Time-share sales provide an opportunity for vacationers to own a resort condo for 1 week (or more) each year forever. The owners may use their week at their own condo or trade the week and vacation elsewhere. Timeshare vacation sales usually require payment in full or financing through the time-share company, and interest rates are usually in the 13% to 18% range. Suppose the cost to buy a 1-week time share in a 3-bedroom condo is $21,773. Also suppose a 10% down payment is required, with the balance financed for 15 years at 17.5%, compounded monthly. (Round your answers to the nearest cent.)
(a) Find the monthly payment.
$ 1
(b) Determine the total cost over the life of the loan.
$ 2
(c) Suppose maintenance fees for this condo are $400 per year. Find the annual cost of the condo over the life of the loan. Assume that the annual maintenance fees remain constant.
$ 3
(d) Use (c) and the 10% down payment to determine the average annual cost for having this vacation condo for 1 week over the life of the loan.
$ 4
To answer the questions, we'll need to perform some calculations. Let's go step-by-step:
(a) To find the monthly payment, we can use the loan formula:
\[ P = \frac{{r(PV)}}{{1 - (1 + r)^{-nt}}} \]
Where:
P = monthly payment
r = monthly interest rate
PV = present value of the loan
n = number of payments per year
t = total number of years
The down payment is 10% of the cost, so the present value of the loan is \( P{V} = 0.9 \times \$21,773 = \$19,595.70 \).
The monthly interest rate is 17.5% divided by 12, or \( r = \frac{{17.5\%}}{{12}} = 0.0146 \).
The loan term is 15 years, so the total number of payments is \( nt = 15 \times 12 = 180 \).
Substituting the values into the loan formula:
\[ P = \frac{{0.0146(19595.70)}}{{1 - (1 + 0.0146)^{-180}}} \]
Calculating this expression gives us the monthly payment.
(b) The total cost over the life of the loan is simply the monthly payment multiplied by the total number of payments (180).
(c) The annual cost of the condo is the sum of the annual maintenance fees ($400) and the annual share of the loan payment.
(d) The average annual cost for having the vacation condo is the total cost over the life of the loan divided by the number of years (15).
Now, we'll calculate the values step-by-step.
(a) To find the monthly payment, we can use the formula for monthly mortgage payments:
M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
where:
M = monthly payment
P = principal amount (loan amount)
r = monthly interest rate (annual interest rate divided by 12)
n = number of monthly payments (loan term in years multiplied by 12)
Let's calculate the monthly payment:
Principal amount (loan amount) = Cost of the time share - Down payment
P = $21,773 - ($21,773 * 0.1) = $21,773 - $2,177.3 = $19,595.7
Monthly interest rate = Annual interest rate / 12
r = 17.5% / 12 = 0.175 / 12 = 0.0145833 (approx.)
Number of monthly payments = Loan term in years * 12
n = 15 * 12 = 180
Using these values in the formula, we can calculate the monthly payment, M:
M = $19,595.7 * (0.0145833 * (1 + 0.0145833)^180) / ((1 + 0.0145833)^180 - 1)
Using a calculator, the monthly payment comes out to be approximately $317.49.
So, the monthly payment is $317.49.
(b) To determine the total cost over the life of the loan, we multiply the monthly payment by the number of monthly payments:
Total cost = Monthly payment * Number of monthly payments
Total cost = $317.49 * 180
Using a calculator, the total cost comes out to be approximately $57,148.20.
So, the total cost over the life of the loan is $57,148.20.
(c) The annual cost of the condo over the life of the loan includes the annual maintenance fees. Since the maintenance fees are $400 per year and we have a loan term of 15 years, the total cost of maintenance fees over the life of the loan is:
Maintenance fees over the life of the loan = Maintenance fees per year * Loan term in years
Maintenance fees over the life of the loan = $400 * 15
The maintenance fees over the life of the loan come out to be $6,000.
So, the annual cost of the condo over the life of the loan is $6,000.
(d) The average annual cost for having this vacation condo for 1 week over the life of the loan can be calculated by dividing the total cost (including the down payment and maintenance fees) by the loan term. Since the down payment is 10% of the cost of the time share, the average annual cost would be:
Average annual cost = (Total cost + Down payment) / Loan term
Average annual cost = ($57,148.20 + ($21,773 * 0.1)) / 15
Using a calculator, the average annual cost comes out to be approximately $6,852.65.
So, the average annual cost for having this vacation condo for 1 week over the life of the loan is $6,852.65.