Quarterly Sales Decomposition and Seasonal Forecasting
A regional wholesaler records quarterly sales in $000 for three years. Year 1: Q1 120, Q2 140, Q3 180, Q4 160. Year 2: Q1 132, Q2 154, Q3 198, Q4 176. Year 3: Q1 145, Q2 169, Q3 218, Q4 194. The sales director wants forecasts for Year 4 Q1 and Q2 and has asked for a seasonal analysis. A junior analyst proposes simply extrapolating the annual totals, arguing that the quarterly pattern is a distraction. You are asked to use a centered four-quarter moving average, calculate average seasonal indices under a multiplicative model, deseasonalize the data, estimate the underlying quarterly trend, and produce the two forecasts. The director also wants a critique of the annual-total shortcut.
1. Calculate the centered four-quarter moving averages, the ratio-to-moving-average values, and the normalized seasonal indices.
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A centered four-quarter moving average for a quarter uses half-weights on the two end quarters. Example for Year 1 Q3: [0.5(120) + 140 + 180 + 160 + 0.5(132)] / 4 = 151.5.
| Quarter | Actual | Centered MA | Ratio |
|---|---|---|---|
| Y1 Q3 | 180 | 151.500 | 1.1881 |
| Y1 Q4 | 160 | 154.750 | 1.0339 |
| Y2 Q1 | 132 | 158.750 | 0.8315 |
| Y2 Q2 | 154 | 163.000 | 0.9448 |
| Y2 Q3 | 198 | 166.625 | 1.1883 |
| Y2 Q4 | 176 | 170.125 | 1.0345 |
| Y3 Q1 | 145 | 174.500 | 0.8309 |
| Y3 Q2 | 169 | 179.250 | 0.9428 |
Average ratios by quarter: Q1 = (0.8315 + 0.8309)/2 = 0.8312; Q2 = (0.9448 + 0.9428)/2 = 0.9438; Q3 = (1.1881 + 1.1883)/2 = 1.1882; Q4 = (1.0339 + 1.0345)/2 = 1.0342. The sum is 3.9975, so multiply by 4/3.9975 = 1.0006 to normalize.
Normalized indices: Q1 0.8317; Q2 0.9444; Q3 1.1890; Q4 1.0349. They sum to 4.0000. Q1 is about 17% below the annual average, Q2 about 6% below, Q3 about 19% above, and Q4 about 3.5% above.
2. Deseasonalize Year 3, estimate the quarterly trend increment, and forecast Year 4 Q1 and Q2.
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Deseasonalized value = actual / seasonal index.
Year 3: Q1 = 145 / 0.8317 = 174.3; Q2 = 169 / 0.9444 = 179.0; Q3 = 218 / 1.1890 = 183.4; Q4 = 194 / 1.0349 = 187.5.
The deseasonalized series rises by about (187.5 − 174.3)/3 = 4.4 per quarter. Extend this trend: Year 4 Q1 trend = 187.5 + 4.4 = 191.9; forecast = 191.9 × 0.8317 = 159.6. Year 4 Q2 trend = 191.9 + 4.4 = 196.3; forecast = 196.3 × 0.9444 = 185.4.
So the forecasts are approximately $159,600 and $185,400. Extending the same trend through Q3 and Q4 gives roughly 238.5 and 212.1, for a Year 4 total near 795.5. The forecasts depend on the trend continuing and on the seasonal pattern remaining stable; with only two observations per quarter, the seasonal indices are imprecise.
3. Critique the junior analyst's proposal to extrapolate annual totals instead of using quarterly decomposition.
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Annual totals are 600, 660, and 726. Each year is exactly 10% higher than the previous year (660/600 = 1.10; 726/660 = 1.10). A naive 10% extrapolation gives Year 4 total = 726 × 1.10 = 798.6.
The seasonal method gives about 795.5, only 0.4% lower. In this dataset the annual shortcut happens to work well because both the trend and the seasonal pattern are very stable. But the shortcut gives no quarterly forecast and cannot reveal changes in seasonality. It would mislead if, for example, Q3's share were falling or if a new competitor changed the quarterly pattern.
Limitations of the decomposition: centered moving averages lose the first two and last two quarters; only two ratios per quarter are available; the method assumes multiplicative seasonality and a stable linear trend; and it produces a point forecast without a prediction interval. The director should treat the forecasts as conditional on stable trading conditions and monitor actual versus forecast values each quarter.