After-Tax Analysis of Alternatives with Unequal Lives A firm must decide between two system designs, S1 and S2, whose estimated cash flows are shown in the following table. The effective income tax rate is 40% and MACRS (GDS) depreciation is used. Both designs have a GDS recovery period of five

Question

After-Tax Analysis of Alternatives with Unequal Lives
A firm must decide between two system designs, S1 and S2, whose estimated cash flows are shown in the following table. The effective income tax rate is 40% and MACRS (GDS) depreciation is used. Both designs have a GDS recovery period of five years. If the after-tax desired return on investment is 10% per year, which design should be chosen?

Design
S2	S1
$200,000	$100,000	Capital investment
6	7	Useful life (years)
$60,000	$30,000	MV at end of useful life
$40,000	$20,000	Annual revenues less expenses

Accepted Answer

Note that the design alternatives have different useful lives. The same basic principles of engineering economy apply to both before-tax and after-tax analyses. Therefore, we must analyze the two system designs over a common period of time. As we discovered in Chapter 6, using the repeatability assumption along with the annual worth method simplifies the analysis of alternatives having unequal lives.
Both alternatives would be depreciated using a five-year GDS recovery period. No adjustments to the GDS rates are required because the useful life of each alternative is greater than or equal to six years of depreciation deductions. Tables 7-10 and 7-11 summarize the calculation of the ATCFs for the design alternatives.

TABLE 7-10

After-Tax Analysis of Design S1, Example 7-20

	(A) BTCF	(B) Depreciation Deduction	(C) = (A) − (B) Taxable Income	(D) = − t(C) Cash Flow for Income Taxes	(E) = (A) + (D) ATCF	PW(10%)

End of Year, k
0	-$100,000				-$100,000	-$100,000
1	20,000	$20,000	$0	$0	20,000	18,182
2	20,000	32,000	-12,000	4,800	24,800	20,495
3	20,000	19,200	800	-320	19,680	14,786
4	20,000	11,520	8,480	-3,392	16,608	11,343
5	20,000	11,520	8,480	-3,392	16,608	10,312
6	20,000	5,760	14,240	-5,696	14,304	8,075
7	20,000	0	20,000	-8,000	12,000	6,158
7	30,000		30,000	-12,000	18,000	9,238
PW[latex]_{S1}[/latex](10%) = -$1,411

TABLE 7-11

After-Tax Analysis of Design S2, Example 7-20

End of Year, k	(A) BTCF	(B) Depreciation Deduction	(C) = (A) - (B) Taxable Income	(D) = —t(C) Cash Flow for Income Taxes	(E) = (A) + (D) ATCF	PW(10%)
0	-$200,000				-$200,000	-$200,000
1	40,000	$40,000	$0	$0	40,000	36,364
2	40,000	64,000	-24,000	9,600	49,600	40,989
3	40,000	38,400	1,600	-640	39,360	29,571
4	40,000	23,040	16,960	-6,784	33,216	22,687
5	40,000	23,040	16,960	-6,784	33,216	20,624
6	40,000	11,520	28,480	-11,392	28,608	16,149
6	50,000		50,000	-20,000	30,000	16,935
PW[latex]_{S2}[/latex](10%) = -$16,681

We can’t directly compare the PW of the after-tax cash flows because of the difference in the lives of the alternatives. We can, however, directly compare the AWs of the ATCFs by using the repeatability assumption from Chapter 6.

AW[latex]_{S1}[/latex](10%) = PW[latex]_{S1}[/latex](A/P, 10%, 7) = −$1,411(0.2054) = −$290
AW[latex]_{S2}[/latex](10%) = PW[latex]_{S2}[/latex](A/P, 10%, 6) = −$16,681(0.2296) = −$3,830
Based on an after-tax annual worth analysis, Design S1 is preferred since it has the greater (less negative) AW. Neither design however makes money, so if a system is not required, don’t recommend either one.

Question 7.20: After-Tax Analysis of Alternatives with Unequal Lives A firm...

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