CGMA BA1

11. Time series model

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Seasonal variation and forecasting methods

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Seasonal variations

The formulas for seasonal variation are:

$S S = \frac{Y}{T} (multiplicative model) = Y - T (additive model)$

Let’s use the data from Example 2 to calculate the seasonal variations used for forecasting.

Year	Quarter	Sales
2022	Q1	1,020
	Q2	1,300
	Q3	1,100
	Q4	1,400
2023	Q1	1,400
	Q2	1,900
	Q3	1,700
	Q4	2,150
2024	Q1	1,700
	Q2	2,500
	Q3	2,100
	Q4	2,790

The trendline is based on the following equation:

$Y = 900 + 140 x$

where $x$ represents the time period. The trend values for all quarters are:

Year	Quarter	Trend
2022	Q1	1,040
	Q2	1,180
	Q3	1,320
	Q4	1,460
2023	Q1	1,600
	Q2	1,740
	Q3	1,880
	Q4	2,020
2024	Q1	2,160
	Q2	2,300
	Q3	2,440
	Q4	2,580

The trend values above come directly from the linear equation. If you’re unsure how to apply the equation, refer back to the previous chapter.

Now calculate the seasonal variation for each quarter using $S = Y / T$ . For example:

$\frac{1 , 020}{1 , 040} = 0.9808$

Arrange the results by quarter across years:

	Q1	Q2	Q3	Q4
2022	0.9808	1.1017	0.8333	0.9589
2023	0.8750	1.0920	0.9043	1.0644
2024	0.7870	0.9130	0.8607	1.0814
Average	0.8809	1.0356	0.8661	1.0349

The average seasonal factor for each quarter is calculated by adding the three values in that column and dividing by 3.

Under the multiplicative model, the four quarterly averages must add up to 4. The current total is 3.8175, so an adjustment is needed:

$4 - 3.8175 0.1825 \div 4 = 0.1825 = 0.045625$

Add 0.045625 to each quarterly average. The adjusted seasonal factors are:

Q1	Q2	Q3	Q4	Total
0.9266	1.0812	0.9117	1.0805	4.0000

These adjusted values are the seasonal variations used for forecasting. When forecasting the next period, you:

calculate the trend
combine the trend with the seasonal variation to get the actual forecast

Example 3

KTA uses the equation $Y = 900 + 140 x$ to forecast sales, where $x$ is the period number and the seasonal variation for Q4 is 1.0805. The sales manager wants the forecast for Q4 2025. Q1 2022 is period 1. Calculate the sales forecast for Q4 2025.

Solution

(spoiler)

Q4 2025 is period 16 (counting from Q1 2022).

Step 1 — Calculate the trend:

$Y Y = 900 + 140 (16) = 3, 140$

Step 2 — Apply the seasonal variation:

$Forecast = T \times S = 3, 140 \times 1.0805 = 3, 392.77$

A question might tell you which method to use (additive or multiplicative). If it doesn’t, use the information provided to choose the appropriate method. In this question, the seasonal variation is given as a multiplying factor, so the multiplicative model is appropriate.

Seasonal variation under additive model

Under the additive model, the four quarterly averages must add up to zero. If they don’t, adjust each quarterly average equally until the total becomes zero.

The formula for seasonal variation is:

$S = Y - T$

For example:

$1, 020 - 1, 040 = - 20$

Calculate the quarterly variations for all years:

	Q1	Q2	Q3	Q4
2022	-20	120	-220	-60
2023	-200	160	-180	130
2024	-460	490	-340	210
Average	-226.67	256.67	-246.67	93.33

Averages are calculated the same way as in the multiplicative model.

The sum of the averages is -123.33. Because the total must be zero, divide by 4 to find the per-quarter adjustment:

$- 123.33 \div 4 = - 30.83$

Add 30.83 to each quarterly average:

$Q 1 : - 226.67 + 30.83 Q 2 : 256.67 + 30.83 Q 3 : - 246.67 + 30.83 Q 4 : 93.33 + 30.83 = - 195.83 = 287.50 = - 215.83 = 124.17$

Check that the adjusted values sum to zero:

$- 195.83 + 287.50 - 215.83 + 124.17 = 0$

These adjusted values are the seasonal variations for the additive model.

Example 4

KTA uses the equation $Y = 900 + 140 x$ to forecast sales, where $x$ is the period number and the seasonal variation for Q4 is $+ 124.17$ . The sales manager wants the forecast for Q4 2025. Q1 2022 is period 1. Calculate the actual sales forecast for Q4 2025 using the additive model.

Solution

(spoiler)

Q4 2025 is period 16 (counting from Q1 2022).

Step 1 — Calculate the trend:

$Y Y = 900 + 140 (16) = 3, 140$

Step 2 — Apply the seasonal variation:

$Forecast = T + S = 3, 140 + 124.17 = 3, 264.17$

A question might tell you which method to use. If it doesn’t, use the information provided to choose the appropriate method. In this question, the seasonal variation is given as a real value, so the additive model is appropriate.

An examination question may ask for a ‘seasonally adjusted figure’, which is the same as the trend. In that case, rearrange the formula to solve for trend:

$T T = Y + S (additive model) = \frac{Y}{S} (multiplicative model)$

Next | Wrapping up

Seasonal variation and forecasting methods

Seasonal variations

The formulas for seasonal variation are:

$S S = \frac{Y}{T} (multiplicative model) = Y - T (additive model)$

Let’s use the data from Example 2 to calculate the seasonal variations used for forecasting.

Year	Quarter	Sales
2022	Q1	1,020
	Q2	1,300
	Q3	1,100
	Q4	1,400
2023	Q1	1,400
	Q2	1,900
	Q3	1,700
	Q4	2,150
2024	Q1	1,700
	Q2	2,500
	Q3	2,100
	Q4	2,790

The trendline is based on the following equation:

$Y = 900 + 140 x$

where $x$ represents the time period. The trend values for all quarters are:

Year	Quarter	Trend
2022	Q1	1,040
	Q2	1,180
	Q3	1,320
	Q4	1,460
2023	Q1	1,600
	Q2	1,740
	Q3	1,880
	Q4	2,020
2024	Q1	2,160
	Q2	2,300
	Q3	2,440
	Q4	2,580

The trend values above come directly from the linear equation. If you’re unsure how to apply the equation, refer back to the previous chapter.

Now calculate the seasonal variation for each quarter using $S = Y / T$ . For example:

$\frac{1 , 020}{1 , 040} = 0.9808$

Arrange the results by quarter across years:

	Q1	Q2	Q3	Q4
2022	0.9808	1.1017	0.8333	0.9589
2023	0.8750	1.0920	0.9043	1.0644
2024	0.7870	0.9130	0.8607	1.0814
Average	0.8809	1.0356	0.8661	1.0349

The average seasonal factor for each quarter is calculated by adding the three values in that column and dividing by 3.

Under the multiplicative model, the four quarterly averages must add up to 4. The current total is 3.8175, so an adjustment is needed:

$4 - 3.8175 0.1825 \div 4 = 0.1825 = 0.045625$

Add 0.045625 to each quarterly average. The adjusted seasonal factors are:

Q1	Q2	Q3	Q4	Total
0.9266	1.0812	0.9117	1.0805	4.0000

These adjusted values are the seasonal variations used for forecasting. When forecasting the next period, you:

calculate the trend
combine the trend with the seasonal variation to get the actual forecast

Example 3

KTA uses the equation $Y = 900 + 140 x$ to forecast sales, where $x$ is the period number and the seasonal variation for Q4 is 1.0805. The sales manager wants the forecast for Q4 2025. Q1 2022 is period 1. Calculate the sales forecast for Q4 2025.

Solution

(spoiler)

Q4 2025 is period 16 (counting from Q1 2022).

Step 1 — Calculate the trend:

$Y Y = 900 + 140 (16) = 3, 140$

Step 2 — Apply the seasonal variation:

$Forecast = T \times S = 3, 140 \times 1.0805 = 3, 392.77$

Seasonal variation under additive model

Under the additive model, the four quarterly averages must add up to zero. If they don’t, adjust each quarterly average equally until the total becomes zero.

The formula for seasonal variation is:

$S = Y - T$

For example:

$1, 020 - 1, 040 = - 20$

Calculate the quarterly variations for all years:

	Q1	Q2	Q3	Q4
2022	-20	120	-220	-60
2023	-200	160	-180	130
2024	-460	490	-340	210
Average	-226.67	256.67	-246.67	93.33

Averages are calculated the same way as in the multiplicative model.

The sum of the averages is -123.33. Because the total must be zero, divide by 4 to find the per-quarter adjustment:

$- 123.33 \div 4 = - 30.83$

Add 30.83 to each quarterly average:

$Q 1 : - 226.67 + 30.83 Q 2 : 256.67 + 30.83 Q 3 : - 246.67 + 30.83 Q 4 : 93.33 + 30.83 = - 195.83 = 287.50 = - 215.83 = 124.17$

Check that the adjusted values sum to zero:

$- 195.83 + 287.50 - 215.83 + 124.17 = 0$

These adjusted values are the seasonal variations for the additive model.

Example 4

KTA uses the equation $Y = 900 + 140 x$ to forecast sales, where $x$ is the period number and the seasonal variation for Q4 is $+ 124.17$ . The sales manager wants the forecast for Q4 2025. Q1 2022 is period 1. Calculate the actual sales forecast for Q4 2025 using the additive model.

Solution

(spoiler)

Q4 2025 is period 16 (counting from Q1 2022).

Step 1 — Calculate the trend:

$Y Y = 900 + 140 (16) = 3, 140$

Step 2 — Apply the seasonal variation:

$Forecast = T + S = 3, 140 + 124.17 = 3, 264.17$

An examination question may ask for a ‘seasonally adjusted figure’, which is the same as the trend. In that case, rearrange the formula to solve for trend:

$T T = Y + S (additive model) = \frac{Y}{S} (multiplicative model)$

Seasonal variations

The formulas for seasonal variation are:

$S S = \frac{Y}{T} (multiplicative model) = Y - T (additive model)$

Let’s use the data from Example 2 to calculate the seasonal variations used for forecasting.

Year	Quarter	Sales
2022	Q1	1,020
	Q2	1,300
	Q3	1,100
	Q4	1,400
2023	Q1	1,400
	Q2	1,900
	Q3	1,700
	Q4	2,150
2024	Q1	1,700
	Q2	2,500
	Q3	2,100
	Q4	2,790

The trendline is based on the following equation:

$Y = 900 + 140 x$

where $x$ represents the time period. The trend values for all quarters are:

Year	Quarter	Trend
2022	Q1	1,040
	Q2	1,180
	Q3	1,320
	Q4	1,460
2023	Q1	1,600
	Q2	1,740
	Q3	1,880
	Q4	2,020
2024	Q1	2,160
	Q2	2,300
	Q3	2,440
	Q4	2,580

The trend values above come directly from the linear equation. If you’re unsure how to apply the equation, refer back to the previous chapter.

Now calculate the seasonal variation for each quarter using $S = Y / T$ . For example:

$\frac{1 , 020}{1 , 040} = 0.9808$

Arrange the results by quarter across years:

	Q1	Q2	Q3	Q4
2022	0.9808	1.1017	0.8333	0.9589
2023	0.8750	1.0920	0.9043	1.0644
2024	0.7870	0.9130	0.8607	1.0814
Average	0.8809	1.0356	0.8661	1.0349

The average seasonal factor for each quarter is calculated by adding the three values in that column and dividing by 3.

Under the multiplicative model, the four quarterly averages must add up to 4. The current total is 3.8175, so an adjustment is needed:

$4 - 3.8175 0.1825 \div 4 = 0.1825 = 0.045625$

Add 0.045625 to each quarterly average. The adjusted seasonal factors are:

Q1	Q2	Q3	Q4	Total
0.9266	1.0812	0.9117	1.0805	4.0000

These adjusted values are the seasonal variations used for forecasting. When forecasting the next period, you:

calculate the trend
combine the trend with the seasonal variation to get the actual forecast

Example 3

KTA uses the equation $Y = 900 + 140 x$ to forecast sales, where $x$ is the period number and the seasonal variation for Q4 is 1.0805. The sales manager wants the forecast for Q4 2025. Q1 2022 is period 1. Calculate the sales forecast for Q4 2025.

Solution

(spoiler)

Q4 2025 is period 16 (counting from Q1 2022).

Step 1 — Calculate the trend:

$Y Y = 900 + 140 (16) = 3, 140$

Step 2 — Apply the seasonal variation:

$Forecast = T \times S = 3, 140 \times 1.0805 = 3, 392.77$

Seasonal variation under additive model

Under the additive model, the four quarterly averages must add up to zero. If they don’t, adjust each quarterly average equally until the total becomes zero.

The formula for seasonal variation is:

$S = Y - T$

For example:

$1, 020 - 1, 040 = - 20$

Calculate the quarterly variations for all years:

	Q1	Q2	Q3	Q4
2022	-20	120	-220	-60
2023	-200	160	-180	130
2024	-460	490	-340	210
Average	-226.67	256.67	-246.67	93.33

Averages are calculated the same way as in the multiplicative model.

The sum of the averages is -123.33. Because the total must be zero, divide by 4 to find the per-quarter adjustment:

$- 123.33 \div 4 = - 30.83$

Add 30.83 to each quarterly average:

$Q 1 : - 226.67 + 30.83 Q 2 : 256.67 + 30.83 Q 3 : - 246.67 + 30.83 Q 4 : 93.33 + 30.83 = - 195.83 = 287.50 = - 215.83 = 124.17$

Check that the adjusted values sum to zero:

$- 195.83 + 287.50 - 215.83 + 124.17 = 0$

These adjusted values are the seasonal variations for the additive model.

Example 4

KTA uses the equation $Y = 900 + 140 x$ to forecast sales, where $x$ is the period number and the seasonal variation for Q4 is $+ 124.17$ . The sales manager wants the forecast for Q4 2025. Q1 2022 is period 1. Calculate the actual sales forecast for Q4 2025 using the additive model.

Solution

(spoiler)

Q4 2025 is period 16 (counting from Q1 2022).

Step 1 — Calculate the trend:

$Y Y = 900 + 140 (16) = 3, 140$

Step 2 — Apply the seasonal variation:

$Forecast = T + S = 3, 140 + 124.17 = 3, 264.17$

An examination question may ask for a ‘seasonally adjusted figure’, which is the same as the trend. In that case, rearrange the formula to solve for trend:

$T T = Y + S (additive model) = \frac{Y}{S} (multiplicative model)$