Understanding IRR results
IRR is most useful when compared to a required rate of return (hurdle rate) or cost of capital. The following table provides general interpretation guidance used in corporate finance.
| Comparison | Conventional interpretation |
|---|---|
| IRR > hurdle rate / WACC | Investment meets or exceeds return threshold; often accepted |
| IRR = hurdle rate / WACC | Investment earns exactly the required return; break-even |
| IRR < hurdle rate / WACC | Investment returns less than the required rate; often rejected |
| IRR does not exist or is negative | Investment destroys value at any positive discount rate; total returns < total outflows |
- The IRR criterion assumes that interim cash flows are reinvested at the IRR itself. This reinvestment-rate assumption can be unrealistic for high IRRs. The Modified Internal Rate of Return (MIRR) addresses this by specifying separate financing and reinvestment rates.
- Projects with unconventional cash flow patterns — where the sign of cash flows changes more than once (e.g., outflow, inflow, outflow) — can yield multiple mathematically valid IRRs. The calculator reports one solution (typically the economically meaningful root near the initial guess of 10%), but users should verify by examining the NPV profile.
- IRR does not account for the scale of an investment. A project with a 50% IRR on a $1,000 investment is not necessarily preferable to one with a 20% IRR on a $1,000,000 investment in terms of absolute value creation. NPV analysis accounts for scale and is generally preferred in academic corporate finance.
- This calculator treats equally spaced periods. For irregular time intervals, the XIRR function (as implemented in spreadsheet software) is more appropriate.
What is the Internal Rate of Return?
The Internal Rate of Return (IRR) is the annualized rate of return at which the net present value of all cash flows from an investment equals zero. Conceptually, it is the discount rate that equates the present value of future inflows with the present value of the initial and subsequent outflows. A project with an IRR exceeding the required rate of return (or hurdle rate, or weighted average cost of capital) is generally considered acceptable under the IRR rule.
Cash flows are entered as a time series: the first value (typically negative) represents the initial investment or outflow at time 0, and subsequent values represent inflows (positive) or additional outflows (negative) at equally spaced intervals (typically annual). For example, an investment of $10,000 today that returns $3,000, $4,000, $5,000, and $3,000 over four years is entered as: -10000, 3000, 4000, 5000, 3000.
The IRR is computed numerically because the NPV function is a polynomial in (1+r), and closed-form solutions exist only for very simple cases. Newton-Raphson iteration — a standard root-finding algorithm — converges to the IRR rapidly from an initial guess, with a bisection fallback for robustness. Be aware that projects with unconventional cash flow patterns (multiple sign changes) can have multiple IRRs or no real IRR; in such cases, the Modified Internal Rate of Return (MIRR) is a more reliable metric.
How to use this IRR calculator
- Enter cash flows as a comma-separated list. The first value is typically a negative number representing the initial investment (outflow). Subsequent values are inflows (positive) or additional outflows (negative) at each period.
- Periods are assumed to be equally spaced. For annual cash flows, the result is an annual IRR. For monthly cash flows, multiply the result by 12 to annualize (approximate).
- Read the IRR as an annualized percentage rate, along with the total invested (sum of negative flows) and total returned (sum of positive flows).
- Compare the IRR to your required return or cost of capital: if IRR exceeds the hurdle rate, the investment meets the return threshold by this metric.
IRR and the NPV equation
The IRR is defined as the rate r that satisfies NPV = 0, where NPV is the sum of all discounted cash flows. Because this equation is a polynomial in (1+r), there is no general closed-form solution. Newton-Raphson iteration updates an initial estimate of r using the ratio of the NPV function to its first derivative with respect to r (the sum of time-weighted discounted flows), repeated until the estimate converges to within a tolerance of 10^-9.
자주 묻는 질문
What is a good IRR for an investment?
Whether an IRR is 'good' depends on the type of investment and the investor's required rate of return. In corporate capital budgeting, a project is typically accepted if its IRR exceeds the company's weighted average cost of capital (WACC). In private equity, target IRRs are often 20% or higher. In real estate, many investors look for IRRs in the 10–15% range. There is no universal threshold; the relevant comparison is always the investor's own hurdle rate or opportunity cost.
Why is the first cash flow negative?
By convention, the initial investment is entered as a negative number because it represents a cash outflow — money leaving the investor's account. Subsequent periods typically show positive values (inflows from the investment). If a later period also requires additional investment, that would also be entered as a negative number.
What is the difference between IRR and NPV?
NPV (Net Present Value) is the sum of all cash flows discounted at a specified required rate of return, expressed in dollars. IRR is the discount rate that makes NPV equal to zero. NPV directly shows how much value an investment creates in absolute dollar terms; IRR shows the investment's rate of return. Most finance textbooks, including Brealey, Myers and Allen, recommend NPV as the primary decision criterion because it is not subject to the reinvestment-rate and multiple-IRR problems that can affect IRR.
Can a cash flow series have more than one IRR?
Yes. Descartes' rule of signs states that the maximum number of positive real IRRs equals the number of times the cash flow series changes sign. A series with two sign changes (e.g., negative, positive, negative) can have zero, one, or two real IRRs. When multiple IRRs exist, the standard IRR rule is unreliable and the Modified Internal Rate of Return (MIRR) or direct NPV analysis should be used instead.
How does this calculator solve for IRR numerically?
This calculator uses Newton-Raphson iteration starting from an initial guess of r = 0.10 (10%). The algorithm updates the estimate by subtracting the NPV divided by the derivative of NPV with respect to r, repeating until the change is less than 10^-9 or until 100 iterations are reached. If Newton-Raphson fails to converge (e.g., because the derivative is near zero or the estimate diverges), the calculator falls back to bisection search over the range −99% to 1000%, which is guaranteed to converge to the root in that interval.
참고 자료
- Brealey RA, Myers SC, Allen F. Principles of Corporate Finance (13th ed.). McGraw-Hill, 2020. Chapter 5: Why NPV Leads to Better Investment Decisions.
- Ross SA, Westerfield R, Jordan BD. Fundamentals of Corporate Finance (12th ed.). McGraw-Hill, 2019. Chapter 9: Net Present Value and Other Investment Criteria.
- Damodaran A. Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.). Wiley Finance, 2012. Chapter 5: Estimating Hurdle Rates.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press, 2007. Chapter 9: Root Finding and Nonlinear Sets of Equations.