Financial Management and Strategic Management · Investment Decisions

Internal Rate of Return (IRR) and MIRR for CA Inter

Q: How do I choose the first rate when calculating IRR by interpolation?

For equal annual inflows, divide the outlay by the annual inflow and find that factor in the annuity table for the project life. For uneven inflows, divide the outlay by the average inflow and use that as a rough guide. Test NPV, then move the rate up or down.

Q: What is the difference between IRR and MIRR?

IRR assumes cash inflows are reinvested at the IRR itself and can produce more than one rate for unconventional cash flows. MIRR compounds inflows at a stated reinvestment rate, discounts outflows at the finance rate, and gives one answer. This makes MIRR a more realistic measure.

Q: Which is better, NPV or IRR?

For mutually exclusive projects where the two methods conflict, NPV is preferred. It shows the absolute increase in wealth and assumes reinvestment at the cost of capital. IRR is useful as a percentage measure, but it can mislead when projects differ in size or timing.

Q: Why can a project have multiple IRRs?

When cash flows change sign more than once, for example outflow, then inflows, then another outflow, the NPV equation can have more than one rate that makes NPV zero. The IRR is then not reliable. Use NPV or MIRR instead.

Updated 4 October 2026 · Fact-checked

IRR is the discount rate at which a project's NPV is zero. Find two rates giving a positive and a negative NPV, then interpolate between them. Accept the project if IRR exceeds the cost of capital. MIRR compounds inflows at a reinvestment rate, discounts outflows at the finance rate, and solves for one rate.

Understand Internal Rate of Return (IRR) and MIRR

Internal Rate of Return (IRR) is the rate of return a project earns on its own cash flows. It is the discount rate at which the present value of cash inflows equals the present value of cash outflows. At that rate, NPV is zero.

Think of it as the project's break-even cost of money. If the firm pays 12% for its funds and the project earns an IRR of 16%, the project leaves a surplus. If the IRR were 10%, it would destroy value.

There is no direct formula for IRR when cash flows are uneven. You use trial and error. Pick a rate, find the NPV, then move up if NPV is positive and down if it is negative. Once you have one rate with a positive NPV and one with a negative NPV, you interpolate to estimate the rate where NPV is zero.

IRR has two known weaknesses. It assumes cash inflows are reinvested at the IRR itself, which can be unrealistic for a very high IRR. It can also give multiple rates, or no meaningful rate, when cash flows change sign more than once. MIRR fixes both problems by using an explicit reinvestment rate and a finance rate, which gives a single answer.

For mutually exclusive projects, NPV and IRR can rank projects differently, usually because of differences in project size or in the timing of cash flows. In that case, NPV is the preferred method because it measures the absolute increase in wealth.

Key rules to remember

IRR definition: Σ [CFt ÷ (1 + r)^t] − Initial outflow = 0; The r that satisfies this is the IRR. t runs from 1 to n.
Interpolation formula: IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L); L is the lower rate (positive NPV) and H is the higher rate (negative NPV). Keep the gap between L and H small, ideally 1% to 5%.
Equal annual inflows: first guess: Factor = Initial outlay ÷ Annual cash inflow; Look up this factor in the annuity table on the row for n years. The column closest to it gives your first trial rate.
Uneven inflows: rough first guess (heuristic only): Factor = Initial outlay ÷ Average annual inflow; This is only a rough heuristic, not a dependable method. It ignores the timing of the inflows and can be well off when they are very uneven. Use the nearest annuity factor only to pick a first trial rate, then check the NPV and adjust.
MIRR: MIRR = (Terminal value of inflows ÷ PV of outflows)^(1/n) − 1; Terminal value is each inflow compounded to the end of year n at the reinvestment rate. Outflows are discounted to year 0 at the finance rate (usually the cost of capital).
IRR decision rule: Accept if IRR > cost of capital; reject if IRR < cost of capital; At IRR equal to cost of capital, the project is at break-even. For mutually exclusive projects, prefer NPV if the two methods conflict.

How to solve Internal Rate of Return (IRR) and MIRR questions

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

1Write the cash flow line: outflow at year 0 and inflows for years 1 to n. Check whether salvage value or working capital recovery belongs in the last year.
2Find the first trial rate. For equal inflows, compute outlay ÷ annual inflow and look it up in the annuity table. For uneven inflows, outlay ÷ average inflow gives only a rough heuristic factor. Use it just to pick a first trial rate, and expect to adjust after checking the NPV.
3Compute NPV at the trial rate using the discount factors. Show each year's present value in a small table.
4If NPV is positive, try a higher rate. If NPV is negative, try a lower rate. Continue until you have one positive NPV and one negative NPV, with rates 1% apart if possible.
5Apply the interpolation formula. Write the formula and the numbers in the answer. State the IRR to two decimal places.
6Compare IRR with the cost of capital or the required rate in the question, and state accept or reject.
7For MIRR, compound each inflow to the end of year n at the reinvestment rate and add them for the terminal value. Take the PV of outflows at the finance rate, apply the MIRR formula, and compare with the cost of capital.

Quickest way: Fast IRR with two trials, and MIRR in three lines

When to use it: Use this in the 70-mark descriptive section when time is short, and for 1 or 2 mark MCQs asking for an IRR or a decision.

For equal inflows, divide outlay by inflow and read the closest columns from the annuity row. The IRR lies between the two columns on either side of the factor, so use them as your trial rates and interpolate.
If the cost of capital is given, test NPV at that rate first. A positive NPV means IRR is above the cost of capital and a negative NPV means it is below. For an accept or reject MCQ, this alone is enough and no interpolation is needed.
For MCQ options, test NPV at the middle option. Its sign tells you whether the IRR is higher or lower, which removes two options at once. With no negative marking, always attempt it.
In written answers, keep a neat table (year, cash flow, factor, present value) for each trial rate. Examiners give step marks for the table, the formula and the final comparison, even if one factor is misread.
For MIRR: (1) terminal value = Σ inflow × (1 + reinvestment rate)^(n − t); (2) divide by PV of outflows; (3) take the nth root and subtract 1. Use the nearest simple trial rate to take the root and state the result as approximate.

Common mistakes in Internal Rate of Return (IRR) and MIRR

Interpolating between two rates that are far apart, such as 10% and 25%.
Students stop as soon as they get one positive and one negative NPV. The NPV curve is not a straight line, so the estimate is poor over a wide gap.
Fix: Narrow the gap to 1% to 5% and re-run the NPV if needed. Always show the two NPVs and the formula.
Writing the interpolation formula with the wrong sign or the wrong NPV in the numerator.
Students memorise the formula without understanding it. The numerator must be the NPV at the lower rate, and the denominator is the difference between the two NPVs.
Fix: Use the logic: you start at L and move towards H by the fraction NPV at L ÷ (total NPV swing). Add this fraction × (H − L) to L. The denominator is NPV at L minus NPV at H, which equals the sum of their absolute values when one is positive and one is negative.
Treating the first trial rate as the answer without checking NPV.
Students read the annuity table, see a near match, and stop. With uneven inflows, the average-based factor is only a rough guide.
Fix: Always test NPV at the trial rate, then adjust up or down.
Using the cost of capital to compound inflows in MIRR when the question gives a separate reinvestment rate.
Students assume one rate fits all. MIRR can use a reinvestment rate for inflows and a finance rate for outflows.
Fix: Read the question for both rates. Compound inflows at the reinvestment rate. Discount outflows at the finance rate. If only one rate is given, use it for both.
Recommending a project on the highest IRR when comparing mutually exclusive projects.
A higher percentage looks better. But a small project can have a high IRR and a low NPV, and the methods may conflict.
Fix: When NPV and IRR conflict for mutually exclusive projects, prefer the higher NPV and say why: it measures the absolute addition to shareholder wealth and assumes reinvestment at the cost of capital.
Applying IRR to cash flows that change sign more than once without a warning.
Students apply the standard method to every cash flow pattern, for example a large outflow in a later year, such as for restoration costs.
Fix: Check the sign pattern first. If it changes more than once, mention that multiple IRRs are possible and use NPV or MIRR instead.

Worked examples

Example 1

A project needs an initial outlay of ₹2,00,000 and gives cash inflows of ₹80,000, ₹90,000 and ₹1,00,000 at the end of years 1, 2 and 3. The cost of capital is 12%. Compute the IRR by interpolation and advise whether to accept the project. Present value factors: at 15%: 0.8696, 0.7561, 0.6575; at 16%: 0.8621, 0.7432, 0.6407.

Show the solution

Average annual inflow = (80,000 + 90,000 + 1,00,000) ÷ 3 = ₹90,000. Factor = 2,00,000 ÷ 90,000 = 2.22. Adding the given 16% factors gives a 3-year cumulative factor of 0.8621 + 0.7432 + 0.6407 = 2.246, which is close to 2.22. So the average-based guess points to a rate near 16%. This is only a rough heuristic. Since the later inflows are larger, the true IRR is expected to be somewhat lower, so trial rates of 15% and 16% are sensible.
NPV at 15%: 80,000 × 0.8696 = ₹69,568; 90,000 × 0.7561 = ₹68,049; 1,00,000 × 0.6575 = ₹65,750. Total PV = ₹2,03,367. NPV = 2,03,367 − 2,00,000 = +₹3,367.
NPV at 16%: 80,000 × 0.8621 = ₹68,968; 90,000 × 0.7432 = ₹66,888; 1,00,000 × 0.6407 = ₹64,070. Total PV = ₹1,99,926. NPV = 1,99,926 − 2,00,000 = −₹74.
Interpolation: IRR = 15% + [3,367 ÷ (3,367 + 74)] × (16% − 15%) = 15% + (3,367 ÷ 3,441) × 1% = 15% + 0.98% = 15.98%.
Decision: IRR of 15.98% exceeds the cost of capital of 12%, so the project adds value.

Answer: IRR ≈ 15.98%. It is above the 12% cost of capital, so accept the project.

Example 2

A project costs ₹1,00,000 now and gives inflows of ₹40,000, ₹50,000 and ₹60,000 at the end of years 1, 2 and 3. The reinvestment rate and the cost of capital are both 10%. Compute the MIRR and state the decision.

Show the solution

Terminal value of the inflows at the end of year 3: ₹40,000 × (1.10)² = 40,000 × 1.21 = ₹48,400.
₹50,000 × 1.10 = ₹55,000.
₹60,000 received at the end of year 3 is not compounded: ₹60,000.
Total terminal value = 48,400 + 55,000 + 60,000 = ₹1,63,400.
PV of outflows = ₹1,00,000 (all at year 0, so no discounting is needed).
MIRR = (1,63,400 ÷ 1,00,000)^(1/3) − 1 = (1.634)^(1/3) − 1. Since 1.178³ ≈ 1.6347 and 1.1778³ ≈ 1.6338, the cube root is about 1.1778. So MIRR ≈ 17.8%.
Decision: MIRR of about 17.8% is above the 10% cost of capital, so accept.

Answer: MIRR ≈ 17.8%, which is more than the 10% cost of capital. Accept the project.

Exam tips

Start every IRR answer with the cash flow table and the first trial rate logic. Even if arithmetic slips, you keep the step marks for method.
Use the discount factors given in the question. Do not recompute them from formulas, as small rounding differences can change the interpolated answer.
Where the question asks only whether to accept, test NPV at the cost of capital and stop. Do not spend 10 minutes interpolating when the decision is already clear.
In theory questions on NPV vs IRR, name the cause of conflict (scale or timing of cash flows), name the reinvestment assumption of each method, and conclude with a recommendation for NPV.
For IRR limitations, learn four points: reinvestment assumption, multiple IRRs with non-conventional cash flows, conflict with NPV for mutually exclusive projects, and unsuitability for comparing projects of different size.

Practice questions from Investment Decisions

Internal Rate of Return (IRR) and MIRR 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 (IRR) and MIRR: frequently asked questions

How do I choose the first rate when calculating IRR by interpolation?

For equal annual inflows, divide the outlay by the annual inflow and find that factor in the annuity table for the project life. For uneven inflows, divide the outlay by the average inflow and use that as a rough guide. Test NPV, then move the rate up or down.

What is the difference between IRR and MIRR?

IRR assumes cash inflows are reinvested at the IRR itself and can produce more than one rate for unconventional cash flows. MIRR compounds inflows at a stated reinvestment rate, discounts outflows at the finance rate, and gives one answer. This makes MIRR a more realistic measure.

Which is better, NPV or IRR?

For mutually exclusive projects where the two methods conflict, NPV is preferred. It shows the absolute increase in wealth and assumes reinvestment at the cost of capital. IRR is useful as a percentage measure, but it can mislead when projects differ in size or timing.

Why can a project have multiple IRRs?

When cash flows change sign more than once, for example outflow, then inflows, then another outflow, the NPV equation can have more than one rate that makes NPV zero. The IRR is then not reliable. Use NPV or MIRR instead.