In benefit to cost analysis

Question

BlackTom AI · Accepted Answer

Benefit-cost ratio equals the present value of benefits divided by the present value of costs, so costs are represented as positive values in the denominator.

1. Relevant knowledge points and concepts
- Benefit-cost analysis (BCA) typically uses the benefit-cost ratio (BCR): BCR = PV(benefits) / PV(costs). Both numerator and denominator are the present values of respective streams.
- Sign conventions: individual cash flows may be negative for outflows, but when computing the BCR we use the magnitude (positive) of total costs as the denominator. Net present value (NPV) uses subtraction (NPV = PV(benefits) − PV(costs)), where costs reduce NPV.
- A BCR > 1 indicates benefits exceed costs and the project is economically justified.

2. Complete reasoning process step by step
- Start with the definition: BCR = PV(benefits) / PV(costs).
- If costs were entered as negative in the denominator, the ratio would be negative or undefined, which contradicts the intended economic interpretation of a ratio comparing magnitudes.
- Therefore BCR uses positive cost amounts (the absolute present value of outflows).

3. Calculations with intermediate steps (example)
- Suppose PV(benefits) = $120, PV(costs) = $80.
- BCR = 120 / 80 = 1.5.
- Interpretation: benefits are 1.5 times costs; project is acceptable (BCR > 1).
- If one mistakenly used costs as −80, BCR = 120 / (−80) = −1.5, which is not meaningful for comparing benefit magnitudes.

4. Why the correct answer is correct
- The formula requires magnitudes to express how many times benefits exceed costs. Using positive costs ensures a meaningful, interpretable ratio. Negative costs would flip signs and break the interpretation.

5. Why other options are incorrect
- "Costs are in the numerator": incorrect because benefits occupy the numerator by definition.
- "Benefits are in the denominator": incorrect for the same reason; benefits are the numerator.
- "Cost is a negative value in the formula": incorrect because, although individual cost cash flows are outflows, the BCR denominator uses the positive present value of total costs.

6. Conclusion confirming final answer
- Thus, the correct choice is: cost is a positive value in the formula. This matches the provided answer.

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