Project Evaluation: The Multiple IRR Problem
The __internal rate of return (IRR)__ is a great tool for calculating the rate of return provided by a project, and then comparing this to a hurdle rate.
Imagine a project that costs $25,000 to start, and is expected to provide cash flows of $15,000 in years 1 and 2. Since $25,000 is going out and then $30,000 is coming in, it's reasonable to expect a decent rate of return. These discounted cash flows, listed in order from year zero to year two, are
No, this is too low. The left side of the equation wouldn't be too close to zero. It would be:
$$\displaystyle -\$ 25{,}000 + \frac{\$ 15{,}000}{(1 + 0.08)} + \frac{\$ 15{,}000}{(1 + 0.08)^2} = \$ 1{,}749 $$.
No, it's possible to obtain this value by recognizing that there is $5,000 more coming in than going out and using the ratio of this cash difference divided by the initial investment; however, this ignores the time value of money.
Try discounting these two future cash flows at 8%, 10%, and 12%. What seems to work best for an IRR?
No, discounting the left-hand side cash flows at 8% should give you -$20.58.
But now try the same thing for 18%, 20%, and 22%. Same project, and same cash flows. Which one of these works as well?
No, discounting the right-hand side cash flows at 12% should give you $12.76.
No, discounting the left-hand side cash flows at 18% should give you $11.49.
To summarize this discussion:
[[summary]]
No, discounting the left-hand side cash flows at 22% should give you -$16.12.
$$ \displaystyle -\$ 25{,}000 + \frac{\$ 15{,}000}{(1 + IRR)} + \frac{\$ 15{,}000}{(1 + IRR)^2} = 0 $$.
What IRR do you estimate for this?
Exactly! A rate of approximately 13% evens out the equation, and is the correct IRR for the project.
[[calc: cf 25000 sign enter down 15000 enter down 2 enter irr cpt , 25000 chs cf0 15000 cfj 2 nj irr]]
So there's really no problem using this method when there's a combination of a cash outflow and a series of cash inflows. But a problem can arise when the sign in the cash flow series changes more than once. The problem is that a second sign change can provide a second possible IRR! It's not guaranteed, but it's possible.
In fact, suppose that you spend $10,000 for a new big project today that will provide a positive cash flow of $23,000 next year. The only problem is that, in year 2, you'll have a residual payment of $13,200 due to a supplier.
To calculate the IRR, set up a similar equation as before:
$$\displaystyle - \$ 10{,}000 + \frac{\$ 23{,}000}{(1 + IRR)} - \frac{\$ 13{,}200}{(1 + IRR)^2} = 0 $$.
Yes! A rate of 10% makes the left-hand side equal to zero, so this is an IRR for the project.
That's right! Using 20% makes the left-hand side equal to zero as well. So there's another IRR.
This is kind of strange, and it can be difficult to use this result logically with a hurdle rate of 14%. The IRRs are both above and below this. Unfortunately, there's little that can be done to solve this problem with IRR alone. Other tools, such as net present value, will be more successful in evaluating projects like this.
It's not too common to run into such an issue, and in fact the changing sign in a cash flow series doesn't mandate the existence of multiple IRRs; it just presents the possibility. So it's good to be aware.
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