Question

Problem 05.023 Alternative Comparison - Different Lives Compare the alternatives C and D on the basis of a present worth analysis using an interest rate of 10% per year and a study period of 10 years. $ 40,000 $-6,000 D $-22.000 $-3,000 Alternative First Cost AOC, per Year Annual Increase in Operating Cost, per Year Salvage Value Life, Years $-200 $-300 $7,000 10 $200 5 The present worth of alternative C is $ and that of alternative D is $ (Click to select) offers the lower present worth

Answer 1

When alternatives under consideration have unequal lives, approach is to use an analysis period that is the least common multiple of the alternative lives. For example, in this case, alternative C has a 10-year life and D has a 5-year life, then a 10-year analysis period is used. In such cases, we assume that each alternative can be identically replaced at the end of its service life.

Calculate the Present Worth (PW), taking into account possible different time horizons and compare the values for each alternative.

The PW for the alternative C is:

Year	Costs ($)	P/F @ 10%	PW ($)
Day 1	40,000	1.0000	40,000
Year 1	6,000	0.9091	5,455
Year 2	6,200	0.8264	5,124
Year 3	6,400	0.7513	4,808
Year 4	6,600	0.6830	4,508
Year 5	6,800	0.6209	4,222
Year 6	7,000	0.5645	3,951
Year 7	7,200	0.5132	3,695
Year 8	7,400	0.4665	3,452
Year 9	7,600	0.4241	3,223
Year 10	7,800	0.3855	3,007
Salvage (10th year)	-7,000	0.3855	-2,699
		Total PW	78,747

The PW for the alternative D is:

Assuming that alternative D is identically replaced at end of year 5.

Year	Costs ($)	P/F @ 10%	PW ($)
Day 1	22,000	1.0000	22,000
Year 1	3,000	0.9091	2,727
Year 2	3,300	0.8264	2,727
Year 3	3,500	0.7513	2,630
Year 4	3,700	0.6830	2,527
Year 5	3,900	0.6209	2,422
Salvage (5th year)	-200	0.6209	-124
Year 5 (End)- Replace	22,000	0.6209	13,660
Year 6	3,000	0.5645	1,693
Year 7	3,200	0.5132	1,642
Year 8	3,400	0.4665	1,586
Year 9	3,600	0.4241	1,527
Year 10	3,800	0.3855	1,465
Salvage (10th year)	-200	0.3855	-77
		Total PW	56,405

The present worth of alternative C is $ 78,747 and alternative D is $ 56,405,

Hence, alternative D offers the lower present worth.