Look at a firm’s monthly sales for three years and you will usually see three things at once: a general direction (up or down), a repeating pattern within each year (high at Christmas, low in the rains), and some untidy noise. A time series analysis pulls these apart.
Once the direction and the pattern are known separately, they can be projected forward and put back together to forecast the future.
This topic covers the nature of a time series, its four components, the additive and multiplicative models, and the technique of moving averages used to estimate the trend, including centred moving averages for an even-length cycle. The next topic turns the trend into a line of best fit and calculates the seasonal variations. The calculations here are mechanical but easy to slip on, so each step is shown in full.
In 30 seconds
A time series is a record of a variable at regular intervals. It has four components: trend (long-term direction), seasonal variation (regular pattern within a year or week), cyclical variation (longer, irregular swings, tied to the economy) and random variation (unpredictable). Additive model: TS = T + S + C + R. Multiplicative (proportional) model: TS = T × S × C × R. Random and cyclical effects are usually ignored.
A moving average averages one full cycle at a time (5 days, 12 months, 4 quarters) to smooth out seasonal effects and reveal the trend. With an even cycle, average the moving averages in pairs to get a centred moving average that lines up with an actual period.
The nature of a time series
A time series is a sequential record of data points collected at regular intervals, such as monthly sales revenue or quarterly profit. It is a structured record of the past and a tool for predicting the future: by studying how a variable behaved over time, accountants can extend the series forward. But first the series must be split into its component parts.
The components of a time series
| Component | Symbol | Meaning | Example |
|---|---|---|---|
| Trend | T | The general, long-term direction of the data: a persistent rise or fall over time. It may reflect economic growth, inflation, technology or expansion | Quarterly profits that rise steadily over five years |
| Seasonal variation | S | Regular fluctuations repeating at fixed intervals within a year (or a week). Caused by weather, holidays or customer habits. In this chapter it means the difference between the actual value in a period and the value predicted by the trend | Retail sales rising every December; cold drinks selling more in the dry season |
| Cyclical variation | C | Swings lasting longer than a year, without a fixed period, linked to the economic or business cycle (boom, recession, recovery) | Sales growing quickly in a boom and falling in a recession, whatever the underlying trend |
| Random (irregular) variation | R | Unpredictable, non-recurring effects of unusual events. Treated as noise | A flood or strike that halts production for a week |
The term “seasonal variation” has a specific meaning in this analysis. It refers to the variation in each period of the cycle, so it could be daily, weekly, monthly or quarterly depending on the data.
What exam questions usually assume
- Questions ask you to identify a trend and seasonal variations.
- Random fluctuations cannot be predicted, so they are normally assumed to be negligible and ignored.
- There is usually too little data to identify cyclical variations, which also need many years of history, so they are ignored.
A time series is drawn as a graph with time along the horizontal axis. The syllabus calls this plot a histogram, but it is more properly a line graph or time series plot. On it, the trend is a smooth line, and the actual values bounce above and below it because of seasonal variation.
The two models
| Additive model | Multiplicative (proportional) model | |
|---|---|---|
| Formula | TS = T + S + C + R | TS = T × S × C × R |
| Assumption | The components are independent, and seasonal effects are about the same size in every period | The components interact, and seasonal effects are proportional to the trend, growing as the trend grows |
| Seasonal variation expressed as | A fixed amount above or below the trend (for example +₦4m or −₦3m) | A percentage or proportion of the trend (for example 1.10 or 0.94) |
| Suits | Businesses where the seasonal swing stays about the same in naira terms | Businesses where bigger sales mean bigger seasonal swings |
| Seasonal variations in a cycle | Add to zero | Average 1 (add to the number of periods, e.g. 4 for quarters) |
With the additive model, the forecast for a period is: trend value + seasonal variation. With the multiplicative model, it is: trend value × seasonal proportion.
Analysing a time series
There are two jobs: estimate the trend line, and calculate the seasonal variations. A forecast then takes the trend value for the future period and adjusts it for the season. Two methods give a trend line: moving averages and linear regression. They can also be used together: moving averages to reveal the trend, and regression to fit a line to the moving averages.
Moving averages
A moving average smooths out fluctuations by averaging small groups of consecutive values. The group size equals the length of the cycle. A seasonal pattern repeating every cycle is cancelled out when exactly one full cycle is averaged, leaving the trend.
| Data | Cycle | Moving average of |
|---|---|---|
| Daily, with a weekly pattern (5-day working week) | 5 days | 5 values |
| Daily, 7-day week | 7 days | 7 values |
| Monthly, with a yearly pattern | 12 months | 12 values |
| Quarterly, with a yearly pattern | 4 quarters | 4 values |
Steps
- Decide the length of the cycle. Number each period 1, 2, 3 and so on, instead of using names or dates.
- Calculate the moving averages. Average the first cycle of values, then slide the group forward one period and average again, until the data is used up. Each average belongs to the middle of the group it covers.
- Centre if needed. If the cycle has an even number of periods, the average falls between two periods (for example, between Q2 and Q3). Average each adjacent pair of moving averages to get a centred moving average that lines up with an actual period.
- Use the results as the trend to find a line of best fit (next topic).
Odd cycle: a five-day example
A shop works five days a week. Daily sales (units) over three weeks, numbering the days 1 to 15, are:
| Mon | Tue | Wed | Thu | Fri | |
|---|---|---|---|---|---|
| Week 1 (days 1-5) | 65 | 76 | 85 | 84 | 85 |
| Week 2 (days 6-10) | 80 | 91 | 100 | 99 | 100 |
| Week 3 (days 11-15) | 95 | 106 | 115 | 114 | 115 |
| Day | Sales | 5-day total | Moving average (÷ 5) |
|---|---|---|---|
| 1 | 65 | ||
| 2 | 76 | ||
| 3 | 85 | 65+76+85+84+85 = 395 | 79 |
| 4 | 84 | 76+85+84+85+80 = 410 | 82 |
| 5 | 85 | 425 | 85 |
| 6 | 80 | 440 | 88 |
| 7 | 91 | 455 | 91 |
| 8 | 100 | 470 | 94 |
| 9 | 99 | 485 | 97 |
| 10 | 100 | 500 | 100 |
| 11 | 95 | 515 | 103 |
| 12 | 106 | 530 | 106 |
| 13 | 115 | 545 | 109 |
| 14 | 114 | ||
| 15 | 115 |
The first moving average (79) is the average of days 1 to 5 and belongs to day 3, the middle day. The next (82) belongs to day 4, and so on. The moving averages rise by exactly 3 each day, so the trend is a straight line rising by 3 units a day.
The process always loses values at the start and end of the series: with a 5-point average, the first two and last two days have no moving average.
An equation for the trend can use day 3 as the anchor: trend = 79 + 3 × (day − 3) = 70 + 3 × day. Forecast for day 20 (a Friday): 70 + 3 × 20 = 130, assuming sales keep growing at 3 a day. A further adjustment for the day of the week is added in the next topic.
Centred moving averages
When the cycle has an even number of periods, a moving average falls between two periods. For quarterly data, the average of Q1 to Q4 sits at “quarter 2.5”, and the average of Q2 to Q5 at “quarter 3.5”. Seasonal variation needs a trend value opposite an actual quarter, so a second average is taken of each adjacent pair. These are the centred moving averages.
Example. Quarterly sales (₦m) over three years, with the quarters numbered 1 to 12:
| Q1 | Q2 | Q3 | Q4 | |
|---|---|---|---|---|
| Year 1 | 19 | 25 | 30 | 26 |
| Year 2 | 27 | 33 | 38 | 34 |
| Year 3 | 35 | 41 | 46 | 42 |
| Quarter | Sales | 4-quarter total | Moving average (÷ 4) | Sum of 2 moving averages | Centred moving average (÷ 2) |
|---|---|---|---|---|---|
| 1 | 19 | ||||
| 2 | 25 | 100 (Q1-Q4) | 25.0 (at 2.5) | ||
| 3 | 30 | 108 (Q2-Q5) | 27.0 (at 3.5) | 52.0 | 26.0 |
| 4 | 26 | 116 | 29.0 | 56.0 | 28.0 |
| 5 | 27 | 124 | 31.0 | 60.0 | 30.0 |
| 6 | 33 | 132 | 33.0 | 64.0 | 32.0 |
| 7 | 38 | 140 | 35.0 | 68.0 | 34.0 |
| 8 | 34 | 148 | 37.0 | 72.0 | 36.0 |
| 9 | 35 | 156 | 39.0 | 76.0 | 38.0 |
| 10 | 41 | 164 | 41.0 | 80.0 | 40.0 |
| 11 | 46 | ||||
| 12 | 42 |
The total in each row is the sum of four consecutive quarters. For example, 19 + 25 + 30 + 26 = 100, and 100 ÷ 4 = 25.0, sitting between Q2 and Q3. The centred value for Q3 is (25.0 + 27.0) ÷ 2 = 26.0. A short cut: centred moving average = (sum of the two 4-quarter totals) ÷ 8. For Q3: (100 + 108) ÷ 8 = 26.0. The first centred value is at Q3 and the last at Q10, so four values are lost, two at each end.
The ideas in more depth
Why averaging a full cycle removes seasonality
Over one complete cycle, the seasonal variations add up to zero (additive model): the highs cancel the lows. So the average of one full cycle contains only the trend (and a little noise). If you averaged fewer or more periods than a full cycle, some seasons would be counted more than others, and the seasonal effect would not cancel.
How many values are lost?
| Cycle length | Values lost at each end | Total lost |
|---|---|---|
| Odd (such as 5 or 7) | (length − 1) ÷ 2, so 2 for a 5-day cycle | 4 (5-day) or 6 (7-day) |
| Even, centred (such as 4 or 12) | Length ÷ 2, so 2 for quarters and 6 for months | 4 (quarterly) or 12 (monthly) |
This matters because the first and last periods of the data have no trend value. Later forecasts extend the trend line beyond the available moving averages.
Using the moving averages for short-term forecasts
A moving average can itself be used for a quick forecast. If the last three months’ sales are 120, 132 and 126, the three-month moving average is 126, a simple forecast for next month. This works for a stable series but lags behind a clear trend, because it averages old values with new. That is why, for trending data, a trend line fitted to the moving averages, followed by seasonal adjustment, is preferred.
Choosing between additive and multiplicative
Plot the data. If the highs and lows around the trend stay about the same size as the trend grows, use the additive model. If they widen as the trend grows, use the multiplicative model. Examiners will normally tell you which model to use.
Common assumptions
- The pattern within the cycle repeats in the same way every cycle.
- The trend continues in a straight line.
- Random variation is negligible and cyclical variation can be ignored.
- The cycle length is correctly identified.
Worked example: building the trend from quarterly data
Scenario. A firm’s quarterly sales (₦000) over three years are: Year 1: 40, 55, 70, 50. Year 2: 48, 63, 78, 58. Year 3: 56, 71, 86, 66. Calculate the four-quarter moving averages and the centred moving averages, describe the trend and find the trend value for quarter 3 of year 1.
- Four-quarter totals.
Q1-Q4: 40 + 55 + 70 + 50 = 215.
Q2-Q5: 55 + 70 + 50 + 48 = 223.
Q3-Q6: 231.
Q4-Q7: 239.
Q5-Q8: 247.
Q6-Q9: 255.
Q7-Q10: 263.
Q8-Q11: 271.
Q9-Q12: 279. - Moving averages (÷ 4). 53.75, 55.75, 57.75, 59.75, 61.75, 63.75, 65.75, 67.75, 69.75.
- Centred moving averages.
Q3: (53.75 + 55.75) ÷ 2 = 54.75.
Q4: 56.75.
Q5: 58.75.
Q6: 60.75.
Q7: 62.75.
Q8: 64.75.
Q9: 66.75.
Q10: 68.75. - Describe the trend. It rises by exactly ₦2,000 per quarter, from 54.75 at Q3 to 68.75 at Q10 (an increase of 14 over 7 quarters).
- Trend value for Q3 of year 1. This is the first centred moving average: ₦54,750.
Seeing the components together
The additive model says that each value is the trend plus the seasonal variation plus random variation. A worked illustration makes the idea concrete. Take four consecutive quarters with these components:
| Quarter | Trend (T) | Seasonal (S) | Random (R) | Actual = T + S + R |
|---|---|---|---|---|
| 1 | 100 | +8 | +1 | 109 |
| 2 | 104 | −3 | −1 | 100 |
| 3 | 108 | −12 | 0 | 96 |
| 4 | 112 | +7 | +2 | 121 |
The seasonal variations sum to zero (8 − 3 − 12 + 7 = 0), as the additive model requires. Looking only at the actual column (109, 100, 96, 121), you could not tell which part of the movement is growth and which is season. The analysis in this chapter works backwards: it starts from the actuals, estimates the trend with moving averages, and obtains the seasonal variations by difference. The random part is what is left over, and it is assumed small.
Using moving averages directly for a quick forecast
Besides estimating a trend, a moving average can be used as a simple forecast, particularly for stable series with no marked seasonality. Take the last three periods of sales: 132, 126 and 138. The simple three-period moving average is (132 + 126 + 138) ÷ 3 = 132. A weighted moving average gives more weight to recent periods. With weights of 1, 2 and 3 (the latest period weighted 3): (132 × 1 + 126 × 2 + 138 × 3) ÷ 6 = 798 ÷ 6 = 133. The weighted figure is higher because the latest sales were the highest.
The longer the average, the smoother the result, but the slower it responds to genuine changes. A 12-month average hardly moves from month to month; a 3-month average is more reactive but also more affected by random noise. Where there is a clear trend, simple moving average forecasts lag behind: they always look backwards.
A twelve-month cycle
With monthly data and a yearly pattern, the cycle is 12 months, an even number, so the same centring step is used. Each moving total covers 12 months and each moving average is placed between months 6 and 7. To centre it, average two adjacent moving averages. A shortcut: add two adjacent 12-month totals and divide by 24.
For example, if the 12-month totals ending in consecutive months are ₦1,440 and ₦1,476 (thousands), the centred moving average is (1,440 + 1,476) ÷ 24 = 121.5. Six values are lost at the start and six at the end, so a record of three years of months (36 values) yields 24 trend values.
Drawing a time series and its trend
- Put time on the horizontal axis and the variable on the vertical axis. Plot each value and join the points.
- Plot each centred moving average against its period. These points should lie on a smooth, nearly straight line.
- Draw the trend line through them, using the equation if one has been calculated.
- The vertical gap between the actual line and the trend line is the seasonal (and random) variation for each period.
- Extend the trend line to the future periods to be forecast.
A graph helps to show whether the additive or the proportional model fits. If the gaps between the actual and trend lines stay the same size as the trend rises, the additive model is suitable. If they widen as the trend rises, the proportional model is better.
Seasonal or cyclical?
Examination questions may ask you to distinguish these, and to say why cyclical variations are usually ignored.
| Seasonal variation | Cyclical variation | |
|---|---|---|
| Length | Within a year (or week) and repeats at fixed intervals | Longer than a year and varies in length |
| Cause | Weather, holidays, customer habits | The business or economic cycle: boom, recession, recovery |
| Predictability | Regular, so can be measured and forecast | Irregular timing, hard to predict |
| Example | School uniform sales peaking before each term | Construction demand growing in a boom and falling in a downturn |
| In exams | Estimated and used | Usually ignored because there are too few years of data |
Common mistakes in the exam
- Using a moving average of the wrong length. It must equal the cycle length.
- Placing the moving average against the first period of the group instead of the middle.
- Forgetting to centre when the cycle is even.
- Dividing the sum of two moving averages by 4 instead of 2, or the sum of two totals by 4 instead of 8.
- Naming the components without explaining them.
- Mixing up the additive model (add, absolute amounts) and the multiplicative model (multiply, proportions).
- Trying to forecast the random component.
Quick revision
- Time series components: trend, seasonal, cyclical, random. Ignore random and cyclical in exams.
- Additive: TS = T + S + C + R. Multiplicative: TS = T × S × C × R.
- Moving average: average one full cycle (5 days, 12 months, 4 quarters). Sits at the middle of the group.
- Even cycle: centre by averaging adjacent pairs (or divide the sum of two totals by 8 for quarters).
- Moving averages lose values at both ends.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer.
Section A style: multiple choice
Question 1. The long-term direction of a time series is its:
- Seasonal variation
- Trend
- Random variation
- Cyclical variation
Show the answer
B. It shows persistent increase or decrease.
Question 2. The additive model is:
- TS = T × S × C × R
- TS = T + S + C + R
- TS = T − S
- TS = T ÷ S
Show the answer
B. The components are added.
Question 3. Daily sales of 80, 84, 88, 92 and 96 give a five-day moving average of:
- 84
- 88
- 92
- 440
Show the answer
B. 440 ÷ 5 = 88, which belongs to the middle day.
Question 4. Four-quarter moving averages of 25.0 and 27.0 give a centred moving average of:
- 25.0
- 26.0
- 27.0
- 52.0
Show the answer
B. (25.0 + 27.0) ÷ 2 = 26.0.
Question 5. Centring is needed when the cycle has:
- An even number of periods
- An odd number of periods
- One period
- No pattern
Show the answer
A. Otherwise the average falls between two periods.
Question 6. For monthly data with a yearly pattern, the moving average should cover:
- 4 months
- 7 months
- 12 months
- 3 months
Show the answer
C. One full cycle.
Question 7. Which component cannot be predicted?
- Trend
- Seasonal variation
- Random variation
- None of these
Show the answer
C. It results from unusual events.
Question 8. Cyclical variations differ from seasonal variations because they:
- Repeat within each year
- Last longer than a year and have no fixed period
- Are random
- Are always zero
Show the answer
B. They follow the economic cycle.
Question 9. The multiplicative model is appropriate when seasonal effects:
- Are constant in naira terms
- Grow in proportion to the trend
- Are zero
- Are random
Show the answer
B. Seasonal swings are proportional to the trend level.
Question 10. The trend is y = 70 + 3x (x = day number). The trend value for day 20 is:
- 130
- 90
- 133
- 60
Show the answer
A. 70 + 3 × 20 = 130.
Section B style: written questions
Question 11 (10 marks). Describe the four components of a time series and distinguish the additive model from the multiplicative model.
Question 12 (20 marks). Quarterly sales (₦000) over three years are: Year 1: 40, 55, 70, 50. Year 2: 48, 63, 78, 58. Year 3: 56, 71, 86, 66.
(a) Calculate the four-quarter moving averages and the centred moving averages. (10 marks)
(b) Using the first and last centred moving averages, find the equation of the trend line, with the first centred value as the starting point. (6 marks)
(c) Forecast the trend value for quarter 16 (Year 4, Q4). (4 marks)
A further written question
Question 13 (10 marks). Sales for six periods are 12, 15, 18, 15, 18, 21, and the seasonal cycle is three periods. (a) Calculate the three-period moving averages. (4 marks) (b) State the trend. (2 marks) (c) Explain why a four-quarter moving average needs a second averaging step. (4 marks)
Model answers
Show the model answer to Question 11
Components. Trend: the general long-term direction, a persistent rise or fall. Seasonal variation: regular fluctuations repeating within a year or week, caused by weather, holidays or customer habits. Cyclical variation: swings longer than a year with no fixed period, tied to the economic or business cycle. Random variation: unpredictable, non-recurring variations from unusual events, treated as noise.
Models. The additive model, TS = T + S + C + R, assumes the components are independent and that seasonal effects are roughly constant in size. The multiplicative (proportional) model, TS = T × S × C × R, assumes the components interact, so seasonal effects are proportional to the trend and grow as it grows. Seasonal variations are absolute amounts adding to zero in the additive model, and proportions averaging one in the multiplicative model.
Show the model answer to Question 12
(a) Four-quarter totals: 215, 223, 231, 239, 247, 255, 263, 271, 279 (Q1-Q4 to Q9-Q12). Moving averages (÷ 4): 53.75, 55.75, 57.75, 59.75, 61.75, 63.75, 65.75, 67.75, 69.75. Centred moving averages: Q3 54.75; Q4 56.75; Q5 58.75; Q6 60.75; Q7 62.75; Q8 64.75; Q9 66.75; Q10 68.75.
(b)
- The first centred value is 54.75 (Q3) and the last is 68.75 (Q10), seven quarters later.
- Slope = (68.75 − 54.75) ÷ 7 = 14 ÷ 7 = 2.
- With Q3 as x = 0, the trend line is y = 54.75 + 2x (₦000).
(c)
- Q16 is 13 quarters after Q3, so x = 13.
- Trend = 54.75 + 2 × 13 = 54.75 + 26 = ₦80,750.
Show the model answer to Question 13
(a)
- Averages.
(12 + 15 + 18) ÷ 3 = 15 (period 2) - (15 + 18 + 15) ÷ 3 = 16 (period 3)
- (18 + 15 + 18) ÷ 3 = 17 (period 4)
- (15 + 18 + 21) ÷ 3 = 18 (period 5).
(b) The moving averages rise by 1 each period, so the trend is upward at 1 unit per period, with trend = 15 at period 2.
(c) With four quarters in the cycle, the moving average of four values falls at the midpoint between the second and third quarters, which is not an actual quarter. Seasonal variation compares the actual value in a quarter with the trend value for that quarter, so a trend value is needed opposite each actual quarter.
Averaging each adjacent pair of moving averages gives a centred moving average that lines up with a real quarter.
Pathfinder on Time series analysis and moving averages
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Part (a), 10 marks, asks how a time series can be analysed. It covers the trend line, moving averages and seasonal variation. Part (b) asks about the two models used to estimate seasonal variation.
Revise first: Analysing a time series
Asks you to explain the two models used to estimate seasonal variation, the additive and the proportional model.
Revise first: The two models
Multiple-choice items can test the components of a time series, the length of a moving average and centring. Check Section A of every diet you practise.
Revise first: Moving averages
Source: ICAN Pathfinder, Management Accounting. Diets covered: May Diet, 2024 (previous syllabus). May Diet, 2025 and November Diet, 2024 are not covered here.
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