Suppose the purchase price of a truck is $90,000, its residual value in four years is certain to be $15,000, and there is no risk that the lessee will default on the lease. Assume that capital markets are perfect and that the risk-free interest rate is 6% per annum, compounded monthly. The monthly lease payments for a four-year fair market value lease of the truck, due at the beginning of each month, are closest to:[Fill in the blank]Single choice
A
a. $1,750.
B
b. $2,115.
C
c. $1,870.
D
d. $1,825.

Question

Suppose the purchase price of a truck is $90,000, its residual value in four years is certain to be $15,000, and there is no risk that the lessee will default on the lease. Assume that capital markets are perfect and that the risk-free interest rate is 6% per annum, compounded monthly. The monthly lease payments for a four-year fair market value lease of the truck, due at the beginning of each month, are closest to:[Fill in the blank]Single choice

A

a. $1,750.

B

b. $2,115.

C

c. $1,870.

D

d. $1,825.

BlackTom AI · Accepted Answer

Monthly rate = 6%/12 = 0.005 (0.5%). Number of months n = 4×12 = 48. For a fair-value lease the lessor’s cost (purchase price) equals the present value of the lease payments (an annuity due because payments are at the beginning of each month) plus the present value of the certain residual value.

1) PV of residual value:
PV_residual = 15,000 / (1 + i)^48
i = 0.005, so (1.005)^48 ≈ 1.2704
PV_residual ≈ 15,000 / 1.2704 ≈ $11,808 (rounded)

2) Amount to be recovered by the annuity due (PV of payments):
PV_payments = Purchase price − PV_residual = 90,000 − 11,808 ≈ $78,192

3) Present value factor for an annuity due:
PV_factor_due = (1 + i) × [1 − (1 + i)^−n] / i
Compute v^n = (1.005)^−48 ≈ 1 / 1.2704 ≈ 0.7874
So [1 − v^n] = 1 − 0.7874 = 0.2126
Divide by i: 0.2126 / 0.005 = 42.52
Multiply by (1 + i) = 1.005: PV_factor_due ≈ 42.52 × 1.005 ≈ 42.74

4) Monthly payment (PMT):
PMT = PV_payments / PV_factor_due ≈ 78,192 / 42.74 ≈ $1,830 (rounded)

This computed payment (about $1,830) is closest to option d. $1,825. Why other choices are incorrect:
- $1,750 and $1,870 differ substantially from the calculated ≈$1,830 (errors of ≈$80 and $40 respectively).
- $2,115 is far too high (≈$285 above the calculated value).
Therefore the closest choice is d. $1,825.