Valuation · Capital Allocation
Internal Rate of Return (IRR): Formula, Calculation & Decision Rule
Internal rate of return is the annualized compound return embedded in a series of cash flows. It is the discount rate that makes net present value equal to zero. Unlike NPV, which gives you a dollar figure, IRR gives you a percentage — making it easy to compare against a hurdle rate or cost of capital. This guide covers the formula, a step-by-step worked example using a stock investment, and the critical cases where IRR misleads and NPV is the better tool.
IRR is the discount rate at which an investment has zero net present value.Internal rate of return (IRR) is the discount rate at which a project has zero net present value — the annualized compound return embedded in a series of cash flows. Decision rule: accept when IRR exceeds the hurdle rate (WACC). IRR is not a rate you set; it is a rate you solve for. It tells you the implied return if every cash flow is reinvested at that same rate.
IRR Formula & How It Works
IRR is defined as the rate r that makes the sum of all discounted cash flows equal to zero. Written out formally for a project with cash flows at times 0 through n:
The IRR equation
0 = CF₀ + CF₁/(1+r) + CF₂/(1+r)² + … + CFₙ/(1+r)ⁿ
Where CF₀ is the initial outlay (a negative number), CF₁…CFₙ are subsequent cash inflows, and r is the IRR you are solving for. Because the equation is a polynomial in r, there is no algebraic closed form. You must solve it iteratively.
Solving by Interpolation: A 3-Step Example
Before financial calculators, analysts used linear interpolation to bracket the answer. The method still builds intuition for what IRR means.
Interpolation example: $1,000 invested today, $1,200 returned in two years
- Step 1 — Try a low rate (10%)
- NPV = -$1,000 + $1,200/(1.10)² = -$1,000 + $991.74 = -$8.26 (still negative at 10%)
- Step 2 — Try a lower rate (9%)
- NPV = -$1,000 + $1,200/(1.09)² = -$1,000 + $1,010.08 = +$10.08 (positive at 9%)
- Step 3 — Interpolate between 9% and 10%
- IRR ≈ 9% + [10.08 / (10.08 + 8.26)] × 1% ≈ 9.54%
The exact answer (solved numerically) is 9.54% — confirming the interpolation. For more complex multi-period cash flows, the same bracketing logic applies; financial calculators and spreadsheets iterate automatically. Use the IRR calculator to solve any cash flow series in seconds.
Worked Example: Stock Investment IRR
IRR is not just for corporate capital budgeting. Any investor who pays a price today, receives cash flows over time, and eventually exits at a terminal price can express that outcome as an IRR. Here is a mid-cap stock holding example.
Cash flow timeline
- Year 0 (purchase price)
- −$80.00 per share
- Year 1 (dividend received)
- +$2.00
- Year 2 (dividend received)
- +$2.50
- Year 3 (dividend + sale proceeds)
- +$2.50 + $110.00 = +$112.50
- Solving for IRR
- 0 = −80 + 2/(1+r) + 2.5/(1+r)² + 112.5/(1+r)³ → r ≈ 15.3%
At a 15.3% IRR, this investment clears a 10% hurdle rate by over 5 percentage points — a wide margin of safety. If your cost of capital is 10%, the NPV of this investment at that discount rate is positive, and the IRR tells you how much buffer you have before the deal breaks even on a time-value basis. See how the numbers flow through a full DCF model in the DCF calculator.
IRR vs. NPV: When They Agree and When They Diverge
IRR and NPV are complementary tools. For a single independent project with conventional cash flows (one sign change — negative outlay followed by positive inflows), they give the same accept/reject verdict. The divergences matter.
| Situation | IRR verdict | NPV verdict |
|---|---|---|
| Single independent project, conventional flows | Accept if IRR > WACC | Accept if NPV > 0 (at WACC). Same decision. |
| Mutually exclusive projects (pick one) | IRR may favor the smaller project (higher %, less absolute value) | NPV picks the project that creates more dollars — use NPV here. |
| Multiple sign changes in cash flows | Multiple IRRs possible; ambiguous | NPV is unambiguous at any single discount rate. |
| Comparing a large vs. small project | IRR ignores scale — 50% on $1 ≠ 50% on $1M | NPV captures absolute value. Use it alongside IRR. |
The Reinvestment Rate Assumption
IRR implicitly assumes that all interim cash flows are reinvested at the IRR itself. For a project with a 25% IRR, that means every dividend and coupon is assumed to compound at 25% until the end — often unrealistic for long-horizon investments. Modified IRR (MIRR) corrects this: you specify a finance rate (cost of borrowing) and a reinvestment rate (your realistic reinvestment opportunity) separately, producing a more conservative and accurate estimate of the true compounded return. When IRR is unusually high, calculate MIRR as a sanity check.
Practical Use for Stock Investors
- Expected total return over a holding period. Input your purchase price as a negative cash flow, model out expected dividends per year, and use a terminal price estimate (based on a P/E multiple or terminal value method) as the final inflow. The IRR that results is your expected annualized total return. If it beats your hurdle rate, the position is worth taking.
- Comparing two investment opportunities. If two stocks both clear your hurdle rate, rank them by IRR when they are roughly the same size bet. If the position sizes differ significantly, compare by NPV instead — a 20% IRR on a $500 position creates less absolute wealth than a 14% IRR on a $50,000 position.
- Buyback yield as a simplified one-period IRR proxy. When a company repurchases its own stock, the implied one-period return is roughly the inverse of P/E (the earnings yield) — a simplified IRR assuming earnings are the "cash flow" returned. This is why share buybacks are most accretive when the stock trades at a low earnings multiple. The WACC guide shows how to set your hurdle rate against which to benchmark any IRR.
Limitations of IRR
- The scale problem. IRR is a percentage, not a dollar figure. A 50% IRR on a $1 investment creates $0.50 of wealth; a 15% IRR on a $1 million investment creates $150,000. When choosing between unequal-sized opportunities, NPV ranks projects by absolute value creation — IRR alone cannot. Always pair IRR with NPV when scale differs.
- Multiple IRRs with non-conventional cash flows. If a project has more than one sign change in its cash flow stream — for example, a construction project with large outflows at both the beginning and the end — the IRR polynomial can have multiple positive roots. Each is mathematically valid, but none is uniquely meaningful. In these cases, NPV at the relevant discount rate is the only reliable criterion.
- Ignores absolute value creation. Two investments with identical IRRs but different durations and cash flow magnitudes create very different amounts of wealth. IRR tells you the percentage return, not the dollar compounding. A five-year project at 15% and a ten-year project at 15% have the same IRR but the longer one creates far more terminal wealth. Consider IRR alongside the absolute NPV and your reinvestment assumptions before deciding.