Question

3-55. The upturned wingtips on jet aircraft reduce drag and increase lift. Wingtips save on fuel consumption, averaging 4% of the total fuel bill. A single set of wingtips can annually save 45,000 gallons of jet fuel costing $3 per gallon. (Fuel is the biggest operating expense of an airline company.) Furthermore, 100 metric tons of carbon dioxide emissions each year will be avoided, and maintenance expenses will be reduced. Wingtips cost $1 million to install. The airline company's MARR is 20%. Considering only the fuel savings, what is the simple payback period for the wingtips? What is the discounted payback period? (5.8)

Answer 1

Annual fuel saving ($) = 45,000 x 3 = 135,000

(1) Simple Payback period (PBP) is the time by when cumulative cash flows becomes zero.

Year	Cash flow ($)	Cumulative Cash Flow ($)
0	-10,00,000	-10,00,000
1	1,35,000	-8,65,000
2	1,35,000	-7,30,000
3	1,35,000	-5,95,000
4	1,35,000	-4,60,000
5	1,35,000	-3,25,000
6	1,35,000	-1,90,000
7	1,35,000	-55,000
8	1,35,000	80,000

PBP lies between years 7 and 8.

PBP = 7 + (Absolute value of cumulative cash flow, year 7 / Discounted cash flow, year 8)

= 7 + (55,000 / 135,000)

= 7.41 years

(2) Discounted payback period (DPBP) is the time by when cumulative discounted cash flows becomes zero.

Year	Cash flow ($)	PV Factor @20%	Discounted Cash Flow ($)	Cumulative Discounted Cash Flow ($)
0	-10,00,000	1.0000	-10,00,000	-10,00,000
1	1,35,000	0.8333	1,12,500	-8,87,500
2	1,35,000	0.6944	93,750	-7,93,750
3	1,35,000	0.5787	78,125	-7,15,625
4	1,35,000	0.4823	65,104	-6,50,521
5	1,35,000	0.4019	54,253	-5,96,267
6	1,35,000	0.3349	45,211	-5,51,056
7	1,35,000	0.2791	37,676	-5,13,380
8	1,35,000	0.2326	31,397	-4,81,983
9	1,35,000	0.1938	26,164	-4,55,820
10	1,35,000	0.1615	21,803	-4,34,016
11	1,35,000	0.1346	18,169	-4,15,847
12	1,35,000	0.1122	15,141	-4,00,706
13	1,35,000	0.0935	12,618	-3,88,088
14	1,35,000	0.0779	10,515	-3,77,573
15	1,35,000	0.0649	8,762	-3,68,811
16	1,35,000	0.0541	7,302	-3,61,509
17	1,35,000	0.0451	6,085	-3,55,424
18	1,35,000	0.0376	5,071	-3,50,354
19	1,35,000	0.0313	4,226	-3,46,128
20	1,35,000	0.0261	3,521	-3,42,607
21	1,35,000	0.0217	2,934	-3,39,672
22	1,35,000	0.0181	2,445	-3,37,227
23	1,35,000	0.0151	2,038	-3,35,189
24	1,35,000	0.0126	1,698	-3,33,491
25	1,35,000	0.0105	1,415	-3,32,076
26	1,35,000	0.0087	1,179	-3,30,896
27	1,35,000	0.0073	983	-3,29,914
28	1,35,000	0.0061	819	-3,29,095
29	1,35,000	0.0051	682	-3,28,412
30	1,35,000	0.0042	569	-3,27,844
31	1,35,000	0.0035	474	-3,27,370
32	1,35,000	0.0029	395	-3,26,975
33	1,35,000	0.0024	329	-3,26,646
34	1,35,000	0.0020	274	-3,26,371
35	1,35,000	0.0017	229	-3,26,143
36	1,35,000	0.0014	190	-3,25,952
37	1,35,000	0.0012	159	-3,25,794
38	1,35,000	0.0010	132	-3,25,661
39	1,35,000	0.0008	110	-3,25,551
40	1,35,000	0.0007	92	-3,25,459
41	1,35,000	0.0006	77	-3,25,383
42	1,35,000	0.0005	64	-3,25,319
43	1,35,000	0.0004	53	-3,25,266
44	1,35,000	0.0003	44	-3,25,221
45	1,35,000	0.0003	37	-3,25,185
46	1,35,000	0.0002	31	-3,25,154
47	1,35,000	0.0002	26	-3,25,128
48	1,35,000	0.0002	21	-3,25,107
49	1,35,000	0.0001	18	-3,25,089
50	1,35,000	0.0001	15	-3,25,074

As is seen, even after 50 years, cumulative discounted cash flows cannot be even nearer to zero (since at the high discount rate of 20%, the discount factor becomes very small). So DPBP can be taken as an infinite period and cannot be calculated precisely.