GSOE9830 Chap.8 IRR and Multiple Return Rates
IRR and Multiple Return Rates
Internal rate of return is a discount rate that makes NPV equal zero. For a conventional cash-flow pattern, the NPV profile typically falls and crosses zero once. A project is accepted by the IRR rule when that rate exceeds the relevant alternative return.
Non-conventional patterns can change sign more than once, creating multiple zero crossings and an ambiguous decision, so NPV at the stated rate remains the reliable value criterion. The zero-NPV equation should be written even when software finds the root, and every reported rate should be substituted back to verify it. Counting cash-flow sign changes warns how many positive roots may be possible but does not guarantee them.
Sampling or plotting the profile reveals actual crossings and the NPV sign around each one. IRR remains a percentage property of the cash-flow pattern, while NPV reports dollar value relative to a selected opportunity cost.
What this chapter covers
- 01
Zero-NPV rate
- 02
NPV profile
- 03
Required return
- 04
Sign changes
- 05
Multiple IRRs
- 06
IRR limitation
Worked example · free
Locate a conventional project's IRR
- +1Write the zero-NPV equation with both discounted receipts.
- +1Testing about 13.07% makes NPV approximately zero.
- +1The IRR is above the 10% required return.
- +1Accept, while recognising NPV measures the value added.
Key terms
- Internal rate
- A discount rate that makes project NPV equal zero.
- NPV profile
- A curve showing project NPV across different discount rates.
- Sign change
- A switch between negative and positive cash flow in sequence.
- Multiple IRRs
- More than one discount rate satisfying the zero-NPV equation.
IRR and Multiple Return Rates FAQ
How is IRR used for a conventional project?
Compare it with the return available on a comparable alternative. An IRR above that required return supports acceptance, while a lower IRR supports rejection under the stated assumptions.
Why can a project have multiple IRRs?
A non-conventional cash-flow pattern can cross between negative and positive more than once, allowing its NPV profile to cross zero at multiple economically relevant discount rates.
What should I do when IRR is ambiguous?
Report the issue and use NPV at the stated opportunity cost for the decision. NPV gives a unique value at that rate even when several internal rates exist.
Exam move
Write the dated cash-flow signs before calling any IRR function. A conventional project normally has one initial negative amount followed by positive receipts. Its NPV profile generally falls as the discount rate rises and crosses zero once. Predict that shape first. If the signs change more than once, mark the pattern as non-conventional and expect that several roots or no useful positive root may occur.
Build the zero-NPV equation explicitly. Show each cash flow divided by one plus the unknown rate to its time power. A numerical function can solve the equation, but the written form demonstrates which dates and signs create the root. Substitute the returned rate back into the expanded NPV and confirm the residual is close to zero. A root that fails substitution is a calculator or input error.
Plot or sample the NPV profile around the root. Evaluate NPV at zero, at the required return and at rates on both sides of the reported IRR. For a conventional profile, the sign should change from positive below the crossing to negative above it. This check connects the percentage to the underlying value calculation and catches a return displayed in the wrong percentage format.
Count sign changes for non-conventional cash flow, ignoring zero entries. Treat the count as an upper bound on possible positive roots, not a guarantee. Search the economically relevant rate interval and record every crossing found. If multiple rates exist, explain that the IRR rule is ambiguous and use NPV at the stated required return for the decision. Compare IRR with opportunity cost only after checking uniqueness.
A conventional project with IRR above the required return is accepted by the IRR rule; one below is rejected. At equality, NPV is zero under the model. Keep the percentage conclusion separate from the dollar value conclusion, because IRR is a property of the pattern and NPV measures value at a chosen rate. Practise project comparisons that produce conflicting IRR and NPV rankings.
Identify whether scale or timing causes the conflict, calculate incremental NPV and use the governing value criterion for mutually exclusive choices. Do not select a smaller project merely because its percentage return looks larger.
Maintain an error log for sign pattern skipped, first software root accepted as unique, rate not substituted back, required return omitted, percentage described as dollars and IRR used to rank incompatible alternatives. Before submission, state the sign pattern, equation, root or roots, comparison rate, NPV implication and decision. That sequence shows both computational control and awareness of the rule’s limits.