Question

A company has purchased equipment? (for ?$54,000?) that will reduce materials and labor costs by ?$12,000 each year for N years. After N? years, there will be no further need for the? machine, and because the machine is specially? designed, it will have no MV at any time. The? IRS, however, has ruled that the company must depreciate the equipment on a SL basis with a tax life of six years. If the effective income tax rate is 45?%, what is the minimum number of years the company must operate the equipment to earn 11?% per year after taxes on its? investment?

Answer 1

Working notes:

(i) Annual depreciation, years 1-6 = Cost / Useful life = $54,000 / 6 = $9,000

(ii) Taxable income (TI) ($), years 1-6 = Annual cost reduction - Annual depreciation = 12,000 - 9,000 = 3,000

[TI from year 7 onwards = $12,000, since depreciation would be zero]

(iii) After-tax income ($), years 1-6 = TI x (1 - tax rate) = 3,000 x (1 - 0.45) = 3,000 x 0.55 = 1,650

[After-tax income from year 7 onwards = $12,000 x (1 - 0.45) = $12,000 x 0.55 = $6,600]

(iv) After-tax cash flow (ATCF) ($), years 1-6 = After-tax income + Annual depreciation = 1,650 + 9,000 = 10,650

[After-tax cash flow from year 7 onwards = After-tax income from year 7 onwards = $6,600]

(v) PV factor @11% = (1.11)^-N

Therefore, minimum number of years is the discounted payback period, which is the time by when the cumulative discounted ATCF is zero.

Year	Cost/ATCF ($)	PV Factor @11%	Discounted ATCF ($)	Cumulative Discounted ATCF ($)
0	-54,000	1.0000	-54,000	-54,000
1	10,650	0.9009	9,595	-44,405
2	10,650	0.8116	8,644	-35,762
3	10,650	0.7312	7,787	-27,974
4	10,650	0.6587	7,015	-20,959
5	10,650	0.5935	6,320	-14,639
6	10,650	0.5346	5,694	-8,945
7	6,600	0.4817	3,179	-5,766
8	6,600	0.4339	2,864	-2,902
9	6,600	0.3909	2,580	-322
10	6,600	0.3522	2,324	2,003

DPBP lies between years 9 and 10.

DPBP = 9 + (absolute value of cumulative discounted ATCF in year 9 / Discounted ATCF in year 10)

= 9 + (322/2,324)

= 9 + 0.14

= 9.14 years