NPV and IRR of a project with even cash flows
Discount an even annuity to find NPV, then interpolate between two discount rates to find the IRR.
Marks
8 marks
Exam time
14 min
Difficulty
Moderate
The question
A project requires an initial investment of ₹10,00,000 and is expected to generate net cash inflows of ₹3,00,000 per year for five years. The company's cost of capital is 10%. Compute the net present value and the internal rate of return, and advise whether the project should be accepted. (PVAF at 10% for 5 years = 3.7908; at 15% = 3.3522; at 16% = 3.2743.)
- Given
- Initial investment = ₹10,00,000Annual cash inflow = ₹3,00,000 for 5 yearsCost of capital = 10%PVAF: 10% → 3.7908, 15% → 3.3522, 16% → 3.2743
Formulas used
- NPV = (Annual cash inflow × PVAF) − Initial investment
- PVAF at IRR = Initial investment ÷ Annual cash inflow
- IRR = LR + [(PVAF at LR − PVAF at IRR) ÷ (PVAF at LR − PVAF at HR)] × (HR − LR)
Step-by-step working
01Discount the inflows at the cost of capital
2PV of inflows = ₹3,00,000 × 3.7908 = ₹11,37,240Because the inflows are equal, use the annuity factor once. Discounting each year separately wastes several minutes and invites arithmetic slips.
02Compute the NPV
1.5PV of inflows ₹11,37,240 Less: Initial investment ₹10,00,000 ────────── Net present value ₹ 1,37,240NPV is positive, so at a 10% cost of capital the project adds value.
03Find the PVAF that corresponds to the IRR
1.5At IRR, NPV = 0, so: PVAF at IRR = ₹10,00,000 ÷ ₹3,00,000 = 3.3333This shortcut works only for even annual cash flows. With uneven flows you must discount year by year and interpolate on NPV instead.
04Interpolate between 15% and 16%
2PVAF at 15% = 3.3522 (above 3.3333) PVAF at 16% = 3.2743 (below 3.3333) IRR = 15% + [(3.3522 − 3.3333) ÷ (3.3522 − 3.2743)] × 1% = 15% + (0.0189 ÷ 0.0779) × 1% = 15% + 0.24% = 15.24% (approximately)Choose the two rates that bracket the target factor. Interpolating outside the bracket produces a number that looks reasonable and is wrong.
05Advise
1NPV = ₹1,37,240 (positive) IRR = 15.24% > cost of capital 10% Both criteria point the same way — accept the project.
The question asks for advice, so a one-line recommendation is a mark. Never leave a capital budgeting question without one.
NPV · IRR · Decision
₹1,37,240 · 15.24% · Accept
Where marks are lost
- Deducting the initial investment before multiplying by the annuity factor.
- Using the present value factor for a single sum instead of the cumulative annuity factor.
- Interpolating between two rates that both lie on the same side of the target PVAF.
- Answering the computation but not the question — the advice line is separately marked.
- Applying the PVAF shortcut to uneven cash flows, where it does not hold.
In the exam
When NPV and IRR disagree — which happens with mutually exclusive projects of different scale or timing — NPV prevails, because it measures absolute value added. Be ready to state that reasoning in one sentence; it is a frequent theory sub-part.
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Talk to usWhy is NPV preferred to IRR when the two conflict?
NPV measures the absolute rupee value added and assumes reinvestment at the cost of capital, which is realistic. IRR is a percentage that implicitly assumes reinvestment at the IRR itself, which usually is not achievable, and it can produce multiple answers when cash flows change sign more than once.
Can I compute IRR without the interpolation formula?
In an exam, no — you interpolate between two given rates. Quote the formula, show the substitution and state that the result is approximate. Examiners accept a small variation in the final figure if the method is right.
Reading a solution is not practice. Writing one is.
Generate fresh numericals on this exact chapter, attempt them under time, and have your written working evaluated step by step.
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