ACCA Applied Knowledge · Management Accounting

Analytical techniques in budgeting and forecasting: formula sheet

Full chapter guide

Key formulas

Linear cost equation: y = a + bx; a = fixed cost, b = variable cost per unit, x = activity level.
High-low variable cost: b = (cost at highest activity − cost at lowest activity) ÷ (highest activity − lowest activity); Choose the points by activity level, not by cost.
High-low fixed cost: a = total cost at either point − (b × activity at that point); Using either point gives the same a.
Regression slope b: b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²); n is the number of data pairs. Calculate b before a.
Regression intercept a: a = (Σy ÷ n) − b × (Σx ÷ n); This is mean of y minus b times mean of x.
Forecast: y = a + b × x; Substitute the planned activity level x.
Range of r: -1 ≤ r ≤ +1; +1 is perfect positive, -1 is perfect negative, 0 means no linear correlation.
Coefficient of determination: r² = r × r; Always between 0 and 1. Shown as a proportion or a percentage of variation in y explained by x.
Correlation coefficient from r²: r = ±√r²; Take the sign from the slope b of the regression line. Positive b gives positive r.
Unexplained variation: 1 - r²; The proportion of variation in y not explained by x.
Correlation coefficient formula: r = [nΣxy - ΣxΣy] ÷ √{[nΣx² - (Σx)²][nΣy² - (Σy)²]}; You can apply it by hand or use the calculator's linear regression mode. Know the structure, but most questions give you r.
Additive model: A = T + S + R; Seasonal variation S is an absolute amount. Seasonal variations should sum to zero.
Multiplicative model: A = T × S × R; S is a proportion or percentage of trend. Seasonal factors should average 1 (sum to the number of periods).
Seasonal variation, additive: S + R = A − T; Calculate for each period, then average by season to remove R.
Seasonal variation, multiplicative: S × R = A ÷ T; Calculate for each period, then average by season to remove R.
Moving average, odd number of periods: Trend = sum of n values ÷ n; Place the result against the middle period.
Centred moving average, even number of periods (e.g. quarters): Trend = (½ first + middle three + ½ last) ÷ 4, or the average of two consecutive 4-period averages; Needed so the trend lines up with an actual period, not between two periods.
Seasonal adjustment of actual data: Additive: A − S. Multiplicative: A ÷ S; Gives the deseasonalised figure, which shows the underlying trend.
Forecast: Forecast = projected trend + S (additive), or projected trend × S (multiplicative); Project the trend first, usually with a regression line or the average change in trend per period.
Trend line: y = a + bx; a is the starting value, b is the change in trend per period, x is the period number. Check how x is numbered in the question.
Additive forecast: Forecast = Trend + Seasonal variation; The seasonal variation is an amount. It may be negative. The variations for one full cycle should sum to about zero.
Multiplicative forecast: Forecast = Trend × Seasonal index; An index of 1.10 means 10% above trend. An index of 0.90 means 10% below trend. Indices over a cycle average 1 (for four quarters they sum to 4).
Seasonal variation (additive): Actual − Trend; Average these figures for each season to get the seasonal variation.
Seasonal index (multiplicative): Actual ÷ Trend; Average these figures for each season to get the index.
Learning curve formula: y = ax^b; y = cumulative average time per unit for x units; a = time for the first unit; x = cumulative number of units; b = the index of learning.
Index of learning: b = log(learning rate as a decimal) ÷ log 2; For an 80% rate, b = log 0.8 ÷ log 2 = -0.3219. b is always negative for a learning rate below 100%.
Total time for x units: Total time = y × x; Cumulative average time multiplied by cumulative units.
Time for additional units: Time for extra units = total time at higher output − total time at lower output; Do this to find the time for a batch or for the marginal unit.
Doubling rule: Cumulative average time at 2x units = learning rate × cumulative average time at x units; Quick method when output is 2, 4, 8, 16 and so on times the starting point.
Simple price index: Price index = (Current price ÷ Base price) × 100; Base period always equals 100. Use the same formula for a quantity index with quantities.
Weighted price index: Weighted index = Σ(index × weight) ÷ Σ weights; Weights could be quantities, expenditure shares or given percentages. Do not divide by the number of items.
Weighted aggregate index: Index = Σ(current price × base quantity) ÷ Σ(base price × base quantity) × 100; Base-weighted (Laspeyres style). Using current quantities as weights gives a current-weighted index instead. Follow the weights the question gives.
Adjusting a value to a new price level: Adjusted value = Original value × (New index ÷ Old index); Use for inflating or deflating a cost to different dates.
Percentage change in an index: % change = (New index − Old index) ÷ Old index × 100; Do not subtract the index points and call it a percentage unless the base is 100.
Changing the base: New index = (Old index ÷ Index of the new base period) × 100; Use when you need a different period to equal 100.
Expected value: EV = Σ(p × x); p is the probability of each outcome and x is its value (profit, cost, sales units). Add up all the products.
Probability check: Σp = 1; All probabilities for one decision must total 1 (100%). Check this before you calculate.
Expected profit from expected units: Expected profit = (expected units × contribution per unit) − fixed costs; This works when the profit is a straight-line function of units. If it is not, calculate the profit for each outcome first, then weight.
Decision rule: Choose the highest EV of profit, or the lowest EV of cost; This rule ignores risk. It assumes the decision maker is risk neutral.

Quick revision

Linear relationship: y = a + bx, where a is fixed cost and b is variable cost per unit.
High-low: b = (cost at high activity − cost at low activity) ÷ (high activity − low activity).
Use activity levels to choose high and low points, not the cost levels.
Correlation r ranges from −1 to +1. Values near ±1 show a strong linear relationship; near 0 shows a weak one.
Coefficient of determination = r². It is the share of variation in y explained by x.
Trend is the long-term direction. Seasonal variation is a repeating short-term pattern.
Additive model: Y = T + S. Multiplicative model: Y = T × S.
Forecast = trend value for the period plus (or times) the seasonal variation.
Cumulative average time learning curve: Y = aX^b, where Y is the cumulative average time per unit for X units, a is the time for the first unit, and b = log learning rate ÷ log 2. Enter the learning rate as a decimal, for example b = log 0.8 ÷ log 2 for an 80% curve, not log 80. ACCA publishes a list of given formulae, so check it before your exam to see exactly what is provided.
Index = (current price ÷ base price) × 100.
Expected value = Σ (probability × outcome). Probabilities must add up to 1.
Forecasts get less reliable the further you extrapolate beyond the data.

Common mistakes

Choosing the highest and lowest cost instead of the highest and lowest activity. Fix: Look only at the activity column when picking the two points. Use the costs that go with those activity levels.
Calculating a before b in regression. Fix: The formula for a needs b. Always work out b first.
Treating r² as the correlation coefficient itself, for example saying r = 0.64 when r² = 0.64. Fix: Read the label. If given r², take the square root to find r, and choose the sign from the slope.
Saying r = -0.9 is a weak correlation because it is negative. Fix: Strength depends on size, ignoring the sign. -0.9 is strong negative.
Not centring a four-point or twelve-point moving average. Fix: Average two consecutive moving averages, or use ½, 1, 1, 1, ½ weights, so the trend lines up with an actual period.
Using A − T when the question asks for the multiplicative model, or the reverse. Fix: Underline the model name in the question. Write 'additive: subtract, multiplicative: divide' beside your working.
Adding a seasonal index instead of multiplying by it in the multiplicative model. Fix: Read the model first. An index like 1.10 or 0.85 is always multiplied.
Using the wrong value of x. Fix: Write the period list down, such as Year 1 Q1 = 1, and count to the target period.
Applying the learning rate to the time of the last unit instead of the cumulative average time. Fix: In this model, only the cumulative average time falls to 80% on each doubling. Work out totals, then subtract.
Treating the batch size as the cumulative output. Fix: Use cumulative units. If 10 units were made first, the second batch takes output from 10 to 30, so find totals at 30 and 10 and subtract.

Exam tips

Read the question for the method named. If it says high-low, do not use regression, and the other way round.
In multiple choice, wrong options are often built from common errors, such as using highest cost points or forgetting to square before adding. Do your own calculation before looking at the options.
For number entry, keep full calculator accuracy until the final step and round only as the question instructs.
Check the units of x and y in the data table before you substitute into the equation.
Be ready for short theory points: regression uses all data, high-low uses two points, and both assume a linear relationship.
In multiple choice, check the sign of r before any calculation. It often removes two options.
If a question gives r² and asks for r, always look for the slope or the description of direction to choose the sign.
Use wording such as 'explained by' for r². Avoid saying 'caused by'.