# Calculate CAGR (examples) – Annual growth rate explained

The key figure of the annual growth rate CAGR – calculation with examples

The CAGR is a key figure of the business analysis with which various business components can be examined and made measurable. CAGR stands for Compound Annual Growth Rate and is also known as the annual growth rate. It can be used to determine how strongly a company or a specific key figure of a company has grown within a defined period of time. The annual growth rate helps, for example, investors to compare several companies in order to make their decisions. However, annual growth also plays a weighty role on the stock market.

## CAGR – definition and explanation

The CAGR ratio indicates the annual growth for a specific period. It can be used to analyze various underlying values, which is why it can be described as particularly flexible. The result of the CAGR is given as a percentage and reflects the value of how much a certain key figure has grown per year over a certain period. The annual growth rate thus describes an average value per year.

It is advantageous to consider a long period of time. A positive annual growth rate over several years reflects the sustainable management of a company. The informative value of the annual growth rate reaches its limits when particularly volatile companies are to be examined. The formula can also deliver unusable results, which will be discussed in more detail later on. There is hardly any limit to the scope of application of the formula, as will become clear from later examples.

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## Calculation of the CAGR

Basically, only two values are needed to calculate the annual growth rate:

The initial value of a key figure to be analyzed
The final value of a key figure to be analyzed

The CAGR is calculated by dividing the final value by the initial value. The result is then converted into a percentage. However, since the annual growth rate ratio is more meaningful when a period of several years is considered, this must be included in the formula. In the case of the number of years, it is the turn of the year that is decisive, which must be counted and is indicated in the formula with the value n. Since the growth is not distributed, the CAGR increases according to the concept of compound interest. In order for the annual growth rate to be calculated, the value n is first required.

n = last\thinspace year – first\thinspaceyear

n = 2020 – 2015 = 5

Thus, six years are included, but the base year may not yet be counted. This is due to the fact that no growth takes place within a year when viewed on an annual basis. Of course, a monthly, weekly or daily consideration can also be made. The formula is versatile applicable.

The value from the end year is divided in the formula with the value from the beginning year. The nth root divides the growth to the individual years. This is necessary because otherwise the compound interest would be disregarded. However, since growth builds on each other and is not distributive, compound interest must be observed.

The formula is:

CAGR=\sqrt[n]{\frac{end value}{start value}}

Subsequently, this number is transformed into a percentage value:

CAGR=(\sqrt[n]{\frac{end value}{start value}}-1)*100

### Example: Calculation of annual growth

Ampion AG is a manufacturer of pharmaceuticals with moderate growth The following table shows the sales in the past years:

 YEAR: SALES: 2014 1.1 billion 2015 1.2 billion 2016 1.6 billion 2017 1.4 billion 2018 1.7 billion 2019 1.9 billion 2020 2.0 billion

To determine the annual growth, the value n must first be calculated.

n = last\thinspace year – first\thinspaceyear

n = 2020 – 2014 = 6

Subsequently, the key figure CAGR can be determined using the formula:

CAGR=(\sqrt[n]{\frac{end value}{start value}}-1)*100

CAGR=(\sqrt[6]{\frac{2,000,000,000}{1,100,000,000}}-1)*100 = 10.5%

From this it can be seen that Ampion AG had an average growth of 10.5% from 2014 to 2020. Already now, a weakness of the CAGR can be seen, as the short growth dip in 2017 is not depicted. This problem will be described in more detail in the further course.

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### The advantages of the CAGR (annual growth rate)

The key figure of annual growth must be interpretedso that conclusions can be drawn from it. The pure ratio by itself has little significance. However, if it is compared with other key figures or with the same key figure from other companies, this changes. By comparing annual growth, different companies can be compared and analyzed.

In addition, the dependency of two key figures within a company can be examined. This in turn can be used to generate measures and process improvements. The annual growth rate offers a great advantage in different areas of application, which applies to investors, companies and private individuals. For more details, please refer to the chapter “Areas of application of the CAGR“.

Another advantage of the ratio is that fluctuations in growth can be balanced outsince it is an average value. The base value to be examined does not have to increase permanently in order to provide a result. How this effect can become a disadvantage at the same time is described in the chapter “Inconstant growth“. A next advantage is the fact that the ratio can show a negative result from the annual growth. If the growth of the examined base value is declining, the result of the formula is represented with a negative sign. In such a case, the ratio says that the annual growth has been steadily decreasing on average over the past years and thus has slipped into the negative range.

#### Example: Calculating negative growth

Ampion AG has only been consistently positive economically since 2014, after a new drug was acquired from the competition. After a successful founding year in 2009, growth declined in the following years. The table shows sales from 2009 to 2013:

 YEAR: SALES: 2009 0.8 billion 2010 0.5 billion 2011 0.4 billion 2012 0.4 billion 2013 0.3 billion

To determine the annual growth, the value n must first be calculated.

n = 2013 – 2009 = 4

The key figure CAGR can then be calculated using the formula:

CAGR=(\sqrt[4]{\frac{300,000,000}{800,000,000}}-1)*100 = -21.74%

Ampion AG had to cope with declining sales in the first four years after its foundation. The average annual growth slipped into negative territory to -21.75% revenue decline per year.

### The disadvantages of the CAGR

As already indicated in the foregoing, the key figure of annual growth is only meaningful to a limited extent. Moreover, it reaches its limits when negative growth is included in the calculation. In such a case, the mathematical calculation is no longer possible.

The interpretation of the ratio becomes even less clear in the case of inconstant growth. Since the formula analyzes only the initial and final values of the period under consideration, the result may deviate significantly from reality. Examples of this are described in the following chapter.

#### Inconstant growth

The CAGR is calculated from the initial value and the final value of a period whose annual growth is to be determined on average. This works very well for constant growth and provides a comprehensible result. Critically, however, the result must be questioned in the case of inconstant growth. In this situation, the percentage can deviate significantly from reality. This is always the case if, by chance, a particularly bad or a particularly good year forms the initial or final value. Two examples will illustrate this.

### Example: Embellishing growth with the CAGR

Hugenschnieder GmbH is a small craft business that has been struggling economically for years. For an investor, the CAGR for the annual profit should be disclosed. Since Hugenschnieder GmbH is very interested in an acquisition, the ratio should be as positive as possible. In 2019, a very high annual profit was reported due to a one-time project, which was used to pay off debt. The following table shows the profits for the years 2014 to 2020:

 YEAR: PROFIT: 2014 + 20.000€ 2015 + 3.000€ 2016 – 32.000€ 2017 – 17.000€ 2018 – 54.500€ 2019 + 84.000€ 2020 – 17.000€

In order for Hugendschnieder GmbH to show a positive CAGR, it works with outdated figures. It uses the year 2015 as the starting value and the year 2019 as the final value.

To determine the annual growth, the value n must first be calculated.

n = 2019 – 2015 = 4

Subsequently, the key figure CAGR can be determined using the formula:

CAGR=(\sqrt[4]{\frac{84,000}{3,000}}-1)*100 = 130%

According to the key figure, Hugenschnieder GmbH has an annual growth in profit of 130 %. Although the value is correct, it strongly distorts the result. If the profits and losses of the individual years are considered as an absolute number, the company has made a loss of 13,500 EUR. The CAGR figure must therefore always be viewed with caution, as this example reveals.

#### Example: Negative CAGR in a solid company

The example of Hugenschnieder GmbH can be turned around and give an economically strong company a negative CAGR, should the final year happen to have been determined by a crisis. As an example, the following list shows the sales of Schnellfürst SE. The company is a successful logistics provider with a constant growth. Due to the Corona pandemic in 2020, the company has to face a crisis year.

 YEAR: SALES: 2014 1.5 billion 2015 1.2 billion 2016 2.2 billion 2017 2.0 billion 2018 2.3 billion 2019 2.5 billion 2020 0.8 billion

Schnellfürst SE is required to provide its current CAGR, taking as base year the year in which a change of corporate form took place.

In order to determine the annual growth, the value n must first be calculated.

n = 2020 – 2014 = 6

Subsequently, the key figure CAGR can be determined using the formula:

CAGR=(\sqrt[6]{\frac{800.000.000}{1.5000.000.000}}-1)*100 = 9,95%

According to the ratio, Schnellfürst SE has a declining annual growth in sales of – 9.95%. Although this value gives correct information about the currently valid growth rate, it does not reflect the good constant growth in any way. This example also shows that the key figure must be questioned if the basic values are inconstant.

### Negative growth

If it is still possible to use the key figure in the case of inconstant growth, but there is a lot of room for interpretation, in the case of negative growth, however, the CAGR cannot be calculated from the outset. There are mathematical reasons for this. A negative number makes the formula presented here unusable. This is because an nth root of negative numbers can be solved mathematically, but would be completely unusable for use in common practice. A calculation with negative growth does not yield a meaningful percentage value.

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## Areas of application of the CAGR (annual growth rate)

The annual growth rate can be used in different areas and can be helpful. These include not only economic sectors, but also the private sector. However, the ratio is mainly used to make two companies comparable for investors. This does not have to involve large corporate takeovers. Even small shareholders can use the CAGR to analyze share performance and make their investments based on it.

### CAGR for investors

An investor can get a good overview of a company he wants to invest in by looking at the annual growth rate of various basic figures. In addition, he has the opportunity to analyze several companies with regard to their growth. The problems already presented, which arise in the case of inconstant growth, must always be taken into account. Investors can refer to the following key figures for an analysis:

Sales
Profit
Dividend
Web page views
Sales figures of certain products
Store openings
etc.

However, it is not only large investors who can use the key figure of the annual growth rate to invest money in companies for return purposes. For example, the CAGR is a very good way of comparing share performance from the past. This makes it possible to find particularly profitable shares.

### Example: Comparing stock performance

The fund operator Greenstone would like to add new shares into the portfolio and wants to profit from the strong chart performance of American technology companies. To help Greenstone compare the best stock trajectories, the company wants to analyze the annual growth rate of several U.S. tech stocks over the past six years. Greenstone has identified the following stock prices for Facebook, Apple, Microsoft and Netflix.

 SHARE: 2015 SHARE PRICE: SHARE PRICE 2021: Apple 23,05€ 108,00€ Facebook 65,50€ 222,80€ Microsoft 38,94€ 180,10€ Netflix 40,80€ 429,30€

The annual growth rates are as follows:

Apple: 29.35
Microsoft: 29.08 %
Netflix: 48.03 %

Netflix’s stock has the best returns, with annual growth of 48.03%. However, this is purely a consideration of the past. It cannot be said for sure that stock trajectories will continue in the future. Nevertheless, the CAGR of stocks is always a good indication.

### CAGR in the company

A company can use the key figure for annual growth itself for its own evaluations. For this purpose, for example, correlations of the annual growth rate of different basic figures can be compared with each other. If there is annual growth in the number of complaints and at the same time the CAGR in the number of orders has turned negative, conclusions can be drawn from this.

The correlation of two values does not necessarily have to have a causal relationship. However, it is always an opportunity to point out errors and make adjustments. However, the consideration is very individual in each company, which is why no blanket suggestions can be made at this point.

### CAGR in the private sector

The key figure of the annual growth can be used as one of the few also in the private sector. An example is the already presented possibility to analyze stock trends and to control the investments of saved money over it. But other calculations can be made, which include, for example, the necessary growth for the redemption of a loan.

#### Example: Calculate necessary capital growth for loan repayment

In 2020, the Hase family moved into a new house and took out a loan to finance it. This loan must be repaid in 2040. In 2020, the family had EUR 26,484 at its disposal, which it must invest profitably on a sustained basis so that the loan of EUR 100,000 can be repaid in 2040.

To determine the annual growth, the value n must first be calculated.

n = 2040 – 2020 = 20

Subsequently, the key figure CAGR can be determined using the formula:

CAGR=(\sqrt[20]{\frac{100,000}{26,484}}-1)*100 = 6.86%

The Hase family must invest its money in such a way that average annual growth of 6.86% is generated from now on, so that the home loan can be paid off in full in 2040 using the initial capital invested.

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