How to Calculate IRR: Internal Rate of Return with Worked Examples for Multiple Projects

Internal rate of return (IRR) is the discount rate at which the net present value (NPV) of a project's cash flows equals zero. It represents the effective annual yield of a project based on its cash flow pattern. Mathematically: NPV = 0 = Σ CFt / (1 + IRR)^t, where CFt is the cash flow in period t. Solve for IRR. The IRR decision rule: accept a project if IRR > required rate of return (hurdle rate, or cost of capital). Reject if IRR < hurdle rate. The intuition is that IRR tells you the project's implicit yield, so you accept only if that yield exceeds what you'd need to fund the project. IRR is popular because it produces a single intuitive number (a percentage rate) that managers can compare to cost of capital or industry benchmarks. A 22% IRR is easy to reason about. A $4.2M NPV requires context (for what size project?). However, IRR has limitations that often cause it to mislead. Non-conventional cash flows (sign changes beyond the initial outflow) can produce multiple IRRs or no real IRR. Mutually exclusive project comparisons can favor the wrong project when projects have different scales or timing. Reinvestment rate assumption is that intermediate cash flows earn the IRR, which is often unrealistic for high-IRR projects. For corporate finance exams, IRR calculations appear frequently and the NPV vs IRR conflict is heavily tested. For real-world investment decisions, NPV is the theoretically correct criterion — use IRR as a supplementary check, not the primary decision rule.

Project cash flows: Year 0: -$100,000 (investment) Year 1: +$40,000 Year 2: +$50,000 Year 3: +$45,000 Find IRR via trial-and-error interpolation. Step 1: try 10%. NPV = -100,000 + 40,000/1.10 + 50,000/1.21 + 45,000/1.331 NPV = -100,000 + 36,364 + 41,322 + 33,809 = +11,495 NPV positive, so IRR > 10%. Step 2: try 17%. NPV = -100,000 + 40,000/1.17 + 50,000/1.3689 + 45,000/1.601613 NPV = -100,000 + 34,188 + 36,525 + 28,099 = -1,188 NPV negative, so IRR < 17%. Step 3: try 15%. NPV = -100,000 + 40,000/1.15 + 50,000/1.3225 + 45,000/1.520875 NPV = -100,000 + 34,783 + 37,807 + 29,589 = +2,179 NPV positive, so IRR > 15%. Step 4: interpolate between 15% and 17%. IRR ≈ 15% + (2,179 / (2,179 + 1,188)) × (17% - 15%) IRR ≈ 15% + (2,179 / 3,367) × 2% IRR ≈ 15% + 1.29% = 16.29% Verify with Excel: =IRR({-100000, 40000, 50000, 45000}) = 16.29%. Decision: if hurdle rate is 10%, accept (16.29% > 10%). If hurdle rate is 18%, reject. This content is for educational purposes only and does not constitute financial advice.

Two mutually exclusive projects (you can only pick one): Project A: -$10,000 / +$6,500 / +$6,500 (IRR 19.4%, NPV at 10% = $1,281) Project B: -$50,000 / +$30,000 / +$30,000 (IRR 13.1%, NPV at 10% = $2,066) IRR picks A (19.4% > 13.1%). NPV picks B ($2,066 > $1,281). Which is right? NPV. In mutually exclusive projects, NPV measures absolute value creation — what shareholders care about. IRR measures percentage return on project capital, but a high percentage on a small project can produce less total value than a lower percentage on a larger project. The key insight: IRR ignores project scale. Project A earns 19.4% on $10,000 = roughly $1,940 of value per year. Project B earns 13.1% on $50,000 = roughly $6,550 of value per year. The per-dollar return is lower, but the total dollars earned is higher. When the two rules conflict, trust NPV. Shareholder wealth is measured in dollars, not percentages. The conflict arises from: (1) scale differences (as above); (2) timing differences (one project's cash flows weighted earlier or later than another's). Both cases favor NPV as the correct criterion. Crossover rate: the discount rate at which two projects have equal NPVs. Below the crossover, one project wins; above, the other wins. Find it by taking the difference in cash flows between the two projects and finding the IRR of that difference. This content is for educational purposes only and does not constitute financial advice.

Conventional cash flows have one sign change: a single outflow at time 0 followed by inflows. The IRR equation has one positive real root — one IRR. Non-conventional cash flows have multiple sign changes. The IRR equation can have multiple real roots — multiple IRRs. Example: mining project. Year 0: -$10M (investment) Years 1-5: +$4M each (operating) Year 6: -$8M (environmental cleanup) Two sign changes. Solving yields two rates: approximately 7.3% and 30.8%. Which IRR is right? Neither individually. The multiple-IRR situation breaks the simple IRR decision rule. Solutions: 1. Use NPV at cost of capital. NPV gives unambiguous signal. If NPV > 0 at your cost of capital, accept. 2. Use Modified IRR (MIRR). MIRR assumes intermediate cash flows are reinvested at the cost of capital (not at IRR). This produces a single rate even for non-conventional cash flows. MIRR formula: MIRR = (FV of positive cash flows at reinvestment rate / |PV of negative cash flows at finance rate|)^(1/n) - 1 Excel: =MIRR(values, finance_rate, reinvest_rate) For the mining project with 10% finance and reinvest rate: PV of negatives: -10 - 8/1.10^6 = -10 - 4.52 = -14.52M FV of positives (at year 6): 4 × ((1.10^5 - 1)/0.10) × 1.10 = 4 × 6.1051 × 1.10 = 26.86M MIRR = (26.86/14.52)^(1/6) - 1 = 10.8% A single, unambiguous rate. MIRR > 10% cost of capital, so accept. This content is for educational purposes only and does not constitute financial advice.

Excel IRR function: =IRR(values, [guess]) Example: =IRR(A1:A5) where A1:A5 contains cash flows starting with negative outflow. Returns the IRR as a decimal (multiply by 100 for percentage). Excel XIRR function: =XIRR(values, dates, [guess]) Use when cash flows occur at irregular dates. Dates must align with values. Essential for real-world projects with non-annual timing. Excel MIRR function: =MIRR(values, finance_rate, reinvest_rate) Specify both the finance rate (cost of negative cash flows) and reinvestment rate (typical = cost of capital). TI BA II Plus financial calculator: 1. Press CF to enter cash flow mode 2. CF0 = -100,000 (outflow is negative), press ENTER, down arrow 3. C01 = 40,000, press ENTER, down arrow 4. F01 = 1 (frequency — set to 1 for each unique cash flow), ENTER, down arrow 5. Repeat for C02, C03, etc. 6. Press IRR, then CPT to compute For cash flows that repeat (e.g., $10,000 for 3 years), use F = 3 to avoid entering the same number three times. HP 12C: press g END first (to set end-of-period convention), enter CF0 with CHS then g CF0, enter CF1 then g CFj, repeat for each cash flow, then press f IRR. Google Sheets has identical IRR, XIRR, and MIRR functions as Excel. This content is for educational purposes only and does not constitute financial advice.

Four common mistakes: 1. Using IRR as primary decision rule for mutually exclusive projects. NPV is correct. IRR can mislead with scale or timing differences. 2. Ignoring the multiple IRR problem. Non-conventional cash flows break the simple rule. Use NPV or MIRR. 3. Interpreting IRR as the actual return you will earn. Standard IRR assumes intermediate cash flows reinvested at IRR — often implausible for high-IRR projects. MIRR addresses this. 4. Including sunk costs or ignoring opportunity costs. Sunk costs should NOT be in cash flow estimates (past spending is irrelevant to future decisions). Opportunity costs SHOULD be included (foregone earnings from alternatives). When IRR works well: independent projects (accept/reject standalone); projects with similar scale and timing; quick screening ('25% IRR is clearly above our 10% hurdle — pursue'). When IRR fails: mutually exclusive projects with different scales or timing; non-conventional cash flows with multiple sign changes; projects where reinvestment at IRR is implausible. For CFA and corporate finance exams, know: - How to calculate IRR via interpolation - Excel IRR, XIRR, MIRR functions - The NPV vs IRR conflict and crossover rate - Multiple IRR problem - NPV is the theoretically correct decision rule This content is for educational purposes only and does not constitute financial advice.