Lesson 6 of 9
Internal rate of return
An IRR is the rate that sets NPV to zero. One set of cash flows here has two of them, both correct, and the rule breaks.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- An IRR is the discount rate at which a stream's NPV is exactly zero. It is solved for, never calculated directly.
- A stream whose sign changes more than once can have more than one IRR, and every one of them is arithmetically correct.
- When IRR and NPV disagree on which of two projects is better, NPV is the one to trust.
NPV asks what a stream is worth at a rate you supply. IRR asks the reverse: which rate would make this stream worth exactly its cost. It is a useful number with two failure modes that are easy to demonstrate and easy to miss.
NPV of one investment across discount rates
- NPV
Show the numbers
| NPV | |
|---|---|
| 8% | $197.81 |
| 10% | $137.24 |
| 12% | $81.43 |
| 15% | $5.65 |
| 18% | −$61.85 |
| 20% | −$102.82 |
Solving for it
There is no formula. You try rates until the NPV is zero, which is what a spreadsheet does behind the function name. For the stream above, 15 percent leaves 5.65 on the table and 16 percent leaves a deficit, so the answer is between them. It is 15.24 percent.
The number means: this stream repays its cost and earns 15.24 percent a year on the balance still outstanding. That is a real and useful statement.
Failure one: more than one answer
An IRR is a root of a polynomial. A stream whose sign changes once has one root. A stream whose sign changes twice can have two, and there is no basis for preferring either.
Failure two: scale and timing
Two projects can rank one way on IRR and the other way on NPV. A small project returning 50 percent on 100 dollars creates less value than a large one returning 15 percent on 10,000. IRR is a rate and takes no account of how much capital the rate is earned on.
The same happens with timing. A short project with a high rate can lose to a long one with a lower rate, because the long one keeps earning. IRR quietly assumes every cash inflow is reinvested at the IRR itself, which for a 50 percent IRR is an assumption nobody would make out loud.
The working rule
Use IRR to describe a stream and NPV to decide between streams. When they disagree, the disagreement is usually about scale, timing or reinvestment, and NPV is the measure that already accounts for all three. Before quoting any IRR, check whether the cash flows change sign more than once. If they do, quote the NPV profile instead.
Check yourself
What is an internal rate of return?
- The average annual profit divided by the outlay
- The discount rate at which the stream's NPV equals zero
- The return the investor is guaranteed to earn
Show the answer
Answer: B.
IRR is solved for rather than computed. It is the rate that makes the present value of the inflows exactly equal the outlay.A project's cash flows are negative, then positive, then negative again. What should you expect?
- An IRR that is always higher than the true return
- That more than one rate can set the NPV to zero
- That the NPV cannot be computed
Show the answer
Answer: B.
Each sign change allows another root. With two sign changes there can be two IRRs, both arithmetically valid and neither meaningful as a return.Project A has a 50 percent IRR on 100 dollars. Project B has a 15 percent IRR on 10,000. Which creates more value?
- A, because the rate is higher
- B, because a rate says nothing about the capital it is earned on
- They are equivalent
Show the answer
Answer: B.
IRR is scale-free. Fifty percent of a small outlay is a smaller amount of value than fifteen percent of a large one, and NPV is the measure that shows it.Sources
- Damodaran, Investment Valuation: the multiple-IRR problem and the reinvestment assumption inside an internal rate of return.
- Williams, The Theory of Investment Value, 1938: the present-value framework that IRR inverts.
What this leaves out
- Both examples are invented arithmetic, built so that every row can be recomputed by hand.
- A modified IRR exists and fixes the reinvestment assumption by specifying a separate reinvestment rate. It does not fix the scale problem.
- Nothing here is advice about any security.