# Internal Rate of Return (IRR) explained with formula and example

Internal rate of return is one of the techniques used in capital budgeting process to evaluate investment decisions (i.e. whether to make investment in long term assets or not).

## Definition

IRR is defined as

“Rate at which the return will be received from the project”

OR mathematically speaking

“The rate (used for discounting) at which 1) present value of cash outflow (initial costs) equals 2) present value of net cash inflows earned over the life of the project.”

OR

“The rate of return at which net present value (NPV) of the project is zero.”

In other words IRR is the rate of return that is actually offered by the project. Why word “internal” is used. Because cash flows (both inflow & outflow) that are internal only to that project are considered.

## Formula

From definition, we can say IRR is rate of discounting at which

 Initial cost of the investment = Present value of future cash inflows & outflows associated with investment

Or, in mathematical terms

 IC  =    ∑ n t=1 FCFt (1 + IRR)t

Where
FCF = Future cash flows
IRR = the discount rate i.e. internal rate of return.
t = are periods (usually year wise) in which cash flows are generated.
n = is the total life of the project / investment; and
IC = Initial cost of starting the project or making the investment.
Formula can best be understood by knowing the method of calculating the npv and with an example.

## How to calculate Internal Rate of Return (IRR)

#### Step 1: Determine the future cash flows of the investment / Project

Predict / forecast all the relevant future cash flows of the investment. For this purpose future cash flows can be divided into following four broader categories.

• Inflows: Identify year wise cash inflows over the life of the project. These inflows are revenues generated from the investment e.g. by installing a new plant, the sales of the company increased. The revenue from these extra sales is inflows.
• Outflows: Identify year wise cash outflows over the life of the project. These outflow could include operating costs but do not include non-cash costs. For example the outflows associated with a new plant will be salaries of staff, repair & maintenance expenses of plant etc. But this will not include depreciation of the plant as it is non-cash cost.
• Taxes: Year wise government claims i.e. taxes etc. will also be predicted and taken as cash outflow for the purpose of determining future cash flows.
• Salvage value at the end of project: At the completion of the project’s life it has some salvage value. This value will be taken as an inflow. For example the plant has 10 years of useful life, after which it will be sold at some salvage value.

#### Step 2: Determine the required rate of return for the investment / project

Ideally required rate of return for each project should be determined separately. However for the sake of simplicity companies use their overall required rate of return for the purpose. This required rate of return (RRR) will be compared with IRR to decide whether to accept or reject the project.
Required rate of return takes into account the riskiness of project and cost of capital. How to calculate the required rate of return is explained separately.

#### Step 3: Solve the equation for IRR

Step 3 involves mathematical skills.
By putting the value of future cash flows of investment (determined in step 1) and initial cost of investment in the formula, solve this equation for value of IRR. For this purpose

• Use financial calculator: the easiest solution
• Use annuity tables: where cash flows are even
• Use trial and error method: where cash flows are uneven

This step requires nothing more than pure mathematical skills.

#### Step 4: Interpret the internal rate of return

Just compare IRR with RRR and decide. It is simple and explained below after the example.

## Example: Calculating IRR using annuity table

Assume project A’s initial cost is \$90,000. Project will return \$ 25,000 annually for 7 years.

• Calculate internal rate of return for the project.
• Comment whether the project should be taken if company’s required rate of return (RRR)is 20%.

#### Step 1: Determine the future cash flows of the investment / Project

Future cash flows, in this example, are easy to calculate i.e. \$25,000 per annual. The complexities of inflows, outflows, taxes and salvage value are not taken under this example. However how to calculate complex cash flows have been discussed on some other pages.

#### Step 2: Determine the required rate of return for the investment / project

RRR is given i.e. 20%.
Required rate of return takes into account the riskiness of project and cost of capital. How to calculate the required rate of return is explained separately.

#### Step 3: Solve the equation for IRR

Formula for IRR is

 Initial cost of the investment = Present value of future cash inflows & outflows associated with investment
 IC = N ∑FCF t t =1 (1 + IRR) t

 90,000 = 25000 + 25000 + 25000 ……… 25000 (1 + IRR) 1 (1 + IRR) 2 (1 + IRR) 3 (1 + IRR) 7
 90,000 = 25,000 * 7 1 ∑ (1 + IRR) t t =1

For applying annuity table, we can further reduce the last impression of the equation as follows:

 90,000 = 25,000 * PVifA i, 7 (present value if annuity i at 7 years)

Dividing both sides with 25,000, we get

 3.60 = PVifA i, 7

In annuity table the value of impression “PVifA i. 7” searching in 7th year equals to 3.60 at 20%.
Thus

 IRR = 20%

#### Step 4: Interpret the internal rate of return

Since IRR is equal to RRR i.e. both are 20%, the project should be accepted.
Detailed Interpretation of IRR is discussed below.

## Interpretation of internal rate of return

The result of irr will be either equal to, less than or greater than RRR i.e. required rate of return. The result will determine whether to take the project or not.

#### When irr is greater than required rate of return i.e. irr > RRR

Accept the project.
This means the projects return will exceed the minimum return required from projects. As a financial manger our goal is to maximize the shareholder’s wealth (value). Projects with irr exceeding RRR fulfill this goal of financial manger and should therefore be accepted.

#### When irr is equal to required rate of return i.e. irr = RRR

Accept the project.
This means project is at least recovering the minimum required rate of return. Therefore project should be accepted.

#### When irr is less than required rate of return i.e. irr < RRR

Reject the project.
This means, project has failed to recover even the cost of the project. Accepting such project will harm the goal of maximizing the shareholder’s wealth and it should therefore be rejected.

• Time of value of money: irr recognizes the important phenomenon “a dollar today, worth more than a dollar after one year.” Therefore irr uses discounting techniques.
• Risks of projects: irr realizes the fact that “some investments are more risky than others and proper weightage should be given to investment according to risk involve.” IRR do this by comparing with RRR which has taken into account appropriate risks.
• Cash not profits: IRR is based on cash flows rather than on accounting profits. Thus it accounts for the “true timing of the costs & benefits” as opposed to accounting where accrual concept of costs & benefits is used.

• Time consuming requiring professionals: IRR is normally calculated by the professionals only. Also it is lengthy process. Cost of employees and cost of time both are valuable resources of business.
• Prediction of future cash flows: predicting the future inflows and outflows associated with an investment is a difficult task.

## Limitations of IRR

The results of IRR become meaningless either of the following two. Namely;

#### Multiple Solutions of IRR

One solution of IRR will only exist if cash flows from the project are conventional i.e. one cash outflow at the start of the project and then cash inflows from the project. If there is cash outflow other than initial outflow, there will exist more than one solution of IRR.

 Conventional Cash Flows Non Conventional Cash Flows ( 100, 000) Initial cost ( 100, 000) Initial cost 50,000 Return of project 50,000 Return of project 50,000 Return of project (50,000) more cost 50,000 Return of project 50,000 Return of project

Thus irr rule will totally break as none of the solutions will be meaningful for the purpose of decision making.

#### 2) Mutually Exclusive Projects

Projects are mutually exclusive when acceptation of one project will require automatic rejection of all other projects. For example a company is considering purchasing one building. Options available are

• A building situated at Location A
• A building situated at Location B
• A building situated at Location C
• A building situated at Location D

By purchasing building at any of these locations, say at location B, all other options of Location A, C & D will automatically be rejected. Thus these projects are mutually exclusive.

If projects are mutually exclusive, the project with highest NPV should be accepted. “IRR will give wrong answer” as it does not consider highest npv. In other words NPV is the best technique for mutually exclusive projects.