Time series and index numbers - Moving averages and trend es...

Learning Outcomes

After reading this article, you will be able to explain the components of time series data, calculate and interpret moving averages, estimate trends using different methods, and apply seasonal and index number adjustments in data analysis. You will also understand how to use index numbers such as Laspeyre and Paasche indices for forecasting, compare results over time, and recognise the benefits and limitations of these approaches for ACCA exam settings.

ACCA Management Accounting (MA) Syllabus

For ACCA Management Accounting (MA), you are required to understand how time series analysis and index numbers support budgeting and forecasting. In particular, you should focus your revision on:

• Identifying the components of a time series: trend, seasonal, cyclical, and random variations
• Calculating and interpreting moving averages to derive trend values
• Using regression for trend estimation in time series
• Determining seasonal variations and constructing seasonal indices
• Explaining, calculating, and applying simple, chain, and weighted index numbers (including Laspeyre and Paasche indices)
• Adjusting historic and forecast data using index numbers for like-for-like comparison
• Recognising the advantages and limitations of time series and index number techniques in business forecasting

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. Which of the following is NOT a typical component of a time series?
   1. Trend
   2. Seasonal variation
   3. Random variation
   4. Opportunity cost
2. A company tracks sales monthly. What is the main benefit of applying a 12-month moving average to these sales figures?
   1. Find the random variation
   2. Identify the fundamental trend
   3. Calculate the index number for each month
   4. Forecast next year's costs directly

3. Calculate the simple index number for 2023 if the sales value is £18,000 and 2020 (base year) sales were £15,000.

4. The Laspeyre price index is best described as using which of the following weights?
   1. Current period quantities
   2. Base period quantities
   3. Average of both periods
   4. No weights at all

Introduction

Many business decisions depend on predicting future sales, costs, or activity levels based on past data. Time series analysis and index numbers provide systematic ways to analyse trends and patterns in historical figures and produce more reliable forecasts. This article explains key methods—such as moving averages and index numbers—used by management accountants to derive trends, handle seasonal effects, and adjust business data, ensuring forecasts are as accurate and relevant as possible for budgeting decisions.

Key Term: time series
A sequence of data points collected or recorded at successive, usually regular, intervals over time.

Key Term: index number
A statistical measure that expresses changes in a variable relative to a designated base value, typically set at 100.

COMPONENTS OF A TIME SERIES

A time series is not just a set of numbers. Each figure may reflect long-term changes, seasonal effects, regular cycles, and unpredictable elements. Separating these helps to clarify which factors drive results and what can be forecast.

Key Components

• Trend: The long-term increase or decrease in data over time.
• Seasonal variation: Regular and predictable changes within specific periods (e.g., spikes in December or drops each summer).
• Cyclical variation: Fluctuations due to broader economic cycles, often spanning several years.
• Random variation: Non-recurring factors that cause unpredictable changes.

Key Term: trend
The fundamental direction in a time series, indicating persistent growth or decline after removing other influences.

Key Term: seasonal variation
A consistent pattern of variation that repeats within each cycle (e.g., annually or quarterly).

Key Term: cyclical variation
Rises and falls in data associated with longer economic or business cycles.

Key Term: random variation
Irregular, erratic changes in a time series that cannot be predicted or planned for.

A time series can be described by an additive model (Value = Trend + Seasonal + Cyclical + Random) or a multiplicative one (Value = Trend × Seasonal × Cyclical × Random). The choice depends on how the variations behave in your data.

MOVING AVERAGES AND TREND ESTIMATION

One of the most widely used methods to reveal the fundamental trend is the moving average. This approach "smooths" fluctuations in the data, making the long-term direction clearer.

Simple Moving Average

A moving average recalculates the average for a set window of periods (such as 3 months or 4 quarters) and moves step by step through the data. This is especially helpful for eliminating seasonal and random variation.

Worked Example 1.1

A business has monthly sales for 5 months: Jan–May: 480, 505, 510, 520, 535. What is the 3-month moving average for March?

Answer:
Sum Feb, Mar, Apr: (505 + 510 + 520) = 1,535.
Divide by 3: 1,535 / 3 = 511.7.
The 3-month moving average for March is 511.7.

When working with an even number of periods (like quarterly data using a 4-period moving average), the calculated averages need to be centered to align the results accurately with the time points.

Purpose and Benefits

• Reduces the impact of seasonal and random noise
• Clarifies the overall trend for more realistic forecasting
• Provides a basis for identifying seasonal deviations

Limitations of Moving Averages

• May conceal sudden shifts in data
• Can "lag" behind real changes, causing trend lines to be slow to react
• Does not capture non-linear or abrupt fluctuations well

Worked Example 1.2

Quarterly sales: Q1–Q4: 200, 240, 260, 220. Calculate the 2-quarter centered moving average for Q2.

Answer:
The average of Q1 and Q2: (200+240)/2 = 220
The average of Q2 and Q3: (240+260)/2 = 250
Center between Q2's two averages: (220+250)/2 = 235
So, the centered moving average for Q2 is 235.

TREND ESTIMATION USING REGRESSION

While moving averages expose trends visually, regression analysis provides a formula to quantify the trend and make forecasts.

A simple linear trend can be identified using:

y = a + bx

• y = forecast value
• x = period number
• a = estimated starting value (intercept)
• b = estimated change per period (slope)

This is usually calculated using the least squares method based on all available data.

Worked Example 1.3

A business records sales: Year 1 = 1,200 units, Year 2 = 1,300, Year 3 = 1,400, Year 4 = 1,480. Estimate the linear trend line.

Answer:
Calculate the slope b: (1,480 - 1,200) / (4 - 1) = 280 / 3 ≈ 93 units per year.
The intercept a: Use Year 1 as x=1, y=1,200 → a = 1,200 - (93×1) = 1,107
The trend equation: y = 1,107 + 93x

Regression can be used for interpolation (forecasting within the provided data range) or extrapolation (beyond the range—less reliable).

SEASONAL VARIATIONS AND INDICES

If seasonal effects are present (e.g., higher sales at year-end), forecasts should be adjusted using seasonal indices.

Calculating Seasonal Indices

1. Estimate the trend (by moving average or regression).
2. Find the difference (additive) or ratio (multiplicative) between actual values and trend for each season.
3. Calculate average seasonal effect per period.

Indices can be applied:

• As an absolute adjustment (additive)
• As a percentage or ratio (multiplicative)

Key Term: seasonal index
A calculated number representing the typical relative size of each period compared to the overall trend.

Worked Example 1.4

Suppose, after calculating trends, April consistently shows sales 150 units above trend (additive model). If forecast trend for next April is 900, the seasonally adjusted forecast is:

Answer:
900 + 150 = 1,050 units.

INDEX NUMBERS: MEANING AND CALCULATION

Index numbers enable quick comparison between different time periods or items by stating values relative to a chosen base period.

Key Term: base year
The starting point of comparison in an index, assigned a value of 100.

A simple index number is calculated as: Index = (Value in period / Value in base period) × 100

Worked Example 1.5

If material cost was £12 per unit in 2019 (base period) and £15 in 2023, what is the index for 2023 compared to 2019?

Answer:
(15 / 12) × 100 = 125.
The cost in 2023 is 25% higher than in 2019.

CHAIN AND WEIGHTED INDEX NUMBERS

A chain index updates the base period each year, linking changes across several periods. Useful when the mix or weight of items changes over time.

A weighted index number applies more importance to items that make a greater contribution to total value.

Key Term: weighted index number
An index number that assigns weights to items based on their value or quantity significance.

LASPEYRE AND PAASCHE INDICES

Laspeyre index: Weights are the quantities from the base period.
Paasche index: Weights are the quantities from the current period.

Key Term: Laspeyre index
A weighted index using base period quantities as weights.

Key Term: Paasche index
A weighted index using current period quantities as weights.

Worked Example 1.6

A company produced 100 units of A (base year), price £2; 200 units of B, price £3. This year: A: 90 units, £2.50; B: 210 units, £4. Calculate the Laspeyre price index.

Answer:
Total spend at current prices using base quantities: (100×2.5) + (200×4) = 250 + 800 = 1,050
Total spend at base prices: (100×2) + (200×3) = 200 + 600 = 800
Laspeyre index = (1,050 / 800) × 100 = 131.25

ADJUSTING VALUES USING INDEX NUMBERS

Index numbers can restate old data in current terms or remove inflation to compare like with like.

Adjusted Value = Original Value × (Comparison Year Index / Original Year Index)

Worked Example 1.7

An item cost £1,000 when the index was 120; if today's index is 150, what would be the cost at today's price level?

Answer:
1,000 × (150/120) = £1,250

USES AND LIMITATIONS OF TIME SERIES AND INDEX NUMBERS

Applications

• Detecting and forecasting trends in sales, production, or costs
• Adjusting for inflation or price changes to allow meaningful comparisons
• Budgeting and making informed business decisions

Limitations

• Patterns may change due to external shocks or new business conditions
• Indices may be biased depending on selected base period and chosen weights
• Moving averages smooth data but may hide sudden changes or turning points
• Index numbers reflect average change; not all items move the same way

Exam Warning

Avoid mechanically applying trend and seasonal adjustments without checking for recent business changes. Real-world events can override statistical patterns.

Revision Tip

Always check whether the question refers to an additive or multiplicative seasonal model, and whether the required index number is simple, chain, or weighted.

Summary

Time series and index numbers are essential tools for detecting trends, adjusting for seasonality, and making reliable forecasts in budgeting and decision-making. Moving averages smooth out fluctuating data to reveal fundamental patterns, while regression quantifies that trend. Seasonal indices ensure forecasts account for regular fluctuations. Index numbers—especially weighted types like Laspeyre and Paasche—standardise data for meaningful comparison and inflation adjustment. These techniques provide a structured approach to planning, but should always be accompanied by managerial judgement.

Key Point Checklist

This article has covered the following key knowledge points:

• The components of a time series: trend, seasonal, cyclical, and random variation
• How to calculate and centre moving averages for trend estimation
• Using regression to calculate a linear trend for forecasting
• Calculation and application of seasonal indices (additive and multiplicative)
• The meaning, calculation, and use of simple, chain, and weighted index numbers
• Understanding and applying Laspeyre and Paasche indices in data analysis
• Adjusting values for inflation using index numbers
• Main strengths and weaknesses of time series and index number techniques

Key Terms and Concepts

• time series
• index number
• trend
• seasonal variation
• cyclical variation
• random variation
• seasonal index
• base year
• weighted index number
• Laspeyre index
• Paasche index