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Corporate Accounting and Financial Management · Capital Budgeting

Discounting Techniques: NPV, PI and IRR Explained

Updated 11 October 2026 · Fact-checked

Discounting techniques value a project's cash flows at the cost of capital. NPV is the present value of inflows minus the cash outlay; accept if it is positive. PI is present value of inflows divided by the outlay; accept if it is above 1. IRR is the rate that makes NPV zero; accept if it exceeds the cost of capital.

Understand Discounting Techniques: NPV, PI and IRR

A rupee today is worth more than a rupee next year. So you cannot add up cash flows from different years directly. Discounting techniques bring every cash flow to its value today, using the firm's required rate of return (the cost of capital or hurdle rate). This is why they are better than payback and ARR, which ignore the time value of money.

Net Present Value (NPV) is the sum of the present values of all cash inflows minus the initial outlay. A positive NPV means the project earns more than the required return, so it adds to shareholder wealth. A negative NPV means it destroys value.

Profitability Index (PI) is the present value of inflows divided by the present value of outflows. It shows the value created per rupee invested. It is useful when you must rank projects of different sizes.

Internal Rate of Return (IRR) is the discount rate at which NPV is exactly zero. It is the project's own break-even return. You compare it with the cost of capital. If IRR is higher, the project is acceptable.

Modified IRR (MIRR) fixes a weakness of IRR. Ordinary IRR assumes cash inflows are reinvested at the IRR itself. MIRR assumes they are reinvested at the cost of capital, which is more realistic. It also gives one single answer, even when cash flows change sign more than once.

For a single project with normal cash flows (one outflow first, then inflows), NPV, PI and IRR all give the same accept or reject decision. Conflicts arise only when you compare mutually exclusive projects.

Key rules to remember

Net Present Value
NPV = Σ [Ct ÷ (1 + k)^t] − C0
Ct is the cash inflow in year t, k is the cost of capital, C0 is the initial outlay. Accept if NPV > 0.
Present value using factors
PV = Cash flow × PV factor at k% for year t
In exams, PV factors are usually given in the question. Use them as given.
Profitability Index
PI = PV of cash inflows ÷ PV of cash outflows
Also written as 1 + (NPV ÷ initial outlay) when the whole outflow occurs at time zero. Accept if PI > 1.
IRR condition
Σ [Ct ÷ (1 + r)^t] = C0, that is NPV at r = 0
r is the IRR. Accept if r > cost of capital.
IRR for equal annual inflows (starting point)
Annuity factor = Initial outlay ÷ Annual inflow
Look up this factor in the annuity table for the given number of years. The rate nearest to it is your trial rate.
IRR by interpolation
IRR = L + [(PV at L − Outlay) ÷ (PV at L − PV at H)] × (H − L)
L is the lower trial rate (PV above outlay), H is the higher trial rate (PV below outlay). The two rates should be 1% to 5% apart.
Modified IRR
MIRR = (Terminal value ÷ PV of outflows)^(1/n) − 1
Terminal value is the future value of all inflows at the cost of capital at the end of year n. Accept if MIRR > cost of capital.
Decision rules
NPV > 0, PI > 1, IRR > k: accept. NPV < 0, PI < 1, IRR < k: reject.
For mutually exclusive projects, NPV is the preferred criterion because it measures absolute wealth added.

How to solve Discounting Techniques: NPV, PI and IRR questions

Use this method for any NPV, PI or IRR question. Work in a neat table so you can earn step marks even if one figure goes wrong.

  1. 1Read the question and list the initial outlay, the yearly net cash inflows, the project life and the discount rate. Use cash flows after tax and before depreciation, as given in the question.
  2. 2Check the timing. Treat the initial outlay as year 0, so it is not discounted. Add salvage value and working capital recovery in the last year if given.
  3. 3Take the PV factor for each year at the given rate. Multiply each cash flow by its factor to get the present value. If inflows are equal each year, use the annuity factor once.
  4. 4Add the present values of inflows. Subtract the outlay to get NPV. Divide total PV of inflows by the outlay to get PI.
  5. 5For IRR, find the annuity factor (or the average-based trial rate for uneven flows). Pick a trial rate and compute NPV. If NPV is positive, try a higher rate. If negative, try a lower rate, until you have one rate with positive NPV and one with negative NPV.
  6. 6Interpolate between the two rates to get the IRR to two decimal places.
  7. 7State the decision by comparing the result with the cost of capital or the benchmark. Write a one-line conclusion such as 'NPV is positive, so the project should be accepted.'
  8. 8If projects are mutually exclusive, rank them by NPV and mention any conflict with IRR or PI.

Quickest way: Table method with annuity shortcut

When to use it: Use it when the question gives PV factor tables and asks for NPV, PI and IRR together, with limited time.

  1. Draw one table with columns: Year, Cash flow, PV factor, PV. Fill it once and reuse the total for NPV and PI.
  2. If inflows are equal, skip the table. Compute annual inflow × annuity factor in one line.
  3. For IRR with equal inflows, divide outlay by annual inflow, read the table row for the project life, and pick the two nearest rates around that factor.
  4. For uneven inflows, take the average annual inflow, divide the outlay by it, and use that factor to choose your first trial rate.
  5. Interpolate with the formula using the two PV totals. Write the final IRR and the decision in one line.

Common mistakes in Discounting Techniques: NPV, PI and IRR

  • Discounting the initial outlay at year 1 instead of treating it as year 0.

    Students apply a PV factor to every number in the table out of habit.

    Fix: The outlay made today has a factor of 1. Subtract it directly from the total PV of inflows.

  • Using accounting profit instead of cash flow, or deducting depreciation from the inflow.

    The question gives profit after depreciation, and students forget that depreciation is non-cash.

    Fix: Add back depreciation to profit after tax to get cash inflow, unless the question already gives net cash flows.

  • Mixing up the decision rules, for example accepting a project with PI below 1.

    NPV is compared with zero while PI is compared with one, and the two get confused.

    Fix: Remember that NPV = 0 is the same as PI = 1. Positive NPV means PI above 1.

  • Interpolating with two rates where both NPVs have the same sign.

    Students stop after one trial and pick a second rate at random.

    Fix: Keep trying until one rate gives positive NPV and the other negative. The IRR must lie between them.

  • Applying the interpolation formula with PV at H and PV at L in the wrong order.

    The formula is memorised without understanding it.

    Fix: Think of it as: how far is the lower-rate PV above the outlay, as a share of the total gap between the two PVs. Then add that share of the rate gap to L.

  • Ignoring the cost of capital when stating the decision, and writing only the IRR value.

    Students think the calculation is the whole answer.

    Fix: Always compare IRR with the required rate and write the accept or reject conclusion.

Worked examples

Example 1

Rathi Textiles Ltd is considering a machine costing ₹10,00,000. Expected net cash inflows are ₹3,00,000, ₹4,00,000, ₹4,00,000 and ₹3,00,000 in years 1 to 4. The cost of capital is 10%. PV factors at 10% for years 1 to 4 are 0.909, 0.826, 0.751 and 0.683. Calculate NPV and profitability index and advise whether to accept the project.

Show the solution
  1. Year 1: ₹3,00,000 × 0.909 = ₹2,72,700.
  2. Year 2: ₹4,00,000 × 0.826 = ₹3,30,400.
  3. Year 3: ₹4,00,000 × 0.751 = ₹3,00,400.
  4. Year 4: ₹3,00,000 × 0.683 = ₹2,04,900.
  5. Total PV of inflows = ₹2,72,700 + ₹3,30,400 + ₹3,00,400 + ₹2,04,900 = ₹11,08,400.
  6. NPV = ₹11,08,400 − ₹10,00,000 = ₹1,08,400.
  7. PI = ₹11,08,400 ÷ ₹10,00,000 = 1.108, or about 1.11.
  8. NPV is positive and PI is above 1, so the project earns more than 10% and adds value.

Answer: NPV = ₹1,08,400 and PI = 1.108 (about 1.11). Accept the project.

Example 2

Mehta Industries Ltd is evaluating a project with an outlay of ₹1,00,000 and equal annual net cash inflows of ₹40,000 for 3 years. The cost of capital is 8%. Annuity factors for 3 years are 2.577 at 8%, 2.531 at 9% and 2.487 at 10%. Find the IRR by interpolation and state the decision.

Show the solution
  1. Required annuity factor = ₹1,00,000 ÷ ₹40,000 = 2.5.
  2. The 3-year row shows 2.531 at 9% and 2.487 at 10%. So the IRR lies between 9% and 10%.
  3. PV at 9% = ₹40,000 × 2.531 = ₹1,01,240. This is above the outlay, so NPV is ₹1,240.
  4. PV at 10% = ₹40,000 × 2.487 = ₹99,480. This is below the outlay, so NPV is −₹520.
  5. IRR = 9% + [(₹1,01,240 − ₹1,00,000) ÷ (₹1,01,240 − ₹99,480)] × (10% − 9%).
  6. IRR = 9% + (₹1,240 ÷ ₹1,760) × 1% = 9% + 0.7045% = 9.70% (approximately).
  7. Check at the cost of capital: PV at 8% = ₹40,000 × 2.577 = ₹1,03,080, so NPV = ₹3,080, which is positive. This agrees with the IRR result.
  8. IRR of 9.70% is higher than the cost of capital of 8%.

Answer: IRR is about 9.70%. It exceeds the 8% cost of capital, so accept the project.

Exam tips

  • Write the full table of year, cash flow, PV factor and PV. Examiners award marks for each step, not only the final figure.
  • Always end with a decision sentence that compares your result with the benchmark: NPV with zero, PI with one, IRR with the cost of capital.
  • In IRR questions, show both trial rates with their PV totals before interpolating. A correct method with a small arithmetic slip still earns most marks.
  • In a theory question on NPV versus IRR, mention three points: NPV gives absolute wealth, IRR gives a percentage; IRR assumes reinvestment at IRR while NPV assumes reinvestment at the cost of capital; IRR can give multiple answers when cash flows change sign.
  • Read whether the question gives profit or cash flow. Convert profit to cash flow by adding back depreciation before discounting.

Practice questions from Capital Budgeting

Discounting Techniques: NPV, PI and IRR in other exams

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

Discounting Techniques: NPV, PI and IRR: frequently asked questions

What is the difference between the NPV and IRR methods?

NPV gives the rupee value a project adds, using a fixed discount rate. IRR gives the percentage return at which NPV becomes zero. NPV assumes reinvestment at the cost of capital, while IRR assumes reinvestment at the IRR itself. For mutually exclusive projects, NPV is generally preferred.

How do I calculate IRR by interpolation?

Find two trial rates, one giving a PV above the outlay and one giving a PV below it. Then apply IRR = L + [(PV at L − Outlay) ÷ (PV at L − PV at H)] × (H − L). The answer lies between the two rates.

What is the profitability index formula and when is it used?

PI = PV of cash inflows ÷ PV of cash outflows. A PI above 1 means accept. It is mainly used to rank projects of different sizes, especially when funds are limited.

Can NPV be positive while IRR is below the cost of capital?

For a normal project with one outflow followed by inflows, no. Both give the same accept or reject result. Differences appear only when ranking mutually exclusive projects, or when cash flows change sign more than once.

Why is modified IRR better than ordinary IRR?

MIRR assumes inflows are reinvested at the cost of capital, which is more realistic than reinvesting at the IRR. It also gives a single answer, avoiding the multiple IRR problem.