Performance Management · Analytical techniques in budgeting and forecasting

Time Series Analysis and Seasonal Variations for ACCA PM

Updated 11 October 2026 · Fact-checked

Time series analysis splits past data into trend, seasonal, cyclical and random parts. You find the trend with moving averages, measure the seasonal variation as the gap (additive) or ratio (multiplicative) between actual and trend, average it by season, then add or multiply it onto a projected trend to forecast.

Understand Time Series Analysis and Seasonal Variations

A time series is a set of figures recorded at regular intervals, such as quarterly sales. Raw figures are hard to read because several things move them at once. Time series analysis separates those effects so you can forecast.

The four components are:
- Trend (T): the underlying long-term direction, up or down.
- Seasonal variation (S): a regular pattern that repeats within a year or shorter cycle, such as high sales every Q4.
- Cyclical variation (C): longer swings, such as business cycles over several years. They are not regular enough to forecast easily.
- Random or residual variation (R): unpredictable one-off effects.

To find the trend, you smooth out the seasonal pattern with a moving average. If the pattern repeats every four quarters, a four-point average covers one full cycle, so the seasonal ups and downs cancel out. With an even number of points, the average falls between two periods. You fix this by averaging two consecutive moving averages, which gives a centred moving average that lines up with an actual period.

Then you compare actual with trend. In the additive model (Y = T + S + R) the seasonal variation is a fixed amount, such as +₹20,000. In the multiplicative model (Y = T × S × R) it is a proportion, such as 1.25 or +25%. Use additive when the swings stay about the same size as the trend grows. Use multiplicative when the swings grow with the trend. In PM exams, the question usually tells you which model to use.

Finally, you forecast. Project the trend forward, usually by extending the average movement per period, then apply the seasonal adjustment for the target period. Forecasts are only reliable if the past pattern continues, and the further ahead you go, the less reliable they are.

Key rules to remember

Additive model: Y = T + S + R; Seasonal variation is a fixed amount. Estimate S = Y − T, then average by season.
Multiplicative model: Y = T × S × R; Seasonal variation is a proportion. Estimate S = Y ÷ T, then average by season.
Four-point moving average (quarterly data): (Q1 + Q2 + Q3 + Q4) ÷ 4; Falls between the middle two periods. Roll forward one period at a time.
Centred moving average (even number of points): (MA₁ + MA₂) ÷ 2, or (sum of two consecutive 4-period totals) ÷ 8; Places the trend on an actual period so you can compare it with the actual figure.
Seasonal adjustment check (additive): Sum of average seasonal variations = 0; If not zero, spread the difference equally across the seasons.
Seasonal adjustment check (multiplicative): Sum of average seasonal factors = number of seasons (4 for quarters, 12 for months); If not, scale each factor by number of seasons ÷ actual sum.
Forecast: Additive: T + S. Multiplicative: T × S; T is the projected trend for the target period.
Seasonally adjusted figure: Additive: Y − S. Multiplicative: Y ÷ S; Strips out seasonality so you can see the underlying trend.

How to solve Time Series Analysis and Seasonal Variations questions

Use this order for any time series question. Follow it and you will not lose track of the periods.

1Check the model (additive or multiplicative) and the length of the cycle (4 for quarters, 12 for months, 7 for days).
2Calculate moving averages of that cycle length. If the cycle is even, centre them by averaging adjacent pairs.
3Match each centred average to its actual period. This is the trend for that period.
4Find the seasonal variation for each period: actual − trend (additive) or actual ÷ trend (multiplicative).
5Group by season (all Q1s, all Q2s and so on) and average. Do not mix seasons.
6Check the total: zero for additive, equal to the number of seasons for multiplicative. Adjust if it is not.
7Project the trend forward. Use the average change per period from the centred moving averages, or the trend line given.
8Apply the seasonal figure for the forecast period: add it or multiply by it. Comment on reliability if asked.

Quickest way: Rolling totals shortcut for quarterly data

When to use it: Use when you must calculate centred moving averages for quarterly data under time pressure. It avoids dividing twice.

Write the data in one column in time order.
Calculate the first four-period total, then roll: add the new figure and subtract the oldest. This is faster and less error-prone than re-adding.
Add each pair of consecutive four-period totals and divide by 8. This gives the centred trend directly.
Put each result against the correct actual period, between the first two and last two periods lost.
Check that the trend changes by a steady amount. A sudden jump usually means a slip in the rolling total.
For forecasting, take the average rise per period from the first and last trend figures: (last − first) ÷ number of gaps.

Common mistakes in Time Series Analysis and Seasonal Variations

Placing the moving average against the wrong period.
A four-point average falls between two periods, and students put it on the first or last period of the group.
Fix: Centre it by averaging two consecutive averages (or dividing consecutive total pairs by 8). The result lines up with the third period of the first total.
Mixing seasons when averaging the variations.
Students average across the table rows or columns inconsistently.
Fix: Set out the variations in a grid with years down and quarters across, then average each quarter column.
Skipping the adjustment so the variations do not total zero (or the number of seasons).
Students forget the check, or round too early.
Fix: Always add up the averages. Additive: share any difference equally, subtracting it per season. Multiplicative: scale each factor.
Using the wrong operation in the forecast.
Students add a multiplicative factor or multiply by an additive amount.
Fix: Additive means add or subtract. Multiplicative means multiply or divide. Write the model at the top of your answer.
Forecasting from the latest actual figure instead of the trend.
The last actual looks like a natural starting point.
Fix: Start from the last trend figure, extend it by the average change per period, then apply the seasonal factor.
Treating the forecast as certain.
Students stop once the number is calculated.
Fix: If the question asks for comment, say the forecast assumes the trend and seasonal pattern continue, ignores cyclical and random effects, and gets less reliable further ahead.

Worked examples

Example 1

Quarterly sales (units, 000) are: Year 1: 40, 60, 80, 50. Year 2: 48, 68, 88, 58. Year 3: 56, 76, 96, 66. Using the additive model, calculate the centred moving average trend, the average seasonal variations and a forecast for Year 4 Quarters 1 and 2.

Show the solution

Four-quarter rolling totals: 230, 238, 246, 254, 262, 270, 278, 286, 294.
Centred trend = sum of consecutive pairs ÷ 8: Y1Q3 468 ÷ 8 = 58.5; Y1Q4 60.5; Y2Q1 62.5; Y2Q2 64.5; Y2Q3 66.5; Y2Q4 68.5; Y3Q1 70.5; Y3Q2 72.5.
Variation = actual − trend: Y1Q3 80 − 58.5 = +21.5; Y1Q4 50 − 60.5 = −10.5; Y2Q1 48 − 62.5 = −14.5; Y2Q2 68 − 64.5 = +3.5; Y2Q3 88 − 66.5 = +21.5; Y2Q4 58 − 68.5 = −10.5; Y3Q1 56 − 70.5 = −14.5; Y3Q2 76 − 72.5 = +3.5.
Average by quarter: Q1 = −14.5; Q2 = +3.5; Q3 = +21.5; Q4 = −10.5. Total = −14.5 + 3.5 + 21.5 − 10.5 = 0, so no adjustment is needed.
The trend rises by 2 per quarter (72.5 − 58.5 = 14 over 7 gaps). Y3Q2 trend = 72.5, so Y3Q3 = 74.5, Y3Q4 = 76.5, Y4Q1 = 78.5, Y4Q2 = 80.5.
Forecast Y4Q1 = 78.5 − 14.5 = 64. Forecast Y4Q2 = 80.5 + 3.5 = 84.

Answer: Average seasonal variations: Q1 −14.5, Q2 +3.5, Q3 +21.5, Q4 −10.5 (000 units). Forecast sales: Year 4 Q1 = 64,000 units; Year 4 Q2 = 84,000 units.

Example 2

A multiplicative model gives these average seasonal factors: Q1 0.80, Q2 1.05, Q3 1.25, Q4 0.90. Latest trend sales (Year 4 Q4) is ₹4,00,000 per quarter, rising by ₹5,000 per quarter. (a) Forecast sales for Year 5 Q3. (b) Year 4 Q1 actual sales were ₹3,10,000. Calculate the seasonally adjusted figure.

Show the solution

Check the factors: 0.80 + 1.05 + 1.25 + 0.90 = 4.00, which equals the number of quarters, so no adjustment is needed.
(a) Year 5 Q3 is three quarters after Year 4 Q4 (Y5Q1, Y5Q2, Y5Q3).
Trend for Y5Q3 = ₹4,00,000 + 3 × ₹5,000 = ₹4,15,000.
Forecast = trend × seasonal factor = ₹4,15,000 × 1.25 = ₹5,18,750.
(b) Seasonally adjusted = actual ÷ factor = ₹3,10,000 ÷ 0.80 = ₹3,87,500.

Answer: (a) Forecast sales for Year 5 Q3 = ₹5,18,750. (b) Seasonally adjusted Year 4 Q1 sales = ₹3,87,500, which is higher than the actual because Q1 is normally a weak quarter.

Exam tips

Read which model is required and the cycle length before calculating anything. Write them at the top of your working.
In objective test questions, one wrong period in the trend gives a wrong answer and no partial marks. Check the first and last trend figures against the pattern of steady change.
Always do the total check on the seasonal figures. Examiners often set a question where the raw average does not sum to zero or to the number of seasons.
In written parts, state the limits: the forecast assumes the trend and seasonality continue, ignores cyclical and random variation, and is weaker for long horizons or with little data.
Count periods carefully when extending the trend. Number each quarter on paper and count the gap from the last trend figure.

Practice questions from Analytical techniques in budgeting and forecasting

Time Series Analysis and Seasonal Variations in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Time Series Analysis and Seasonal Variations: frequently asked questions

What is the difference between additive and multiplicative time series models?

The additive model (Y = T + S + R) treats seasonal variation as a fixed amount added to the trend. The multiplicative model (Y = T × S × R) treats it as a proportion of the trend. Use multiplicative when the seasonal swings get bigger as the trend rises.

Why do you need a centred moving average?

With an even cycle, such as four quarters, a simple moving average falls between two periods. Averaging two consecutive moving averages puts the trend on an actual period. Only then can you compare it with the actual figure.

How do you calculate a seasonal variation?

Find the centred trend for each period that has one. Then subtract it from the actual (additive) or divide actual by it (multiplicative). Group the results by season and average them, then check the total.

Why do the seasonal variations need to sum to zero?

In the additive model, seasonal effects should net out over a full cycle, otherwise they would distort the trend. If the average variations do not sum to zero, share the difference equally across the seasons. For the multiplicative model the factors should sum to the number of seasons.