Calculate NPV, IRR, payback and profitability index, and explain why NPV is the primary rule.
NPV = the present value of all future cash flows minus the initial investment, discounted at the project's cost of capital. Accept projects with positive NPV. NPV measures the dollars of value a project adds today.
A project costs $1,000 and returns $450 a year for three years. The cost of capital is 10%.
| Year | Cash flow | Discount factor at 10% | Present value |
|---|---|---|---|
| 0 | (1,000) | 1.0000 | (1,000.00) |
| 1 | 450 | 0.9091 | 409.09 |
| 2 | 450 | 0.8264 | 371.90 |
| 3 | 450 | 0.7513 | 338.09 |
| NPV | 119.08 |
IRR is the discount rate that makes NPV equal zero. Accept when IRR exceeds the cost of capital. Here IRR is about 16.6%, above the 10% hurdle, so both rules agree.
The time needed to recover the initial outlay: 1,000 / 450 = 2.22 years. Payback is simple but ignores the time value of money and every cash flow after the cut-off. Discounted payback fixes the first flaw only.
PI = PV of future cash flows / initial investment = 1,119.08 / 1,000 = 1.12. Useful for ranking projects when capital is rationed.
When NPV and IRR rank mutually exclusive projects differently, choose the higher NPV. Modified IRR (MIRR) assumes reinvestment at the cost of capital and removes the multiple-solution problem.
A percentage is easy to communicate, and payback is a rough check on liquidity and risk. Surveys of chief financial officers find most large firms compute both NPV and IRR.
Two mutually exclusive projects each cost 500. A pays 650 in one year; B pays 900 in four years. The cost of capital is 10%.
| Project A | Project B | |
|---|---|---|
| NPV at 10% | 650 / 1.10 minus 500 = 90.91 | 900 / 1.4641 minus 500 = 114.71 |
| IRR | 30.0% | 15.8% |
IRR prefers A; NPV prefers B. At 10% B creates more value, because its larger payoff, even though later, is worth more than A's quick 30% return on a small base. The two NPVs are equal at the crossover rate where 650 / (1 + r) = 900 / (1 + r)^4, which is about 11.5%. Above that rate A is better; below it, B.
MIRR compounds the interim cash flows forward at the cost of capital and finds the single rate that links the outlay to that terminal value. For the module's project (cost 1,000, three payments of 450) at 10%: terminal value = 450 x 1.21 + 450 x 1.10 + 450 = 1,489.50, so MIRR = (1,489.50 / 1,000)^(1/3) minus 1 = 14.2%. It is lower than the 16.6% IRR because it does not assume reinvestment at 16.6%.
Comparing a 3-year machine with a 6-year machine by NPV favours the longer one unfairly if both will be replaced. The equivalent annual annuity (EAA) converts each NPV to a level annual amount:
Machine A has an NPV of 120 over 3 years; at 10% the annuity factor is 2.4869, so EAA = 48.25. Machine B has an NPV of 180 over 6 years; the factor is 4.3553, so EAA = 41.33. If each can be repeated, A creates more value per year.
When a firm can spend only a fixed budget this year, rank projects by profitability index and fund down the list until the money runs out, checking that the combination chosen has the highest total NPV. Permanent rationing is itself a warning: a firm that has many positive-NPV projects should usually be able to raise capital for them.
Many firms set hurdle rates above their WACC, partly to allow for optimistic forecasts. The better fix is to correct the forecasts, but a modest cushion is common. Using a single high hurdle for everything rejects safe projects that would add value.
Comparing actual results with the original forecast, two or three years later, improves future forecasts and discourages managers from inflating them to win approval.
Build a spreadsheet with two mutually exclusive projects: A costs 1,000 and pays 1,300 in one year; B costs 1,000 and pays 1,750 in four years. Compute NPV at 5%, 10% and 15% and the IRR of each. Find the discount rate where the ranking flips.
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