ACCA Applied Skills · Performance Management

Analytical techniques in budgeting and forecasting: formula sheet

Full chapter guide

Key formulas

Variable cost per unit: Variable cost per unit = (Cost at highest activity − Cost at lowest activity) ÷ (Highest activity − Lowest activity); Choose the highest and lowest ACTIVITY levels, not the highest and lowest costs.
Fixed cost: Fixed cost = Total cost at a level − (Variable cost per unit × Activity at that level); Gives the same answer using the high or low point.
Cost function: Total cost = Fixed cost + (Variable cost per unit × Activity); Use it to forecast cost, only within the relevant range.
Mixed cost: y = a + bx; y = total cost, a = fixed cost, b = variable cost per unit, x = activity.
Regression line: y = a + bx; a = fixed cost, b = variable cost per unit of activity, x = activity level.
Slope b: b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²); n is the number of pairs of data. Calculate b before a. Note that (Σx)² is the square of the total, not Σx².
Intercept a: a = (Σy − bΣx) ÷ n; Equivalent to a = ȳ − b x̄, where ȳ and x̄ are the means.
Correlation coefficient: r = (nΣxy − ΣxΣy) ÷ √[(nΣx² − (Σx)²)(nΣy² − (Σy)²)]; Ranges from −1 to +1. Close to +1 or −1 means a strong linear relationship.
Coefficient of determination: r² = proportion of variation in y explained by variation in x; If r = 0.9, then r² = 0.81, so 81% of the variation in y is explained by x. The rest is due to other factors.
Correlation coefficient: r = (nΣxy − ΣxΣy) ÷ √[(nΣx² − (Σx)²)(nΣy² − (Σy)²)]; This is given in the ACCA formulae sheet, but you must know how to use it. n is the number of pairs of data.
Coefficient of determination: r² = r × r; Gives the proportion of variation in y explained by x. Express it as a percentage when you comment.
Range of r: -1 ≤ r ≤ +1; +1 is perfect positive correlation, -1 is perfect negative, 0 means no linear correlation.
Unexplained variation: 1 − r²; The proportion of variation in y caused by other factors or chance.
Regression line (for linking): y = a + bx; The sign of b matches the sign of r.
Additive model: Y = T + S + R; Seasonal variation is a fixed amount. Estimate S = Y − T, then average by season.
Multiplicative model: Y = T × S × R; Seasonal variation is a proportion. Estimate S = Y ÷ T, then average by season.
Four-point moving average (quarterly data): (Q1 + Q2 + Q3 + Q4) ÷ 4; Falls between the middle two periods. Roll forward one period at a time.
Centred moving average (even number of points): (MA₁ + MA₂) ÷ 2, or (sum of two consecutive 4-period totals) ÷ 8; Places the trend on an actual period so you can compare it with the actual figure.
Seasonal adjustment check (additive): Sum of average seasonal variations = 0; If not zero, spread the difference equally across the seasons.
Seasonal adjustment check (multiplicative): Sum of average seasonal factors = number of seasons (4 for quarters, 12 for months); If not, scale each factor by number of seasons ÷ actual sum.
Forecast: Additive: T + S. Multiplicative: T × S; T is the projected trend for the target period.
Seasonally adjusted figure: Additive: Y − S. Multiplicative: Y ÷ S; Strips out seasonality so you can see the underlying trend.
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 = index of learning.
Index of learning: b = log r ÷ log 2; r is the learning rate as a decimal. It is negative because time falls. Use b = −0.152 for 90%, −0.322 for 80%, −0.234 for 85% and −0.415 for 75%. The formula sheet normally gives the formula; check what your exam provides.
Doubling rule: Cumulative average time after doubling output = previous cumulative average × r; Use this when output doubles exactly (1, 2, 4, 8, 16 and so on). No logs needed.
Total time: Total time for X units = Y × X; Y is the cumulative average time, not the time of one unit.
Incremental time: Time for units (m+1) to n = total time for n units − total time for m units; Always subtract totals, never averages.
Steady state: Total time = time for learning-period units + (extra units × steady-state time per unit); Use only after the point at which learning is stated to stop.
Expected value: EV = Σ (probability × outcome); Add the products for every possible outcome. Use the same measure throughout, such as profit or contribution.
Probability check: Σ probabilities = 1; If the probabilities do not add to 1, a case is missing or a figure is wrong. Check this before calculating.
Joint probability (independent events): P(A and B) = P(A) × P(B); Use this only when the events are independent, for example price level and an unrelated cost change.
Maximin rule: Choose the option with the highest of the minimum outcomes; Ignores probabilities. Reflects a risk-averse decision maker.
Maximax rule: Choose the option with the highest of the maximum outcomes; Ignores probabilities. Reflects an optimist or risk seeker.
Index number: Index = (Current price or value ÷ Base price or value) × 100; The base period always equals 100.
Updating a figure: New value = Old value × (Index at new date ÷ Index at old date); Use for budgets, cost updates and price changes. The index must suit the item.
Real terms (deflating): Real value = Cash value × (Base index ÷ Current index); Removes inflation so different years are compared on the same price basis.
Percentage change from an index: % change = (Later index ÷ Earlier index − 1) × 100; Do not subtract the indices and call it a percentage unless the earlier index is 100.
Compound inflation: Future value = Current value × (1 + i)ⁿ; i is the annual rate and n the number of years.
Weighted index: Weighted index = Σ(index × weight) ÷ Σ weights; Use when a cost is made up of several items of different importance.

Quick revision

High-low: variable cost per unit = (cost at highest activity − cost at lowest activity) ÷ (highest activity − lowest activity). Then fixed cost = total cost − variable cost.
High-low uses only the highest and lowest activity levels, so it can be distorted by unusual points.
Regression line: y = a + bx, where b is the variable cost per unit (or slope) and a is the fixed cost (or intercept).
The correlation coefficient r runs from −1 to +1. Values near +1 or −1 mean a strong linear relationship. Near 0 means a weak one.
Coefficient of determination = r². It is the share of the variation in y explained by x.
Do not forecast far outside the range of your data. Reliability falls the further you extrapolate.
Time series: trend + seasonal variation (additive model), or trend × seasonal factor (multiplicative model). Seasonal variations should sum to zero in the additive model.
Learning curve: cumulative average time per unit falls by a fixed percentage each time cumulative output doubles. Check which version the question uses.
Expected value = Σ (probability × outcome). Probabilities must add up to 1.
Expected value is a long-run average. It may not match any single outcome.
Index number = (current value ÷ base value) × 100. Use it to convert figures to a common price level.
Objective test answers score all or nothing, so check units, signs and rounding before you submit.

Common mistakes

Choosing the highest and lowest costs instead of activity levels. Fix: Look only at the activity column. Take the costs that go with those activity levels.
Forgetting to deduct the variable part when finding fixed cost. Fix: Always compute fixed cost = total cost − rate × activity.
Confusing Σx² with (Σx)². Fix: Σx² is the total of each x squared. (Σx)² is the total of x, then squared. Calculate them separately and label them.
Calculating a before b. Fix: The formula for a needs b. Always find b first.
Saying r² = 0.64 means 64% correlation. Fix: Say that 64% of the variation in y is explained by x. Then r = 0.8 (or -0.8).
Losing the negative sign when taking the square root of r². Fix: Take the sign from the slope b or from the numerator nΣxy − ΣxΣy.
Placing the moving average against the wrong period. Fix: Centre it by averaging two consecutive averages (or dividing consecutive total pairs by 8). The result lines up with the third period of the first total.
Mixing seasons when averaging the variations. Fix: Set out the variations in a grid with years down and quarters across, then average each quarter column.
Treating the time for the second unit as a × r, or treating the average as the time of the last unit. Fix: Remember Y is an average. Multiply by X to get a total, then subtract totals to get the time for specific units.
Subtracting cumulative averages instead of cumulative totals to find the time for later units. Fix: Always convert to totals first. For units 5 to 8, use total for 8 units minus total for 4 units.

Exam tips

In objective questions, find the highest and lowest ACTIVITY first. Wrong points give a wrong answer and score zero.
Read for inflation or price indices. Restate costs to one price level before subtracting.
In written parts, state the limitations: only two points, possible outliers, assumes linearity, and valid only within the relevant range.
Show the check of fixed cost from the second point. It catches arithmetic slips and earns method marks.
If a forecast is outside the observed range, say that it is an extrapolation and less reliable.
In Section A and B objective questions, the sums are usually given. Go straight to the formulas for b and then a. Wrong answers score zero, so check arithmetic once.
In Section C, show the formula, the substituted numbers and the result. Method marks are available even if you make an arithmetic slip.
Always interpret a and b in words with units. Examiners often reward the statement that a is fixed cost and b is variable cost per unit.