# The Internal Rate of Return

The internal rate of return is the discount rate that sets the present value of all cash inflows of a project equal to the present value of all cash outflows of the same project. In other words, it is the effective rate of return that makes a project have a net present value of zero. Thus:

NPV = 0 if r = IRR, for any given project.

Or

PV outgo = PV income when r = IRR

The internal rate of return method of project appraisal assumes that all proceeds from the project can be re-invested immediately, and in projects offering returns equal to the IRR, until maturity. A higher IRR indicates a more “profitable” project.

## Interpreting the Internal rate of Return

You should note that the IRR need not be positive – It can be zero or even negative. A positive return indicates that the project makes money for the investor. A zero return indicates that the investor is neither profitable nor loss-making. Lastly, if the IRR is negative, the investor loses money. However, if the IRR is less than -1, it means the yield or rather the return is undefined.

The internal rate of return is usually compared to the cost of capital, usually the weighted average cost of capital, WACC. A project whose IRR is above its WACC increases the shareholders’ wealth. Otherwise, it would be unwise to borrow cash at an interest rate, say, 10% and then invest the money in a project with a rate of return less than 10%. The borrower would be unable to service the loan.

Usually, candidates cannot solve questions involving IRR directly and you may need to carry out linear extrapolation. Working with a spreadsheet or calculator is also a better, easier approach. However, you should aim to understand the manual approach first. It will then be easier to use a calculator.

## Question

Suppose we have a project with the following cash flows;

Outgo: $150,000 at t = 0,$250,000 at t = 1, and some more $250,000 at t = 2 Income:$1 million at t = 3

Find the IRR of the project.

A. 25.2%

B. 0%

C. 23%

Solution

The correct answer is A.

We need to find the rate r such that:

$$-150,000 – 250,000(1 + r)^{-1} – 250,000(1 + r)^{-2} + 1000,000( 1 + r )^{-3} = 0$$

$$\text{At } 20\%, -150,000 – 250,000 * 1.2^{-1} – 250,000 * 1.2^{-2} + 1000,000 * 1.2^{-3} = 46,800$$

$$\text{At } 25\%, -150,000 – 250,000 * 1.2^{-1} – 250,000 * 1.2^{-2} + 1000,000 * 1.2^{-3} = 2,000$$

We can approximate r by linear extrapolation using the two values:

$$r = 20\% + \cfrac {(0 – 46,800)}{(2 – 46,800)} * (25\% – 20\%) = 25.2\%$$

Suppose the WACC is 20% in the example above. What would be your advice to investors? Since WACC (20%) is less than IRR (25.2%), the project is economically viable and would increase the investors’ wealth.

Conclusion: A capital project should be accepted if its IRR is greater than the cost of capital.

Reading 7 LOS 7a (Part 2)

Calculate and interpret the net present value (NPV) and the internal rate of return (IRR) of an investment.

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