Internal rate of return

The internal rate of return, or IRR, is the discount rate at which a project's NPV is exactly zero. It answers a different question from NPV: not how many dollars the project adds at a given rate, but what rate it earns on the money tied up.

Finding it

There is no formula for IRR in general. It is found by trial: try a rate, compute NPV, move the rate until NPV reaches zero. Spreadsheets do the trials with the IRR function. In Northside's espresso machine case, the NPV is $6,081.39 at 10%, $665.90 at 20% and −$1,349.25 at 25%, so the IRR lies between 20% and 25%. Trial gives 21.6%, and at that rate NPV is zero.

The decision rule is to accept a project when its IRR exceeds the required rate. For a single project with the usual shape, an outlay now followed by inflows, IRR and NPV give the same answer.

When IRR and NPV disagree

They can disagree when a business must choose between two projects, because IRR ignores scale. A small project can earn a high rate on a small sum. A larger one can earn a lower rate on more, and add more dollars.

Hartley can fund one of two things. A hydro-jetting unit for drain work costs $30,000 and returns 16,000 a year for 3 years. A second service van, fully fitted, costs $60,000 and returns 25,000 a year for 4 years.

Hartley: two ways to spend on growth, discounted at 10%

	A: hydro-jetting unit	B: second service van
Year 0	(30,000)	(60,000)
Year 1	16,000	25,000
Year 2	16,000	25,000
Year 3	16,000	25,000
Year 4	0	25,000
NPV at 10%	9,789.63	19,246.64
IRR	27.8%	24.1%

IRR ranks A first at 27.8% against 24.1%. NPV at 10% ranks B first, at $19,246.64 against $9,789.63. When the two disagree on a choice between projects, NPV is the guide, because it measures the value added at the rate the business can reinvest at.

Looking at the extra spend shows why. Choosing B over A means spending $30,000 more and getting the cash flow difference:

The extra spend on B over A

	B minus A $
Year 0	(30,000)
Year 1	9,000
Year 2	9,000
Year 3	9,000
Year 4	25,000
IRR of the difference	21.5%
NPV at 10%	9,457.00

That extra $30,000 earns 21.5%, above the 10% required, so it is worth spending. The rate at which the two projects have equal NPV is 21.5%, the IRR of the difference. Below that rate B has the higher NPV. Above it A does.

NPV of each project at different rates

Discount rate	A $	B $
0%	18,000.00	40,000.00
10%	9,789.63	19,246.64
15%	6,531.60	11,374.46
20%	3,703.70	4,718.36
25%	1,232.00	(960.00)
30%	(942.19)	(5,843.98)

Unusual cash flows

IRR assumes cash flows change sign once: outflow first, then inflows. If a project has a large outflow at the end, there can be more than one IRR, or none. Hartley considers taking over a leased unit. It pays $10,000 now, receives $50,000 in year 1, and must pay $60,000 in year 2 to restore the premises.

Lease-restoration project: NPV at different rates

Discount rate	NPV $
10%	(14,132.23)
50%	(3,333.33)
100%	0.00
150%	400.00
200%	0.00
300%	(1,250.00)

NPV is zero at both 100% and 200%, so the project has two IRRs. It is positive between them and negative either side. At 10% the NPV is $-14,132.23, so the project should be rejected even though two rates are called "IRR". A project with an end-of-life cleanup cost, such as restoring leased premises, needs NPV.

Reinvestment

IRR implicitly assumes each inflow is reinvested at the IRR itself, which is rarely true. NPV assumes reinvestment at the discount rate, a more realistic figure. The modified IRR corrects for this by compounding inflows at a stated reinvestment rate. For the espresso machine, compounding the inflows at 10% gives a modified IRR of 16.6%, well under the 21.6% shown by plain IRR.