ACCA PM — Performance Management

The PM formula sheet,
actually explained

Six formulas. Three areas. Every one with a live calculator you can type your own numbers into.

You get these formulas in the exam. Everybody does. Knowing the formula was never the hard part — knowing which number goes where is.

Start with regression

What's on the sheet

PM gives you formulae only — and unlike FM, no discount or annuity tables. Three areas, six formulas, and that is the lot.

1 · Regression equation 2 · Correlation coefficient 3 · Learning curve 4 · Learning index 5 · Demand curve 6 · Marginal revenue What it doesn't give you
Area 1 of 3

Regression analysis

Splitting a mixed cost into its fixed and variable parts — properly, rather than with high-low.

Formula 1

The regression equation

As given in the ACCA exam y = a + bx
a = (Σy)/n − b(Σx)/n
b = [ nΣxy − ΣxΣy ] ÷ [ nΣx² − (Σx)² ]

What each bit is

  • y — total cost (the thing you're predicting)
  • x — activity level (units, hours, machine hours)
  • a — fixed cost: the intercept, where the line crosses at x = 0
  • b — variable cost per unit: the gradient
  • n — number of paired observations

Live calculator — type your own numbers

b — variable cost per unit—
a — fixed cost—
The line—
Predicted y at x = —

Worked example

Six months of output and total cost. n = 6, Σx = 1,350 units, Σy = $13,250, Σxy = 3,262,500, Σx² = 347,500.

  1. Numerator of b: (6 × 3,262,500) − (1,350 × 13,250) = 19,575,000 − 17,887,500 = 1,687,500
  2. Denominator of b: (6 × 347,500) − 1,350² = 2,085,000 − 1,822,500 = 262,500
  3. So b = 1,687,500 ÷ 262,500 = $6.43 per unit
  4. Then a = (13,250 ÷ 6) − 6.4286 × (1,350 ÷ 6) = 2,208.33 − 1,446.43 = $761.90
  5. Cost equation: y = 761.90 + 6.43x — fixed costs $761.90 a month, $6.43 a unit on top

Where the marks go

  • Σx² is not (Σx)². Square each x then add, versus add all the x's then square. They are wildly different numbers and the formula uses both — one in each half of the denominator. This single confusion costs more marks than anything else on the sheet.
  • Find b first, always. The formula for a contains b. Students who start with a end up stuck or invent a value.
  • a is the fixed cost, b is the variable cost per unit. Getting these the wrong way round produces an answer that is internally consistent and completely wrong — and the examiner cannot award follow-through for it.
  • Don't round b early. Carry it to at least two decimals into the calculation of a, or your fixed cost drifts.
Sense check: b should look like a plausible cost per unit and a should be positive in almost every exam scenario. A negative fixed cost, or a variable cost of $4,000 a unit on a $6,500 total, means you have swapped a term somewhere.
Formula 2

The correlation coefficient

As given in the ACCA exam r = [ nΣxy − ΣxΣy ] ÷ √{ [ nΣx² − (Σx)² ] [ nΣy² − (Σy)² ] }

What it tells you

  • r — how closely the points sit to a straight line, from −1 to +1
  • r = +1 — perfect positive linear relationship
  • r = 0 — no linear relationship at all
  • r² — the coefficient of determination: the proportion of the variation in y explained by variation in x

Live calculator

r — correlation coefficient—
r² — coefficient of determination—
In words—

Worked example

Same six months, plus Σy² = 31,072,500.

  1. Top: (6 × 3,262,500) − (1,350 × 13,250) = 1,687,500
  2. First bracket: (6 × 347,500) − 1,350² = 262,500
  3. Second bracket: (6 × 31,072,500) − 13,250² = 186,435,000 − 175,562,500 = 10,872,500
  4. Bottom: √(262,500 × 10,872,500) = 1,689,388
  5. r = 1,687,500 ÷ 1,689,388 = 0.999 — a very strong positive correlation, so the line is reliable for forecasting within this range of output

Where the marks go

  • The square root covers the whole product of both brackets — not just the first one. Rooting one bracket and multiplying is a common and expensive slip.
  • r² is not r squared as a percentage of something else. If r = 0.9, then r² = 0.81, meaning 81% of the variation in cost is explained by the variation in activity. Say it in those words and the mark is yours.
  • Interpretation carries marks of its own. A number alone rarely scores. State the direction, the strength, and — if asked — whether the regression line is therefore reliable for forecasting.
  • Correlation is not causation, and a high r says nothing about whether extrapolating beyond the observed range is safe. Examiners ask for that caveat routinely.
Sense check: r must land between −1 and +1. If it doesn't, you have either mis-keyed a sum of squares or lost the square root. There is no such thing as r = 3.4.
Area 2 of 3

The learning curve

The most examinable two lines on the whole sheet — and the ones most often applied to the wrong quantity.

Formula 3

Y = axb

As given in the ACCA exam Y = axb

What each bit is

  • Y — the cumulative average time per unit for x units
  • a — the time taken for the very first unit
  • x — the cumulative number of units produced
  • b — the index of learning (see formula 4)

Live calculator

b — index of learning—
Y — cumulative average per unit—
Total time for all x units—
Time for unit x alone (incremental)—

Worked example

First unit takes 100 hours. 80% learning curve. How long for 8 units in total, and how long does the 8th unit alone take?

  1. b = log 0.8 ÷ log 2 = −0.09691 ÷ 0.30103 = −0.3219
  2. Y = 100 × 8^−0.3219 = 100 × 0.512 = 51.2 hours — that's the average per unit
  3. Total for 8 units: 51.2 × 8 = 409.6 hours
  4. For the 8th unit alone, do the same for 7 units: Y₇ = 100 × 7^−0.3219 = 53.45 hrs → total 374.1 hrs
  5. Incremental 8th unit: 409.6 − 374.1 = 35.5 hours

Where the marks go

  • Y is the average, not the total, and not the last unit. This is the single biggest error on the paper. If the question asks for total time you must multiply by x. If it asks for the time of the nth unit you must do the calculation twice and subtract.
  • b is negative. Always. If your b is positive, times are increasing with experience, which is not learning. Check the minus sign before going any further.
  • Read what's being costed. Learning applies to labour hours — and therefore to labour cost and to variable overhead absorbed on labour hours. It does not apply to materials.
  • Watch for the steady state. Once learning ends, every further unit takes the same time as the last one produced — you no longer use the formula, you use that flat rate. Questions signal this with a phrase like "after the 50th unit no further learning occurs".
Sense check: the cumulative average must be lower than a and must fall as x rises. On an 80% curve, doubling output from 1 to 2 units should give exactly 80 hours average, from 2 to 4 exactly 64. If doubling doesn't produce the learning rate, your b is wrong.
Formula 4

The index of learning

As given in the ACCA exam b = log LR ÷ log 2

What each bit is

  • LR — the learning rate as a decimal (80% → 0.8)
  • log — either base works, as long as you use the same one top and bottom
  • b — always between −1 and 0

Live calculator — works both ways

b — index of learning—
Check: average after doubling—

Worked example

  1. 90% curve: b = log 0.9 ÷ log 2 = −0.04576 ÷ 0.30103 = −0.152
  2. 80% curve: b = log 0.8 ÷ log 2 = −0.322
  3. 70% curve: b = log 0.7 ÷ log 2 = −0.515

Note the pattern — the steeper the learning (lower percentage), the more negative b becomes.

Where the marks go

  • LR goes in as a decimal. Typing log 80 rather than log 0.8 gives b = +6.3, and everything downstream collapses.
  • It's log 2 on the bottom, not log of anything in the question. The 2 is there because a learning curve is defined on doubling output. It never changes.
  • Don't confuse the learning rate with the rate of reduction. An 80% curve means each doubling takes 80% of the previous average — a 20% reduction. Questions sometimes quote the reduction, so read carefully.
Sense check: b for any sensible learning rate sits between about −0.15 (90%) and −0.52 (70%). Anything outside that range, or anything positive, is a keying error.
Area 3 of 3

The demand curve

Pricing questions live or die on getting a and b right before you do anything clever.

Formula 5

P = a − bQ

As given in the ACCA exam P = a − bQ
b = (change in price) ÷ (change in quantity)
a = price when Q = 0

What each bit is

  • P — selling price per unit
  • Q — quantity demanded at that price
  • a — the theoretical price at which demand falls to zero
  • b — how much price must fall to sell one more unit

Live calculator

b — slope—
a — price at zero demand—
Demand equation—
Price at Q = —

Worked example

At $50, demand is 10,000 units. Every $2 reduction in price sells another 500 units.

  1. b = 2 ÷ 500 = 0.004
  2. a = 50 + (10,000 ÷ 500) × 2 = 50 + 40 = $90
  3. Demand equation: P = 90 − 0.004Q
  4. Check it against what you were told: at Q = 10,000 → P = 90 − 40 = $50 ✓

Where the marks go

  • b is price change per one unit, not per batch. If $2 shifts 500 units, b is 0.004 — not 2, and not 500. Nearly every lost mark here starts by putting the batch size into b.
  • Build a from a known point. Take the current price and add b × current quantity. Don't guess it and don't assume it's the highest price in the question.
  • Always verify. Substitute the quantity you were given back into your equation. If it doesn't return the price you were given, your a or b is wrong — and you have thirty seconds to find out for free.
  • Mind the sign. The minus is already in the formula, so b itself is entered as a positive number. Making b negative as well flips the curve upward.
Sense check: a must be higher than any selling price in the question, and b is usually a small decimal. If a comes out below your current price, you have subtracted where you should have added.
Formula 6

MR = a − 2bQ

As given in the ACCA exam MR = a − 2bQ

What it's for

Profit is maximised where marginal revenue = marginal cost. Set MR = MC, solve for Q, then put that Q back into the demand equation to get the price to charge.

Live calculator — profit-maximising price

Optimal quantity—
Optimal price—
MR at that quantity—
Contribution at optimum—

Worked example

Using P = 90 − 0.004Q, with marginal cost of $30 a unit.

  1. MR = 90 − 0.008Q
  2. Set MR = MC: 90 − 0.008Q = 30
  3. 0.008Q = 60 → Q = 7,500 units
  4. Price from the demand curve: P = 90 − 0.004 × 7,500 = 90 − 30 = $60

Where the marks go

  • Put Q back into the demand curve, not into MR. Substituting into MR gives you $30 — the marginal cost — and students hand it in as the selling price surprisingly often. MR finds the quantity; P = a − bQ finds the price.
  • The 2 only appears in MR. Half the errors here are writing 2b in the demand equation or forgetting it in MR.
  • Marginal cost means variable cost. Fixed costs and absorbed overhead are irrelevant to the optimum. If the question hands you a full absorption cost, strip it back first.
  • Check the optimum is achievable — capacity constraints or a maximum market size can make the theoretical answer impossible, and saying so earns the mark.
Sense check: the optimal price should sit comfortably between marginal cost and a. If your price is below marginal cost, or above a, re-check which equation you substituted into.
Worth knowing

PM gives you no maths tables

FM candidates get present value and annuity tables. PM candidates get formulae and nothing else — no discount factors, no annuity factors, nothing for logs either.

In practice this matters twice: you will be computing log 0.8 ÷ log 2 on your calculator rather than looking it up, and any discounting that appears in a PM question comes with its factors supplied inside the question itself. If they aren't there, you aren't meant to discount.

The bit nobody tells you

What the sheet doesn't give you

Six formulas are given to you. The rest of PM is examined just as hard, and every one of these has to come out of your own head.

  1. Every variance. Material price and usage, labour rate and efficiency, variable and fixed overhead, sales price and volume — none of them are on the sheet.
  2. Mix and yield variances, materials and sales, including the individual-versus-weighted-average methods.
  3. Planning and operational splits — the revised-standard logic that decides which half is which.
  4. Breakeven point, margin of safety and target profit, single-product and multi-product.
  5. The C/S ratio and the weighted average C/S ratio for a product mix.
  6. Throughput accounting ratio, return per factory hour and cost per factory hour.
  7. ABC cost driver rates — pool cost divided by driver volume.
  8. Target cost and the cost gap, and life-cycle cost per unit.
  9. ROI and residual income, including the imputed interest charge.
  10. Transfer pricing rules — the minimum and maximum transfer price logic.
  11. Expected values, and the decision rules: maximin, maximax, minimax regret.
  12. Price elasticity of demand, plus mark-up and margin conversions.

Knowing it isn't the same as scoring it

Every formula above turns up in real PM questions. Go and meet them where it counts — in past exam questions, with the marks attached.

Practise PM questions