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Work out a project's internal rate of return and net present value, and explain each

CFA® Level I · Corporate Finance · Capital Investments and Capital Allocation · CI.7.b

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, read as a compound return, implies 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

From the question bank, with the reason each wrong option is wrong.

What does a positive net present value indicate?

  1. The project recovers its cost and adds value at the required rate of return - correct
  2. The project has a short payback periodWhy not: Payback measures speed of recovery and ignores everything after it, including the cost of capital.
  3. 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.

Net present value (NPV) shows how much value a project adds in today's money, after paying for the capital it uses. Formula: NPV = Sum of Cash flow / (1 + r)^t - Initial cost, where r is the required rate of return and t is the year. A positive NPV means the discounted cash coming in is more than the cost, so the project earns more than the return investors demand. Payback period only measures how fast the cost is recovered and ignores the cost of capital.

A harder one

Delmar Components can install only one of two mutually exclusive machines. Machine A costs $100,000 and generates after-tax cash flows of $70,000 at the end of each of the next two years (IRR 25.7%). Machine B costs $250,000 and generates $160,000 at the end of each of the next two years (IRR 18.2%). Delmar's required rate of return is 10%. Which machine should Delmar choose, and on what basis?

  1. Machine B, ranked by NPV - correct
  2. Machine A, ranked by IRRWhy not: The higher percentage return has been preferred. IRR ignores scale, so a small project with a high rate can add less value than a larger one that earns well above the required return.
  3. Machine A, ranked by paybackWhy not: The faster recovery of the outlay has been preferred (1.4 years against 1.6). Payback ignores the cash flows after recovery and the time value of money, so it does not measure value added.

Only one machine can be bought, so pick the one that adds more value, which is the NPV. Formula: NPV = CF1 / (1 + r) + CF2 / (1 + r)^2 - outlay. Machine A: $70,000 / 1.10 + $70,000 / 1.21 - $100,000 = $63,636 + $57,851 - $100,000 = $21,488. Machine B: $160,000 / 1.10 + $160,000 / 1.21 - $250,000 = $145,455 + $132,231 - $250,000 = $27,686. B adds $6,198 more. IRR is only a percentage and ignores size, so a smaller project can win on IRR yet add less value. Payback ignores later cash and the time value of money.

The hardest tier

What unrealistic assumption does the internal rate of return embed?

  1. That the cash flows are received annuallyWhy not: A convention of the arithmetic, and easily adjusted for by changing the period.
  2. That the discount rate is constantWhy not: Net present value makes the same simplification, so it cannot be what distinguishes the two.
  3. That interim cash flows are reinvested at the IRR - correct

IRR quietly assumes that every cash flow the project pays out along the way can be reinvested at the IRR itself. NPV instead assumes reinvestment at the required return, which is usually more realistic. For example, a project with a 40% IRR assumes each interim cash flow keeps earning 40% until the end, which few firms could achieve. Idea: the higher the IRR, the less believable the reinvestment assumption.

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