Question4 Multiple IRRs are most likely when:Select one alternative: The discount rate is very high Cash flows are conventional (cash outflow in year 0 and cash-inflows thereafter) Cash flows change sign multiple times The project life is long ResetMaximum marks: 1 Flag question undefined

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BlackTom AI · Accepted Answer

Multiple internal rates of return (IRRs) occur when cash flows change sign more than once.

1) Relevant concepts:
- IRR is the discount rate r that makes NPV = Σ Ct/(1+r)^t = 0.
- When cash flows are non‑conventional (signs change more than once), the NPV equation becomes a polynomial in (1+r) of degree equal to the number of periods; a polynomial can have multiple positive roots.
- Descartes’ Rule of Signs: the maximum number of positive real roots of a polynomial is equal to the number of sign changes in its sequence of coefficients (or less by an even integer). This explains why multiple IRRs are possible only with multiple sign changes.

2) Step‑by‑step reasoning:
- If cash flows are conventional (one sign change: outflow at t=0, inflows thereafter), the NPV equation has one positive root (one IRR).
- If cash flows change sign multiple times, the NPV polynomial can cross zero multiple times as r varies, producing multiple IRRs.

3) Example and intermediate calculations:
Consider CF0 = −100, CF1 = +230, CF2 = −132 (signs: −, +, −; two sign changes).
NPV = −100 + 230/(1+r) − 132/(1+r)^2 = 0.
Multiply by (1+r)^2: −100(1+r)^2 + 230(1+r) − 132 = 0.
Expand: −100(1 + 2r + r^2) + 230 + 230r − 132 = 0
⇒ −100 − 200r − 100r^2 + 230 + 230r − 132 = 0
Combine terms: −100r^2 + 30r − 2 = 0
Multiply by −1: 100r^2 − 30r + 2 = 0
Discriminant D = (−30)^2 − 4·100·2 = 900 − 800 = 100, sqrt = 10
Roots: r = (30 ± 10)/(200) → r1 = 40/200 = 0.20 (20%), r2 = 20/200 = 0.10 (10%).
Thus two positive IRRs exist.

4) Why the correct answer is correct:
Multiple sign changes create a polynomial NPV function that can have multiple positive roots; therefore multiple IRRs are most likely when cash flows change sign multiple times.

5) Why other options are incorrect:
- A very high discount rate does not create multiple IRRs; it only affects NPV levels at high r.
- Conventional cash flows (single sign change) yield a single IRR.
- Long project life alone does not imply multiple IRRs; length without sign changes still gives one IRR.

6) Conclusion:
The correct choice is "Cash flows change sign multiple times," which matches the provided answer.

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