Forecasting Techniques
1 Introduction
This chapter covers the mathematical techniques used to prepare financial forecasts. There are three, and you are expected to be able to calculate and to interpret the results of all of them:
the HIGH-LOW method;
REGRESSION ANALYSIS (least squares); and
TIME SERIES analysis.
The first two do the same job — separating a semi-variable cost into its fixed and variable elements, so that costs can be predicted at any level of activity. The third does a different job: it takes a series of past observations recorded at regular intervals and separates the underlying trend from the seasonal pattern, so that sales can be predicted for a particular future period.
1.1 Forecasting and budgeting
Forecast and budget are not the same thing
A forecast is a prediction of what WILL happen, given the information available. A budget is a plan of what the organisation INTENDS to make happen. The budget may deliberately be set above the forecast, as a target, or below it, as a prudent basis for committing cash.
The relationship runs one way: the forecast comes first and the budget is built on it. Chapter 11 §3 explains that the budget cascade begins with the principal budget factor, and that for most organisations this is the level of sales demand. That sales figure has to come from somewhere, and the techniques in this chapter are where it comes from.
A SALES forecast, usually from time series analysis (section 5), starts the cascade: it drives the production budget, and through that the materials, labour and overhead budgets.
A COST forecast, from high-low or regression (sections 3 and 4), turns a level of activity into a budgeted cost. It is what allows a semi-variable cost to be budgeted at all, and it is what makes a FLEXED budget possible — you cannot flex a budget for a cost whose fixed and variable elements you have not separated.
The forecast is also what a budget is later STRESS TESTED against (chapter 11 §8.4): the assumptions being flexed in a what-if analysis are the outputs of this chapter.
The consequence for accuracy. Every weakness of a forecasting technique becomes a weakness of the budget built on it, and then a weakness of every variance calculated against that budget. A cost forecast built on an extrapolation outside the observed range does not merely give a poor budget; it gives variances that measure the forecasting error rather than the manager’s performance, which is exactly what the planning variances in chapter 10 exist to separate out.
2 Semi-variable costs
The first of this chapter’s three recordings. It covers sections 2 and 3 — semi-variable costs, why they have to be split, and the high-low method on Example 1 — reaching $75 a unit and $32,500 a period with the check at the low point, then the method’s two weaknesses: extreme observations and forecasting outside the range. One point before you play it.
Select on activity, not on cost: high-low takes the periods with the highest and lowest output and reads off their costs. If you hear the two periods chosen by their cost, follow the activity rule instead: in Example 1 both pick the same pair, so the answer stands, but section 3 shows a data set where they do not.
From your earlier studies you will be familiar with the behaviour of fixed and variable costs. Many costs are neither: they are SEMI-VARIABLE, with a fixed element and a variable element.
A telephone bill is the standard example. The line rental is fixed — it is known in advance and does not change with the number of calls made. The call charges are variable, because they are driven by usage and rise broadly in proportion to it. Electricity behaves the same way: lighting the factory costs the same whether one unit or a thousand is produced, while running the machines does not.
2.1 Why the split matters
As a management accountant you need accurate cost information as a reliable basis for budgets and forecasts. Beyond statutory and reporting requirements, better cost information supports better planning and better decisions in areas such as product pricing — and none of it is possible until a semi-variable cost has been separated into its two elements.
Total cost = Fixed cost + (Variable cost per unit × Number of units)
2.2 Historic data
Historic cost data can be obtained from the accounting records. Plotted against output, a semi-variable cost looks like this — broadly linear, but not exactly so, because in the real world nothing is perfectly linear:
Individually the observations are not much use in deciding which element is fixed and which is variable. The high-low method and regression analysis both take this scatter of points and estimate the straight line that best represents it.
Note which variable is on which axis, because it changes the arithmetic. Cost depends on output, so cost is the DEPENDENT variable and goes on the vertical axis as y; output is the INDEPENDENT variable and goes on the horizontal axis as x. Getting them the wrong way round produces a regression line that predicts output from cost, which is not what anybody wants.
3 The high-low method
The high-low method is a very quick and simple way of identifying the fixed and variable elements of a semi-variable cost. It assumes a linear relationship, and it uses just two of the observations: those at the highest and the lowest ACTIVITY levels within a normal range of business activity.
The rule, and why it is the activity level
High-low splits a semi-variable cost into its fixed and variable parts. Cost is the dependent variable: output drives cost, not the other way round. So the two observations are chosen by activity level — the highest and the lowest output in the data — and the costs belonging to those two periods are then read off. They are read off; they are never searched for.
Selecting the highest and lowest cost instead is a method error, not a harmless shortcut. It happens to pick the same two periods whenever cost rises steadily with output, which is true of most textbook data sets and of Example 1 below — which is exactly why the error survives unnoticed until it meets a data set where it does not hold.
Where the two rules part company.
Take five periods: 100 units at $41,000, 250 units at $52,000, 400 units at $88,000, 550 units at $70,000 and 700 units at $85,000. The 400-unit period carried a one-off charge, so it is the most expensive period in the table even though it is not the busiest.
Selecting on activity gives the correct pair — 700 units at $85,000 and 100 units at $41,000. Variable cost = (85,000 − 41,000) ÷ (700 − 100) = $73.33 a unit, and fixed cost = 85,000 − (700 × 73.33) = $33,667.
Selecting on cost gives 400 units at $88,000 and 100 units at $41,000. That yields (88,000 − 41,000) ÷ (400 − 100) = $156.67 a unit and a fixed cost of 88,000 − (400 × 156.67) = $25,333 — more than double the variable rate and a quarter off the fixed cost. Every forecast taken from that line is wrong, and nothing in the arithmetic itself gives any sign of it.
A period that costs more than a busier one is not an oddity to be tidied away. It is usually a stepped fixed cost, a price change or a one-off, and section 3.1 is where that is dealt with. It is never a reason to change which two periods high-low selects.
Do not sort the data by cost.
Sorting the table by cost and taking the top and bottom rows is the commonest way this error is made, because it looks like the same operation as sorting by output. Sort by output, or read the highest and lowest output straight off the table as given.
Between those two points the fixed cost is the same, by definition. So the whole of the difference in total cost is the variable cost of the difference in output:
Variable cost per unit = (Total cost at the highest activity − Total cost at the lowest activity) ÷ (Units at the highest activity − Units at the lowest activity)
Substituting the total variable cost back into either of the two total costs then gives the fixed cost.
The following table shows the total costs recorded at different activity levels over seven periods.
Output (units) | Total cost ($) |
100 | 40,000 |
400 | 65,000 |
200 | 45,000 |
700 | 85,000 |
600 | 70,000 |
500 | 70,000 |
300 | 50,000 |
Required
Use the high-low technique to estimate the variable cost per unit and the fixed cost per period.
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3.1 The limitations of high-low
It is fast, and it is easy to get right under exam pressure. It has two real weaknesses.
IT USES ONLY THE TWO EXTREME OBSERVATIONS and throws away everything in between. If either extreme happens to be unrepresentative — an unusually expensive month, a one-off cost — the whole estimate is distorted, and there is nothing in the method to reveal it. The data must therefore be checked first: anomalies and one-off events should be removed before the method is applied.
IT ASSUMES LINEARITY over the whole range, and any estimate made outside the observed range assumes the line continues. Neither is guaranteed.
The first of those is what regression analysis fixes, by using every observation instead of two.
4 Regression analysis
Regression analysis — also called the LEAST SQUARES method — is used in the same circumstances as the high-low method, where costs are believed to follow a linear relationship. Instead of using only the two extreme points it incorporates every observation, and it therefore produces an estimate based on far more of the available information.
What it does is find the straight line that most nearly passes through all the points: specifically, the line that minimises the sum of the squares of the vertical distances of the points from it. That is a line of best fit through the scatter in section 2.2.
4.1 The equations
The line has the usual form:
y = a + bx
and the two coefficients are found from:
Symbol | In a costing context |
y | the DEPENDENT variable — total cost, because cost depends on the level of output |
x | the INDEPENDENT variable — the level of activity |
a | the intercept — the FIXED COST, being the value of y when x is zero |
b | the gradient — the VARIABLE COST PER UNIT, being the increase in y for each additional unit of x |
n | the number of pairs of observations |
Σ | “the sum of” whatever follows it |
b must be calculated before a, because the formula for a contains b.
This formula is examinable and is NOT provided in the objective test examination. You have to know it.
The following table shows the number of units produced in each of seven periods and the total cost incurred. (It is the same data as Example 1.)
Output (units) | Total cost ($) |
100 | 40,000 |
400 | 65,000 |
200 | 45,000 |
700 | 85,000 |
600 | 70,000 |
500 | 70,000 |
300 | 50,000 |
Required
Calculate the regression line, y = a + bx.
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4.2 Using the equation to forecast
Once a linear expression for the data has been obtained it can be used to estimate the total cost y for any given activity level x. Forecasting beyond the range of the observed data, or applying the equation to circumstances materially different from those in which the data was collected, should be avoided.
Required
Using the regression line from Example 2, calculate the estimated total cost if the forecast output level is 650 units.
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4.3 The correlation coefficient
Regression will fit a line to any set of points, including points that are plainly not linear. It is therefore worth having a measure of HOW linear the relationship is, and that measure is Pearson’s correlation coefficient, r.
You are not required to calculate r or to remember its formula for P1, but you must be able to interpret a value you are given. The formula is shown for completeness only:
The result always falls between −1 and +1, and it is interpreted as follows.
Value of r | Interpretation |
r = +1 | Perfect POSITIVE linear correlation. The points lie exactly on an upward-sloping straight line: y increases directly with x — costs rise directly with output. |
r = −1 | Perfect NEGATIVE linear correlation. The points lie exactly on a downward-sloping straight line: y falls as x rises. |
r = 0 | No linear correlation. The two variables show no straight-line relationship at all. |
In between | The closer r is to +1 or to −1, the more nearly the points lie on a straight line and the more confidence can be placed in a forecast made from the regression line. An r of +0.91, for example, indicates a very strong positive linear relationship in the data observed. |
For the data in Example 2, r works out at +0.977 — a very strong positive linear relationship, which is why the regression line is a reasonable basis for the forecast in Example 3. The square of the correlation coefficient, r² = 0.955, is the coefficient of determination: it says that about 95% of the variation in total cost is explained by the variation in output, leaving 5% explained by something else.
These techniques can be applied to scenarios outside management accounting.
Required
State what type of correlation you would expect to find in the examples below — positive, negative, or none.
The number of hours spent studying and the likely exam score.
The number of pages printed and the ink remaining in a printer.
Shoe size and level of disposable income.
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4.4 Problems with regression analysis
Regression is better than high-low, because it uses all the observations rather than two. Four weaknesses remain, and all four are examinable.
IT ASSUMES A LINEAR RELATIONSHIP. The formula will produce a line for any data, including data that plainly lies on a curve. If the underlying relationship is not linear the line is meaningless, however carefully it was calculated. The correlation coefficient is the check.
FORECASTING OUTSIDE THE OBSERVED RANGE IS DANGEROUS. Even where the relationship is strongly linear over the observed range, there is no reason to assume it continues beyond it. Costs commonly step or curve at higher volumes — an extra shift, overtime premiums, a second machine. The observations in Example 2 run from 100 to 700 units, so a forecast for 1,500 units would be an extrapolation with nothing behind it.
THE NUMBER OF OBSERVATIONS MATTERS. Two points always lie exactly on a straight line and give perfect correlation, which proves nothing whatever. Seven points that look linear are reasonable evidence; the more observations, the more confidence a forecast deserves.
THE DATA IS HISTORIC, AND CORRELATION IS NOT CAUSATION. The relationship held in the past under past conditions. Inflation, a change of supplier, a new production method or a change in product mix can all break it, and a strong correlation between two variables that are not causally connected will forecast confidently and wrongly.
5 Time series analysis
The third recording, on sections 5.4 and 5.5: the moving averages and centred trend of Example 5, then the seasonal variations under the additive and the multiplicative models (Examples 5 and 6) and how a forecast uses them. Three points before you play it.
The adjustment: the recording notes that the additive variations should sum to zero and the multiplicative ones to 400%, but it does not adjust them. The adjustment is required: quarter 2, for example, is +2.953 after adjustment, not +3.125, and 102.95%, not 103.00% (Answers 5 and 6). Section 5.6 forecasts with the adjusted figures.
Check quarter 1 against Answer 6: 20X2 quarter 1 is 99.3% of its trend, and the quarter 1 average is 100.00% before the adjustment.
Historigram: if you hear the chart of quarterly sales called a histogram, the term is historigram — a graph of a time series. A histogram is a frequency chart (section 5.1).
Management accountants can use historic data to identify trends and patterns that inform their forecasts. Those patterns are frequently not visible in the individual observations and become apparent only when the general movement over a period of time is examined.
Where high-low and regression are used to forecast COSTS from a level of activity, time series analysis is used to forecast the level of activity itself — most often sales volumes, which is where the budget cascade begins.
5.1 Key definitions
Time series
A time series is a set of observations recorded at equal intervals over a period of time — for example monthly, quarterly or annually.
A time series is displayed on a time-series graph, with time on the horizontal axis — years, quarters or months — and the observed value on the vertical axis. Older texts call such a graph a historigram.
A historigram is a graph of a TIME SERIES. A histogram is something different — a chart of a FREQUENCY DISTRIBUTION, in which the area of each bar represents the frequency of a class. The two words are easily confused.
5.2 The components of a time series
Component | What it is | Examinable in P1? |
Trend (T) | The underlying general movement of the data over time, once the shorter-term movements have been removed. | Yes |
Seasonal variation (SV) | A regular variation that is calendar-related and repeats WITHIN one year: any predictable pattern that recurs within a year is described as seasonal. | Yes |
Cyclical variation (C) | Variation repeating over a cycle LONGER than one year — typically the economic cycle of boom and recession. | No — you are not required to calculate it |
Random or residual variation (R) | Unpredictable, irregular movement. Because it is unpredictable it is of no use in forecasting. | No — you are not required to calculate it |
5.3 The additive and the multiplicative models
The four components can be combined in two ways.
Model | Form | Assumes | Use it when |
Additive | TS = T + SV + C + R | the components are independent of each other, so the seasonal variation is a constant AMOUNT | the seasonal swing stays about the same size as the trend rises or falls |
Multiplicative | TS = T × SV × C × R | the components move in line with the trend, so the seasonal variation is a constant PROPORTION | the seasonal swing grows as the trend grows — usually the more realistic assumption for a growing business |
Regression analysis can also be used to estimate a trend line for a time series. The x variable is then the time period, with period 1 being the first observation, and the resulting line can be extended to forecast the trend for a future period.
5.4 Finding the trend: moving averages
To separate the trend from the seasonal variation we use moving averages, a smoothing technique that averages away the seasonal pattern and the random fluctuations. With quarterly data the average is taken over four quarters, so that every quarter of the year is represented once.
A four-quarter average falls BETWEEN two quarters rather than on one, so it is centred: each pair of consecutive moving averages is averaged again, and the result is the trend value for the quarter that sits between them. This is why the first two and the last two quarters of the series have no trend value.
Set out below are the sales per quarter, in thousands of units, of a company over the last three years.
Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |
20X1 | 80 | 87 | 82 | 90 |
20X2 | 90 | 95 | 93 | 102 |
20X3 | 105 | 112 | 103 | 116 |
Required
Identify the trend and calculate the average seasonal variation, using the additive model.
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5.5 The multiplicative model
The seasonal variations in Example 5 were calculated as absolute amounts. If the trend is rising, it is often more sensible to expect the seasonal swing to rise with it: a quarter that runs 3% above the trend when sales are 90,000 would be expected to run 3% above the trend when sales are 180,000, not 2,950 units above it.
The multiplicative model deals with this by measuring the seasonal variation as the actual value expressed as a PERCENTAGE of the trend.
Required
Using the data from Example 5, together with the trend already calculated, calculate the average seasonal variation using the multiplicative model.
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5.6 Using the model to forecast
Neither model is any use until it is used to produce a number. Forecasting from a time series is a two-step operation:
FORECAST THE TREND for the period required. The trend has been smoothed, so it can be extended by eye from a graph, by regression on the trend values, or — most simply — by taking the average movement per period.
ADJUST IT BY THE SEASONAL VARIATION for that quarter: ADD the adjusted seasonal variation under the additive model, or MULTIPLY by the adjusted seasonal index under the multiplicative model.
Worked through, on the data in Examples 5 and 6. The trend rose from 86.000 in 20X1 quarter 3 to 107.250 in 20X3 quarter 2 — an increase of 21.250 over 7 quarters, so an average of 3.036 per quarter. Quarter 2 of 20X4 is four quarters after 20X3 quarter 2, so the forecast trend is 107.250 + (4 × 3.036) = 119.393 thousand units.
Model | Adjustment for quarter 2 | Forecast (000 units) |
Additive | 119.393 + 2.953 | 122.346 |
Multiplicative | 119.393 × 102.95% | 122.915 |
So the forecast for quarter 2 of 20X4 is about 122,000 to 123,000 units. The two models differ by about 600 units here, and the gap widens the further the trend moves from the level at which the seasonal variations were measured — which is exactly the reason for preferring the multiplicative model where the trend is rising strongly.
The forecast is only as good as the trend extrapolation, and every warning in section 4.4 applies to it. Extending a trend four quarters beyond the data assumes the trend continues; it also assumes the seasonal pattern is stable, that the past three years are representative, and that nothing has changed in the market. State those assumptions when you are asked to comment on a forecast.
6 Forecasting considerations
Required
Discuss the limitations of using past data as a prediction of future results.
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7 Test your knowledge
Two quick checks before you move on: work through the flashcards to fix this chapter’s key terms and definitions, then sit the objective questions for exam-style practice. Both mark themselves and explain the answers as you go.
Forecasting Techniques
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