Financial Management and Business Data Analytics · Capital Budgeting

Internal Rate of Return and Modified IRR

Updated 10 October 2026 · Fact-checked

The internal rate of return (IRR) is the discount rate at which a project's NPV is zero. Find it by trying two rates that give a positive and a negative NPV, then interpolating. Accept the project if IRR exceeds the cost of capital. Modified IRR (MIRR) assumes reinvestment at the cost of capital.

Understand Internal Rate of Return and Modified IRR

Every project has a discount rate at which the present value of its cash inflows exactly equals the present value of its outflows. That rate is the internal rate of return (IRR). At the IRR, NPV = 0. Think of it as the project's own yearly return on the money invested.

The decision rule is simple. If IRR is higher than the cost of capital (the required rate of return), the project earns more than it costs, so accept it. If IRR is lower, reject it. If IRR equals the cost of capital, you are indifferent.

For even cash flows, you can find the IRR from the annuity table. For uneven cash flows, there is no direct formula. You use trial and error, then interpolation: pick a low rate that gives a positive NPV and a high rate that gives a negative NPV, and assume NPV moves in a straight line between them. This gives an approximate IRR, which is what exams expect.

The IRR method has a hidden assumption: cash inflows are reinvested at the IRR itself. For a project with a very high IRR, this is often unrealistic. The modified IRR (MIRR) fixes this. It compounds all inflows forward to the end of the project at the cost of capital (or a stated reinvestment rate). It then finds the rate that turns the present value of outflows into this terminal value.

NPV and IRR usually agree for a single, conventional project. They can conflict when projects are mutually exclusive and differ in size or timing of cash flows. NPV assumes reinvestment at the cost of capital and measures the gain in rupees. IRR gives a percentage. When they conflict, NPV is generally preferred because it shows the addition to wealth. A project with sign changes in its cash flows more than once can also have more than one IRR. MIRR avoids this problem.

Key rules to remember

IRR definition: Σ [Cash inflow(t) ÷ (1 + IRR)^t] − Initial outlay = 0; IRR is the rate at which NPV = 0.
Payback-style factor (even inflows): Annuity factor = Initial outlay ÷ Annual cash inflow; Look up this factor in the annuity table along the row for the project's life. The rate it falls at is the IRR.
IRR by interpolation: IRR = LR + [NPV at LR ÷ (NPV at LR − NPV at HR)] × (HR − LR); LR is the lower rate with a positive NPV. HR is the higher rate with a negative NPV. Use the sign of NPV carefully: the denominator is the sum of the two NPVs ignoring signs.
Terminal value for MIRR: TV = Σ [Cash inflow(t) × (1 + r)^(n − t)]; r is the reinvestment rate, usually the cost of capital. n is the project life. The last year's inflow is not compounded.
Modified IRR: MIRR = (TV ÷ PV of outflows)^(1/n) − 1; Use the present value of outflows at the cost of capital. If the only outflow is at time 0, it is the initial outlay.
Decision rule: Accept if IRR (or MIRR) > cost of capital; Reject if lower. For mutually exclusive projects, compare with NPV before finalising.

How to solve Internal Rate of Return and Modified IRR questions

Use this method for any IRR or MIRR question. It works for even and uneven cash flows.

1List the initial outlay and the cash inflows by year. Note the cost of capital given in the question.
2If inflows are even, divide the outlay by the annual inflow to get the annuity factor. Find the two nearby rates in the table for the project life.
3If inflows are uneven, compute the average annual inflow, divide the outlay by it, and use that factor to guess a starting rate.
4Calculate NPV at a trial rate. If NPV is positive, try a higher rate. If negative, try a lower rate. Stop when you have one positive NPV and one negative NPV at rates 1 or 2 percentage points apart.
5Apply the interpolation formula using both NPVs and both rates. Show the working.
6Compare IRR with the cost of capital and state accept or reject.
7For MIRR, compound each inflow to the end of the project at the reinvestment rate, add them to get TV, then apply (TV ÷ PV of outflows)^(1/n) − 1.
8Write a one-line conclusion. If asked, comment on NPV versus IRR or the reinvestment assumption.

Quickest way: Factor lookup for even inflows, smart first guess for uneven

When to use it: Use when time is short and the question gives a present value table. It saves trial rounds.

For even inflows, compute outlay ÷ annual inflow and find the two table rates around it. You get both NPV points straight from the table.
For uneven inflows, use the average inflow to get a factor and pick the starting rate. Your first trial is then usually within 1 to 2 points of the answer.
Compute NPV at only two rates, one on each side of zero. A third trial wastes time.
Plug into the interpolation formula. Cross-check: the answer must lie between the two trial rates.
For MIRR, compute TV first. Then take the nth root with the calculator power key, or use the table for a close value.

Common mistakes in Internal Rate of Return and Modified IRR

Interpolating with two rates that both give a positive NPV (or both negative).
Students stop after the first trial and use a guess for the second rate.
Fix: Keep trying until NPV changes sign. The IRR must lie between the two rates, so the answer cannot be outside that range.
Writing the interpolation formula with the wrong denominator, such as NPV at LR − NPV at HR with signs mishandled.
The negative NPV is subtracted without noticing the double negative.
Fix: Treat the denominator as the total gap between the two NPVs. A positive NPV of 1,132 and a negative of −3,542 give a gap of 4,674.
Forgetting to deduct the initial outlay when finding NPV at the trial rates.
Students compute only present value of inflows and compare it with nothing.
Fix: Always write NPV = PV of inflows − outlay. The IRR is where this is zero.
Compounding the last year's inflow in MIRR.
Students apply (1 + r)^n to every inflow by habit.
Fix: Compound each inflow for (n − t) years. The year-n inflow is multiplied by 1.
Accepting a project because IRR is high without checking NPV in a conflict between mutually exclusive projects.
IRR is a percentage and looks easy to compare.
Fix: If the question asks to choose between projects and the two methods disagree, state that NPV is the better guide and explain the reinvestment assumption.
Using the cost of capital as the discount rate for the IRR trials.
Students mix up the required rate with the rate being solved for.
Fix: The cost of capital is only the benchmark for the final decision. Trial rates are chosen to make NPV zero.

Worked examples

Example 1

Ganga Textiles Ltd is considering a machine costing ₹2,00,000. It will give net cash inflows of ₹60,000 a year for 5 years. The cost of capital is 12%. Present value annuity factors for 5 years: 15% = 3.3522; 16% = 3.2743. Calculate the IRR by interpolation and advise whether to accept.

Show the solution

Inflows are even, so annuity factor = 2,00,000 ÷ 60,000 = 3.3333.
In the 5-year row, 3.3333 lies between 3.3522 (15%) and 3.2743 (16%). So IRR is between 15% and 16%.
NPV at 15% = 60,000 × 3.3522 − 2,00,000 = 2,01,132 − 2,00,000 = +₹1,132.
NPV at 16% = 60,000 × 3.2743 − 2,00,000 = 1,96,458 − 2,00,000 = −₹3,542.
IRR = 15 + [1,132 ÷ (1,132 + 3,542)] × (16 − 15) = 15 + 1,132 ÷ 4,674 = 15 + 0.24 = 15.24% (approx).
IRR of 15.24% is higher than the cost of capital of 12%.

Answer: IRR is approximately 15.24%. It exceeds the 12% cost of capital, so accept the machine.

Example 2

Kaveri Foods Ltd is evaluating a project with an outlay of ₹1,00,000 now and cash inflows of ₹40,000, ₹50,000 and ₹60,000 at the end of years 1, 2 and 3. The cost of capital and reinvestment rate is 10%. Calculate the modified IRR and state the decision.

Show the solution

Compound each inflow to the end of year 3 at 10%.
Year 1 inflow: 40,000 × (1.10)^2 = 40,000 × 1.21 = ₹48,400.
Year 2 inflow: 50,000 × (1.10)^1 = ₹55,000.
Year 3 inflow: ₹60,000, with no compounding.
Terminal value = 48,400 + 55,000 + 60,000 = ₹1,63,400.
PV of outflows = ₹1,00,000, all at time 0.
MIRR = (1,63,400 ÷ 1,00,000)^(1/3) − 1 = (1.634)^(1/3) − 1.
The cube root of 1.634 is about 1.1778, so MIRR is about 17.8%.
MIRR of 17.8% is above the 10% cost of capital.

Answer: MIRR is approximately 17.8%. It is higher than 10%, so accept the project.

Exam tips

In the MCQ section, expect questions on the definition of IRR (NPV = 0), the decision rule, or the reinvestment assumption. Learn these one-liners.
In written answers, show both trial NPVs and the interpolation formula. Step marks are given even if the final decimal differs slightly.
If the present value table is given, use only those factors. Do not recompute factors by calculator unless told to, or your answer may differ from the model answer.
For a question that asks for the difference between NPV and IRR, write four points: measure (rupees vs percentage), reinvestment assumption, ranking conflicts, and multiple IRRs.
End every capital budgeting answer with a clear accept or reject statement linked to the cost of capital.

Practice questions from Capital Budgeting

Internal Rate of Return and Modified IRR in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Internal Rate of Return and Modified IRR: frequently asked questions

How do I calculate IRR by interpolation?

Find a lower rate where NPV is positive and a higher rate where NPV is negative. Then use IRR = LR + NPV at LR ÷ (NPV at LR − NPV at HR) × (HR − LR). The answer lies between the two rates and is an approximation.

What is the difference between NPV and IRR?

NPV gives the rupee gain at the cost of capital, while IRR gives the project's own percentage return. NPV assumes inflows are reinvested at the cost of capital. IRR assumes reinvestment at the IRR itself. For mutually exclusive projects with conflicting rankings, NPV is generally preferred.

What is the MIRR formula?

MIRR = (Terminal value of inflows ÷ PV of outflows)^(1/n) − 1. The terminal value compounds each inflow to the end of the project at the reinvestment rate. The last year's inflow is not compounded.

Why is MIRR considered better than IRR?

MIRR uses a more realistic reinvestment rate, usually the cost of capital, instead of the IRR. It also gives a single answer, whereas a project with changing cash flow signs can have several IRRs.