An unconventional cash flow is any sequence of project or investment cash flows in which the sign (inflow vs. outflow) changes more than once over the life of the project. In contrast, a conventional cash flow has a single sign change: typically an initial outflow (the investment) followed by a series of inflows (project returns).
Key practical consequences
– Multiple IRRs: More than one change of sign can produce multiple positive internal rates of return (IRRs), making the IRR decision rule ambiguous.
– NPV is unambiguous: Net present value (NPV) always gives a unique, economically meaningful result for a given discount rate.
– Reinvestment assumptions matter: The conventional IRR assumes interim cash inflows are reinvested at the IRR, which is unrealistic especially when multiple IRRs exist. Modified IRR (MIRR) replaces that assumption and yields a unique rate.
Source for definitions and background: Investopedia — “Unconventional Cash Flow” .
Why unconventional cash flows arise (common real-world examples)
– Large projects with construction, operating, major scheduled maintenance or refurbishment cash outlays years after start (e.g., power plants, pipelines).
– Projects with staged investments or decommissioning costs (e.g., environmental remediation at end of life).
– Projects that produce early large positive cash flows followed by significant future costs (warranty, royalties, buyback obligations).
Mathematical note
– NPV(r) = Σ[CFt / (1 + r)^t], where CFt may be positive (inflow) or negative (outflow).
– IRR is any r that solves NPV(r) = 0. Because NPV(r) is a polynomial in (1 + r), multiple positive roots can occur when cash flow signs change multiple times. (Descartes’ Rule of Signs explains the maximum number of positive roots equals the number of sign changes.)
Practical steps for analyzing projects with unconventional cash flows
1. Identify sign changes
• List all cash flows by period and mark inflows (+) and outflows (–). Count sign changes. If more than one, proceed with caution.
2. Compute NPV at the firm’s discount/hurdle rate
• Use NPV as the primary decision metric: if NPV > 0 (at your required rate), the project adds value. NPV is unique for a given discount rate and avoids the multiple-IRR ambiguity.
3. Calculate IRR(s) and inspect the NPV profile
• Use software or a financial calculator to compute IRR; be aware it may return only one root.
• Plot the NPV profile: compute NPV for a range of discount rates (e.g., 0%–30%) and graph NPV vs. r. The number of times the curve crosses the horizontal axis shows how many IRRs exist. This visual helps explain multiple IRRs to stakeholders.
4. Compute MIRR (Modified Internal Rate of Return)
• MIRR assumes:
• Negative cash flows are financed at the finance (borrow) rate, and
• Positive cash flows are reinvested at the firm’s reinvestment or cost-of-capital rate.
• MIRR yields a unique rate and is often more realistic than IRR. In Excel: =MIRR(values, finance_rate, reinvest_rate). Formula (for n years): MIRR = (TVpos / |PVneg|)^(1/n) − 1, where TVpos is the terminal value of positive cash flows compounded at reinvest_rate and PVneg is the present value of negative cash flows discounted at finance_rate.
5. Rely on NPV for ranking and acceptance
• When comparing mutually exclusive projects or under capital rationing, use NPV (or Profitability Index if budget-constrained). NPV aligns with shareholder wealth maximization and is robust to sign-change issues.
6. Break the project into phases (if practical)
• If the project can be segmented (construction, operation, refurbishment), analyze each phase separately and evaluate optionality: accept phase 1 then decide on future phases contingent on outcomes. This can remove ambiguity and highlight managerial flexibility.
7. Run sensitivity and scenario analysis
• Test how NPV and MIRR respond to variations in discount rates, timing, and magnitude of the late outflows. Estimate break-even discount rates and probabilities of adverse outcomes.
8. Consider financing and tax timing separately
• Distinguish operating cash flows from financing cash flows (debt/interest repayments) and tax effects. Sometimes treating financing separately eliminates sign changes in the operating cash-flow stream.
9. Document assumptions transparently
• Make reinvestment rate, financing rate, maintenance schedules, and end-of-life costs explicit when presenting MIRR and NPV results. This prevents misunderstandings and allows stakeholders to verify conclusions.
10. Communicate results clearly
• Present: NPV at the company’s required rate, MIRR (with stated reinvest/finance rates), IRR(s) if multiple (with NPV profile), and sensitivity/scenario outputs. Recommend a decision rule (preferably NPV-based) and the rationale.
Illustrative (high-level) example
– Suppose CF0 = –1000, CF1 = +2300, CF2 = –1320. There are two sign changes (– to +, then + to –). The IRR equation could have two positive roots (two IRRs). However, at a required cost of capital of 10% the NPV might be negative or positive — that single NPV result is the correct guide for creating value. Compute MIRR and NPV to determine accept/reject rather than relying on ambiguous IRR comparisons.
Tools and Excel tips
– IRR: =IRR(range_of_CFs) — may return one root only.
– XIRR: =XIRR(values, dates) — for irregular timing.
– MIRR: =MIRR(values, finance_rate, reinvest_rate) — yields unique rate.
– NPV: =NPV(rate, range_of_future_CFs) + initial_CF (remember to add CF0 if it’s outside the NPV range).
– NPV profile: calculate NPV for a grid of discount rates (e.g., 0%, 2%, 4% … 30%) and plot NPV vs. rate to see how many times it crosses zero.
Decision checklist for managers
– If project cash flows are conventional (one sign change): IRR is usually reliable as a supplementary metric; still use NPV for primary decision.
– If unconventional: prefer NPV at the firm’s discount rate; compute MIRR for a single-rate summary; show the NPV profile and scenarios; consider breaking the project into phases; document reinvestment and financing assumptions.
– For mutually exclusive projects: rank by NPV (not IRR or MIRR) unless there are other constraints or strategic reasons.
When multiple IRRs can still be used
– If stakeholders insist on IRR, present all positive IRRs and explain their dependence on timing of cash flows. Use the NPV profile to show which IRR would be “relevant” under different required rates, but emphasize the ambiguity and recommend an NPV-based decision.
Further reading
– Investopedia, “Unconventional Cash Flow”:
– Brealey, Myers, and Allen, Principles of Corporate Finance (standard textbook discussing IRR, MIRR, and NPV concepts)
Summary
Unconventional cash flows (multiple sign changes) generate multiple IRRs and can mislead decision-makers if IRR is used alone. Use NPV as the primary decision criterion, compute MIRR when a single rate summary is needed, plot the NPV profile, run sensitivity/scenario analysis, and clearly document assumptions. Breaking projects into phases and separating operating from financing cash flows often simplifies analysis and leads to clearer investment decisions.