan amount is $550,000 - $110,000 = $440,000.
Step 2: Calculate the monthly interest rate
The stated annual interest rate is 6.1%, compounded monthly. To calculate the monthly interest rate, divide the annual interest rate by 12 and convert it to a decimal:
Monthly interest rate = (6.1% / 12) / 100 = 0.00508333
Step 3: Calculate the number of monthly payments
The mortgage term is 30 years, so the total number of monthly payments will be 30 years * 12 months = 360 months.
Step 4: Calculate the monthly payment
To calculate the monthly payment, we can use the formula for an amortizing loan:
Monthly Payment = P * (r(1+r)^n) / ((1+r)^n -1)
Where:
P = Loan amount
r = Monthly interest rate
n = Number of monthly payments
Substituting the given values:
Monthly Payment = $440,000 * (0.00508333 * (1+0.00508333)^360) / ((1+0.00508333)^360 - 1)
Calculating this using a financial calculator or spreadsheet, we find that the monthly payment is approximately $2,723.08.
Step 5: Calculate the remaining balance at the end of Year 8
After making payments for 8 years, Audrey will have made 8 * 12 = 96 monthly payments. To calculate the remaining balance, we need to determine the present value of the remaining payments.
Using the formula for the present value of an annuity:
Present Value = C * [(1 - (1 + r)^(-n))/r]
Where:
C = Monthly payment
r = Monthly interest rate
n = Number of remaining payments (in months)
Substituting the given values:
Present Value = $2,723.08 * [(1 - (1 + 0.00508333)^(-264))/0.00508333]
Calculating this using a financial calculator or spreadsheet, we find that the remaining balance at the end of Year 8 is approximately $326,264.84.
Therefore, the balloon payment in eight years will be $326,264.84.