Justine decides to enter into a 6-year car lease agreement.The purchase price of the car is $24,995,and the expected residual value after 5 years is $9,500.If there is no risk of default and the risk-free rate is 7% APR with monthly compounding,what is the monthly lease payment for a 6-year lease in a perfect capital market?

Question

Quizplus · Accepted Answer

To calculate the monthly lease payment, we need to find the present value of the lease payments and the residual value at the end of year 6.

1. Calculate the present value of the lease payments:

We can use the formula for the present value of an annuity:

PV = PMT × [1 – (1 + r/12)^-n] / (r/12)

where PV is the present value, PMT is the monthly lease payment, r is the annual interest rate, and n is the total number of payments.

In this case, n = 6 years × 12 months per year = 72 months. The interest rate is 7% APR, which means the monthly interest rate is r/12 = 0.07/12 = 0.005833.

The lease payments are the same for each month, and we can subtract the residual value at the end of year 5 from the total amount paid over the lease term to get the present value of the lease payments:

PMT × [1 – (1 + r/12)^-72] / (r/12) = $24,995 – $9,500 = $15,495

PMT = $317.74

2. Calculate the present value of the residual value:

We can use the formula for the present value of a single amount:

PV = FV / (1 + r/12)^n

where PV is the present value, FV is the future value, r is the monthly interest rate, and n is the number of months.

In this case, the future value is $9,500 at the end of year 5, and we need to discount it for one more year at the same interest rate. Therefore, n = 12 months and r = 0.005833.

PV = $9,500 / (1 + 0.005833)^12 = $7,898.52

3. Add the present values of the lease payments and the residual value:

Total PV = $15,495 + $7,898.52 = $23,393.52

Therefore, the monthly lease payment is $23,393.52 / 72 months = $317.74.

The correct answer is C.

Justine Decides to Enter into a 6-Year Car Lease Agreement