Solution by Hand
(a) Table 7-6 applies the format illustrated in Figure 7-5 to calculate the BTCF and ATCF for this example. In column D, the effective income tax rate is very close to 0.38 [from Equation (7-15)] based on the information just provided.
(7-15)
t = State rate + Federal rate(1 − State rate), or
TABLE 7-6 ATCF Analysis of Example 7-15
| (E)= (A)+(D) |
(D)= − 0.38(C) |
(C)= (A)− (B) |
Depreciation Deduction |
(B) |
(A) |
|
| ATCF |
Cash Flow for
Income Taxes |
Taxable
Income |
= Deduction |
GDS Recovery Rate |
Cost Basis × |
BTCF |
End of
Year, k |
| -$180,000 |
|
|
— |
— |
— |
$180,000 |
0 |
| 36,000 |
0 |
0 |
= $36,000 |
0.2000 |
$180,000× |
36,000 |
1 |
| 44,208 |
+8,208 |
− 21,600 |
= 57,600 |
0.3200 |
180,000× |
36,000 |
2 |
| 35,453 |
− 547 |
1,440 |
= 34,560 |
0.1920 |
180,000× |
36,000 |
3 |
| 30,200 |
− 5,800 |
15,264 |
= 20,736 |
0.1152 |
180,000× |
36,000 |
4 |
| 30,200 |
− 5,800 |
15,264 |
= 20,736 |
0.1152 |
180,000× |
36,000 |
5 |
| 26,260 |
− 9,740 |
25,632 |
= 10,368 |
0.0576 |
180,000× |
36,000 |
6 |
| 22,320 |
− 13,680 |
36,000 |
0 |
|
0 |
36,000 |
7–10 |
| 18,600 |
− 11,400^{b} |
30,000^{a} |
|
|
|
30,000 |
10 |
| Total $130,201 |
|
|
|
|
|
$210,000 |
Total |
| PW (10%)= $17,208 |
^{a} Depreciation recapture= MV_{10} − BV_{10}= $30,000− 0= $30,000 (gain on disposal).
^{b} Tax on depreciation recapture= $30,000(0.38)= $11,400.
After-tax IRR: Set PW of column E= 0 and solve for i′ in the following equation:
0= − $180,000+$36,000(P/F, i′ , 1)+$44,208(P/F, i′ , 2)+$35,453(P/F, i′ , 3)+$30,200(P/F, i′ , 4)+$30,200(P/F, i′ , 5)+$26,260(P/F, i′ , 6) +$22,320(P/A, i′ , 4)(P/F, i′ , 6)+$18,600(P/F, i′, 10); IRR= 12.4%.
(b) The before-tax IRR is computed from column A:
0 = −$180,000 + $36,000(P/A, i′%, 10) + $30,000(P/F, i′%, 10).
By trial and error, we find that i′ = 16.1%.
The entry in the last year is shown to be $30,000 because the machinery will have this estimated MV. The asset, however, was depreciated to zero with the GDS method. Therefore, when the machine is sold at the end of year 10, there will be $30,000 of recaptured depreciation, or gain on disposal [Equation (7-16)], which is taxed at the effective income tax rate of 38%. This tax entry is shown in column D (EOY 10).
t = Federal rate + (1 − Federal rate)(State rate). (7-16)
By trial and error, the after-tax IRR for Example 7-15 is found to be 12.4%.
(c) When MARR = 10% per year is inserted into the PW equation at the bottom of Table 7-6, it can be determined that the after-tax PW of this investment is $17,208.
Spreadsheet Solution
Figure 7-6 displays the spreadsheet solution for Example 7-15. This spreadsheet uses the form given in Figure 7-5 to compute the ATCFs. The spreadsheet also illustrates the use of the VDB (variable declining balance) function to compute MACRS (GDS) depreciation amounts.
Cell B9 contains the cost basis, cells B10:B19 contain the BTCFs, and the year 10 MV is given in cell B20. The VDB function is used to determine the MACRS (GDS) depreciation amounts in column C. Cell B7 contains the DB percentage used for a five-year property class (see Table 7-4).
Note that in Figure 7-6, there are two row entries for the last year of the study period (year 10). The first entry accounts for expected revenues less expenses, while the second entry accounts for the disposal of the asset. To use the NPV and IRR financial functions, these values must be combined into a net cash flow for the year. This is accomplished by the adjusted ATCF and adjusted BTCF columns in the spreadsheet.
TABLE 7-4
MACRS (GDS) Property Classes and Primary Methods for Calculating Depreciation Deductions
| Special Rules |
Class Life (Useful Life) |
GDS Property Class and Depreciation Method |
| Includes some race horses and tractor units for over-the-road use. |
Four years or less |
3-year, 200% DB with switchover to SL |
Includes cars and light trucks,
semiconductor manufacturing
equipment, qualified technological equipment, computer-based central office switching equipment, some renewable and biomass power facilities, and research and development property. |
More than 4 years to less than 10 |
5-year, 200% DB with switchover to SL |
| Includes single-purpose agricultural and horticultural structures and railroad track. Includes office furniture and fixtures, and property not assigned to a property class. |
10 years to less
than 16 |
7-year, 200% DB with switchover to SL |
Includes vessels, barges, tugs,
and similar water transportation
equipment. |
16 years to less than 20 |
10-year, 200% DB with switchover to SL |
| Includes sewage treatment plants, telephone distribution plants, and equipment for two-way voice and data communication. |
20 years to less than 25 |
15-year, 150% DB with switchover to SL |
| Excludes real property of 27.5 years or more. Includes municipal sewers. |
25 years or more |
20-year, 150% DB with switchover to SL |
| Residential rental property. |
N/A |
27.5 year, SL |
| Nonresidential real property. |
N/A |
39-year, SL |
Source: IRS Publication 946. Depreciation. Washington, D.C.: U.S. Government Printing Office, for 2006 tax returns.
