Impact of a Longer Depreciation Schedule on After-Tax PW and IRR
Suppose the machinery in Example 7-15 had been classified in the 10-year MACRS (GDS) property class. Calculate the new after-tax PW and after-tax IRR. Why are these results different than the results of Example 7-15?
If the machinery in Example 7-15 had been classified in the 10-year MACRS (GDS) property class instead of the five-year property class, depreciation deductions would be slowed down in the early years of the study period and shifted into later years, as shown in Table 7-7. Compared with entries in Table 7-6, entries in columns C, D, and E of Table 7-7 are less favorable, in the sense that a fair amount of ATCF is deferred until later years, producing a lower after-tax PW and IRR. For instance, the \mathrm{PW} is reduced from \$ 17,208 in Table 7-6 to \$ 9,136 in Table 7-7. The basic difference between Table 7-6 and Table 7-7 is the timing of the ATCF, which is a function of the timing and magnitude of the depreciation deductions. In fact, the curious reader can confirm that the sums of entries in columns A through E of Tables 7-6 and 7-7 are nearly the same (except for the half-year of depreciation only in year 10 of Table 7-7). The timing of cash flows does, of course, make a difference!
Depreciation does not affect BTCF. Fast (accelerated) depreciation produces a larger PW of tax savings than does the same amount of depreciation claimed later in an asset’s life.
| TABLE 7-6 ATCF Analysis of Example 7-15 | |||||||||
| (A) | (B) | Depreciation Deduction | (C) = (A)−(B) | (D) = −0.38(C) | (E) = (A) + (D) | ||||
| End of Year, k |
BTCF | Cost Basis |
X | GDS Recovery Rate |
= | Deduction | Taxable Income |
Cash Flow for Income Taxes |
ATCF |
| 0 | -$180,000 | — | — | — | -$180,000 | ||||
| 1 | 36,000 | $180,000 | X | 0.2000 | = | $36,000 | 0 | 0 | 36,000 |
| 2 | 36,000 | 180,000 | X | 0.3200 | = | 57,600 | -21,600 | +8,208 | 44,208 |
| 3 | 36,000 | 180,000 | X | 0.1920 | = | 34,560 | 1,440 | -547 | 35,453 |
| 4 | 36,000 | 180,000 | X | 0.1152 | = | 20,736 | 15,264 | -5,800 | 30,200 |
| 5 | 36,000 | 180,000 | X | 0.1152 | = | 20,736 | 15,264 | -5,800 | 30,200 |
| 6 | 36,000 | 180,000 | X | 0.0576 | = | 10,368 | 25,632 | -9,740 | 26,260 |
| 7-10 | 36,000 | 0 | 0 | 36,000 | -13,680 | 22,320 | |||
| 10 | 30,000 | 30,000^{a} | -11,400^{b} | 18,600 | |||||
| Total | $210,000 | Total | $130,201 | ||||||
| PW (10%) = | $17,208 | ||||||||
| { }^{a} Depreciation recapture =\mathrm{MV}_{10}-\mathrm{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 \mathrm{E}=0 and solve for i^{\prime} in the following equation: \begin{aligned}0=&-\$ 180,000+\$ 36,000\left(P / F, i^{\prime}, 1\right)+\$ 44,208\left(P / F, i^{\prime}, 2\right)+\$ 35,453\left(P / F, i^{\prime}, 3\right)+\$ 30,200\left(P / F, i^{\prime}, 4\right)+\$ 30,200\left(P / F, i^{\prime}, 5\right)+\$ 26,260\left(P / F, i^{\prime}, 6\right) \\ &+\$ 22,320\left(P / A, i^{\prime}, 4\right)\left(P / F, i^{\prime}, 6\right)+\$ 18,600\left(P / F, i^{\prime}, 10\right) ; \operatorname{IRR}=12.4 \%\end{aligned}. |
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| TABLE 7-7 ATCF Analysis of Example 7-16 [Reworked Example 7-15 with Machinery in the 10-Year MACRS (GDS) Property Class] |
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| (A) | (B) | Depreciation Deduction | (C) = (A)−(B) | (D) = −0.38(C) | (E) = (A) + (D) | ||||
| End of Year, k |
BTCF | Cost Basis |
X | GDS Recovery Rate |
= | Deduction | Taxable Income |
Cash Flow for Income Taxes |
ATCF |
| 0 | -$180,000 | — | — | — | -$180,000 | ||||
| 1 | 36,000 | $180,000 | x | 0.1000 | = | $18,000 | $18,000 | -$6,840 | 29,160 |
| 2 | 36,000 | 180,000 | x | 0.1800 | = | 32,400 | 3,600 | -1,368 | 34,632 |
| 3 | 36,000 | 180,000 | x | 0.1440 | = | 25,920 | 10,080 | -3,830 | 32,170 |
| 4 | 36,000 | 180,000 | x | 0.1152 | = | 20,736 | 15,264 | -5,800 | 30,200 |
| 5 | 36,000 | 180,000 | x | 0.0922 | = | 16,596 | 19,404 | -7,374 | 28,626 |
| 6 | 36,000 | 180,000 | x | 0.0737 | = | 13,266 | 22,734 | -8,639 | 27,361 |
| 7 | 36,000 | 180,000 | x | 0.0655 | = | 11,790 | 24,210 | -9,200 | 26,800 |
| 8 | 36,000 | 180,000 | x | 0.0655 | = | 11,790 | 24,210 | -9,200 | 26,800 |
| 9 | 36,000 | 180,000 | x | 0.0656 | = | 11,808 | 24,192 | -9,193 | 26,807 |
| 10 | 36,000 | 180,000 | x | 0.0655/ 2 | = | 5,895 | 30,105 | -11,440 | 24,560 |
| 10 | 30,000 | 18,201^{a} | -6,916 | 23,084 | |||||
| Total | $130,196 | ||||||||
| PW (10%) \simeq | $9,136 | ||||||||
| IRR = | 11.2% | ||||||||
| { }^{a} \text { Gain on disposal }=\mathrm{MV}_{10}-\mathrm{BV}_{10}=\$ 30,000-\left(\frac{0.0655}{2}+0.0328\right)(\$ 180,000)=\$ 18,201. | |||||||||