1.26 The Internal Rate of Return (IRR)
Definition of IRR: The IRR is the theoretical discount rate which causes the Initial Outflow to be equal to the PV of Future Net Inflows; in other words, the IRR is the point at which the NPV = 0. This does not mean that the NPV is actually zero. The “true” NPV for a project is based on the cost of capital as the discount rate. (For further definition of IRR, see “What does ‘IRR’ Mean” below.)
The decision rule for IRR is to choose only those (independent) projects whose IRR exceeds the firm’s cost of capital, which was used as the discount rate in calculating the NPV. Choose the mutually exclusive project whose IRR is greatest, provided the IRR exceeds the firm’s cost of capital, i.e., the project’s “hurdle rate.” The hurdle rate is the minimum expected return management will accept in order to invest in a given project, based on the IRR method.
If the project IRR is less than the cost of capital it means that the project will lose money.
If the IRR is greater than the cost of capital (which is the discount rate for the NPV), the NPV will be positive, i.e., greater than zero. Mathematically, if the NPV is positive, the IRR must exceed the cost of capital or discount rate – because the rate must be increased in order to arrive at an NPV of zero.
Illustration of IRR
Note:
Question: Given a series of cash flows, how would you know whether to initially choose a large or a small discount rate in order to discover the IRR? The possibilities are virtually infinite! Here is a problem to work on:
| Initial Outlay: | $1,000 | Cash Flow 1: | $452 |
| Cash Flow 2: | $500 | Cash Flow 3: | $278 |
Answer: Let’s first assume an IRR discount rate of zero. The simple sum of the inflows is $1,230. This means that the simple, non-discounted, (internal-) rate of return is (1,230 ÷ 1,000) – 1 = 23%. Indeed, this would hold true if the discount rate were 0%! Here is the formal calculation: [452 / (1 + 0)1] + [500 / 1.02] + [278 / 1.03] ÷ 1,000 – 1 = 0.23.
(Note that this calculation is similar to the Holding Period Return.)
The first guess for the IRR must, therefore, be greater than zero in order to decrease the NPV (and hence arrive at the IRR), and also less than 23% – because we shall be discounting each of the numbers, resulting in a smaller outcome. (The NPV is $230.) Let’s try 15% for our first iteration; it’s right in the middle of our new range i.e., 0% to 23%). Notice how substantially we have narrowed the range of our guesses! (With a discount rate of zero, the Rate of Return = 23% and the PI = 1.23.)
1st Iteration: k = 15%
| Year | Cash Flow | PVF | PVCF |
| 0 | ($1,000) | ||
| 1 | 452 | ||
| 2 | 500 | ||
| 3 | 278 | ||
| NPV= |
Since this clearly does not provide us with the answer we are looking for, we need to try again. (We are looking for the single discount rate, which will yield: NPV = 0. Should we raise or lower the discount rate from 0.23? Let’s see.
Since the first guess was no good, here is the next try – o