Work out a project's internal rate of return and net present value, and explain each
The idea
Net present value discounts a project's incremental after-tax cash flows at the required rate and subtracts the outlay; a positive figure means the project adds that much value in today's terms. The internal rate of return is the discount rate at which NPV is zero. For a single conventional project the two rules agree, but for mutually exclusive projects of different size or timing they can conflict, and NPV is the rule to follow because it measures value added rather than a percentage. IRR also assumes reinvestment at the IRR itself, which is rarely realistic.
Formulas
Net present value of a project
NPV = Σ [CFₜ ÷ (1 + r)ᵗ] − outlay
- CFₜ
- the after-tax incremental cash flow in year t
- r
- the required return, usually the WACC for a project of the firm's usual risk
- t
- the year the cash flow arrives in
- outlay
- the money spent at the start
Incremental cash flows only: sunk costs are ignored and opportunity costs are counted. A positive NPV is value added in today's money, which is why it is the rule to follow when it disagrees with IRR.
Internal rate of return
Σ [CFₜ ÷ (1 + IRR)ᵗ] = outlay
- CFₜ
- the cash flow in year t
- IRR
- the rate that makes the present value of the inflows equal the outlay
- t
- the year the cash flow arrives in
- outlay
- the initial investment
The discount rate at which NPV is zero. It assumes cash is reinvested at the IRR itself, and it can give two answers or none when the cash flows change sign more than once.
A practice question
What does a positive net present value indicate?
- The project recovers its cost and adds value at the required rate of return — correct
- The project has a short payback periodWhy not: Payback measures speed of recovery and ignores everything after it, including the cost of capital.
- The project's internal rate of return is below the cost of capitalWhy not: That describes a negative net present value; the two measures agree in sign for a conventional project.
NPV is value added in today's money after charging for the capital used. Positive means the project earns more than the required return.
A harder one
A project costs 500 and returns 300 at the end of each of years 1 and 2. At a 10% required rate, what is its approximate NPV?
- Approximately 100Why not: Adds the two receipts undiscounted and subtracts the cost, ignoring the time value entirely.
- Approximately 61Why not: Discounts too lightly. At 10% the receipts are worth 272.73 and 247.93, giving about 21; a figure of 61 corresponds to a rate nearer 4%.
- Approximately 21 — correct
300/1.1 = 272.7 and 300/1.21 = 247.9, totalling 520.6 against an outlay of 500, so NPV is about 21. The 100 figure ignores discounting altogether and the 61 discounts only the second year.
The hardest tier
Two mutually exclusive projects rank differently under NPV and IRR. Which should be chosen and why?
- The higher IRR, because it measures efficiency of capital useWhy not: A percentage cannot distinguish a large gain from a small one, and it is the absolute value added that accrues to shareholders.
- The higher NPV, because it measures the absolute value added to the company — correct
- Neither, because conflicting rules indicate a flawed analysisWhy not: The conflict is a known consequence of differing scale and timing rather than a sign of error.
A percentage return says nothing about scale: a 40% return on a small outlay can add less value than a 15% return on a large one. NPV is denominated in money and is the rule to follow when they disagree.
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