Capital budgeting: methods, comparison and risk analysis
Capital budgeting judges long-term projects. NPV is the gold standard: accept if it is positive. IRR, PI and payback help, but can mislead on mutually exclusive projects.
The capital budgeting process
Use cash flows, not accounting profits: incremental and after tax. Ignore sunk costs; include opportunity costs and changes in working capital.
Methods compared
| Method | Formula and rule | Time value? | Main weakness |
|---|---|---|---|
| Payback | Outlay ÷ annual inflow (even flows); accept if within the target | No | Ignores time value and cash flows after payback |
| Discounted payback | Years for cumulative PV of inflows to cover the outlay | Yes | Still ignores cash flows after payback |
| ARR (accounting rate of return) | Average profit after tax ÷ average investment | No | Uses accounting profit, not cash flow |
| NPV | PV of inflows − outlay; accept if > 0 | Yes | Needs a known discount rate; gives an amount, not a rate |
| IRR | Rate at which NPV = 0; accept if > cost of capital | Yes | Multiple IRRs possible; assumes reinvestment at the IRR |
| PI | PV of inflows ÷ outlay; accept if > 1 | Yes | Can mis-rank projects of different sizes |
Worked example 1: NPV, PI, payback and discounted payback
A project costs ₹1,00,000. Cash inflows are ₹40,000, ₹40,000, ₹30,000 and ₹30,000 in years 1 to 4. The cost of capital is 10%. Find NPV, PI, payback and discounted payback. (PVs are rounded to the nearest rupee.)
- PVs: 40,000 ÷ 1.1 = 36,364; 40,000 ÷ 1.21 = 33,058; 30,000 ÷ 1.331 = 22,539; 30,000 ÷ 1.4641 = 20,490.
- Total PV = 36,364 + 33,058 + 22,539 + 20,490 = ₹1,12,451.
- NPV = 1,12,451 − 1,00,000 = ₹12,451. PI = 1,12,451 ÷ 1,00,000 = 1.12.
- Payback: 40,000 + 40,000 = 80,000 after 2 years; the remaining 20,000 comes from year 3's 30,000, so payback = 2 + 20,000 ÷ 30,000 = 2.67 years.
- Discounted payback: adding the rounded PVs, cumulative PV is 69,422 after 2 years and 91,961 after 3 years. Remaining 8,039 ÷ 20,490 = 0.39, so 3.39 years.
NPV ₹12,451 (accept), PI 1.12, payback 2.67 years, discounted payback 3.39 years. Discounted payback is always longer than simple payback.
Worked example 2: IRR by interpolation, and ARR
A machine costs ₹1,00,000 and yields ₹30,000 a year (after tax, before depreciation) for 5 years, with no salvage value. The cost of capital is 10%. Find payback, NPV, IRR and ARR (straight-line depreciation).
- Payback = 1,00,000 ÷ 30,000 = 3.33 years.
- NPV = 30,000 × PVIFA(10%, 5) − 1,00,000 = 30,000 × 3.7908 − 1,00,000 = 1,13,724 − 1,00,000 = ₹13,724.
- IRR: find the 5-year PVIFA values around the payback figure 3.3333: 3.3522 at 15% and 3.2743 at 16%.
- IRR = 15% + (3.3522 − 3.3333) ÷ (3.3522 − 3.2743) × 1% = 15% + (0.0189 ÷ 0.0779) × 1% = 15.24%.
- ARR: depreciation = 1,00,000 ÷ 5 = 20,000; average profit = 30,000 − 20,000 = 10,000; average investment = (1,00,000 + 0) ÷ 2 = 50,000; ARR = 10,000 ÷ 50,000 = 20%.
Payback 3.33 years, NPV ₹13,724, IRR ≈ 15.24% (above 10%, so accept), ARR 20%.
NPV vs IRR: when they conflict
For independent projects with conventional cash flows, NPV, IRR and PI always give the same accept or reject decision.
Handling risk and uncertainty
| Technique | How it works |
|---|---|
| Sensitivity analysis | Change one variable at a time (sales, price, cost) and watch NPV; shows the critical variables |
| Scenario analysis | Change several variables together under pessimistic, most likely and optimistic cases |
| Certainty equivalent | Multiply each risky cash flow by a coefficient α (0 to 1, lower for riskier flows), then discount at the risk-free rate |
| Risk-adjusted discount rate | Discount at the risk-free rate + a risk premium; penalises distant cash flows more |
| Decision tree | Maps sequential decisions and chance events with probabilities; roll back to find the expected NPV |
| Simulation | Monte Carlo: draw values for many variables at random to get a distribution of NPV (David Hertz, 1964) |
| Statistical measures | Expected NPV, standard deviation and coefficient of variation of NPV |
Worked example 3: certainty equivalent
A project costs ₹80,000 and is expected to give ₹50,000 in each of years 1 and 2. The certainty-equivalent coefficients are 0.9 and 0.8, and the risk-free rate is 6%. Accept or reject?
- Certain cash flows: 50,000 × 0.9 = 45,000; 50,000 × 0.8 = 40,000.
- PV at 6%: 45,000 ÷ 1.06 = 42,453; 40,000 ÷ 1.1236 = 35,600.
- NPV = 42,453 + 35,600 − 80,000 = −₹1,947.
Reject. Ignoring risk and discounting ₹50,000 a year at 6% would wrongly show a positive NPV of ₹11,670.
NET asks quick numericals (payback, NPV, PI, ARR, IRR by interpolation), which method ignores time value, the NPV-IRR conflict and reinvestment assumption, and matching risk techniques with their logic.
Typical MCQ: “Which two methods ignore the time value of money?” Simple payback and ARR. Trap: discounted payback does use time value, but it still ignores cash flows after payback.
Typical MCQ: “For two mutually exclusive projects, NPV favours X and IRR favours Y. Which do you choose?” X, the higher NPV: NPV measures the addition to shareholders' wealth and assumes a realistic reinvestment rate.
Typical MCQ: “Under the certainty-equivalent method, which rate discounts the adjusted cash flows?” The risk-free rate, because risk is already removed from the cash flows. Using a risk-adjusted rate as well would count risk twice.
NPV > 0 ⇔ PI > 1 ⇔ IRR > k for a normal project. Payback and ARR are the two 'no time value' methods. In a clash, NPV wins.