Question5 A project’s cash flows change sign more than once over its life. Which statement iscorrect?Select one alternative: The IRR is still unique and should be used to rank the project The payback rule is guaranteed to give the value-maximising decision Multiple IRRs (or none) may exist, so NPV at the required return is the safest criterion The profitability index removes the multiple-sign-change problem ResetMaximum marks: 2 Flag question undefined

Question

BlackTom AI · Accepted Answer

When cash flows change sign more than once, the IRR equation becomes a polynomial in (1 + r) and can therefore have more than one positive root, so multiple IRRs (or none) may occur.

1. Relevant knowledge points and concepts
- IRR is the rate r that solves NPV(r) = Σ CFt / (1 + r)^t = 0. If cash flows change sign multiple times, this equation can be a higher-order polynomial with multiple positive roots (Descartes’ rule of signs explains the possibility of multiple positive roots).
- NPV evaluated at a given required return is a single-valued measure: it discounts all cash flows at the required rate and yields a unique present-value number.
- Payback ignores time value and cash flows after the payback horizon. Profitability index (PI) is PV of inflows divided by PV of outflows but can be ambiguous with nonconventional cash flows.

2. Step-by-step reasoning
Use a simple numeric example to show multiple IRRs: CF0 = -100, CF1 = +230, CF2 = -132. Solve NPV(r) = 0:
NPV(r) = -100 + 230/(1 + r) − 132/(1 + r)^2 = 0.

3. Calculations and intermediate steps
Multiply by (1 + r)^2:
-100(1 + r)^2 + 230(1 + r) − 132 = 0
Expand: -100(1 + 2r + r^2) + 230 + 230r − 132 = 0
Combine terms: -100r^2 + 30r − 2 = 0
Multiply by −1: 100r^2 − 30r + 2 = 0
Discriminant: Δ = (−30)^2 − 4·100·2 = 900 − 800 = 100
Roots: r = [30 ± sqrt(100)] / (200) = [30 ± 10]/200
Thus r1 = 40/200 = 0.20 (20%), r2 = 20/200 = 0.10 (10%) — two positive IRRs.

If a firm’s required return is, say, 15%, compute NPV(15%):
NPV = −100 + 230/1.15 − 132/1.15^2 ≈ −100 + 200 − 99.84 ≈ +0.16
This single NPV value uniquely indicates marginally positive value at 15%.

4. Why the correct answer is correct
Because multiple sign changes make the IRR equation potentially have multiple positive roots, IRR can be ambiguous. NPV at the required return uses a given discount rate and gives a unique dollar value for the project’s contribution to firm value, making it the safest decision criterion.

5. Why the other options are incorrect
- “The IRR is still unique…”: incorrect — as shown, multiple IRRs can exist.  
- “The payback rule is guaranteed to give the value-maximising decision”: incorrect — payback ignores TVM and cash flows after cutoff, so it can mis-rank projects.  
- “The profitability index removes the multiple-sign-change problem”: incorrect — PI can be ambiguous or misleading with nonconventional flows and does not resolve multiple-root IRR issues; NPV remains preferred.

6. Conclusion
Multiple IRRs (or none) may exist when cash flows change sign more than once, so NPV at the required return is the safest criterion. This matches the provided correct answer.

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