Your firm is considering leasing a new computer. The lease lasts for 9 years. The lease calls for 10 payments of $1,000 per year with the first payment occurring immediately. The computer would cost $7,650 to buy and would be straight-line depreciated to a zero salvage over 9 years. The actual salvage value is negligible because of technological obsolescence. The firm can borrow at a rate of 8%. The corporate tax rate is 30%.
a) What is the after-tax cash flow from leasing relative to the after-tax cash flow from purchasing in years 1-9?
b) What is the after-tax cash flow from leasing relative to the after-tax cash flow from purchasing in year 0?
c) What is the NPV of the lease relative to the purchase?
d) What would the after-tax cash flow in year 9 be if the asset had a residual value of $500 (ignoring any possible risk differences)?
a) To find the after-tax cash flow from leasing relative to the after-tax cash flow from purchasing in years 1-9, we consider two components: The after-tax cash flow from lease payments and the after-tax cash flow from depreciation tax shield.
First, we need to calculate the after-tax lease payments:
- Lease payments = $1,000
- Tax benefit from lease payments = $1,000 × 30% = $300
- After-tax lease payments = $1,000 - $300 = $700
Second, we calculate the after-tax cash flow from the depreciation tax shield:
- Annual depreciation = Total cost of the asset ÷ useful life = $7,650 ÷ 9 = $850
- Tax savings from depreciation expense = $850 × 30% = $255
Now, we can compare the leasing and purchasing cash flows:
- After-tax cash flow from leasing = -$700 (negative because it's an outflow)
- After-tax cash flow from purchasing = +$255 (positive because it's a tax saving)
The difference between the two is:
- After-tax cash flow from leasing relative to purchasing = -$700 - $255 = -$955 in years 1-9.
b) In year 0, the after-tax cash flow from leasing would be -$1,000 (the first lease payment) whereas the after-tax cash flow from buying the asset would be -$7,650. The difference between the two is:
- After-tax cash flow from leasing relative to purchasing = -$1,000 - (-$7,650) = +$6,650 in year 0.
c) To calculate the NPV of the lease relative to the purchase, we need to discount the cash flows and calculate the present value:
- Years 1-9 cash flows discount factor: (1 - (1 + 8%)^(-9)) / 8% = 5.985
- NPV of after-tax leasing cash flows relative to purchasing year 1-9: -$955 * 5.985 = -$5,715.03
- NPV of after-tax leasing cash flows relative to purchasing year 0: $6,650
Total NPV of the lease relative to the purchase:
- NPV of the lease = $6,650 - $5,715.03 = $934.97
d) If the asset had a residual value of $500, the after-tax cash flow in year 9 would be:
- After-tax cash flow from selling the asset = $500 × (1 - 30%) = $350.
So, the after-tax cash flow in year 9 would be $350 if the asset had a residual value of $500 (ignoring any possible risk differences).
a) To calculate the after-tax cash flow from leasing relative to the after-tax cash flow from purchasing in years 1-9, we need to compare the cash flows of the two options in each year.
For leasing:
The annual lease payment is $1,000 for 10 years. However, since the first payment occurs immediately, in year 1, the cash flow is -$1,000.
In years 2-9, the cash flow is also -$1,000 per year.
For purchasing:
The computer costs $7,650 to buy, and since it would be straight-line depreciated to a zero salvage over 9 years, the annual depreciation expense is $7,650/9 = $850 per year.
In years 1-9, the cash flow from purchasing is -$850 (depreciation expense) plus possible tax savings on the depreciation expense. To calculate the tax savings, we need to multiply the depreciation expense by the corporate tax rate of 30%, so the tax savings per year is $850 * 0.3 = $255.
In summary, the after-tax cash flows for leasing relative to purchasing in years 1-9 are as follows:
Year 1: -1,000 (leasing) vs. -850 (purchasing) + 255 (tax savings)
Years 2-9: -1,000 (leasing) vs. -850 (purchasing) + 255 (tax savings)
b) To compare the after-tax cash flow from leasing relative to the after-tax cash flow from purchasing in year 0, we need to consider the initial cash flows of the two options.
For leasing:
The initial cash flow is -$1,000 for the first payment.
For purchasing:
The initial cash flow is the cost of buying the computer, which is -$7,650.
c) To calculate the net present value (NPV) of the lease relative to the purchase, we need to discount the cash flows to their present value using the borrowing rate of 8% and consider the tax effects.
For leasing:
Since the lease payments are made annually, we can discount them using the borrowing rate of 8%.
Using a present value (PV) formula, the present value of the lease payments can be calculated as follows:
PV lease = $1,000 / (1 + 8%) + $1,000 / (1 + 8%)^2 + ... + $1,000 / (1 + 8%)^9
For purchasing:
The cash flows from purchasing in years 1-9 need to be adjusted for tax savings. The present value of these cash flows can be calculated as follows:
PV purchase = (-$850 + $255) / (1 + 8%) + (-$850 + $255) / (1 + 8%)^2 + ... + (-$850 + $255) / (1 + 8%)^9
The NPV of the lease relative to the purchase is calculated as follows:
NPV = PV lease - PV purchase
d) If the asset had a residual value of $500 in year 9, we need to consider this additional cash flow in the calculation.
For purchasing:
The cash flow in year 9 would be the difference between the tax-adjusted salvage value and the tax-adjusted depreciation expense.
The tax-adjusted salvage value is $500 * (1 - tax rate) = $500 * (1 - 0.3) = $350.
The tax-adjusted depreciation expense is $850 - $255 = $595.
The cash flow in year 9 for purchasing is $350 - $595 = -$245.
To answer these questions, we need to calculate the after-tax cash flows for leasing and purchasing, as well as the net present value (NPV) of the lease relative to the purchase.
a) To calculate the after-tax cash flow from leasing relative to purchasing in years 1-9, we need to consider the tax shield from depreciation tax shields and the tax savings from deducting lease payments.
For leasing:
The annual lease payment is $1,000.
Since the lease is tax-deductible, the after-tax lease payment can be calculated as (1 - tax rate) * lease payment.
After-tax lease payment = (1 - 0.30) * $1,000 = $700.
For purchasing:
To calculate the annual depreciation, we divide the purchase cost by the lease term, which is 9 years.
Annual depreciation = Purchase cost / Lease term = $7,650 / 9 = $850.
Since the salvage value is negligible due to technological obsolescence, the total depreciation over the 9 years will be $7,650.
Tax shield = Total depreciation * Tax rate = $7,650 * 0.30 = $2,295.
The after-tax depreciation is: Total depreciation - Tax shield = $7,650 - $2,295 = $5,355.
The after-tax cash flow from leasing relative to purchasing is the difference between the after-tax lease payment and the after-tax depreciation:
After-tax cash flow from leasing = After-tax lease payment - After-tax depreciation
After-tax cash flow from leasing = $700 - $5,355 = -$4,655 (negative value indicates a cash outflow).
b) To calculate the after-tax cash flow from leasing relative to purchasing in year 0, we need to consider the initial cash outlay for purchasing and the savings from deducting lease payments.
For leasing, the initial cash outlay is zero since the first lease payment occurs immediately.
For purchasing, the initial cash outlay is the cost of the computer, $7,650.
After-tax cash flow from leasing relative to purchasing in year 0 = Initial cash outlay of purchasing - Initial cash outlay of leasing
After-tax cash flow from leasing relative to purchasing in year 0 = $7,650 - $0 = $7,650.
c) To calculate the NPV of the lease relative to the purchase, we need to discount the after-tax cash flows to their present value using the borrowing rate, which is 8%.
For leasing:
NPV of leasing = Sum of discounted after-tax cash flows from leasing - Initial cash outlay of leasing
NPV of leasing = (Discounted $700 for 9 years) - $0
Discounted cash flow = Cash flow / (1 + r)^n, where r is the discount rate and n is the year.
NPV of leasing = ($700 / (1 + 0.08)^1) + ($700 / (1 + 0.08)^2) + ... + ($700 / (1 + 0.08)^9) - $0
For purchasing:
NPV of purchasing = Sum of discounted after-tax cash flows from purchasing - Initial cash outlay of purchasing
NPV of purchasing = (Discounted -$4,655 for 1 year) + (Discounted $850 for 2 years) + ... + (Discounted $850 for 9 years) - $7,650
NPV of the lease relative to the purchase = NPV of leasing - NPV of purchasing
d) If the asset had a residual value of $500 in year 9, the after-tax cash flow in year 9 for purchasing would change. The after-tax depreciation would be calculated as the purchase cost minus the residual value, considering straight-line depreciation.
After-tax depreciation = (Purchase cost - Residual value) / Lease term
After-tax depreciation = ($7,650 - $500) / 9 = $778.
Other calculations for leasing and the NPV remain the same.