Internal Rate of Return (IRR)
Reference for IRR — the discount rate where NPV equals zero.
Explains iterative solving, comparison with WACC, and examples for capital budgeting.
The Formula
The IRR is the interest rate at which the present value of all future cash flows equals the initial investment. If the IRR exceeds your required return, the project is worth pursuing.
Variables
| Symbol | Meaning |
|---|---|
| IRR | Internal Rate of Return (percentage) |
| CF_t | Cash flow in period t |
| C₀ | Initial investment |
| t | Time period |
Note: IRR cannot be solved algebraically. It requires iteration (trial and error) or a financial calculator.
Example 1
Invest $5,000, receive $2,000/year for 3 years. Estimate the IRR.
Try r = 10%: NPV = 2000/1.1 + 2000/1.21 + 2000/1.331 - 5000 = -27 (slightly negative)
Try r = 9.7%: NPV ≈ 0
IRR ≈ 9.7%
Example 2
Invest $20,000, returns: Year 1 = $8,000, Year 2 = $10,000, Year 3 = $12,000
Try r = 15%: NPV = 6957 + 7561 + 7890 - 20000 = +2,408
Try r = 25%: NPV = 6400 + 6400 + 6144 - 20000 = -1,056
IRR is between 15% and 25%
IRR ≈ 21.2% (by interpolation or calculator)
When to Use It
Use the IRR when:
- Comparing projects of different sizes on an equal basis
- Determining if a project meets the minimum required return
- Ranking investment alternatives
- Communicating returns to stakeholders in a simple percentage
Key Notes
- IRR is the discount rate where NPV = 0: There is no closed-form formula for IRR — it is found by solving 0 = Σ[CF_t / (1+IRR)^t] iteratively (using trial and error, Newton's method, or a financial calculator).
- Decision rule: Accept a project if IRR > the required rate of return (hurdle rate). Reject it if IRR < hurdle rate. This mirrors the NPV rule: positive NPV ↔ IRR above the hurdle rate.
- IRR accounts for time value of money: Unlike simple ROI, IRR weights cash flows by when they occur. An early cash flow is worth more than an equal late cash flow at the same IRR.
- Multiple IRRs are possible: If the cash flow stream changes sign more than once (e.g., investment → return → more investment), there can be multiple mathematically valid IRR solutions. In this case, use NPV analysis instead.
- Modified IRR (MIRR) solves reinvestment bias: Standard IRR assumes reinvestment of interim cash flows at the IRR itself — often unrealistically high. MIRR uses a separate reinvestment rate (typically the cost of capital), giving a more realistic comparison.