NPV and IRR in capital allocation

How does a company decide which projects to fund?

Capital allocation is the process of deciding where a company puts its long-term money. It moves through four steps, and the NPV and IRR calculations belong to the second.

Capital allocationfrom idea to review

• 1. Idea generation Memorize rooted in the competitive landscape
  • Firm capabilities what it can do well
  • Prior-year plans weigh heavily
• 2. Investment analysis Memorize forecast cash flows, compute NPV and IRR
  • Amount, timing, duration, risk of cash flows
• 3. Planning and prioritization Memorize pick the most value on a risk-adjusted basis
  • Interactions and financing limits and ESG
  • Return unused capital to shareholders
• 4. Monitoring and post-investment review Memorize actual versus forecast
  • Catch systematic optimism and enforce discipline
  • Scale up or exit and seed new ideas

1. Idea generation. Ideas come from managers close to the business and need a good grasp of the competitive landscape and the firm’s own capabilities. Capital spending is highly correlated from one year to the next, so last year’s plans weigh heavily.
2. Investment analysis. Managers forecast the amount, timing, duration and risk of a project’s cash flows and test whether it is a wise use of capital.
3. Planning and prioritization. Management selects the set of projects that adds the most value on a risk-adjusted basis. A project that looks good alone can look worse next to existing operations, other proposals or a financing limit; environmental, social and governance (ESG) factors can matter here. When value-creating ideas run out, spare capital goes back to shareholders to redeploy.
4. Monitoring and post-investment review. Actual results are compared with the forecasts. This validates assumptions, reveals systematic optimism, keeps managers disciplined and suggests new ideas. Management can scale up winners and shrink or sell weak areas.

An outside analyst never sees the project-by-project forecasts. The analyst still needs to understand the logic of NPV and IRR, and uses an aggregate measure, the next lesson’s return on invested capital (ROIC), on published statements.

How do you calculate net present value?

The net present value (NPV) of a project is the present value of its expected after-tax cash inflows less the present value of its outflows. Cash flow at time 0 (usually the outlay) is not discounted.

$$NPV = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t}$$

Here $CF_t$ is the after-tax cash flow at time $t$ and $r$ is the required rate of return : what investors could earn on other investments of similar risk. It is the project’s opportunity cost. The decision rule is simple. Invest if NPV ≥ 0: the project earns at least what investors require, so it adds value (NPV is the value added, in currency). A positive NPV is necessary but not sufficient, since competing projects and financing limits still matter.

What is the internal rate of return?

The internal rate of return (IRR) is the discount rate that makes the NPV equal to zero. It is the project’s own rate of return on the money invested.

$$0 = \sum_{t=0}^{T} \frac{CF_t}{(1+IRR)^t}$$

You cannot rearrange this to isolate the IRR, so you find it by trial or, in the exam, with the calculator. The decision rule compares the IRR with the hurdle rate , which is the required rate of return used in the NPV calculation. Invest if IRR ≥ the hurdle rate.

For an ordinary project (an outlay first, then inflows), the two rules always agree. NPV is positive at the required return exactly when the IRR is above it, which is why the IRR is where the NPV curve crosses zero.

Go deeper The reinvestment assumption

Each measure carries a reinvestment assumption about the interim cash flows. The IRR is the rate of return on the investment if interim cash flows are reinvested at the IRR itself. NPV assumes they are reinvested at the required rate of return, which is usually more realistic: a company that earns 40% on one small project cannot usually find more projects offering 40%. This difference is one reason the two can rank projects differently, and it connects to reinvestment risk in fixed income.

The IRR is also the same calculation as the money-weighted return in the quantitative methods topic, with a portfolio’s contributions and withdrawals playing the role of project cash flows.

Go deeper Uneven dates, and other rules you may meet

If cash flows are not evenly spaced, spreadsheets offer XNPV and XIRR, which discount by the actual dates. The calculator’s cash flow worksheet assumes evenly spaced periods, so a cash flow at an odd date (say 1.5 years) has to be discounted on its own: $CF \div (1+r)^{1.5}$.

Other texts also teach payback, discounted payback and the profitability index. The curriculum names only NPV, IRR and ROIC, so keep your effort there.

What if NPV and IRR disagree?

For ordinary projects NPV and IRR agree on accept or reject. Trouble arises in two cases.

Mutually exclusive projects

Mutually exclusive projects are alternatives where choosing one rules out the others. The IRR is a rate, so it hides project size and timing. NPV is a currency amount, so it tells you how much value each project adds.

Go deeper The crossover rate

The two NPV curves cross at one discount rate, called the crossover rate. Below it, L has the higher NPV; above it, S does. For S and L it is the IRR of the difference in their cash flows (L minus S: −90, −14, 0, +160), which is about 16.17%. At the firm’s 10% required return, which is below the crossover, L wins. If the required return were above 16.17%, S would add more value. This is why the answer depends on the required return, not on the rate each project quotes.

Cash flows that change sign more than once

Ordinary cash flows change sign once. If a project has an outlay, then inflows, then a large closing cost (decommissioning a mine, for example), the signs change twice and there can be multiple IRRs : more than one discount rate at which NPV is zero.

NPV and IRR disagree or misleadNPV is the safer guide

• Mutually exclusive projects Use it only one can be chosen
  • Different size or timing IRR and NPV rank differently
  • Choose the higher NPV it adds the most value
• Unconventional cash flows Use it signs change more than once
  • Multiple IRRs possible a calculator shows only one
  • Use NPV one clear answer

Exam tip Mutually exclusive, or signs flip twice? Go to NPV. Otherwise the two rules agree.