Business Finance · Cost of capital and evaluating investment projects

NPV and IRR Investment Appraisal Explained

Updated 11 October 2026 · Fact-checked

NPV is the present value of a project's relevant future cash flows, discounted at the cost of capital, minus the initial outlay. Accept if NPV is above zero. IRR is the discount rate that makes NPV zero. Accept if IRR exceeds the required return. Solve by listing relevant cash flows, discounting, then deciding.

Understand NPV and IRR Investment Appraisal

A company has money to invest in projects. The question is whether a project adds value for shareholders. Cash received later is worth less than cash received today, so you cannot just add up the cash flows. You must discount them.

Net present value (NPV) does this directly. You discount every relevant cash flow to today at the project's required rate of return (the cost of capital for a project of that risk). You then add them up. A positive NPV means the project earns more than the required return, so it adds value. The NPV is the rupee amount of value added.

Internal rate of return (IRR) asks a different question: what discount rate would make the NPV exactly zero? It is the project's own break-even return. If the IRR is higher than the required return, the project is worth doing. IRR gives a percentage, which is easy to compare, but it can mislead.

Only relevant cash flows go in. These are future, incremental cash flows that happen because of the project. Sunk costs, depreciation (a non-cash item) and the financing cost itself (interest is already reflected in the discount rate) are left out. Opportunity costs, extra working capital and cash effects on other parts of the business are put in.

For a single project with an outlay first and inflows after, NPV and IRR give the same accept or reject answer. They can disagree when you rank mutually exclusive projects, or when cash flows change sign more than once. In those cases, NPV is the more reliable rule because it measures the value added in rupees.

Key rules to remember

Net present value: NPV = Σ [ C_t ÷ (1 + r)^t ], t = 0 to n; C_t is the net relevant cash flow at time t (outflows negative). r is the required return per period. The initial outlay at t = 0 is not discounted.
NPV decision rule: Accept if NPV > 0; reject if NPV < 0; For mutually exclusive projects, choose the highest positive NPV. NPV = 0 means the project just earns the required return.
IRR definition: Σ [ C_t ÷ (1 + IRR)^t ] = 0; Solved by trial and error, interpolation, or a calculator or spreadsheet function.
IRR decision rule: Accept if IRR > required return r; Reliable for a single conventional project. Use with care when ranking projects or when there are several sign changes.
Linear interpolation for IRR: IRR ≈ r1 + (r2 − r1) × NPV1 ÷ (NPV1 − NPV2); Use two rates giving NPVs of opposite sign, one positive (NPV1 at r1) and one negative (NPV2 at r2). The result is an estimate. Keep the two rates close together.
Annuity discount factor: a(n) = [1 − (1 + r)^(−n)] ÷ r; Use for level cash flows at the end of each period for n periods.
Perpetuity present value: PV = C ÷ r; For a level cash flow C at the end of each period, forever.

How to solve NPV and IRR Investment Appraisal questions

Use this method for any NPV or IRR question, whether it is multiple-choice or written.

1Read the question and list the cash flows by year. Decide the timing: assume cash flows arise at year ends unless told otherwise.
2Strike out what is not relevant: sunk costs, depreciation, allocated overheads that do not change, and interest or dividends paid to funders.
3Add what is relevant: opportunity costs, working capital (outflow at the start, recovery at the end), scrap value, and tax effects if the question gives tax details.
4Make the cash flows and the discount rate consistent. Use nominal cash flows with a nominal rate, or real cash flows with a real rate.
5Discount each net cash flow at the required rate and add them to get the NPV. Show the discount factor or the working for each year.
6If asked for IRR, find two rates with NPVs of opposite sign and interpolate. State that the answer is an estimate.
7State the decision using the rule, compare with the required return, and add one line on any limits, such as multiple IRRs or mutually exclusive projects.

Quickest way: Net cash flow line and annuity shortcut

When to use it: Use when the exam is timed and the cash flows are level or follow a clear pattern.

Write one row of net cash flows for t = 0, 1, 2, ... and cross out irrelevant items as you go.
If inflows are level for several years, use the annuity factor once instead of discounting each year.
For IRR, make the first guess from the ratio outlay ÷ annual inflow, which equals the annuity factor you need. Then test two close rates.
In multiple-choice questions, test the required return first. If NPV is positive at that rate, the IRR must be higher for a conventional project, which may remove two options at once.

Common mistakes in NPV and IRR Investment Appraisal

Including sunk costs, such as money already spent on research.
The cost feels like part of the project because it is connected to it.
Fix: Ask whether the cash flow changes depending on the decision. If it has already been spent and cannot be recovered, leave it out.
Deducting depreciation or interest as a cash flow.
Students copy figures from a profit statement instead of building cash flows.
Fix: Use cash only. Depreciation is non-cash. Financing costs are covered by the discount rate, so deducting interest double counts them.
Forgetting working capital.
It is mentioned in a side note and does not look like an investment.
Fix: Treat the working capital increase as an outflow when it is needed, and the recovery as an inflow at the end. Only the change in working capital is a cash flow.
Discounting the initial outlay at t = 0.
Students apply the discount factor to every row out of habit.
Fix: The factor at t = 0 is 1. The outlay at the start counts at full value.
Choosing between mutually exclusive projects by the higher IRR.
A percentage looks like a better measure than a rupee amount.
Fix: Rank by NPV at the required return. IRR ignores project size and can rank differently from NPV. Say this in your answer.
Assuming IRR is always one unique number.
Most textbook examples have one outlay followed by inflows.
Fix: If cash flows change sign more than once, there can be several IRRs, or none. Then rely on NPV.

Worked examples

Example 1

A company is considering a project costing ₹10,00,000 now. It will bring net relevant cash inflows of ₹4,00,000 at the end of year 1, ₹5,00,000 at the end of year 2 and ₹6,00,000 at the end of year 3. The required return is 10% a year. Calculate the NPV and state whether the project should be accepted.

Show the solution

Discount factors at 10%: year 1 = 1 ÷ 1.1; year 2 = 1 ÷ 1.21; year 3 = 1 ÷ 1.331.
PV of year 1 = 4,00,000 ÷ 1.1 = ₹3,63,636.
PV of year 2 = 5,00,000 ÷ 1.21 = ₹4,13,223.
PV of year 3 = 6,00,000 ÷ 1.331 = ₹4,50,789.
Total PV of inflows = 3,63,636 + 4,13,223 + 4,50,789 = ₹12,27,648.
NPV = 12,27,648 − 10,00,000 = ₹2,27,648. This is positive.

Answer: NPV ≈ ₹2,27,648. It is positive, so the project earns more than the 10% required return and should be accepted.

Example 2

A project costs ₹1,00,000 now and gives net cash inflows of ₹60,000 at the end of each of the next two years. The required return is 12% a year. Estimate the IRR by linear interpolation using 10% and 15%, and state the decision.

Show the solution

At 10%: annuity factor for 2 years = 1 ÷ 1.1 + 1 ÷ 1.21 = 0.90909 + 0.82645 = 1.73554. PV = 60,000 × 1.73554 = ₹1,04,132. NPV = +₹4,132.
At 15%: factor = 1 ÷ 1.15 + 1 ÷ 1.3225 = 0.86957 + 0.75614 = 1.62571. PV = 60,000 × 1.62571 = ₹97,543. NPV = −₹2,457.
Interpolate: IRR ≈ 10 + 5 × 4,132 ÷ (4,132 + 2,457) = 10 + 5 × 0.627 = 10 + 3.14 = 13.1%.
Check: the exact IRR solves 60,000 × a(2) = 1,00,000, so a(2) = 1.6667. This happens at about 13.1%, so the estimate is close.
Compare with the required return: 13.1% > 12%.

Answer: IRR ≈ 13.1%. This is above the required return of 12%, so accept the project. The NPV at 12% is positive, which agrees with this.

Exam tips

Always lay out the cash flows in a row with the year above each one. Method marks in written answers depend on a clear layout.
State your assumptions: cash flows at year ends, no tax unless given, and the discount rate used. Examiners give credit for stated assumptions.
For a question that asks you to compare NPV and IRR, give three points: NPV measures value added in rupees, IRR can give several answers when signs change more than once, and IRR can mis-rank mutually exclusive projects.
In multiple-choice questions, check the sign and size of the NPV with rough numbers first. Many wrong options come from forgetting working capital or discounting the outlay.
When a question mentions a sunk cost, a depreciation charge or interest, it is usually testing relevant cash flows. Name the item and say why it is excluded.

Practice questions from Cost of capital and evaluating investment projects

NPV and IRR Investment Appraisal in other exams

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

NPV and IRR Investment Appraisal: frequently asked questions

What is the difference between NPV and IRR?

NPV is a rupee amount: the value a project adds after discounting at the required return. IRR is a percentage: the discount rate at which NPV is zero. For a single conventional project both give the same accept or reject decision, but NPV is more reliable for ranking projects.

Why can IRR give multiple answers?

If the cash flows change sign more than once, for example an outlay, then inflows, then a large closing cost, the NPV equation can have more than one solution. Each solution is a valid IRR, so the rule becomes unclear. Use NPV at the required return instead.

What are relevant cash flows in project appraisal?

They are future cash flows that change because the project goes ahead. Include incremental revenues and costs, opportunity costs, working capital changes and any cash effect on other activities. Exclude sunk costs, non-cash items such as depreciation, and financing charges that are covered by the discount rate.

Which discount rate should I use for NPV?

Use the required return for a project of that risk, which is usually the cost of capital, such as the WACC if the project has similar risk to the firm. Keep the rate consistent with the cash flows: nominal with nominal, real with real.

How do I find IRR in the exam without a spreadsheet?

Pick two rates that give NPVs of opposite sign and use linear interpolation. Choose rates close together for a better estimate. Say clearly that the result is an approximation.