Task
The following data is available:
Determine by basic and chain methods :
– absolute increase
- growth rate, %
– growth rate, %
– average annual growth rate, %
Carry out calculations of all indicators, summarize the calculation results in a table. Draw conclusions by describing each indicator in the table in comparison with the previous or baseline indicator.
The result of this work is a detailed conclusion.
Let's give the calculations.
1. Absolute increase, units
Chain method:
In 1992: 120500–117299=3201
In 1993: 121660–120500=1160
In 1994: 119388–121660=-2272
In 1995: 119115–119388=-273
In 1996: 126388–119115=7273
In 1997: 127450–126388=1062
In 1998: 129660–127450=2210
In 1999: 130720–129660=1060
In 2000: 131950–130720=1230
In 2001: 132580–131950=630
Basic method:
In 1991: 117299–116339=960
In 1992: 120500–116339=4161
In 1993: 121660–116339=5321
In 1994: 119388–116339=3049
In 1995: 119115–116339=2776
In 1996: 126388–116339=10049
In 1997: 127450–116339=11111
In 1998: 129660–116339=13321
In 1999: 130720–116339=14381
In 2000: 131950–116339=15611
In 2001: 132580–116339=16241
2. Growth rate, %
Chain method:
In 1992: 120500/117299*100%=102.7%
In 1993: 121660/120500*100%=100.9%
In 1994: 119388/121660*100%=98.1%
In 1995: 119115/119388*100%=99.7%
In 1996: 126388/119115*100%=106.1%
In 1997: 127450/126388*100%=100.8%
In 1998: 129660/127450*100%=101.7%
In 1999: 130720/129660*100%=100.8%
In 2000: 131950/130720*100%=100.9%
In 2001: 132580/131950*100%=100.4%
Basic method:
In 1991: 117299/116339*100%=100.8%
In 1992: 120500/116339*100%=103.5%
In 1993: 121660/116339*100%=104.5%
In 1994: 119388/116339*100%=102.6%
In 1995: 119115/116339*100%=102.3%
In 1996: 126388/116339*100%=108.6%
In 1997: 127450/116339*100%=109.5%
In 1998: 129660/116339*100%=111.4%
In 1999: 130720/116339*100%=112.3%
In 2000: 131950/116339*100%=113.4%
In 2001: 132580/116339*100%=113.9%
3. Rate of increase, %
Chain method:
In 1992: (120500–117299)/117299*100%=2.7%
In 1993: (121660–120500)/120500*100%=0.9%
In 1994: (119388–121660)/121660*100%=-1.8%
In 1995: (119115–119388)/119388*100%=-0.2%
In 1996: (126388–119115)/119115*100%=6.1%
In 1997: (127450–126388)/126388*100%=0.8%
In 1998: (129660–127450)/127450*100%=1.7%
In 1999: (130720–129660)/129660*100%=0.8%
In 2000: (131950–130720)/130720*100%=0.9%
In 2001: (132580–131950)/131950*100%=0.4%
Basic method:
In 1991: (117299–116339)/116339*100%=0.8%
In 1992: (120500–116339)/116339*100%=3.5%
In 1993: (121660–116339)/116339*100%=4.5%
In 1994: (119388–116339)/116339*100%=2.6%
In 1995: (119115–116339)/116339*100%=2.3%
In 1996: (126388–116339)/116339*100%=8.6%
In 1997: (127450–116339)/116339*100%=9.5%
In 1998: (129660–116339)/116339*100%=11.4%
In 1999: (130720–116339)/116339*100%=12.3%
In 2000: (131950–116339)/116339*100%=13.4%
In 2001: (132580–116339)/116339*100%=13.9%
4. Average annual growth rate, %
Chain method:
Tr =100,9%*100,4% = 102,9%
Basic method:
113,4%*113,9% = 109,9%
Let's summarize the obtained data in a table.
Dynamics of indicators of absolute increase (decrease), growth rate (decrease), rate of increase (decrease) in the presence of stolen motorcycles in Arkhangelsk in the period from 1990 to 2001, calculated by the basic and chain methods
| № | Years | Presence of stolen motorcycles, units | Absolute increase (decrease) in the number of stolen motorcycles, units | Growth (decrease) rate of stolen motorcycles, % | Rate of increase (decrease) in the number of stolen motorcycles, % | |||
| Chain method | Basic method | Chain method | Basic method | Chain method | Basic method | |||
| 1 | 1990 | 116339 | - | - | - | 100,0 | - | 100,1 |
| 2 | 1991 | 117299 | 960 | 960 | 100,8 | 100,8 | 0,8 | 0,8 |
| 3 | 1992 | 120500 | 3201 | 4161 | 102,7 | 103,5 | 2,7 | 3,5 |
| 4 | 1993 | 121660 | 1160 | 5321 | 100,9 | 104,5 | 0,9 | 4,5 |
| 5 | 1994 | 119388 | -2272 | 3049 | 98,1 | 102,6 | -1,8 | 2,6 |
| 6 | 1995 | 119115 | -273 | 2776 | 99,7 | 102,3 | -0,2 | 2,3 |
| 7 | 1996 | 126388 | 7273 | 10049 | 106,1 | 108,6 | 6,1 | 8,6 |
| 8 | 1997 | 127450 | 1062 | 11111 | 100,8 | 109,5 | 0,8 | 9,5 |
| 9 | 1998 | 129660 | 2210 | 13321 | 101,7 | 111,4 | 1,7 | 11,4 |
| 10 | 1999 | 130720 | 1060 | 14381 | 100,8 | 112,3 | 0,8 | 12,3 |
| 11 | 2000 | 131950 | 1230 | 15611 | 100,9 | 113,4 | 0,9 | 13,4 |
| 12 | 2001 | 132580 | 630 | 16241 | 100,4 | 113,9 | 0,4 | 13,9 |
In 1990, the number of stolen motorcycles in Arkhangelsk amounted to 116,339 units.
In 1991, the number of stolen motorcycles in Arkhangelsk amounted to 117,299 units. The absolute increase in the number of motorcycles stolen in Arkhangelsk by chain and base methods in 1991 compared to 1990 amounted to 960 units. The growth rate of the availability of stolen motorcycles in Arkhangelsk by chain and basic methods in 1991 compared to 1990 was 100.8 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk by chain and basic methods in 1991 compared to 1990 was 0.8 percent.
In 1992, the number of stolen motorcycles in Arkhangelsk amounted to 120,500 units. The absolute increase in the number of motorcycles stolen in Arkhangelsk using the chain method in 1992 compared to 1991 was 3,201 units. The absolute increase in the number of stolen motorcycles in Arkhangelsk using the base method in 1992 compared to 1990 was 4,161 units. The growth rate of the presence of stolen motorcycles in Arkhangelsk using the chain method in 1992 compared to 1991 was 102.7 percent. The growth rate of the presence of stolen motorcycles in Arkhangelsk in the basic way in 1992 compared to 1990 was 103.5 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk by chain method in 1992 compared to 1991 was 2.7 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk on a base basis in 1992 compared to 1990 was 3.5 percent.
In 1993, the number of stolen motorcycles in Arkhangelsk amounted to 121,660 units. The absolute increase in the number of motorcycles stolen in Arkhangelsk using the chain method in 1993 compared to 1992 was 1,160 units. The absolute increase in the number of stolen motorcycles in Arkhangelsk using the base method in 1993 compared to 1990 was 5,321 units. The growth rate of the presence of stolen motorcycles in Arkhangelsk using the chain method in 1993 compared to 1992 was 100.9 percent. The growth rate of the presence of stolen motorcycles in Arkhangelsk in the basic way in 1993 compared to 1990 was 104.5 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk using the chain method in 1993 compared to 1992 was 0.9 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk on a base basis in 1993 compared to 1990 was 4.5 percent.
In 1994, the number of stolen motorcycles in Arkhangelsk amounted to 119,388 units. The absolute decrease in the number of motorcycles stolen in Arkhangelsk using the chain method in 1994 compared to 1993 amounted to 2,272 units. The absolute increase in the number of stolen motorcycles in Arkhangelsk using the base method in 1994 compared to 1990 was 3,049 units. The rate of decrease in the number of stolen motorcycles in Arkhangelsk in a chain manner in 1994 compared to 1993 was 98.1 percent. The growth rate of the presence of stolen motorcycles in Arkhangelsk in the basic way in 1994 compared to 1990 was 102.6 percent. The rate of decrease in the number of stolen motorcycles in Arkhangelsk in a chain manner in 1994 compared to 1993 was 1.8 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk on a base basis in 1994 compared to 1990 was 2.6 percent.
In 1995, the number of stolen motorcycles in Arkhangelsk amounted to 119,115 units. The absolute decrease in the number of motorcycles stolen in Arkhangelsk by chain method in 1995 compared to 1995 was 273 units. The absolute increase in the number of stolen motorcycles in Arkhangelsk using the base method in 1995 compared to 1990 was 2,776 units. The rate of decrease in the number of stolen motorcycles in Arkhangelsk in a chain manner in 1995 compared to 1994 was 99.7 percent. The growth rate of the presence of stolen motorcycles in Arkhangelsk in the basic way in 1995 compared to 1990 was 102.3 percent. The rate of decrease in the number of stolen motorcycles in Arkhangelsk in a chain manner in 1995 compared to 1994 was 0.2 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk on a base basis in 1995 compared to 1990 was 2.3 percent.
In 1996, the number of stolen motorcycles in Arkhangelsk amounted to 126,388 units. The absolute increase in the number of motorcycles stolen in Arkhangelsk using the chain method in 1996 compared to 1995 was 7,273 units. The absolute increase in the number of stolen motorcycles in Arkhangelsk using the base method in 1996 compared to 1990 was 10,049 units. The growth rate of the presence of stolen motorcycles in Arkhangelsk using the chain method in 1996 compared to 1995 was 106.1 percent. The growth rate of the presence of stolen motorcycles in Arkhangelsk in the basic way in 1996 compared to 1990 was 108.6 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk by chain method in 1996 compared to 1995 was 6.1 percent. The rate of increase in the number of stolen motorcycles in Arkhangelsk on a base basis in 1996 compared to 1990 was 8.6 percent.
The growth rate is used when analyzing any series of dynamics. The growth rate formula is often used in statistics and economics in conjunction with an indicator such as the growth rate (as a percentage).
DEFINITION
Growth rate shows how many times the indicator has changed in comparison with the base one, and rate of increase reflects how much the studied value has changed.
If the result of the calculation is a positive value, then we can talk about an increasing growth rate, but with a negative value, the rate of the value under study decreases when compared with the previous (base) period.
The growth rate formula is often used in analysis investment projects. This indicator is also often used by municipal organizations in calculations:
- calculation of population growth;
- future building needs;
- volumes of service provision, etc.
Growth rate formula
To calculate the growth rate, you need to find the ratio of the indicator being studied to the previous (basic) one, then subtract one from the result obtained. The final result is multiplied by 100 to express the result as a percentage. The growth rate formula using the first method looks like this:
Tp=((Pip/Pbp)-1)*100%
Here Tp is the growth rate,
In the case when instead actual value For the analyzed indicators, only the value of the absolute increase is known, an alternative formula is used. In this case, the percentage ratio of the absolute increase to the level in comparison with which it was calculated is found.
Тп=((Pip-Pbp)/Pbp)*100%
Here Tp is the growth rate,
Pbp – indicator of the base period,
Pip is an indicator of the period under study.
A big challenge for students is the difference between the rate of growth and the rate of increase. Let us highlight several provisions in which the difference between these values lies:
- The growth rate formula and the growth rate formula are calculated using different methods.
- The growth rate reflects the percentage of one indicator relative to another, and the growth rate shows how much it has grown.
- Based on calculations using the growth rate formula, the growth rate can be calculated, while the growth rate is not calculated using the growth rate formula.
- The growth rate does not take a negative value, while the growth rate can be either positive or negative.
Examples of problem solving
EXAMPLE 1
| Exercise | For the enterprise Severmet LLC, the following indicators are presented for 2015 and 2016: Enterprise profit 2015 – 120 million. rubles, 2016 – 110.4 million. rubles It is known that in 2017 the amount of income increased by 25 million rubles compared to 2016. |
| Solution | Let's determine the growth rate in percentage for 2015 and 2016, for which we need the growth rate formula: Tr=P 2016 /P 2015 Here Tp is the growth rate, P2015 – indicator for 2015, P2016 – indicator for 2016. Tr=110.4 million. rub./120 million rub. * 100% = 92% Growth rate denotes percentage changes in value in the current period compared to the previous one. To calculate, you need a growth rate formula: Tp=((P 2016 -P 2015)/P 2015)*100% Тп=((110.4-120)/120)*100%=-8% Or the second way: Tp=((P 2016 /P 2015)-1)*100% Тп=((110.4/120)-1)*100%=-8% Let's calculate the figures for 2017 Tr = (120 million rubles + 25 million rubles)/120 million rubles = 1.21 (or 121%) Тп=(145 million rubles/120 million rubles)-1=0.208 (or 20.8%) Conclusion. We see that the growth rate when comparing 2015 and 2016 was 92%. This means that the company’s profit in 2016 decreased by 92% compared to 2015. When calculating the growth rate, the result was a negative value (-8%), which indicates that the company’s profit in 2016 (when compared with 2015) decreased by 8%. In 2017, the profit was 121% compared to 2016. When calculating the growth rate, we see that it was 20.8%. A positive value indicates an increase in profit by exactly this amount of percent. |
| Answer | When comparing 2015 and 2016, Tr = 92%, TP = 8%, when comparing 2016 and 2017, Tr = 121%, TP = 20.8%. |
Instructions
Growth rates are expressed as percentages. If we calculate the average annual growth rate, the analyzed period under consideration is from January 1 to December 31. It coincides not only with the calendar, but also with the usually taken into account financial year. It is most convenient to take the value of the base indicator for which the growth rate will be determined as 100%. Its meaning in in absolute terms must be known by January 1st.
Determine the absolute values of the indicators at the end of each month of the year (APi). Calculate the absolute values of the increase in indicators (Pi) as the difference between two compared, one of which will be the base value of the indicators as of January 1 (To), the second - the values of the indicators at the end of each month (Pi):
APi = Po – Pi,
You should have twelve such absolute values of monthly growth, according to the number of months.
Add up all the absolute values of the increase for each month and divide the resulting amount by twelve - the number of months in a year. You will receive the average annual growth rate in absolute units (P):
P = (AP1 + AP2 + AP3 +…+ AP11 + AP12) / 12.
Determine the average annual base growth rate of KB:
Kb = P / Po, where
By - the value of the base period indicator.
Express the average annual base growth rate as a percentage and you will get the average annual growth rate (ARg):
TRsg = Kb * 100%.
Using indicators of average annual growth rates over several years, you can track the intensity of their changes over the long-term period under consideration and use the obtained values to analyze and forecast the development of the situation, industry, financial sector.
In analytical calculations, both coefficients and growth rates are equally often used. They have identical essence, but are expressed in different units of measurement.
Sources:
- business growth rate
- Let's calculate the average annual growth rate
To determine the intensity of changes in any indicators over a certain period of time, a set of characteristics is used, which are obtained by comparing several levels of indicators measured at different points on the time scale. Depending on how the measured indicators are compared with each other, the resulting characteristics are called growth coefficient, growth rate, growth rate, absolute growth or the absolute value of 1% growth.
Instructions
Determine which indicators and how should be compared with each other in order to obtain the desired value of absolute growth. Proceed from the fact that this should show the absolute rate of change of the thing under study and be calculated as the difference between the current level and the level taken as .
Subtract from the current value of the indicator under study its value measured at that point on the time scale that is taken as the base. For example, let's say that the number of workers employed in production at the beginning of the current month is 1549 people, and at the beginning of the year, which is considered the base period, it was equal to 1200 workers. In this case, for the period from the beginning of the year to the beginning of the current month it was 349 units, since 1549-1200=349.
If you need not only this indicator for one last period, but also to determine the average value of absolute growth over several periods, then you need to calculate this value for each time mark in relation to the previous one, then add the resulting values and divide them by the number of periods. For example, let’s say that we need to calculate the average value of the absolute increase in the number of people employed in production by current year. In this case, subtract the corresponding value for the beginning of January from the indicator value as of the beginning of February, then perform similar operations for the pairs March/, /March, etc. Having finished with this, add up the resulting values and divide the result by the serial number of the last month of the current year participating in the calculation.
The term " pace growth» used in industry, economics, and finance. This is a statistical quantity that allows you to analyze the dynamics of ongoing processes, the speed and intensity of the development of a particular phenomenon. For determining pace ov growth it is necessary to compare values obtained at certain intervals.
Instructions
Determine the period of time for which you need an average pace growth. Usually this period is taken calendar year or its multiple. This allows us to eliminate the influence of such factors as seasonality, caused by changing climatic conditions. In the case when the period under study is equal to a year, we speak of average annual pace Oh growth.
Rates of growth− is the ratio of the levels of a series of one period to another.
Growth rates can be calculated as base rates when all levels of the series refer to the level of the same period, taken as the base:
T R = y i /y 0 − base growth rate
and as chain ones, this is the ratio of each level of the series to the level of the previous period:
T R = y i /y i-1− chain growth rate.
Growth rates can be expressed as a ratio or percentage.
Basic growth rates characterize a continuous line of development, and chain rates characterize the intensity of development in each individual period, and the product of chain rates is equal to the base rate. And the quotient of dividing the base rates is equal to the intermediate chain rate.
8.3 Growth and growth rate. Absolute value of 1% increase.
There is a distinction between the concepts of absolute and relative growth. The absolute increase is calculated as the difference between the levels of the series and expressed in units of measurement of the indicators of the series.
If the previous level is subtracted from the subsequent level, then we have a chain absolute increase:
If the same level, the base one, is subtracted from each level, then this is the base absolute increase:
The following relationship exists between chain and basic absolute increases: the sum of successive chain increases is equal to the corresponding basic increase, which characterizes the total increase for the entire relevant period of time.
Relative score the values of absolute growth compared to the initial level give indicators of the growth rate ( T ∆ i). It is defined in two ways:
As the ratio of absolute growth (chain) to the previous level:
This is a chain growth rate.
As the ratio of the base absolute increase to the base level:
This is the base growth rate.
2 As the difference between the growth rate and one, if the growth rate is expressed by a coefficient:
T ∆ = T R-1, or
T ∆ = T R- 100 if the growth rate is expressed as a percentage.
Rate of increase shows by what percentage the size of the phenomenon increased over the period under study. If the growth rate has a minus sign, then we talk about the rate of decline.
Absolute value of 1 percent increase equal to the ratio of absolute growth (chain) to the chain growth rate, expressed as a percentage:
A i= 0.01x U i;
8.4 Calculation of average dynamics indicators
The average level of the series is called the chronological average.
Average chronological− this average value from indicators that change over time.
In an interval series with equal intervals average level series is determined by the simple arithmetic mean formula.
The average level of a series in an interval dynamics series requires that it be indicated for what period of time it was calculated (monthly average, annual average, etc.).
Example 1
Calculate the average monthly turnover for the first quarter.
Because We are given an interval series with equal intervals; we apply the simple arithmetic mean formula:
If the interval series has different intervals, then it first needs to be reduced to a series with equal intervals, and then it will be possible to use the simple arithmetic average formula.
Example 2 The following data on trade turnover are available, monetary units:
Since the indicators of moment series do not have the property of totality, the average cannot be calculated using the simple arithmetic average formula, due to the fact that the balances changed continuously throughout the month, and the data are given for a specific day.
Therefore, we will use an approximate method based on the assumption that the phenomenon under study changed uniformly throughout each month. The shorter the series interval, the less error will be made when using this assumption.
We get the formula:
This formula is used to calculate average level in moment series with equal intervals.
Example 3 There is data on the balances of building materials at the beginning of the month, den. units:
Determine the average balance for the 1st quarter.
.
If the intervals in the moment series are not equal, then the average level of the series is calculated using the formula:
where is the average level in the intervals between dates,
t- time period (series interval)
Example 4 There is data on the balances of raw materials and supplies, den. units
Find the average monthly balances of raw materials and supplies for the first half of the year.
We apply the formula:
Average absolute increase calculated in two ways:
1 As the simple arithmetic average of annual (chain) increases, i.e.
2 As the quotient of base growth divided by the number of periods:
Calculation of the average absolute value of 1% increase over several years is produced using the simple arithmetic average formula:
When calculating the average annual growth rate You cannot use a simple arithmetic average, because the sum of the annual rates will not make sense. In this case, the geometric mean is used, i.e.:
Where Tr i− annual chain growth rates;
n− number of tempos.
Since the product of chain rates is equal to the base rate, then average tempo growth can be calculated as follows:
Error: Reference source not found
When calculating using this formula, it is not necessary to know the annual growth rate. The average tempo will depend on the ratio of the initial and final levels of the series.
Example 5 The nominal wages of workers in the national economy of the Republic of Belarus are characterized by the data presented in Table 1.
Table 1 – Nominal wages of workers in the national economy of the Republic of Belarus
To analyze dynamics wages define:
average annual salary for 8 years;
annual and basic absolute increases, growth rates and wage increases;
absolute value of 1% increase;
average annual absolute growth;
average annual growth rate and average annual growth rate;
average 1% increase.
Present the results in a table and draw conclusions.
Solution
1 We determine the average annual salary using the simple arithmetic average formula
2 Annual (chain) absolute growth () is determined by the formula
where , is the value of the indicator, respectively, in the th period and the period preceding it.
For example, for 2005, thousand rubles, i.e. wages in 2005 compared to 2004 increased by 64.1 thousand rubles; for 2006 thousand R. etc.
The basic absolute increase () is determined by the formula
where , is the value of the indicator in the th and base (2004) periods, respectively.
For example, for 2005, thousand rubles; for 2006 thousand rubles, i.e. wages in 2006 compared to 2004 increased by 130.3 thousand rubles. etc.
The chain growth rate is determined by the formula
For example, for 2005, i.e. wages in 2001 compared to 2004 increased by 108.8%; for 2006, etc.
The base growth rate is determined by the formula
For example, for 2001; for 2002, i.e. wages in 2002 compared to 2000 increased by 221.2%, etc.
We find the growth rate using the formula
So, the chain growth rate
for 2005: ;
for 2006: .
Base growth rate
for 2005: ;
for 2006: .
3 The absolute value of 1% growth () is found using the formula
This indicator can also be calculated as one hundredth of the previous level:
For example, for 2005, thousand rubles; for 2006 thousand R.
Calculations of indicators for points 1, 2, 3 will be presented in Table 2
Table 2 - Indicators of wage dynamics for 2004-2011.
|
wages, |
Absolute increase, thousand rubles |
Growth rate, % |
Growth rate, % |
Absolute value of 1% increase, thousand rubles |
|||||||
|
basic |
basic |
basic | |||||||||