The internal rate of return (IRR) is the discount rate which will give a net present value (NPV) of zero when applied to a series of cash flows.

## IRR Calculation

Consider as an example, the following cash flow diagram. At the start of year 1 (today) there is a cash out flow of 2,000 representing an investment in a project. For simplicity, with no further investment, the amount of 2,500 is returned in 2 years time at the end of year 2.

The business decides that the appropriate discount rate to use is 8%. The discount rate is the value the business places on its money.

The net present value of this project is the sum of the present values of each of the cash flows. Further details on the calculation of NPV can be found in our net present value tutorial.

NPV = -2,000 + 2,500 /(1 + 8%)^{2}NPV = -2,000 + 2,143.35 NPV = 143.35

The NPV is greater than zero so this project must have a rate of return greater than 8%.

If the same calculation is carried at at a discount rate of 14% the net present value is:

NPV = -2,000 + 2,500 /(1 + 14%)^{2}NPV = -2,000 + 1,923.67 NPV = -76.33

The NPV is less than zero, so this project must have a rate of return less than 14%.

If we vary the discount rate, at a given value the NPV will be equal to zero, and this is the rate of return for that project, this rate is referred to as the internal rate of return or IRR.

In this simple case to solve for the value of the discount rate we would need to solve the equation

NPV = 0 = -2,000 + 2,500 /(1 + i)^{2}(1 + i)^{2}= 2,500/2,000 1 + 2i + i^{2}= 5 / 4 Solving this quadratic equation for i and taking the only positive value i = 11.80%

At a rate of 11.80% the NPV of this project will be zero, and we can say that the internal rate of return or IRR of the project is 11.80%.

## IRR Calculation With Multiple Cash Flows

In the simple example above it was possible to solve for i, in cases where there are multiple cash flows at later stages in the projects life, it is not possible to solve directly for i and iterative techniques need to be applied.

Consider a project with the following cash flow diagram. In this case the initial cash is paid at at the start of period 1 (today). This is followed by a further payment at the end of period 1, and then receipts at the end of period 2 and the end of period 3.

Calculating the NPV at a discount rate of 10% we get

NPV = -5,000 + 3,000 / (1 + 10%)^{2}+ 7,000 / (1 + 10%)^{3}NPV = -5,000 + 2,479.34 + 5,259.20 NPV = 2,738.54

The net present value of the project at 10% is positive, so the return must be greater than the 10%.

The same calculation at 40% gives the following:

NPV = -5,000 + 3,000 / (1 + 40%)^{2}+ 7,000 / (1 + 40%)^{3}NPV = -5,000 - 1,530.61 + 2,551.02 NPV = -918.37

The net present value of the project at 40% is negative, so the return must be less than the 40%.

This process can now be repeated at values between 10% and 40% until a solution is reached. At 29.62% the calculations show the following:

NPV = -5,000 + 3,000 / (1 + 29.62%)^{2}+ 7,000 / (1 + 29.62%)^{3}NPV = -5,000 - 1,785.62 + 3,214.39 NPV = -0.00

The iterative process can also be carried out using a financial calculator or the Excel IRR function

## Using the IRR

Once the IRR has been calculated it can used used together with a few simple rules as follows:

- If the IRR of the project is greater than the required rate of return (discount rate), it should be accepted.
- If the IRR of the project is less than the required rate of return (discount rate), it should be rejected.
- If only one project can be chosen, then the project with the highest IRR should be accepted.

## Limitations and Disadvantages of IRR

The IRR calculation formula will evaluate a project correctly is there is an initial payment (negative cash flow) followed by cash receipts (positive cash flows). If the cash flows are reversed so that the project has a cash receipt (positive cash flow) followed by cash payments (negative cash flow) then the IRR method can be less accurate and care needs to be taken with the solutions produced.

In the event that the project has multiple cash flows which change sign throughout the project, then internal rate of return equation will produce multiple IRRs for the project. The method does not distinguish between the answers, and there is no way of deciding which is the correct solution.

Finally, calculating the internal rate of return tells an investor nothing about the absolute size of a project, it will give the same rate of return irrespective of whether the project is small or large in scale.

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a degree from Loughborough University.