The concept of time series analysis and extrapolation

Time series analysis is a technique used to forecast sales by relying solely on historical sales information. It involves organising past sales data in chronological order to spot patterns and make predictions.

Key features of time series data

Sales figures are recorded over specific periods, such as quarters or months, and arranged by date to form a time series. This data can be plotted on a graph to visualise trends and fluctuations, helping businesses anticipate future sales.

The process of extrapolation

Extrapolation involves projecting future sales based on patterns observed in historical data.

How extrapolation works:

• Actual sales figures are plotted on a graph, and a trend line is drawn through these points.
• This line is then extended forward to estimate future values, assuming that past patterns will continue.
• It works best when sales trends are consistent and unaffected by major changes.

Assumptions and limitations of extrapolation

Extrapolation relies on the idea that sales behaviours remain steady over time. However, it becomes unreliable if external factors, such as economic shifts or market disruptions, alter these patterns.

Understanding moving averages and trends

Moving averages provide a more advanced approach to sales forecasting than basic extrapolation. This method smooths out data to reveal underlying influences on sales.

Factors identified by moving averages

• The trend - The overall long-term direction of sales, such as a steady increase or decrease.
• Seasonal fluctuations - Regular changes tied to seasons, like higher sales during holidays.
• Cyclical fluctuations - Broader patterns linked to economic cycles, such as booms and recessions.
• Random fluctuations - Unpredictable variations caused by one-off events, like sudden weather impacts.

Selecting the period for moving averages

The timeframe for calculating moving averages should match the data's natural cycle.

Examples of appropriate periods:

• For quarterly data with seasonal patterns, use a four-period total.
• For monthly data, a 12-period total may be suitable.
• For daily sales with weekly cycles, a seven-period total could apply.

Calculating moving averages

A moving average is found by summing sales figures over a chosen period and dividing by the number of periods. This helps to average out short-term fluctuations and highlight the trend. For example, a four-quarter moving total adds up sales from four consecutive quarters. Dividing by four gives the average, but it may not align perfectly with a specific quarter.

Centred moving averages

To align the average with a particular period, centre it by combining two consecutive moving totals (e.g., creating an eight-quarter total) and dividing by the total number of periods (e.g., 8). This centred value represents the trend more accurately by smoothing seasonal effects.

Calculating seasonal variations and forecasts

Seasonal variations measure how actual sales deviate from the trend due to predictable seasonal factors. These are used to refine forecasts.

Formula for seasonal variation

$\text{Seasonal variation}=\text{actual sales revenue}-\text{moving average trend}$

Where:

• Actual sales revenue = Recorded sales for a specific period (£)
• Moving average trend = Centred average representing the underlying trend (£)

A negative result shows sales below the trend for that period, often due to seasonal lows.

Calculating average seasonal variation

Average seasonal variation (ASV) provides a typical adjustment for each period.

Steps to calculate ASV:

• Sum the seasonal variations for the same period across multiple cycles (e.g., all Q1 variations).
• Divide by the number of results to get the ASV for that period.

Steps for forecasting with moving averages

1. Plot the centred moving average (trend) on a time series graph.
2. Extrapolate the trend line into the future, keeping projections short-term for better accuracy.
3. Identify the trend value for the forecast period either:
   • From a plotted trend line on a time series graph
   • Or by mathematical extrapolation, by taking the most recent centred moving average and adjust by the known trend rate multiplied by the number of periods ahead
4. Add or subtract the ASV for that specific period to adjust for seasonal effects.

Worked example - Calculating seasonal variation and forecast

A business has the following quarterly sales data: Q1 2022: £110,000, Q2 2022: £140,000, Q3 2022: £170,000, Q4 2022: £130,000, Q1 2023: £120,000, Q2 2023: £150,000, Q3 2023: £180,000, Q4 2023: £140,000. Calculate the centred moving average for Q2 2023, the seasonal variation for Q2 2023, and forecast sales for Q2 2024 assuming the trend increases by £5,000 per quarter.

Step 1: Calculate four-quarter moving totals

Step 2: Calculate centred moving average for Q2 2023

Combine two moving totals around Q2 2023: (£580,000 + £590,000) / 8 = £146,250

Step 3: Calculate seasonal variation for Q2 2023

Seasonal variation = £150,000 - £146,250 = +£3,750

Step 4: Forecast for Q2 2024

Assume trend at Q2 2024 is £146,250 + (4 × £5,000) = £166,250 (accounting for four quarters ahead).

Adjust by ASV (assuming +£3,750 for Q2): £166,250 + £3,750 = £170,000

Benefits and limitations of the moving average method

The moving average approach offers structured insights for sales forecasting but has constraints, especially for long-term predictions.

Benefits of using moving averages

• Identifies and quantifies seasonal patterns, allowing adjustments in forecasts.
• Provides reasonable accuracy for near-future predictions in steady market conditions.
• Calculates average variations for each period, aiding in resource planning like stock levels.

Limitations of using moving averages

• Accuracy decreases for longer-term forecasts, as they depend entirely on historical patterns.
• Ignores external changes, such as new competitors or economic shifts, which can make past data irrelevant.
• Focuses only on quantitative data, overlooking qualitative factors like consumer trends.
• For extended planning, methods incorporating current market insights may be more suitable.