Introduction

Before we measure South Africa’s economic growth rate, we need to deal with the issue of annualised rates of change.

An annualised rate is a rate of change for a given period that is less than a year, but it is calculated as if the rate were for a full year.  In other words, an annualised growth rate is  adjusted to reflect what would have happened if that pace were sustained for a full 12 months. The main benefit of an annualised rate is that it is easily comparable to other annualised data.

The following formula can be used to calculate an annualised rate between two successive periods:

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl)^f - 1
\Biggl]
\times 100
$$

Where:

I = Variable in question
t – 1 = Initial period
t = Current period
f = Frequency of time series

Meaning of and formula for calculating annualised rates of change: Activities

Monthly data

1.This activity deals with calculating the annualised rate based on monthly data.

You are given the following information on employment in the manufacturing sector:

January 2017:   1 252 163
February 2017:  1 263 165

Use the formula for calculating an annualised rate between two successive periods to calculate the annualised growth rate for employment in the manufacturing sector based on monthly data.

a. What is the value of It?

Think again.

It is the value of the current period. In this case the current period is the employment for February 2017, which is equal to 1 263 165.

Correct.

It is the value of the current period. In this case the current period is the employment for February 2017, which is equal to 1 263 165.

b. What is the value of It - 1?

Correct.

It-1 is the value of the initial period. In this case the initial period is the employment for January 2017, which is equal to 1 252 163.

Think again.

It-1 is the value of the initial period. In this case the initial period is the employment for January 2017, which is equal to 1 252 163.

c. What is the value of f?

Correct.

The frequency of this time series is monthly.  In other words, it occurs 12 times a year.  The value of f  is therefore 12/1 = 12.

Think again.

The frequency of this time series is monthly.  In other words, it occurs 12 times a year.  The value of f  is therefore 12/1 = 12.

Think again.

The frequency of this time series is monthly.  In other words, it occurs 12 times a year.  The value of f  is therefore 12/1 = 12.

d. The annualised growth rate (one decimal place) for employment in the manufacturing sector is …

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl)^f - 1
\Biggl]
\times 100
$$

%

Correct.

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl)^f - 1
\Biggl]
\times 100
$$

Where:

I = Value of employment in manufacturing
t – 1 = Initial period = 1 252 165
t = Current period  = 1 263 165
f = Frequency of time series = 12/1 = 12

$$
\Biggl[
\Bigl(
\frac{\text{1 263 165}}{\text{1 252 163}}
\Bigl)^{12} - 1
\Biggl]
\times 100
$$

= (1,00878612 – 1) x 100
= (1,110684 -1) x 100
= 0,110684 x 100
= 11,1%

[TODO:insert video clip]

Incorrect.

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl)^f - 1
\Biggl]
\times 100
$$

Where:

I = Value of employment in manufacturing
t – 1 = Initial period = 1 252 165
t = Current period  = 1 263 165
f = Frequency of time series = 12/1 = 12

$$
\Biggl[
\Bigl(
\frac{\text{1 263 165}}{\text{1 252 163}}
\Bigl)^{12} - 1
\Biggl]
\times 100
$$

= (1,00878612 – 1) x 100
= (1,110684 -1) x 100
= 0,110684 x 100
= 11,1%

Quarterly data

  1. This activity is based on the formula for calculating an annualised rate between two successive periods.

You are given the following quarterly data for real GDP (R billions) for South Africa:

Quarter 1 of 2017: 3 093 404
Quarter 2 of 2017: 3 115 727

Use the formula for calculating an annualised rate between two successive periods to calculate the annualised growth rate for real GDP based on quarterly data.

a. What is the value of It?

Think again.

This is the value for the initial period  It-1 . It is the value of the current period.  In this case the current period is the data for quarter 2, which is equal to 3 115 727.

Correct.

It is the value of the current period.  In this case the current period is the data for quarter 2, which is equal to 3 115 727.

b. What is the value of It - 1?

Correct. 

It-1 is the value of the initial period.  In this case the initial value is the data for quarter 1, which is equal to 3 093 404.

Think again.

This is the value for the current period It.  It-1 is the value of the initial period.  In this case the initial value is the data for quarter 1, which is equal to 3 093 404.

c. What is the value of f?

Think again.

If the frequency is 6, it indicates that the frequency of the data is twice a year (every six months): 12/2 = 6.

Correct.

This data is published every three months and the frequency is therefore 12/3 = 4.

Think again.

If the frequency is 3, it indicates that the data is published every four months: 12/4 = 3.

d. The annualised growth rate for real GDP for the second quarter is …

$$ \Biggl[ \Bigl( \frac{It}{It-1} \Bigl)^f - 1 \Biggl] \times 100 $$
%

Correct.

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl)^f - 1
\Biggl]
\times 100
$$

Where:

I = Quarterly GDP
t – 1 = Initial period = 3 093 404
t = Current period  = 3 115 727
f = Frequency of time series = 12/3 = 4

$$
\Biggl[
\Bigl(
\frac{\text{3 115 727}}{\text{3 093 404}}
\Bigl) - 1
\Biggl]
\times 100
$$

= (1,007216)4  –  1) x 100
= (1,029179 – 1) x 100
= 0,029179 x 100
= 2,9%

Incorrect.

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl)^f - 1
\Biggl]
\times 100
$$

Where:

I = Quarterly GDP
t – 1 = Initial period = 3 093 404
t = Current period  = 3 115 727
f = Frequency of time series = 12/3 = 4

$$
\Biggl[
\Bigl(
\frac{\text{3 115 727}}{\text{3 093 404}}
\Bigl) - 1
\Biggl]
\times 100
$$

= (1,007216)4  –  1) x 100
= (1,029179 – 1) x 100
= 0,029179 x 100
= 2,9%

e. What was the percentage increase in real GDP between the two periods?

$$ \Biggl[ \Bigl( \frac{It}{It-1} \Bigl) - 1 \Biggl] \times 100 $$

Think again.

This is the annualised rate of change. To calculate the percentage change between the two periods, use the following formula:

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl) - 1
\Biggl]
\times 100
$$

$$
\Biggl[
\Bigl(
\frac{\text{3 115 727}}{\text{3 093 404}}
\Bigl) - 1
\Biggl]
\times 100
$$

= (1,007216 – 1) x 100
= 0,007216 x 100
= 0,7

Correct.

This is the percentage change between the two quarters.

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl) - 1
\Biggl]
\times 100
$$

$$
\Biggl[
\Bigl(
\frac{\text{3 115 727}}{\text{3 093 404}}
\Bigl) - 1
\Biggl]
\times 100
$$

= (1,007216 – 1) x 100
= 0,007216 x 100
= 0,7%

f. To obtain an annual growth rate from data for two successive quarters, the rate of change between the two quarters simply has to be multiplied by 4.

Incorrect.  The statement is false.

Based on the data for the above example, the percentage change between quarter 1 and quarter 2 is 0,7%.  By multiplying it by 4, the answer is 2,8%.  Using the formula for calculating annualised changes, the answer is 2,9%.

Correct.  The statement is indeed false.

Based on the data for the above example, the percentage change between quarter 1 and quarter 2 is 0,7%.  By multiplying it by 4, the answer is 2,8%.  Using the formula for calculating annualised changes, the answer is 2,9%.

Growth rate for successive annual totals

If you need to calculate the growth rate between successive annual totals, the formula is:

$$
\Biggl[
\Bigl(
\frac{It}{It-1}
\Bigl) - 1
\Biggl]
\times 100
$$

Where:

I = Variable in question
t – 1 = Initial period
t = Current period

3. Calculate the growth rate between real GDP for 2016 and 2017:

Real GDP for 2016:  3 076 465
Real GDP for 2017:  3 119 984

a. The growth rate is …

%

Correct.

$$
\Biggl[
\Bigl(
\frac{\text{3 119 985}}{\text{3 076 465}}
\Bigl) - 1
\Biggl]
\times 100
$$

= (1,014146 – 1)  100
= 0,014146    100
= 1,4%

Incorrect.

$$
\Biggl[
\Bigl(
\frac{\text{3 119 985}}{\text{3 076 465}}
\Bigl) - 1
\Biggl]
\times 100
$$

= (1,014146 – 1)  100
= 0,014146  x  100
= 1,4%

Calculate the average annual growth rates for periods spanning more than one year

The general formula is as follows:

$$
\Biggl[
\Bigl(
\frac{It}{It - k}
\Bigl)^{1/k} - 1
\Biggl]
\times 100
$$

Where:

I = Variable in question
t = Final period (year)
t – k = Initial period (year
k = Number of years over which the rate is calculated

  1. Use the following information to calculate the average annual growth rate for the period 2013 to 2018:

Real GDP for 2013:  2 973 175
Real GDP for 2018:  3 144 539

 

a. What is the value of It?

Think again.

This is the value for the initial period It-1.  It is the value of the current period.  In this case the current period is the real GDP for 2018, which is 3 144 539.

Correct.

It is the value of the current period.  In this case the current period is the real GDP for 2018, which is 3 144 539.

b. What is the value of It-k?

Correct.

It-k is the value of the initial period.  In this case the initial period is the real GDP for 2013, which is 2 973 175.

Think again.

It-k is the value of the initial period.  In this case the initial period is the real GDP for 2013, which is 2 973 175.

c. What is the value of k?

Think again.

k is the number of years over which the rate is calculated.  In this case it is for 2013 to 2018, which is 5 years.

Correct.

k is the number of years over which the rate is calculated.  In this case it is for 2013 to 2018, which is 5 years.

d. The average annual growth rate for the period 2013 to 2018 is …

$$
\Biggl[
\Bigl(
\frac{It}{It - k}
\Bigl)^{1/k} - 1
\Biggl]
\times 100
$$

%

Correct.

$$
\Biggl[
\Bigl(
\frac{\text{3 144 539}}{\text{2 973 175}}
\Bigl)^{1/5} - 1
\Biggl]
\times 100
$$

= (1,0576371/5 – 1) x 100
= (1,01127  – 1) x 100
= 0,01127  x 100
= 1,1%

Incorrect.

$$
\Biggl[
\Bigl(
\frac{\text{3 144 539}}{\text{2 973 175}}
\Bigl)^{1/5} - 1
\Biggl]
\times 100
$$

= (1,0576371/5 – 1) x 100
= (1,01127  – 1) x 100
= 0,01127  x 100
= 1,1%