1. The Mathematical Definitions
NPV = ∑ [ CF_t / (1 + r)^t ] - Initial Outlay (CF_0)
Internal Rate of Return (IRR):
The discount rate 'r' at which Net Present Value equals exactly zero:
0 = ∑ [ CF_t / (1 + IRR)^t ] - CF_0
2. Why NPV and IRR Can Conflict
For standalone, independent projects, NPV and IRR yield identical accept/reject decisions (if IRR > hurdle rate, NPV > 0). However, when choosing between mutually exclusive projects, conflicts frequently arise due to two structural differences:
- Project Scale (Dollar Value vs. Percentage): A project generating a 100% IRR on a $1,000 investment adds only $1,000 of value, while a project generating a 20% IRR on a $1,000,000 investment adds hundreds of thousands of dollars of real shareholder wealth.
- The Reinvestment Rate Assumption: IRR mathematically assumes all interim cash inflows are reinvested at the IRR itself (which may be unrealistically high, like 35%). NPV assumes interim cash flows are reinvested at the firm's actual cost of capital (WACC), which is far more realistic.
A manufacturing plant must select either Project A (Automation Upgrade) or Project B (Facility Expansion). The company's hurdle rate (WACC) is 10.0%:
- Project A: Initial Cost = $100,000 • Cash Inflows: Year 1 = $50,000, Year 2 = $50,000, Year 3 = $40,000.
NPV @ 10% = $16,829 • IRR = 19.44% - Project B: Initial Cost = $300,000 • Cash Inflows: Year 1 = $130,000, Year 2 = $130,000, Year 3 = $130,000.
NPV @ 10% = $23,328 • IRR = 14.34% - The Decision Conflict:
If ranked by IRR: Project A wins (19.44% vs 14.34%).
If ranked by NPV: Project B wins ($23,328 vs $16,829).
Correct Decision: Choose Project B! It increases absolute shareholder wealth by $23,328, compared to only $16,829 for Project A. Managers cannot deposit percentages in a bank account.
Frequently Asked Questions
What is Modified Internal Rate of Return (MIRR)?
MIRR resolves the reinvestment rate problem by explicitly reinvesting all positive cash flows at the firm's actual cost of capital (WACC) to the end of the project, then finding the discount rate that equates the terminal value to the initial investment outlay.
What causes multiple IRRs in a single project?
Under Descartes' Rule of Signs, every time the sign of the cash flow changes (e.g. negative initial outlay, positive operating cash flow, negative decommissioning cost), an additional mathematical IRR root can exist.