Multiple IRRs

Multiple internal rates of return (multiple IRRs) occur when one cash-flow stream makes net present value equal zero at more than one discount rate. The problem can arise when cash flows change sign more than once, such as an initial investment followed by operating inflows and a later cleanup, decommissioning, or contract-exit cost.

Key Takeaways

• More than one cash-flow sign change creates the possibility of multiple IRRs; it does not guarantee them.
• Each reported IRR is only a zero crossing on the project’s net-present-value profile.
• A spreadsheet may return different roots when the starting guess changes.
• Net present value at a supportable required return is usually the clearer decision measure.
• Modified IRR can provide one summary rate, but it requires explicit financing and reinvestment-rate assumptions.

Why Multiple IRRs Occur

IRR is any rate (r) that solves:

Where (C_t) is the cash flow at time (t). For a conventional investment, the usual pattern is one initial negative cash flow followed by positive cash flows. That pattern generally produces one economically relevant IRR.

A nonconventional pattern can change from negative to positive and then back to negative. After the NPV equation is rearranged, it behaves like a polynomial that can have more than one real root. More than one sign change is a warning that multiple roots may exist; it does not determine how many economically relevant IRRs the project has.

Worked Example

Consider a two-year project with these cash flows:

The IRR equation is:

Multiplying through by ((1+r)^2) produces a quadratic equation with two economically possible solutions:

Both rates make NPV equal zero. Reporting “the project IRR” without further analysis would therefore be ambiguous.

Read the NPV Profile

The project’s NPV at several discount rates is:

The NPV is positive only between the two IRRs. This breaks the simple rule that a project is acceptable whenever IRR exceeds the hurdle rate. At a 5% required return, both IRRs exceed the hurdle rate, yet NPV is negative. At 15%, NPV is slightly positive; at 25%, it is negative again.

The decision should therefore be based on NPV at the project’s risk-appropriate discount rate, supported by sensitivity analysis. The roots show where the profile crosses zero; they do not independently identify the correct required return.

How to Detect the Problem

1. List every cash flow in chronological order, including terminal obligations.
2. Count sign changes in the full cash-flow series.
3. Calculate NPV over a wide range of discount rates rather than relying on one IRR output.
4. Plot or tabulate the NPV profile and identify each zero crossing.
5. Run the spreadsheet IRR function with different starting guesses.
6. Base the decision on NPV at a documented required return and test the uncertain cash flows.

Microsoft documents that Excel’s IRR function uses an iterative calculation starting from an optional guess. If a model has multiple roots, changing that guess can lead the solver toward a different valid result. A single spreadsheet output is not evidence that the cash-flow stream has only one IRR.

IRR, NPV, and MIRR

MIRR can be helpful when readers need a single return measure. It compounds positive cash flows at a reinvestment rate and discounts negative cash flows at a finance rate. Those inputs must be stated and tested; MIRR does not eliminate forecasting risk.

Common Mistakes

Assuming two sign changes mean exactly two IRRs. Sign changes flag a potential problem, but they do not establish the count of economically useful IRRs. Candidate roots may be negative, complex, repeated, or irrelevant to the decision.

Accepting the first spreadsheet answer. Iterative software can converge on one root while another remains undisclosed.

Applying the ordinary IRR decision rule. With multiple roots, “accept if IRR exceeds the hurdle rate” can give the wrong answer.

Omitting terminal obligations. Environmental remediation, lease restoration, decommissioning, and contract-close costs can create the later negative cash flow that changes the analysis.

Using MIRR without disclosing assumptions. Finance and reinvestment rates affect the result and should match the project and currency being analyzed.

Authoritative and Educational Sources

• New York University capital-budgeting chapter: Explains nonconventional cash flows, multiple IRRs, and use of the NPV profile.
• Microsoft Excel IRR function: Documents the iterative method and optional starting guess.
• Microsoft Excel MIRR function: Documents the separate finance-rate and reinvestment-rate inputs.

This article is educational. Project evaluation depends on forecast quality, timing, taxes, financing, risk, and a supportable discount rate; it is not personalized investment advice.

Related Terms

• Internal Rate of Return: The discount rate at which NPV equals zero.
• Net Present Value: The value of forecast cash flows at a chosen required return.
• Modified Internal Rate of Return: A single-rate alternative with explicit financing and reinvestment assumptions.
• Hurdle Rate: The required return used to evaluate a project.

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