
Question 10

You are interested in a new Ford Taurus. After visiting your Ford dealer, doing your research on the best leases available, you have three options. (i) Purchase the car for cash and receive a $1,800 cash rebate from Dealer A. The price of the car is $18,000. (ii) Lease the car from Dealer B. Under this option, you pay the dealer $500 now and $225 a month for each of the next 36 months (the first $225 payment occurs 1 month from today). After 36 months you may buy the car for $10,100. (iii) Purchase the car from Dealer C who will lend you the entire purchase price of the car for a zero interest 36-month loan with monthly payments. The car price is $18,000. Suppose the market interest rate is 6%. What is the net cost today of the cheapest option? (Enter just the number in dollars without the $ sign or a comma and round off decimals to the closest integer, i.e., rounding $30.49 down to $30 and rounding $30.50 up to $31.)

We have three options for purchasing or leasing the Ford Taurus, and we're asked to find the net cost today of the cheapest option, taking into account the time value of money at a 6% market interest rate. Let's break down each option.

### Option 1: Purchase for Cash from Dealer A
- Price of the car: $18,000
- Cash rebate: $1,800
- Net cost today: 
\[
18,000 - 1,800 = 16,200
\]

### Option 2: Lease from Dealer B
- Upfront payment: $500
- Monthly payment: $225 for 36 months
- Option to buy after 36 months for $10,100

The net cost today is the present value (PV) of the payments and the purchase option.

The formula for the present value of an annuity is:
\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]
where:
- $ P = 225 $ (monthly payment)
- $ r = 0.06 / 12 = 0.005 $ (monthly interest rate)
- $ n = 36 $ (number of months)

Let's compute the present value of the monthly payments.

Next, we calculate the present value of the $10,100 payment after 36 months. The formula is:
\[
PV = \frac{F}{(1 + r)^n}
\]
where:
- $ F = 10,100 $ (future payment)
- $ r = 0.005 $ (monthly interest rate)
- $ n = 36 $

Lastly, we sum up the upfront payment, the present value of the monthly payments, and the present value of the $10,100 to find the total cost today for this option.

### Option 3: Purchase from Dealer C (Zero Interest Loan)
- The price is $18,000.
- Zero-interest loan, so we simply divide the price into 36 equal monthly payments.
\[
\text{Monthly payment} = \frac{18,000}{36} = 500
\]
We calculate the present value of these payments using the same annuity formula as above:
\[
PV = 500 \times \frac{1 - (1 + 0.005)^{-36}}{0.005}
\]

Once we have the present values, we will compare the net costs of all three options to find the cheapest. Let's compute this now.The net cost today of the cheapest option is **$16,200**, which corresponds to purchasing the car for cash from Dealer A after applying the $1,800 rebate.

Would you like more details on how the other options compare or have any other questions?

Here are 5 related questions for deeper understanding:
1. How does the present value formula help compare different payment plans?
2. Why is the time value of money important in these kinds of financial decisions?
3. How would the calculation change if the interest rate were different?
4. What happens if you extend the lease term to more than 36 months?
5. How would financing with a down payment affect the overall cost in Option 3?

**Tip:** Always compare the present value of different financial options when future payments are involved, as money now is worth more than the same amount in the future due to interest.

You are interested in a new Ford Taurus. After visiting your Ford dealer, doing your research on the best leases available, you have three options. (i) Purchase the car for cash and receive a $1,800 cash rebate from Dealer A. The price of the car is $18,000. (ii) Lease the car from Dealer B. Under this option, you pay the dealer $500 now and $225 a month for each of the next 36 months (the first $225 payment occurs 1 month from today). After 36 months you may buy the car for $10,100. (iii) Purchase the car from Dealer C who will lend you the entire purchase price of the car for a zero interest 36-month loan with monthly payments. The car price is $18,000. Suppose the market interest rate is 6%. What is the net cost today of the cheapest option?

This example breaks down the cost of purchasing or leasing a Ford Taurus with three different financial options. Learn to use present value formulas, annuity calculations, and interest rate considerations to determine the net cost today of the cheapest option. Suitable for finance students and car buyers interested in understanding the time value of money.

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