Averaging and calculating the average is one of the data summarization indices. The average of mean is often used as a substitute for values so that a data set can be summarized into a single number. In this article from Arman Computer Magazine, we want to discuss a special type of average value called the average growth rate, and by citing examples, show math operations for determine how to get it. In this article, we discuss the fundamental concepts and calculations of growth rate. In financial texts, growth rate and CAGR plays a crucial role in investing.
Growth rate
Growth rate shows how much the values of a variable are increasing or decreasing. The growth rate between two years (consecutive or non-consecutive) can also be used for the average growth rate. In this way, the data between the initial and final values are also estimated.
Suppose $a_0$, $a_1$ to $a_n$ are the values that we want to calculate the average growth rate for (n+1) years. The growth rate for each year will be calculated as follows.
$$ r_1 = \dfrac{a_1}{a_0} , r_2 = \dfrac{a_2}{a_1},\ldots, r_n = \dfrac{a_n }{a_{n-1}} $$
In this way, the geometric mean of $r_i$ values will be as follows.
$$GM = (r_1 \times r_2 \times \ldots \times r_n)^{\frac{1}{n}} =(\dfrac{ a_1}{a_0} \times \dfrac{a_2}{a_1} \times \ldots \times \dfrac{a_n}{a_{n-1}})^{\frac{1}{n}} =(\dfrac {a_0}{a_n})^{\frac{1}{n}} $$
It is known that the form of each fraction can be simplified with the denominator of the previous fraction.
Therefore, the geometric mean can be considered the same as the growth rate with a little modification, and based on the two initial and final values, a numerical sequence of $n$ can be obtained, marked by A and B, respectively. In this way, the average growth rate (or growth rate) is obtained as follows.
$$ r = (\dfrac{B}{A})^{\tfrac{1}{n-1}} -1 $$
The above relationship is sometimes called compound annual growth rate or CAGR.
Note: As you can see, this calculation is very similar to the geometric mean calculation. Only instead of multiplying two values together, their ratio has been calculated, then the (n-1)th root shows the average growth rate.
Example: Let’s assume that in 2007 the income of a company is 1200 and its income in 2012 is 2400. We want to determine its growth rate.
Using the above formula, we get the average growth rate. The number of years from 2007 to 2012 is six years. But the number of ratios that should be calculated according to the contents of the previous section will be equal to 5.
$$ r = (\dfrac{2400}{1200})^{\tfrac{1}{(6-1)}} – 1 = 2 ^{\tfrac{1}{5}} – 1 = 0.15$$
In this way, we can consider the average growth rate to be around 15%.
If we want to predict the company’s income between 2008 and 2011 (or compare with actual values), we do the following. Suppose we show the amount of income in 2009 as $x_{09}$. In this way, we do the calculation as follows.
$$x_{09}= x_{07}*(r+1)^{09-07}$$
Therefore, the predicted value for 2009 will be as follows.
$$x_{09}=1200*(0.15+1)^{2} = 1587$$
Note: To test the above formula, we also try to predict the amount of income in the year 2012. We expect the predicted value to be equal to or at least very close to the actual value. Just note that the distance between 2012 and 2007 is determined based on 5 ratios.
$$x_{2012}= x_{07}*(r+1)^{5} = 1200 * (1.15)^5 = 2413.63$$
As expected, the predicted value for the year 2012 is very close to the actual value. Of course, the reason for the difference is the rounding of the growth rate.
Calculating the average growth rate using Excel
In Excel, you can use the RRI or rate of return for growth investment function to calculate the annual compound growth rate. It is clear that the parameters of this function should be the same as the initial value, the final value and the number of periods, as in the CAGR calculation.
You can see the window related to the parameters of this function in Excel in the image below. It is known that Nper is the number of periods and PV is the initial value of the period and FV is the final value of the period.
At the bottom of this window, you can see the calculated value for RRI or CAGR, which is 0.148698355, and if we round it, we will reach the same value of 0.15.
