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Plate 3 of 19 · Finance

Choosing between projects

About 7 minutes to read, then three questions that count toward your mastery.

NPV, IRR, payback and the profitability index often agree. When they don’t, NPV is the one to trust, and knowing why is most of capital budgeting.

The idea

The net present value of a project is every cash flow discounted at the cost of capital, outlay included. A positive NPV is the value the project adds in today’s dollars, so fund it; between alternatives, fund the one with the highest NPV.

The internal rate of return is the discount rate at which NPV is exactly zero. For one ordinary project it gives the same yes or no as NPV. Between alternatives it can mislead: a percentage ignores size and rewards early cash. And when cash-flow signs change more than once, as with a mine that needs cleaning up, there can be several IRRs.

The payback period counts years until the outlay comes back and ignores everything after. The profitability index, the present value per dollar invested, helps when the budget is short, but with projects that can’t be split, ranking by it can miss the best combination.

The formula

NPV = CF0 + CF1(1 + r) + … + CFN(1 + r)N, PI = NPV + outlayoutlay

CF0
the outlay today, a negative number
CFt
cash flow at the end of year t
r
the cost of capital for projects of this risk
IRR
the discount rate r at which NPV is exactly zero
PI
profitability index: present value of the inflows per dollar of outlay
crossover rate
the rate at which two projects have the same NPV: the IRR of the difference between their cash flows

A worked example

Plate 3

You can fund one of two projects, each costing $1,000k, at a 10% cost of capital. A pays $100k, $350k and $1,000k over three years; B pays $900k, $250k and $100k.

  1. NPV of A: −1,000 + 100 ÷ 1.1 + 350 ÷ 1.1² + 1,000 ÷ 1.1³ = −1,000 + 90.9 + 289.3 + 751.3 = $131.5k.
  2. NPV of B: −1,000 + 818.2 + 206.6 + 75.1 = $99.9k. Both add value, but A adds $31.6k more.
  3. IRR ranks them the other way: 18.3% for B against 15.4% for A. B also pays back sooner, in 1.4 years against 2.55.
  4. The difference between them (A minus B) has an IRR of 12.5%: the crossover rate. Below it A is worth more, above it B. At 10% you are below it.

Fund A. B’s higher IRR and quicker payback reward early cash, but at your 10% cost of capital A’s larger later payments are worth more in today’s money.

Try it

Projects A and B from the example. Move the cost of capital past the crossover rate, or change one cash flow, and watch the choice.

10.0%
$1,000k
$900k

Early cash or late cash?

Fund Aat a 10.0% cost of capital
NPV of A
$131k
NPV of B
$100k
IRR of A, B
15.4%, 18.3%
Crossover rate
12.5%

IRR would pick B and give up $32k of value.

The common mistake

Ranking alternatives by IRR

Choosing B because 18.3% beats 15.4% gives up $31.6k of value. IRR answers “what rate would make this break even?”, not “which project adds the most?”. Use it to accept or reject a single ordinary project; use NPV to choose.

Remember

  • Fund positive-NPV projects; between alternatives, the highest NPV.
  • IRR and payback favour early cash and small projects. Where two NPV lines cross, the ranking flips.
  • More than one sign change can mean more than one IRR: use NPV, or MIRR. With a fixed budget and whole projects, check combinations rather than ranking by PI.

Check yourself

Three questions from the Learn room’s bank. Your answers update your mastery of this skill, the same as in Practise.

Use it on real numbers

Open Compare projects, in Finance › Investment decisions and try the idea on your own numbers.