## What is the internal rate of return (IRR)?

The internal rate of return, or IRR, is a measurement used to estimate the profitability of a project or investment. It is used when companies need to decide between different ways of using their money.

The IRR of the investment is determined by anticipating the profit a project will produce in the future and finding out its value today.

## Why is it important

- A way of judging an investment; the higher the IRR, the better the investment
- Used to compare the profitability of projects of different sizes
- Shows the return of the original financial investment
- Simple way of calculating potential investments and cost-saving opportunities

## Importance for businesses

Businesses typically either distribute surplus profits to shareholders in the form of dividends or reinvest them into the company. Reinvesting can mean:

- Using the money to pursue projects that will expand the company or bring in additional revenue
- Investing in securities that will provide investment income
- Purchasing or upgrading equipment that will increase productivity

It is impossible to know in advance which reinvestment will be the most profitable, so businesses use the IRR calculation to compare options.

## The IRR rule

Companies want their projects and investments to be profitable, but the money they have to spend on them comes at a price, which is called the “cost of capital.”

- Using debt, like loans, means paying interest
- Using equity from selling shares of stock requires meeting shareholder expectations for growth and paying dividends

In general, the IRR Rule states:

- If the estimated IRR of a given option is higher than the cost of the capital required to fund it, it should be pursued.
- When several projects or investments are available, the option with the highest IRR is typically the best choice.

## The IRR formula

** **There isn’t a specific formula for calculating the IRR. Instead, the formula for the Net Present Value, or NPV, of a project is used:

in which:

N = The number of periods (usually years) during which the investment will produce income

C_{n} = The net income produced by the investment during a given period

I = The initial cost of the investment; this is negative because it is an expenditure

R = The rate of return of the investment

Important notes:

- Σ, or sigma, is a mathematical symbol that means we must add all the values for each year the investment produces income. If it will provide income for five years, for example, five numbers will be used to calculate and sum.
- N=0 simply means that the formula incorporates all profit and costs from the beginning of the project.

The NPV formula uses the estimated rate of return, R, of a given project or investment to determine its total value over time. The sum of a project’s income over time must be at least equal to its cost, or I, to be worth pursuing.

By setting the value of NPV to 0, we can use this formula to solve for an unknown R, or IRR, which will be equal to the rate of return that a project must produce, at minimum, to be worth the cost of pursuing it. This requires considerable trial and error, but corporate finance software is built to do this automatically.

## NPV vs. IRR

## Why IRR is complicated

- It’s trial and error
- Investments and projects typically produce profit or loss for many years, which makes this formula long and difficult to do by hand
- The true rate of return is unknowable

## Example

The IRR calculation can seem very intimidating. The best way to understand how it works is with a very simple example.

Assume project ABC will cost $400,000. It will only provide value for one year (making our calculation simple), but it is expected to generate profit of $1 million.

This means that:

- N = 1
- C
_{n}= 1,000,000 - I = -400,000
- R = ?

To find the IRR of project ABC, we plug in these numbers into the NPV formula, setting NPV to 0:

Therefore:

Since there is only one year of income, we can do away with the Σ:

Since raising (1+R) to the power of 1 doesn’t change it, we can eliminate that as well:

Now it’s a simple matter of algebra to solve for R. First, subtract 400,000 from both sides:

Then divide both sides by 1,000,000 to get (1+R) alone:

Next, multiply both sides by (1+R), so it’s no longer a denominator:

Divide both sides by 0.4:

And subtract 1 to solve for R:

Since the rate of return is typically expressed as a percentage, we can also say that:

R = 150%

Since we know the project cost $400,000 but generated $1 million in revenue, we know this value is correct because the $600,000 profit is equal to 150% of the cost:

($1,000,000 – $400,000) / $400,000 =

$600,000 / $400,000 =

1.5, or 150%

## References

- http://www.investopedia.com/terms/i/irr.asp
- http://www.investinganswers.com/financial-dictionary/investing/internal-rate-return-irr-2130
- http://accounting-simplified.com/management/investment-appraisal/internal-rate-of-return-irr.html
- http://www.investopedia.com/ask/answers/021115/what-formula-calculating-net-present-value-npv-excel.asp?ad=dirN&qo=investopediaSiteSearch&qsrc=0&o=40186
- http://www.investopedia.com/ask/answers/032615/what-formula-calculating-net-present-value-npv.asp?ad=dirN&qo=investopediaSiteSearch&qsrc=0&o=40186
- http://smallbusiness.chron.com/internal-rate-return-important-organization-67279.html
- https://www.codecogs.com/latex/eqneditor.php