Growth Rates And How To Calculate Them
Growth rates and how to calculate them.
Growth rates can be tricky to calculate and interpret and many people get confused. So here’s how to get ahead of everyone.
Let’s start with a time series where we know the answer. In the example below, X starts at 100, grows
3%, then falls back again, then grows 3% again. So over the three years, it has grown from 100 to 103.
1
Year
2000
2001
2002
2003
Average
CAGR
2
3
4
X
Growth X
DlnX
100
103
0.03 0.0295588
100 -0.0291262 -0.0295588
103
0.03 0.0295588
0.01029126 0.00985293
0.00990163
Column 3 calculates by Xt/Xt-1 -1 and column 4 by ln(Xt)-ln(Xt-1), click on the spreadsheet to check.
What can we say?
1. look at column 3. Growth 2000 to 2001 is +0.03, but from 01-02 is -0.029. That means that using this growth rate, X does not quite return to where it was in 2000. This is clearly wrong, since column 2 shows that it does. This is a problem in using growth rates, due to the fact that the base year is 100 in the 00-01 and 103 in the 01-02 period, so the sum of the growth rates does not get you back to where you started.
2. now look at column 4. Using the natural logs solves this problem: the rise from 100 to 103 is exactly the same as the fall from 103 to 100, so you end up where you started. So this is a good property. However, as you can see, the change in natural logs does not quite give the same answer as the growth rate: so between 2000 and 2001, the change in natural logs is 2.95588% whereas the correct answer is 3%. Its close, but strictly speaking is different.
1
3. what can we say about the average growth rate over the whole period? Using natural logs we can just take an simple average of the per year growth rates, which is very convenient. Since X has gone from 100 to 103 in the three years, we get an average DlnX of pretty close to 1%.
4. What if we took an average of the growth rates? We get an average of 1.02%. Is this right?
No. The reason is this. If X grows at g% per year then after one year X(t+1)=
Growth rates can be tricky to calculate and interpret and many people get confused. So here’s how to get ahead of everyone.
Let’s start with a time series where we know the answer. In the example below, X starts at 100, grows
3%, then falls back again, then grows 3% again. So over the three years, it has grown from 100 to 103.
1
Year
2000
2001
2002
2003
Average
CAGR
2
3
4
X
Growth X
DlnX
100
103
0.03 0.0295588
100 -0.0291262 -0.0295588
103
0.03 0.0295588
0.01029126 0.00985293
0.00990163
Column 3 calculates by Xt/Xt-1 -1 and column 4 by ln(Xt)-ln(Xt-1), click on the spreadsheet to check.
What can we say?
1. look at column 3. Growth 2000 to 2001 is +0.03, but from 01-02 is -0.029. That means that using this growth rate, X does not quite return to where it was in 2000. This is clearly wrong, since column 2 shows that it does. This is a problem in using growth rates, due to the fact that the base year is 100 in the 00-01 and 103 in the 01-02 period, so the sum of the growth rates does not get you back to where you started.
2. now look at column 4. Using the natural logs solves this problem: the rise from 100 to 103 is exactly the same as the fall from 103 to 100, so you end up where you started. So this is a good property. However, as you can see, the change in natural logs does not quite give the same answer as the growth rate: so between 2000 and 2001, the change in natural logs is 2.95588% whereas the correct answer is 3%. Its close, but strictly speaking is different.
1
3. what can we say about the average growth rate over the whole period? Using natural logs we can just take an simple average of the per year growth rates, which is very convenient. Since X has gone from 100 to 103 in the three years, we get an average DlnX of pretty close to 1%.
4. What if we took an average of the growth rates? We get an average of 1.02%. Is this right?
No. The reason is this. If X grows at g% per year then after one year X(t+1)=