Management Accounting · Analytical techniques in budgeting and forecasting

Time Series Analysis: Trend and Seasonal Variations in ACCA MA

Updated 11 October 2026 · Fact-checked

A time series is a set of values recorded over equal time intervals. You split it into trend, seasonal variation, cyclical variation and random variation. Use moving averages to find the trend, subtract it (additive) or divide by it (multiplicative) to get seasonal variations, then average them and adjust so they cancel out.

Understand Time Series Analysis: Trend and Seasonal Variations

A time series is a set of figures recorded at regular intervals, such as quarterly sales over four years. Managers use it to see what is happening over time and to forecast the future.

A time series has up to four components:

Trend (T): the underlying long-term direction, up or down.
Seasonal variation (S): a regular pattern that repeats within a year or other fixed period, such as higher sales every December.
Cyclical variation (C): longer swings, such as the economic cycle, that last over several years. These are hard to predict and are rarely calculated in MA.
Random variation (R): unpredictable one-off effects such as a strike or bad weather.

The actual figure is built from these parts. In the additive model: A = T + S + R. Seasonal variations are fixed amounts, such as +$20,000 in quarter 4. In the multiplicative model: A = T × S × R. Seasonal variations are proportions, such as 1.10 or 110% of trend. Use the multiplicative model when seasonal swings grow as the trend grows. The question will normally tell you which model to use.

To find the trend, you smooth out the seasonal pattern with a moving average. If you average one full cycle (for example four quarters), the highs and lows cancel and what is left is the trend. Then you compare each actual figure with its trend value. The difference (or ratio) is the seasonal variation for that period. You average these across cycles, because each one also contains some random variation.

Finally, you check the seasonal variations sum to zero (additive) or average 1, so they sum to the number of periods (multiplicative). If they do not, you adjust them. You can then forecast by extending the trend and adding or multiplying by the seasonal variation. Forecasts are only reliable if the past pattern continues.

Key formulas to remember

Additive model: A = T + S + R; Seasonal variation S is an absolute amount. Seasonal variations should sum to zero.
Multiplicative model: A = T × S × R; S is a proportion or percentage of trend. Seasonal factors should average 1 (sum to the number of periods).
Seasonal variation, additive: S + R = A − T; Calculate for each period, then average by season to remove R.
Seasonal variation, multiplicative: S × R = A ÷ T; Calculate for each period, then average by season to remove R.
Moving average, odd number of periods: Trend = sum of n values ÷ n; Place the result against the middle period.
Centred moving average, even number of periods (e.g. quarters): Trend = (½ first + middle three + ½ last) ÷ 4, or the average of two consecutive 4-period averages; Needed so the trend lines up with an actual period, not between two periods.
Seasonal adjustment of actual data: Additive: A − S. Multiplicative: A ÷ S; Gives the deseasonalised figure, which shows the underlying trend.
Forecast: Forecast = projected trend + S (additive), or projected trend × S (multiplicative); Project the trend first, usually with a regression line or the average change in trend per period.

How to solve Time Series Analysis: Trend and Seasonal Variations questions

Follow this order for any question that asks you to find the trend, seasonal variations or a forecast.

1Read which model is asked for (additive or multiplicative) and how many periods make one cycle (4 for quarters, 12 for months, 7 for days of the week).
2Calculate the moving average. For an even cycle length, centre it so it lines up with an actual period. This is the trend.
3For each period with a trend value, find A − T (additive) or A ÷ T (multiplicative).
4Group these results by season (all quarter 1s together, and so on) and take the average of each group.
5Check the totals. Additive averages should sum to zero. Multiplicative averages should sum to the number of periods. If not, spread the difference equally across the seasons.
6If asked for a forecast, extend the trend to the future period using the average change in trend per period (or the given trend line).
7Apply the seasonal variation: add it (additive) or multiply by it (multiplicative). Check your answer is sensible against the actual data.

Quickest way: Shortcut for objective test questions

When to use it: Use when the question gives you trend values or seasonal variations and asks for one number only, which is typical in Section A.

Do not rebuild the whole table. Find only the figures the question needs.
If given the trend and the seasonal variation, apply the model directly: add for additive, multiply for multiplicative.
To project the trend, work out the average change per period from the first and last trend values: (last − first) ÷ number of intervals. Count intervals, not values.
Quick sense check: a quarter with a positive additive variation should give a forecast above trend, and a multiplicative factor above 1 should too.
For a missing seasonal variation, use the total rule: additive variations sum to zero, multiplicative sum to the number of periods.

Common mistakes in Time Series Analysis: Trend and Seasonal Variations

Not centring a four-point or twelve-point moving average.
Students average four numbers and write the result against one period, but it really sits halfway between two periods.
Fix: Average two consecutive moving averages, or use ½, 1, 1, 1, ½ weights, so the trend lines up with an actual period.
Using A − T when the question asks for the multiplicative model, or the reverse.
Students rush and apply the model they practised last.
Fix: Underline the model name in the question. Write 'additive: subtract, multiplicative: divide' beside your working.
Forgetting to adjust the seasonal variations so they sum to zero (or to the number of periods).
The averages look neat, so students move on.
Fix: Always add up your averaged variations. If the total is off, divide the error by the number of seasons and correct each one.
Putting trend values against the wrong periods.
The first trend figure belongs to the third period for a centred four-quarter average, not the first.
Fix: Write the period labels next to every trend figure. The first and last two periods have no trend value.
Counting values instead of intervals when finding the average trend change.
Students divide by the number of trend figures.
Fix: Divide by the number of gaps. From trend values in periods 3 to 10, there are 7 intervals.
Treating the seasonal variation as the full forecast.
Students stop once they have calculated S.
Fix: A forecast needs both parts: projected trend and the seasonal adjustment.

Worked examples

Example 1

Additive model. Quarterly sales ($000) are: Year 1 Q1 100, Q2 120, Q3 140, Q4 110; Year 2 Q1 112, Q2 132, Q3 152, Q4 122. Calculate the trend for Year 1 Q3 and the seasonal variation (including random variation) for that quarter.

Show the solution

Centred trend = (½×100 + 120 + 140 + 110 + ½×112) ÷ 4. Q1 and the next Q1 each get half weight. This is the same as averaging Y1 Q1–Q4 and Y1 Q2–Y2 Q1.
First average: (100 + 120 + 140 + 110) ÷ 4 = 470 ÷ 4 = 117.5.
Second average: (120 + 140 + 110 + 112) ÷ 4 = 482 ÷ 4 = 120.5.
Centred trend for Y1 Q3 = (117.5 + 120.5) ÷ 2 = 119.0.
Seasonal variation plus random = A − T = 140 − 119 = 21.

Answer: Trend for Year 1 Q3 is $119,000 and the seasonal variation (including random) is +$21,000.

Example 2

Multiplicative model. The trend for quarterly demand is projected to be 2,000 units in Quarter 3 and 2,050 units in Quarter 4 of next year. The seasonal factors are Q1 0.90, Q2 1.05, Q3 1.20 and Q4 missing. Forecast demand for Quarter 3 and for Quarter 4.

Show the solution

Multiplicative factors must sum to 4 for four quarters.
Sum of known factors: 0.90 + 1.05 + 1.20 = 3.15.
Q4 factor = 4 − 3.15 = 0.85.
Quarter 3 forecast = trend × factor = 2,000 × 1.20 = 2,400 units.
Quarter 4 forecast = trend × factor = 2,050 × 0.85 = 1,742.5 units.

Answer: Missing Q4 factor is 0.85 (15% below trend). Quarter 3 forecast demand is 2,400 units and Quarter 4 forecast demand is 1,742.5 units.

Exam tips

Check the model in the question first. It decides whether you subtract or divide, and whether the total check is 0 or the number of periods.
In Section A, number entry questions often give you the trend and the seasonal figure. Apply them directly and enter the number in the format asked, for example whole units or one decimal place.
Use the sum rule to find a missing seasonal variation. It is much faster than rebuilding the table.
For multiple response questions on components, remember cyclical variation covers several years, while seasonal repeats within a year.
State the limits of forecasts if asked: they assume the past pattern continues, and the further ahead you forecast, the less reliable the result.

Practice questions from Analytical techniques in budgeting and forecasting

Time Series Analysis: Trend 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: Trend and Seasonal Variations: frequently asked questions

What is the difference between additive and multiplicative models?

The additive model treats seasonal variation as a fixed amount added to the trend, for example +$20,000. The multiplicative model treats it as a proportion of the trend, for example 1.10. Multiplicative suits data where seasonal swings grow with the trend.

Why do I need a centred moving average?

With an even number of periods, such as four quarters, a simple average falls between two periods. Centring, by averaging two consecutive averages, places the trend against an actual period so you can compare it with the actual figure.

Why must seasonal variations add up to zero?

Seasonal variations are meant to show only the pattern within the year, not any overall rise or fall. Over a full cycle the highs and lows must cancel. If your averages do not sum to zero, adjust them equally.

How do I forecast using time series analysis?

First project the trend to the future period, often using the average change in trend per period or a regression line. Then add the seasonal variation for an additive model, or multiply by the seasonal factor for a multiplicative model.