Under certain conditions, a project may have more than one IRR. One such condition is when, in addition to the initial investment at time = 0, a negative cash flow (or cost) occurs at the end of the project's life.

Question

BlackTom AI · Accepted Answer

Multiple IRRs arise when the signs of a project's cash flows change more than once; an extra negative cash flow at the end creates such nonconventional cash flows and can produce more than one IRR.

1) Relevant concepts
- Internal Rate of Return (IRR) is any discount rate r that makes NPV = 0.
- For conventional cash flows (one initial negative outflow followed by all positive inflows), the IRR is unique.
- If cash flows change sign more than once (e.g., negative at time 0, positive intermediate inflows, and a negative outflow at the end), the NPV equation becomes a polynomial in (1+r) with potentially multiple positive real roots. Descartes’ Rule of Signs explains that the number of positive real roots is at most the number of sign changes in cash flows.

2) Step-by-step reasoning
- With an end-of-life negative cash flow, cash-flow signs go negative → positive → negative: two sign changes.
- A polynomial with two sign changes can have up to two positive roots for r (or roots that correspond to positive or negative r), so multiple IRRs are possible.
- Therefore the statement that “one such condition is when, in addition to the initial investment at time = 0, a negative cash flow occurs at the end” is correct.

3) Example calculation (intermediate steps)
Take CF0 = -100, CF1 = 230, CF2 = -120. Set NPV = 0:
-100 + 230/(1+r) – 120/(1+r)^2 = 0. Let x = 1/(1+r).
Then -100 + 230x – 120x^2 = 0 → 120x^2 – 230x + 100 = 0.
Compute discriminant: D = 230^2 – 4·120·100 = 52,900 – 48,000 = 4,900; sqrt(D) = 70.
Roots: x = (230 ± 70)/(240) → x1 = 300/240 = 1.25, x2 = 160/240 = 0.6667.
Convert to r: for x2 = 0.6667, r = (1/x2) – 1 = 0.5 (50%). For x1 = 1.25, r = (1/1.25) – 1 = -0.2 (-20%). Two distinct IRRs exist (one positive, one negative).

4) Why the correct answer is correct
The added negative terminal cash flow makes the cash-flow stream nonconventional (multiple sign changes), which allows the NPV equation to have more than one discount rate solution (multiple IRRs).

5) Why the other option is incorrect
Choosing “False” would deny that nonconventional cash flows can create multiple IRRs; that contradicts the polynomial-root behavior and Descartes’ Rule of Signs, so False is incorrect.

6) Conclusion
The correct answer is True, which matches the provided answer.

Under certain conditions, a project may have more than one IRR. One such condition is when, in addition to the initial investment at time = 0, a negative cash flow (or cost) occurs at the end of the project's life.判断题

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