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# Payback Methods, Even and Uneven Cash Flows

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Payback methods, even and uneven cash flows You have the opportunity to expand your business by purchasing new equipment for \$159,000. The equipment has a useful life of nine years. You expect to incur cash fixed costs of \$96,000 per year to use this new equipment, and you expect to incur cash variable costs in the amount of 10% of cash revenues. Your cost of capital is 12%.

Required
1. Calculate the payback period and the discounted payback period for this investment, assuming you will generate \$140,000 in cash revenues every year.

2. Assume instead that you expect the following cash revenue stream for this investment:
Year 1 \$ 90,000
Year 2 115,000
Year 3 130,000
Year 4 155,000
Year 5 170,000
Year 6 180,000
Year 7 140,000
Year 8 125,000
Year 9 110,000
Based on this estimated revenue stream, what are the payback and discounted payback periods for this investment?

Solution:

1.Payback problem:
Annual revenue \$140,000
Annual costs
Fixed \$96,000
Variable 14,000 110,000
Net annual cash inflow \$ 30,000

Discounted Payback Period with even cash flows:
Period Year Cash Revenues Fixed Costs Variable Costs Net Cash Inflows Disc Factor (12%) Discounted Cash Savings Cumulative Disc. Cash Savings Un-recovered Investment
0 \$159,000
1 \$140,000 \$96,000 \$14,000 \$30,000 .893 \$26,790 \$ 26,790 \$132,210
2 \$140,000 \$96,000 \$14,000 \$30,000 .797 \$23,910 \$ 50,700 \$108,300
3 \$140,000 \$96,000 \$14,000 \$30,000 .712 \$21,360 \$ 72,060 \$ 86,940
4 \$140,000 \$96,000 \$14,000 \$30,000 .636 \$19,080 \$ 91,140 \$ 67,860
5 \$140,000 \$96,000 \$14,000 \$30,000 .567 \$17,010 \$108,150 \$ 50,850
6 \$140,000 \$96,000 \$14,000 \$30,000 .507 \$15,210 \$123,360 \$ 35,640
7 \$140,000 \$96,000 \$14,000 \$30,000 .452 \$13,560 \$136,920 \$ 22,080
8 \$140,000 \$96,000 \$14,000 \$30,000 .404 \$12,120 \$149,040 \$ 9,960
9 \$140,000 \$96,000 \$14,000 \$30,000 .361 \$10,830 \$159,870

\$9,960/\$10,830 = .92
Discounted Payback Period = 8.92 years

2.
Year Revenue
(1) Cash Fixed Costs
(2) Cash
Variable Costs
(3) Net Cash Inflows
(4) = (1) − (2) − (3) Cumulative
Amounts
1 \$ 90,000 \$ 96,000 \$ 9,000 \$(15,000) \$(15,000)
2 115,000 96,000 11,500 7,500 (7,500)
3 130,000 96,000 13,000 21,000 13,500
4 155,000 96,000 15,500 43,500 57,000
5 170,000 96,000 17,000 57,000 114,000
6 180,000 96,000 18,000 66,000 180,000
7 140,000 96,000 14,000 30,000 210,000
8 125,000 96,000 12,500 16,500 226,500
9 110,000 96,000 11,000 3,000 229,500

The cumulative amount exceeds the initial \$159,000 investment for the first time at the end of year 6. So, payback happens in year 6.

Using linear interpolation, a more precise measure is that payback happens at:
5 years +

Discounted Payback Period with uneven cash flows:

Year Cash Revenues Fixed Costs Variable Costs Net Cash Inflows Disc Factor (12%) Discounted Cash Savings Cumulative Disc. Cash Savings Un-recovered Investment
0 \$159,000
1 \$ 90,000 \$96,000 \$ 9,000 \$(15,000) .893 (\$13,395) (\$13,395) \$172,395
2 \$115,000 \$96,000 \$11,500 \$ 7,500 .797 \$ 5,978 (\$ 7,417) \$166,417
3 \$130,000 \$96,000 \$13,000 \$ 21,000 .712 \$14,952 \$ 7,535 \$151,465
4 \$155,000 \$96,000 \$15,500 \$ 43,500 .636 \$27,666 \$ 35,201 \$123,799
5 \$170,000 \$96,000 \$17,000 \$ 57,000 .567 \$32,319 \$ 67,520 \$ 91,480
6 \$180,000 \$96,000 \$18,000 \$ 66,000 .507 \$33,462 \$100,982 \$ 58,018
7 \$140,000 \$96,000 \$14,000 \$ 30,000 .452 \$13,560 \$114,542 \$ 44,458
8 \$125,000 \$96,000 \$12,500 \$ 16,500 .404 \$ 6,666 \$121,208 \$ 37,792
9 \$110,000 \$96,000 \$11,000 \$ 3,000 .361 \$ 1,083 \$122,291 \$ 36,709

At a 12% rate of return, this project does not generate sufficient cash flows to ever recoup the investment under the discounted payback method.

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