Suppose that you have the option to lease a new car, which you otherwise intend to purchase for $21,000. The lease terms: $3000 down and payments of $298 per month for 48 months, at the beginning of each month. Upon termination, you can purchase the car for an addition payment of $7000 at lease expiration. If your financing rate is 5.4% APR, and you discount the lease-purchase option using that same rate, how much will pay to buy car (in present-value terms) using the lease-purchase option

The amount that will be paid to buy the car is $18,539.43.

Explanation:

This can be calculated using the following 3 steps:

Step 1: Calculation of the present of the monthly payment

Since the payments are made at the beginning of each month, this can be calculated using the formula for calculating the present value (PV) of annuity due given as follows:

PVM = P * ((1 - (1 / (1 + r))^n) / r) * (1 + r) (1)

Where;

PVM = Present value monthly payments = ?

P = Monthly withdraw = $298

r = monthly financing rate = Financing rate / Number of months in a year = 5.4% / 12 = 0.054 / 12 = 0.0045

n = number of months = 48

Substitute the values into equation (1), we have:

PVM = $298 * ((1 - (1 / (1 + 0.0045))^48) / 0.0045) * (1 + 0.0045) = $12,896.55

Step 2: Calculation of the present of the purchase amount at lease expiration

This can be calculated using the present value formula as follows:

PVP = P / (1 + r)^n   (2)

Where;

PVP = Present value of the purchase amount at lease expiration = ?

P = Purchase amount at lease expiration = $7000

r = monthly financing rate = Financing rate / Number of months in a year = 5.4% / 12 = 0.054 / 12 = 0.0045

n = number of months = 48

Substitute the values into equation (2), we have:

PVP = $7000 / (1 + 0.0045)^48 = $5,642.88

Step 3: Calculation of the amount that will be paid to buy the car

This can be calculated as follows:

Amount to pay to buy car = PVM + PVP (3)

Where:

PVM = Present value monthly payments = $12,896.55

PVP = $5,642.88

Substitute the values into equation (3), we have:

Amount to pay to buy car = $12,896.55 + $5,642.88 = $18,539.43

Therefore, the amount that will be paid to buy the car is $18,539.43.

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