# 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 âtrueâ NPV for a project is based on the cost of capital as the discount rate. (For further definition of IRR, see âWhat does âIRRâ Meanâ below.)

The decision rule for IRR is to choose only those (independent) projects whose IRR exceeds the firmâ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 firmâs cost of capital, i.e., the projectâs âhurdle rate.â 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 â because the rate must be increased in order to arrive at an NPV of zero.

Illustration of IRR

Note:

Calculating the IRR is accomplished via an iterative, i.e., trial and error process. There is no mathematical solution for IRR (except in the sole instance involving only one future negative cash flow, in which case there is a solvable, quadratic solution).

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:Â Â Letâs first assume an IRR discount rate of zero. The simple sum of the inflows is \$1,230.Â  This means that the simple, non-discounted, (internal-) rate of return is (1,230 Ăˇ 1,000) â 1 = 23%. Indeed, this would hold true if the discount rate were 0%!Â  Here is the formal calculation: [452 / (1 + 0)1] + [500 / 1.02] + [278 / 1.03] Ăˇ 1,000 â 1 =Â 0.23.

(Note that this calculation isÂ similar toÂ theÂ 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.) Letâs try 15% for our first iteration; itâ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.)

1stÂ Iteration: k = 15%

 Year Cash FlowÂ 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? Letâs see.

Since the first guess was no good, here is the next try â or iteration.

We must lower the discount rate in order to raise the NPV (from the negative solution at which we arrived).

2ndÂ Iteration: k = 12%

 Year Cash FlowÂ PVF PVCF 0 (\$1,000) 1 452 2 500 3 278 NPV=

Some Difficulties with the IRR

Project Scale:Â We could have two competing projects, each with radically different costs, and potentially choose one based on the IRR solution alone â without giving any heed to the very different costs involved. From a business perspective, the choice may not be optimal.

Negative Future Cash Flows: In some projects, it is possible that we could get an interim, future cash flow that is negative. This may be due to having to engage in a major and expensive equipment overhaul at a specific point during the projectâs life. Cases wherein there may be ânegative interim FCFsâ may yield a âquadraticâ solution. As you may recall from high school mathematics, such cases make possible two equally valid solutions. This presents a problem for the decision maker, i.e., which solution, if either, should be used in making the decision? See the pages below for analysis.

Reinvestment Rate Assumption:Â As we will soon see, the IRR implies that the pro-forma cash flows are â themselves â being reinvested as received at the calculated IRR; this is a kind of mathematical redundancy, which, for numerous reasons, cannot be valid. See below for our in-depth analysis.

The following are the solutions to the two iterations attempted above.

1stÂ iteration: 15%.

 Year Cash FlowÂ PVF PVCF 0 (\$1,000) 1.0000 (\$1,000) 1 452 .8696 393.06 2 500 .7561 378.05 3 278 .6575 182.79 NPV= (\$46.11)

2ndÂ iteration: 12%.

 Year Cash FlowÂ PVF PVCF 0 (\$1,000) 1.0000 (\$1,000) 1 452 .8929 403.59 2 500 .7972 398.60 3 278 .7118 197.88 NPV= —

Notes:

• When IRR > k, we accept the projectÂ –Â byÂ mathematicalÂ rule.
• If IRR > k, the NPV calculated at kÂ mustÂ be positive; the project is projected to be profitable, i.e., to add to firm wealth.
• IfÂ the NPVÂ exceedsÂ zero,Â thenÂ the IRRÂ discount rate must beÂ increasedÂ in order toÂ makeÂ the NPV = 0.Â And vice versa.Â That is one of our basic TVM rules.
• Thus, for any given project, the IRR and NPVs rules will be consistent with one another.