# Internal Rate of Return (IRR): Definition, Formula, Use, Problems, Example, and Analysis

## What is the Internal Rate of Return?

Internal Rate of Return is the rate or cost of capital that makes a project or investment’s Net Present Value exactly zero. The internal Rate of Return is quite important for management in decision-making for new investment proposals and performance appraisal.

It is also used in performance appraisal of existing projects or companies. This rate is also used to assess the new investment proposal or project whether it should go ahead or stop.

Normally, the Internal Rate of Return is different from the Required Rate of Return. Required Rate of Return is that rate set by management and it is normally higher than or equal IRR. If the project or investment is higher than IRR, that project or investment should be accepted or go ahead.

In decision making, if the projects’ Internal Rate of Return is greater than the cost of capital or target cost of capital, then those projects should be accepted.

IRR is calculated using the calculator or as follows using interpolation of a low discount rate with positive NPV and a high discount rate with negative NPV.

## Internal Rate of Return Formula:

Here is the internal rate for the return formula, and we will learn every aspect of the formula as it is very important for your full understanding of how IRR works.

Remember, the internal rate of return is using the interpolation technique to calculate it and it is very important to understand this concept so that you can get a better understanding of how IRR works.

Here is the formula,

Or IRR = a + [(NPVa / NPVa – NPVb)(b-a)]%

Check out here what does each element of this formula mean,

• a: is the lower of two rates of return that we use in our calculation and it will return a positive net present value.
• b: is the higher of two rates of return that we use in our calculation and it will return a negative net present value.
• NPVa : is the positive net present value as the result of using a lower rate of return a.
• NPVb: is the negative net present value as the result of using a higher rate of return b.
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We will explain in detail NPV, negative NPV, and Positive NPV in this article.

Recommended book:
Investment Appraisal: A Simple Introduction (Simple Introductions)

### What is NPV?

Net Present Value is the value of the net cash flow after discounting. To calculate NPV, you need to know the cash inflow and cash outflow of the projects or company for the period of time; for example, five years.

Once you know the net cash flow, you need to know the discount factor of present value. For example, 10%. This factor is sometimes we use the cost of capital.

### Positive NPV

Positive NPV is happening when there is a low rate of return or there is a large amount of positive net cash flow. We use a low rate of return to find the rate that could lead to positive NPV in order o calculate IIR.

### Zero NPV

Zero NPV is exactly come up from the using IRR in discounting. Once you find the IRR, you test if the NPV for the project you are assessing got zero value or not.

### Negative NPV

Well, basically the negative NPV result from using a large amount of rate of return or discount factor. We find the rate that could make the negative NPV so that we could get the rate that makes zero NPV

We hope you understand the relationship between these groups of factors and elements. And, yes, you still doubt about it, let a comment below so that we could try to make sure that our reader will get the point.

Now, we move to our three easy steps in the calculation of IRR, and we hope that these three steps will bring you to the next level of understanding of IRR.

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Here is what IRR look like,

Okay, here we go…

## Calculating Internal Rate of Return in 3 Steps

### Step 1: Calculate NPV using the Company’s cost of capital.

The first thing you need to do in these steps is to calculate the net present value using the company’s cost of capital.

For example, your projects generate an annual net cash flow of 50,000 for five years and the cost of capital of your company is 10%. The initial investment is USD 150,000. Then you can calculate the Net Present Value of this cash flow right. Don’t tell us you don’t know how to calculate this.

Well, if you don’t, here is the Net Present Value of this

So, the net cash flow will be 50,000 annually in five years right.

NPV for this cash flow would be 50,000 * 3·791 (From Annuity table) = USD 189,550, and the NPV for this projects at 10% of cost of capital would be 189,000 – 150,000 = USD39,000.

Now as you can see, this project generates a positive net present value of 10%.

Let move to the next step,

### Step 2: Calculate the NPV of the company using greater or less than the cost of capital.

What you need to do in this step is that you need to check what the NPV of the first step is, and then follow this rule.

1. If the NPV is positive, use a second rate that is greater than the first-rate.
2. If the NPV is negative, use a second rate that is less than the first-rate.

As we know, in the first step we got a positive NPV. So now what we need to do is calculate net NPV by using the greater rate of return from the first one.

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Now let assume, new rate of return on 20%, and what we got is 50,000 * 2.991 = USD 149,550

This project’s NPV at is 149,550 – 150,000 = USD (450). Negative right?

Okay, now we got both positive and negative NPV. Now we are able to move to step three which is the final step.

### Step 3: calculate IRR:

Use the two NPV values to estimate the IRR. The formula to apply is as follows.

IRR = a + [(NPVa / NPVa – NPVb)(b-a)]%

• a = 10%
• b = 20%
• NPVa = USD39,000
• NPVb = USD (450)

Now we got everything we want right?

IRR = 10% + [(39,000 / 39,000 + 450)(20-10)  = 19.886 %

Now let confirm if this rate is actually make NPV become zero,

Here we go,

• R = 19.886 %
• N = 5

Then, discount value annuity is 2.9981 rounded to 3

• 50,000 * 3 = 150,000
• Then 150,000 – 150,000 = 0 (Congratulation)

Easy right ?

## Use of IRR (Rule)

So now we move to the way IRR is used and what are the important things that we need to know and how to use it. Basically, IRR is the tool used to assess whether the return of the proposed investment project or the current investment project is acceptable or not.

For the proposed investment project, it is used to compare with a required rate of return or cost of capital. Normally, if the IRR of this project is higher than the required rate of return, then the project should be accepted. However, if the IRR is below the required rate of return. Then, the project should not continue.

IRR is also used to assess the current investment projects whether the expected future cash flow is similar to the cash flow during the proposal and the expected IRR is still at the agreed rate of return.