Four mutually exclusive alternatives are being evaluated, and their costs and revenues are itemized below.
Mutually Exclusive Alternatives II III IV
Capital Investment 100,000 152,000 184,000 220,000
Annual revenues less expenses 15,200 31,900 35,900 41,500
Market Value 10,000 0 15,000 20,000
Useful life (years) 12 12 12 12
a. If the MARR is 15% per year and the analysis period is 12 years, use the PW method to determine which alternatives are economically acceptable and which one should be selected.
b. If the total capital investment budget available is $200,000, which alternative should be selected?
c. If the total capital investment budget available is $150,000, which alternative should be selected?
d. If the total capital investment budget available is $200,000, which alternative should be selected (if the alternatives are independent)?
The Correct Answer and Explanation is:
To analyze these four mutually exclusive alternatives using the Present Worth (PW) method at a Minimum Attractive Rate of Return (MARR) of 15% over a 12-year period, we calculate the present worth of each alternative using the formula: PW=−Capital Investment+(Annual Net Cash Flow×P/A,15%,12)+Market Value×(P/F,15%,12)PW = – \text{Capital Investment} + (\text{Annual Net Cash Flow} \times P/A, 15\%, 12) + \text{Market Value} \times (P/F, 15\%, 12)
Where:
- P/AP/A is the present worth factor for an annuity
- P/FP/F is the present worth factor for a single future amount
- (P/A,15%,12)=5.091(P/A, 15\%, 12) = 5.091
- (P/F,15%,12)=0.1827(P/F, 15\%, 12) = 0.1827
a. PW Calculations (Mutually Exclusive Alternatives)
| Alt | Investment | Annual Net | Salvage | PW Calculation | PW Value ($) |
|---|---|---|---|---|---|
| I | 100,000 | 15,200 | 10,000 | -100,000 + (15,200 × 5.091) + (10,000 × 0.1827) | -100,000 + 77,383.2 + 1,827 = -20,789.8 |
| II | 152,000 | 31,900 | 0 | -152,000 + (31,900 × 5.091) | -152,000 + 162,412.9 = 10,412.9 |
| III | 184,000 | 35,900 | 15,000 | -184,000 + (35,900 × 5.091) + (15,000 × 0.1827) | -184,000 + 182,766.9 + 2,740.5 = 1,507.4 |
| IV | 220,000 | 41,500 | 20,000 | -220,000 + (41,500 × 5.091) + (20,000 × 0.1827) | -220,000 + 211,276.5 + 3,654 = -5,069.5 |
Answer for (a):
- Alternatives II and III have positive PW, so they are economically acceptable.
- Since Alternative II has the highest PW ($10,412.9), it should be selected.
b. Budget = $200,000 (Mutually Exclusive)
Still choose the one with the highest PW under budget:
- Select Alternative II
c. Budget = $150,000 (Mutually Exclusive)
Only Alternative I is within budget, but PW = –$20,789.8 → not acceptable.
- None of the alternatives should be selected.
d. Budget = $200,000 (Independent Alternatives)
Select combination of alternatives with maximum total PW under budget.
- Alternative II: Investment = $152,000, PW = $10,412.9
- Remaining budget = $48,000 → can’t afford any other alternative.
- Alternative III: Investment = $184,000, PW = $1,507.4
- Remaining budget = $16,000 → still can’t afford any other.
Better to choose Alternative II alone.
Explanation
When evaluating investment alternatives, the Present Worth (PW) method is a reliable tool for comparing mutually exclusive and independent projects based on time value of money. In this case, four alternatives are analyzed over a 12-year period with a 15% Minimum Attractive Rate of Return (MARR).
For mutually exclusive alternatives, only one can be selected. We calculate the PW for each alternative by discounting the annual net cash flows and salvage value, then subtracting the initial investment. Alternatives II and III yield positive PW values, while Alternatives I and IV result in negative PW. Among them, Alternative II has the highest PW of $10,412.90 and is therefore the most economically desirable.
Under a $200,000 budget for mutually exclusive alternatives (part b), Alternative II remains the optimal choice because it fits within budget and offers the highest economic return. However, under a tighter $150,000 budget (part c), only Alternative I fits the constraint—but its negative PW means it’s not economically viable, so no investment should be made.
In part (d), assuming the alternatives are independent (i.e., more than one can be chosen if funds allow), the goal shifts to maximizing total PW within budget. Evaluating combinations under the $200,000 limit shows that Alternative II alone provides the best return, as combining it with any other would exceed the budget. Alternative III, while feasible, offers a significantly lower PW than Alternative II.
Thus, the analysis concludes that Alternative II is the best choice under both mutually exclusive and independent investment scenarios with a $200,000 budget.
