## **Learning Outcomes**

After reading this article, you will be able to explain seasonal variations in time series, distinguish between additive and multiplicative models, calculate seasonal factors, and apply both models to forecasting using trend and seasonality. You will also interpret seasonal indices and understand their role in index numbers. These skills are essential for ACCA exam questions on budgeting, forecasting, and data analysis.

## **ACCA Management Accounting (MA) Syllabus**

For ACCA Management Accounting (MA), you are required to understand how seasonal patterns affect time series analysis and forecasting. You should focus your revision on:

- The components of time series: trend, seasonal, cyclical, and random variations
- The difference between additive and multiplicative seasonal models
- How to calculate seasonal factors and indices using both methods
- Applying seasonal corrections to forecast future values
- The use of index numbers in adjusting for seasonal effects
- Recognising when each seasonal variation model is appropriate

## **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. What are the key differences between the additive and multiplicative seasonal models in time series forecasting?
2. A trend value for March is 600 units. The additive seasonal adjustment for March is –40. What is the seasonally adjusted forecast for March?
3. For a trend of 1,200 units and a multiplicative seasonal index of 0.97, what is the adjusted forecast?
4. True or false: A seasonal index below 1.00 in the multiplicative model means the actual value is lower than the trend for that period.

## **Introduction**

Time series analysis is concerned with data observed or measured at regular time intervals—for example, monthly sales, quarterly output, or yearly costs. These data often display repeating patterns due to seasonality, such as higher ice cream sales in summer or increased utility costs in winter. Management accountants analyse and adjust for these seasonal patterns to prepare reliable forecasts and effective budgets.

> **Key Term: time series**
> A sequence of data points collected or measured at equally spaced time intervals, used to study trends and patterns over time.

## Components of a Time Series

A time series consists of several identifiable elements:

- **Trend:** The fundamental, long-term increase or decrease in data over time.
- **Seasonal Variation:** Short-term, regular patterns that repeat in each cycle (often monthly, quarterly, or weekly).
- **Cyclical Variation:** Medium- or longer-term fluctuations usually related to economic or business cycles.
- **Random Variation:** Unpredictable, irregular differences with no visible pattern.

For most budgeting and forecasting purposes, trend and seasonal variation are the most important.

> **Key Term: seasonal variation**
> Consistent, predictable changes in data that repeat at regular intervals, linked to calendar, climate, or institutional factors.

## Seasonal Variation Models

Seasonal effects can be described using either the additive or multiplicative model. Each approach handles seasonality differently.

### Additive Model

With the additive model, the seasonal effect is a fixed amount added to (or subtracted from) the basic trend value for each period. The assumption is that the magnitude of the seasonal variation does not change as the overall trend rises or falls.

Observed Value = Trend + Seasonal Adjustment

Additive seasonal adjustments are best used when the fluctuation is approximately constant in size, regardless of the general level.

> **Key Term: additive model**
> A time series model that describes seasonal variation as a constant value to be added or subtracted from the trend in each period.

### Multiplicative Model

The multiplicative model treats seasonality as a factor that increases or decreases in proportion to the trend. Here, the trend value is multiplied by a seasonal index (usually a ratio or percentage).

Observed Value = Trend × Seasonal Index

This is most appropriate when the size of the seasonal variation grows or shrinks along with the overall trend.

> **Key Term: multiplicative model**
> A time series model that expresses seasonal variation as a percentage or ratio of the trend, multiplied to calculate the actual value in each period.

## Calculating Seasonal Factors

To use seasonal adjustments, you need to measure the effect for each period in the cycle (e.g. each month, quarter, or week):

### Additive Model Calculation

1. Find the trend value for each period using moving averages or regression.
2. Subtract the trend from the actual value for that period to get the seasonal deviation.
3. For each season (e.g. all Januaries, all Q1s) calculate the average deviation.

Additive Seasonal Adjustment = Actual Value – Trend Value

### Multiplicative Model Calculation

1. Calculate trend values.
2. Divide the actual value by the trend to get the raw index for each period.
3. Average the indices for each season.

Multiplicative Seasonal Index = Actual Value ÷ Trend Value

### **Worked Example 1.1**

A retailer records sales of 640 units for April. The trend value for April (from moving averages or regression) is 600.

**a) What is the additive seasonal adjustment for April?**  
**b) What is the multiplicative seasonal index?**

> **Answer:**
>
> a) Additive: 640 – 600 = +40  
> b) Multiplicative: 640 ÷ 600 = 1.07

This means April sales are 40 units above trend in the additive model, or 7% above trend multiplicatively.

## Using Seasonal Factors in Forecasting

Adjust future forecasts by combining predicted trend data with the appropriate seasonal effect.

### Additive Model (Forecasting)

Forecast = Trend + Seasonal Adjustment

Example: If the trend for July is forecast at 2,900 units, and the July seasonal adjustment is –200, the forecast is 2,900 + (–200) = 2,700 units.

### Multiplicative Model (Forecasting)

Forecast = Trend × Seasonal Index

Example: If the September trend is 450 units and the September seasonal index is 1.15, the forecast is 450 × 1.15 = 517.5 units.

### **Worked Example 1.2**

The expected trend for December output is 5,000 units. If the additive seasonal adjustment is –150 and the multiplicative seasonal index is 0.96, what is the forecast December output using each model?

> **Answer:**
>
> Additive: 5,000 + (–150) = 4,850 units  
> Multiplicative: 5,000 × 0.96 = 4,800 units

## Interpreting Seasonal Indices

- Additive: An adjustment of zero means the period is 'average' relative to the trend.
- Multiplicative: An index of 1.00 means the period matches the trend. Over 1.00 means above trend; under 1.00 means below.
- The sum of additive adjustments across a full cycle (e.g., 12 months or 4 quarters) should be zero. The average of multiplicative indices should be 1.00.

### **Worked Example 1.3**

A small firm’s quarterly sales trend and actual data:

| Quarter | Trend | Actual |
| ------- | ----- | ------ |
| Q1      | 400   | 440    |
| Q2      | 420   | 400    |
| Q3      | 440   | 470    |
| Q4      | 460   | 450    |

a) What is the Q1 additive seasonal adjustment?  
b) What is the Q3 multiplicative index?

> **Answer:**
>
> a) Additive (Q1): 440 – 400 = +40  
> b) Multiplicative (Q3): 470 ÷ 440 = 1.068

## Deciding Between Additive and Multiplicative Models

- Use the additive model if the seasonal difference appears constant in size as the overall trend changes.
- Use the multiplicative model if the seasonal fluctuation increases or decreases as the trend rises or falls.

### **Exam Warning**

> The choice of model affects your forecast accuracy. If the size of the seasonal swing changes as sales/trend increases, the additive model can miss significant effects. Always inspect actual vs trend differences before selecting the model.

### **Revision Tip**

> When calculating seasonal adjustments, organise your data by season (e.g., all Januaries together). That makes it easier to average each season’s effect.

## Index Numbers and Seasonality

Index numbers often measure changes in variables like price or output over time. Seasonal patterns distort comparisons. To compare periods fairly, data should be seasonally adjusted, using either the additive or multiplicative approach.

> **Key Term: index number**
> A statistic expressing data as a percentage (or factor) of a base period, used to compare changes over time or between series.

## Advantages and Limitations of Time Series Analysis

### Advantages

- Provides forecasts and budgets using objective, historical data
- Identifies trend and regular seasonal effects for practical adjustment

### Limitations

- Assumes past seasonal patterns will continue without change
- May not account for exceptional events, new products, or major policy shifts

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Distinction between trend, seasonal, cyclical, and random time series components
- The concepts and formulas for additive and multiplicative seasonal models
- Calculation and application of seasonal adjustments in both models
- Selection of the appropriate seasonal model based on historical data features
- The use of index numbers in removing or comparing seasonal influences
- Strengths and weaknesses of time series and seasonal adjustment models

## **Key Terms and Concepts**

- time series
- seasonal variation
- additive model
- multiplicative model
- index number

Time series and index numbers - Seasonal variations (additive and multiplicative)