# Average annual growth rate

Average annual growth rate |
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The **average annual growth rate** is a measure of the growth rate of a specific value, such as investment return or population count, over a given period of time expressed in years. It is calculated by taking the arithmetic mean of the yearly growth rates over the period. The formula to calculate the average annual growth rate is as follows:

Average annual growth rate is an important measure used to determine the rate of return for investments, the rate of population growth, and the rate of inflation. It is also used to determine the rate of growth of a company's sales and profits over time.

In summary, the average annual growth rate is a measure of the growth rate of a specific value over a given period of time expressed in years. It is used to measure the rate of return for investments, the rate of population growth, the rate of inflation, and the rate of growth of sales and profits. It is calculated by taking the arithmetic mean of the yearly growth rates over the period.

## Example of Average annual growth rate

The average annual growth rate can be used to measure the rate of return for investments. For example, if an investor put $1000 into a stock in January 2019 and after one year the value of the stock is $1100, then the average annual growth rate of the investment would be\[\begin{equation} AG = \frac{(1100 - 1000)}{1000 \cdot 1} = 0.1 \end{equation}\]

Which is equal to 10%. In other words, the investment yielded a 10% return over the period.

In summary, the average annual growth rate can be used to measure the rate of return for investments. In this example, the rate of return was 10%, calculated by taking the difference between the start and end value of the investment and dividing it by the start value multiplied by the number of years.

## Formula of Average annual growth rate

\(\begin{equation} AG = \frac{\sum_{i=1}^n (V_i - V_{i-1})}{V_0 \cdot n} \end{equation}\)

Where AG is the average annual growth rate, V_i is the value of the variable at the end of year i, V_0 is the value of the variable at the start of the period and n is the number of years in the period. In other words, it is the average of the yearly growth rates of the variable over the period.

## When to use Average annual growth rate

Average annual growth rate is used in a variety of situations, including:

**Analyzing the rate of return for investments**: Average annual growth rate is used to measure the rate of return for investments over a given period of time. This can help investors determine the performance of their investments over time.**Measuring the rate of population growth**: Average annual growth rate is used to measure the rate of population growth over a given period of time. This can help governments and organizations plan for the future and allocate resources accordingly.**Analyzing the rate of inflation**: Average annual growth rate is used to measure the rate of inflation over a given period of time. This can help investors understand the impact of inflation on their investments and plan accordingly.**Measuring the rate of growth of sales and profits**: Average annual growth rate is used to measure the rate of growth of sales and profits over a given period of time. This can help companies assess the performance of their business and make informed decisions about future investments.

## Types of Average annual growth rate

**Simple Average Annual Growth Rate**: This is the simplest form of the average annual growth rate, which is calculated by taking the arithmetic mean of the yearly growth rates over the period.**Compound Average Annual Growth Rate**: This measure is calculated by taking the geometric mean of the yearly growth rates over the period. It is a more accurate measure of the true average annual growth rate, as it takes into account the compounding effect of the growth rate.

## Steps of Average annual growth rate

1. Calculate the growth rate of the value for each year in the period\[\begin{equation} GR_i = \frac{V_i - V_{i-1}}{V_{i-1}} \end{equation}\]

Where GR_i is the growth rate of the value for year i and V_i and V_{i-1} are the values of the variable at the end of year i and the start of the period respectively.

2. Calculate the average annual growth rate by taking the arithmetic mean of the yearly growth rates over the period\[\begin{equation} AG = \frac{\sum_{i=1}^n GR_i}{n} \end{equation}\]

Where AG is the average annual growth rate and n is the number of years in the period.

In summary, the steps to calculate the average annual growth rate is to first calculate the growth rate of the value for each year in the period and then calculate the arithmetic mean of the yearly growth rates over the period.

## Advantages of Average annual growth rate

- The Average annual growth rate is an easy-to-understand measure of the growth of a value over a given period of time.
- It can be used to compare growth rates in different periods or for different variables.
- It is useful for long-term forecasting and for understanding the rate of return for investments.

## Limitations of Average annual growth rate

- Average annual growth rate does not take into account the volatility of the investment or the population. This means that it does not account for the fact that the value of the investment or the population may vary greatly from year to year.
- Average annual growth rate does not take into account compounding, which is the effect of reinvesting returns to increase the value of the investment.
- Average annual growth rate also does not account for changes in the rate of growth over time. For example, if the rate of growth of the population increases or decreases over time, this change is not reflected in the average annual growth rate calculation.

**Compound Annual Growth Rate (CAGR)**: The Compound Annual Growth Rate (CAGR) is a measure of the rate at which a given value has grown over a period of time. It is calculated by taking the geometric mean of the yearly growth rates over the period. It is a more accurate measure than average annual growth rate as it takes into account the compounding effect of the growth over the period.**Simple Annual Growth Rate (SAGR)**: The Simple Annual Growth Rate (SAGR) is a measure of the rate at which a given value has grown over a period of time. It is calculated by taking the arithmetic mean of the yearly growth rates over the period. It is a less accurate measure than average annual growth rate as it does not take into account the compounding effect of the growth over the period.

In summary, there are two related approaches to calculating average annual growth rate, compound annual growth rate and simple annual growth rate. Compound annual growth rate is a more accurate measure as it takes into account the compounding effect of the growth over the period, while simple annual growth rate is a less accurate measure as it does not take into account the compounding effect of the growth over the period.

## Suggested literature

- Nations, H. G. G., & Nations, L. G. G.
*Average Annual Growth Rate 1973-1984*(Percentage). - Aghajanian, A. (1966).
*https://www.unescap.org/sites/default/files/APPJ-Vol-10-No-1.pdf Year Population (millions) Average annual growth rate (%)]*. Indicators, 1976, 1986. - van Genuchten, M., & Hatton, L. (2012). Compound annual growth rate for software. IEEE software, 29(04), 19-21.