A granary has two options for a conveyor used in the manufacture of grain for transporting, filling, or emptying. One conveyor can be purchased and installed for $70,000 with $3,000 salvage value after 16 years. The other can be purchased and installed for $110,000 with $4,000 salvage value after 16 years. Operation and maintenance for each is expected to be $18,000 and $14,000 per year, respectively. The granary uses MACRS-GDS depreciation, has a marginal tax rate of 40%, and has a MARR of 9% after taxes.
A) Determine which alternative is less costly, based upon comparison of after-tax annual worth.
Show the AW values used to make your decision:
Conveyor 1: $ ?
Conveyor 2: $ ?
B) What must the cost of the second (more expensive) conveyor be for there to be no economic advantage between the two?
Cost of the second conveyor: $ ?
A) To determine which alternative is less costly based on after-tax annual worth, we need to calculate the annual worth (AW) for each conveyor.
For Conveyor 1:
Initial cost = $70,000
Salvage value after 16 years = $3,000
Operation and maintenance cost = $18,000
Depreciation calculation:
- Depreciable cost = Initial cost - Salvage value = $70,000 - $3,000 = $67,000
- Depreciation schedule for 16 years using MACRS-GDS: 0.1975, 0.3161, 0.1786, 0.1018, 0.0897, 0.0897, 0.0797, 0.0649, 0.0558, 0.0558, 0.0507, 0.0456, 0.0456, 0.0406, 0.0365, 0.0365
- Annual depreciation = Depreciable cost * Depreciation schedule = $67,000 * (0.1975 + 0.3161 + 0.1786 + 0.1018 + 0.0897 + 0.0897 + 0.0797 + 0.0649 + 0.0558 + 0.0558 + 0.0507 + 0.0456 + 0.0456 + 0.0406 + 0.0365 + 0.0365) = $67,000 * 1.2258 = $82,229
Tax benefit from depreciation = Annual depreciation * Tax rate = $82,229 * 0.4 = $32,891.60
Net annual cost = Initial cost - Salvage value + Operation and maintenance cost - Tax benefit from depreciation
= $70,000 - $3,000 + $18,000 - $32,891.60
= $52,108.40
To calculate the AW, we need to find the present worth (PW) at the MARR of 9%. Assuming the costs occur at the end of the year, the AW is given by AW = PW(MARR, n)/A(MARR, n), where n is the number of years.
PW at MARR of 9% = Net annual cost * A(P/A, MARR, n - 1) + Salvage value * P/F(MARR, n)
= $52,108.40 * A(P/A, 0.09, 16 - 1) + $3,000 * P/F(0.09, 16)
= $52,108.40 * 8.3477 + $3,000 * 0.3339
= $435,998.18 + $1,001.70
= $436,999.88
AW for Conveyor 1 = PW at MARR of 9% * A(MARR, n) = $436,999.88 * A(P/A, 0.09, 16)
Let's calculate this value:
AW for Conveyor 1 = $ ?
For Conveyor 2:
Using similar calculations, we find:
AW for Conveyor 2 = PW at MARR of 9% * A(MARR, n) = ($110,000 - $4,000 + $14,000 * A(P/A, 0.09, 16 - 1) + $4,000 * P/F(0.09, 16)) * A(P/A, 0.09, 16)
Let's calculate this value:
AW for Conveyor 2 = $ ?
B) To find the cost of the second conveyor that would make the two options economically equal, we set the AW for Conveyor 1 equal to the AW for Conveyor 2. Let's solve for the cost:
AW for Conveyor 1 = AW for Conveyor 2
$436,999.88 * A(P/A, 0.09, 16) = ($x - $4,000 + $14,000 * A(P/A, 0.09, 16 - 1) + $4,000 * P/F(0.09, 16)) * A(P/A, 0.09, 16)
Now, we can solve for x:
$x - $4,000 + $14,000 * A(P/A, 0.09, 16 - 1) + $4,000 * P/F(0.09, 16) = $436,999.88 * A(P/A, 0.09, 16)
$x = ($436,999.88 * A(P/A, 0.09, 16) + $4,000 * P/F(0.09, 16) - $14,000 * A(P/A, 0.09, 16 - 1)) / A(P/A, 0.09, 16)
Let's calculate this value:
Cost of the second conveyor = $ ?
To determine which alternative is less costly based on the comparison of after-tax annual worth, we need to calculate the annual worth for each conveyor.
Let's start with Conveyor 1:
Initial Cost: $70,000
Salvage Value: $3,000
Depreciation Period: 16 years
Annual Operation and Maintenance: $18,000
Using the MACRS-GDS depreciation method, the annual depreciation expense can be calculated as follows:
Year 1: Depreciation expense = Initial cost * MACRS depreciation rate
Year 2-16: Depreciation expense = Initial cost * MACRS depreciation rate
Since Conveyor 1 has a 16-year depreciation period, we can use the MACRS Table to find the depreciation rate for each year.
Year 1: Depreciation expense = $70,000 * MACRS depreciation rate for 16 years
For Year 2-16: Depreciation expense = $70,000 * MACRS depreciation rate for 16 years
To calculate the annual after-tax cash flow, we subtract the annual depreciation expense and add the salvage value, and then multiply by (1 - tax rate):
Year 1: Cash flow = Annual operation and maintenance - Depreciation expense + Salvage value * (1 - tax rate)
Year 2-16: Cash flow = Annual operation and maintenance - Depreciation expense
We can calculate the after-tax cash flow for each year and then find the present worth using the formula:
Present worth = ∑(Cash flow / (1 + MARR)^year)
Where ∑ is the summation symbol.
Now let's calculate the after-tax annual worth for Conveyor 1.
Using the MACRS Table, the annual depreciation rates for a 16-year property are as follows:
Year 1: Depreciation rate = 0.119
Year 2-16: Depreciation rate = 0.119
Calculating the present worth:
Year 1: Cash flow = -$18,000 - ($70,000 * 0.119) + ($3,000 * (1 - 0.4))
Year 2-16: Cash flow = -$18,000 - ($70,000 * 0.119)
Present worth for Conveyor 1 = ∑(Cash flow / (1 + 0.09)^year)
Now let's move on to Conveyor 2:
Initial Cost: $110,000
Salvage Value: $4,000
Depreciation Period: 16 years
Annual Operation and Maintenance: $14,000
Using the same approach as Conveyor 1, we can calculate the after-tax annual worth for Conveyor 2.
Using the MACRS Table, the annual depreciation rates for a 16-year property are as follows:
Year 1: Depreciation rate = 0.119
Year 2-16: Depreciation rate = 0.119
Calculating the present worth:
Year 1: Cash flow = -$14,000 - ($110,000 * 0.119) + ($4,000 * (1 - 0.4))
Year 2-16: Cash flow = -$14,000 - ($110,000 * 0.119)
Present worth for Conveyor 2 = ∑(Cash flow / (1 + 0.09)^year)
Now let's calculate the after-tax annual worth for each conveyor:
Conveyor 1: $ ?
Conveyor 2: $ ?
To compare the two alternatives, we can simply compare the after-tax annual worth values.
B) To find the cost of the second (more expensive) conveyor for there to be no economic advantage between the two, we need to equate the after-tax annual worth values of the two conveyors. In other words, we need to find the cost of Conveyor 2 such that the after-tax annual worth for Conveyor 2 equals the after-tax annual worth for Conveyor 1.
To determine which alternative is less costly based on the comparison of after-tax annual worth, we need to calculate the after-tax annual worth (AW) for each conveyor option.
First, let's calculate the after-tax annual worth for Conveyor 1:
1. Determine the net cash flows (annual costs and benefits) for Conveyor 1:
- Initial cost: $70,000
- Salvage value after 16 years: $3,000
- Annual operation and maintenance cost: $18,000
Net annual cash flow for Conveyor 1 = Salvage value - Initial cost + Annual operation and maintenance cost
= $3,000 - $70,000 + $18,000
= -$49,000 (negative sign indicates an outflow)
2. Depreciation expense using MACRS-GDS method:
MACRS-GDS allows you to depreciate an asset over a specified period using accelerated depreciation rates.
To calculate the annual depreciation expense using MACRS-GDS, you can refer to the MACRS depreciation tables provided by the IRS (Internal Revenue Service). Look up the depreciation rate for the asset's recovery period, which is 16 years in this case. Let's assume the depreciation rate for year 1 is 20%.
Depreciation expense for year 1 = Initial cost * Depreciation rate
= $70,000 * 20%
= $14,000
3. Tax shield on depreciation:
Tax shield = Depreciation expense * Tax rate
= $14,000 * 40%
= $5,600
4. After-tax annual worth (AW) for Conveyor 1:
AW = Net annual cash flow + Tax shield on depreciation
= -$49,000 + $5,600
= -$43,400
Therefore, the AW for Conveyor 1 is -$43,400.
Now let's calculate the after-tax annual worth for Conveyor 2:
Follow the same steps as for Conveyor 1, but with the respective values for Conveyor 2.
1. Net annual cash flow for Conveyor 2:
= Salvage value - Initial cost + Annual operation and maintenance cost
= $4,000 - $110,000 + $14,000
= -$92,000
2. Depreciation expense for year 1 for Conveyor 2:
Let's assume the depreciation rate for year 1 is 20% (same as Conveyor 1).
Depreciation expense for year 1 = Initial cost * Depreciation rate
= $110,000 * 20%
= $22,000
3. Tax shield on depreciation for Conveyor 2:
= Depreciation expense * Tax rate
= $22,000 * 40%
= $8,800
4. After-tax annual worth (AW) for Conveyor 2:
AW = Net annual cash flow + Tax shield on depreciation
= -$92,000 + $8,800
= -$83,200
Therefore, the AW for Conveyor 2 is -$83,200.
A) Based on the comparison of after-tax annual worth values:
Conveyor 1: AW = -$43,400
Conveyor 2: AW = -$83,200
Since the AW for Conveyor 1 is higher (closer to zero) than the AW for Conveyor 2, Conveyor 1 is less costly based on after-tax annual worth.
B) To determine the cost of the second (more expensive) conveyor for there to be no economic advantage between the two options, we need to find the cost at which the AW for both conveyors is equal.
Let's assume the cost of the second conveyor is represented by C.
1. Net annual cash flow for Conveyor 2:
= $4,000 - C + $14,000 = -$49,000 (assuming the net cash flow is the same as Conveyor 1)
Solving for C:
C = $4,000 + $14,000 + $49,000
C = $67,000
Therefore, the cost of the second (more expensive) conveyor should be $67,000 for there to be no economic advantage between the two options.