Cost Segregation and the High-Low Method

Most real costs are a mixture. A phone bill has a line rental that never changes and call charges that depend on use. To budget, to price and to judge performance, a manager needs to know how much of a cost is fixed and how much varies.

Cost segregation is the job of splitting a mixed cost into those two parts, and the high-low method is the simplest way of doing it.

This is one of the most reliably examined techniques in the paper. It can appear as a straightforward calculation, as one half of a longer question next to regression analysis, or in a harder form with a step in fixed costs or a change in variable cost per unit.

This page covers every version the syllabus shows, with the reasoning behind each step, so that you can handle whatever the examiner sets.

Cost segregation and the cost function

The total costs of a business, or of part of it such as a production line, are the sum of its fixed and variable costs. In other words, the total is semi-variable in nature. Cost segregation is the process of splitting total costs into fixed and variable parts. When it can be done, total cost can be written as a formula:

On a graph with activity on the horizontal axis and total cost on the vertical axis, this equation is a straight line. The line starts at height a (the fixed cost) and slopes upwards, and the gradient of the line is b, the variable cost per unit.

Once the equation is known, it can be used for cost estimation: working out the likely total cost at any planned activity level. Segregation and estimation are both very useful in forecasting and decision making. Two techniques build the equation: high-low analysis and linear regression analysis. This topic covers the first. The next topic covers the second.

The high-low method

High-low analysis can estimate fixed cost and variable cost per unit whenever three conditions hold:

1. total cost figures are available for at least two different levels of activity;
2. fixed costs can be assumed to be the same in total at each level; and
3. the variable cost per unit can be assumed constant at both levels.

The method uses two historical records: the highest recorded activity and its total cost, and the lowest recorded activity and its total cost. It assumes that these two records are representative of costs at all levels. Because fixed costs are the same in total at both levels, the difference in total cost between the high and low points is entirely variable cost.

The four steps

1. Identify the highest and lowest activity levels and note the total cost at each.
2. Variable cost per unit = (high cost − low cost) ÷ (high units − low units).
3. Fixed cost: substitute b into either the high or the low figures. Fixed cost = total cost − (variable cost per unit × units).
4. Write the cost function y = a + bx.

Important: choose the high and low points by activity level, not by cost. The highest cost does not always occur at the highest activity.

Worked example

A print shop records the number of jobs completed and its total monthly cost over six months.

1. Step 1. The highest activity is March (6,000 jobs, ₦51,000). The lowest is January (3,000 jobs, ₦33,000). Notice that the highest cost and highest activity happen to coincide here, but you must still pick by activity.
2. Step 2.
   Difference in activity = 6,000 − 3,000 = 3,000 jobs.
   Difference in cost = ₦51,000 − ₦33,000 = ₦18,000.
   Variable cost per job = ₦18,000 ÷ 3,000 = ₦6.
3. Step 3.
   Using the high point: fixed cost = ₦51,000 − (6,000 × ₦6) = ₦51,000 − ₦36,000 = ₦15,000.
   Using the low point gives the same: ₦33,000 − (3,000 × ₦6) = ₦15,000.
4. Step 4.
   Total cost = ₦15,000 + ₦6x.

It does not matter which point you substitute into, and doing both is a good check on your arithmetic. The function will not reproduce the middle months exactly. At 5,200 jobs it gives ₦15,000 + (5,200 × ₦6) = ₦46,200 against an actual ₦46,500. That small gap is the price of using only two observations.

Using the cost function to forecast

The equation can estimate cost at any planned level, including levels outside the observed range. For 4,000 jobs: total cost = ₦15,000 + (₦6 × 4,000) = ₦15,000 + ₦24,000 = ₦39,000.

When the data is given as a capacity percentage

A workshop budgets 72,000 machine hours, which is 80% of capacity, at a total cost of ₦468,000. At 100% capacity the total cost would be ₦540,000. First convert to hours at full capacity: 72,000 ÷ 0.80 = 90,000 hours. High: 90,000 hours, ₦540,000. Low: 72,000 hours, ₦468,000. Difference: 18,000 hours and ₦72,000, so variable cost per hour = ₦72,000 ÷ 18,000 = ₦4. Fixed cost = ₦540,000 − (90,000 × ₦4) = ₦540,000 − ₦360,000 = ₦180,000. Check with the low point: ₦468,000 − (72,000 × ₦4) = ₦468,000 − ₦288,000 = ₦180,000.

A service example

A pump repair unit records, over five months, the number of pumps repaired and its repair costs: 18 pumps ₦21,400; 30 pumps ₦28,600; 12 pumps ₦17,800; 25 pumps ₦25,100; 20 pumps ₦22,000. The high is 30 pumps (₦28,600) and the low is 12 pumps (₦17,800). Difference: 18 pumps and ₦10,800, so variable cost per pump = ₦10,800 ÷ 18 = ₦600. Fixed cost per month = ₦17,800 − (12 × ₦600) = ₦17,800 − ₦7,200 = ₦10,600. Estimated cost for 22 pumps = ₦10,600 + (22 × ₦600) = ₦10,600 + ₦13,200 = ₦23,800.

When fixed costs have a step

High-low analysis still works if fixed costs step up between the low and high activity levels, provided the size of the step is known. The idea is to put both observations on the same fixed-cost footing before comparing them.

Step given in naira

1. Identify the highest and lowest activity levels.
2. Adjust one of the totals: add the step to the cost at the lower level, or deduct the step from the cost at the higher level.
3. Variable cost per unit = (difference in adjusted costs) ÷ (difference in units).
4. Find the fixed cost by substitution, using the unadjusted pair.
5. Write the cost function for the unadjusted level.
6. Write the function for the other side of the step by reversing the adjustment to the fixed cost.

Example. A caterer’s fixed costs rise by ₦12,000 once activity reaches 8,000 meals or more. High: 11,000 meals, ₦95,000. Low: 6,000 meals, ₦58,000.

• Adjust the low.
  ₦58,000 + ₦12,000 = ₦70,000.
• Difference: 5,000 meals and ₦25,000 (₦95,000 − ₦70,000).
  Variable cost = ₦25,000 ÷ 5,000 = ₦5 per meal.
• Fixed cost above the step.
  ₦95,000 − (11,000 × ₦5) = ₦95,000 − ₦55,000 = ₦40,000.
  For 8,000 meals or more, total cost = ₦40,000 + ₦5x.
• Below the step.
  fixed cost = ₦40,000 − ₦12,000 = ₦28,000, so total cost = ₦28,000 + ₦5x.
  (Check with the original low point: ₦58,000 − (6,000 × ₦5) = ₦28,000.)
• For 9,000 meals (above the step): ₦40,000 + (9,000 × ₦5) = ₦85,000.

Step given as a percentage

If the step is a percentage, you cannot adjust a total in naira because you do not know the fixed cost yet. You need a third activity level. Choose two levels on the same side of the step, so they share the same fixed cost, and use them to find the variable cost per unit.

Then find the fixed cost on that side, and work out the fixed cost on the other side from the percentage.

Example. A laundry’s fixed costs rise by 25% when activity reaches 9,000 loads or more. Variable cost per load is constant. High: 14,000 loads, ₦187,000. Middle: 10,000 loads, ₦155,000. Low: 6,000 loads, ₦108,000.

• The high and middle points are both above 9,000, so they share the same fixed cost.
• Difference.
  4,000 loads and ₦32,000, so variable cost = ₦32,000 ÷ 4,000 = ₦8 per load.
• Fixed cost above the step.
  ₦187,000 − (14,000 × ₦8) = ₦187,000 − ₦112,000 = ₦75,000.
  Function at 9,000 loads or more: ₦75,000 + ₦8x.
• Fixed cost below the step.
  the higher figure is 125% of the lower one, so lower = ₦75,000 ÷ 1.25 = ₦60,000.
  Function below 9,000 loads: ₦60,000 + ₦8x.
  (Check with the low point: ₦108,000 − (6,000 × ₦8) = ₦60,000.)

When the variable cost per unit changes

The variable cost per unit may fall at high volumes, for instance because of bulk discounts. The same approach applies: use a third level so that two levels share the same variable cost per unit.

Example. A packaging firm’s fixed costs are constant. Variable cost per unit falls by 20% for all units once activity exceeds 15,000 units. High: 22,000 units, ₦182,000. Middle: 18,000 units, ₦158,000. Low: 12,000 units, ₦140,000.

• High and middle are both above 15,000, so they share the same variable cost per unit.
• Difference.
  4,000 units and ₦24,000, so variable cost above 15,000 units = ₦24,000 ÷ 4,000 = ₦6.
• Below 15,000 the unit variable cost is higher.
  If ₦6 is 80% of the original, the original is ₦6 ÷ 0.80 = ₦7.50.
• Fixed cost.
  ₦182,000 − (22,000 × ₦6) = ₦182,000 − ₦132,000 = ₦50,000.
• Function above 15,000 units: ₦50,000 + ₦6x. Below 15,000 units: ₦50,000 + ₦7.50x.
  (Check with the low point: ₦50,000 + 12,000 × ₦7.50 = ₦140,000.)

The ideas in more depth

Why the difference is all variable

Suppose total cost is ₦48,700 at the high point and ₦40,300 at the low point. Both totals contain the same fixed cost. When you subtract one from the other, the fixed cost cancels, and what is left is the extra variable cost caused by the extra units. Dividing by the extra units gives the cost of one more unit.

Strengths and weaknesses

Checking your answer

Always test the cost function on the point you did not use. If you built it from the high point, substitute the low point and see whether you reproduce its cost. If you get a different figure, you have made an arithmetic error or the data does not fit the method.

Common ways the question is set

• A table of months with units and costs. Pick the highest and lowest units.
• Capacity percentages, as in the 85% example. Convert them to units first.
• A given cost function to use for a flexible budget.
• Paired with regression, where you are asked to compare the two answers.

Worked example: a bakery

1. High and low by activity. High: 5,600 loaves, ₦71,800. Low: 3,000 loaves, ₦51,000.
2. Variable cost per loaf.
   (₦71,800 − ₦51,000) ÷ (5,600 − 3,000) = ₦20,800 ÷ 2,600 = ₦8.
3. Fixed cost.
   ₦71,800 − (5,600 × ₦8) = ₦71,800 − ₦44,800 = ₦27,000.
   Check: ₦51,000 − (3,000 × ₦8) = ₦51,000 − ₦24,000 = ₦27,000.
4. Cost function. y = 27,000 + 8x.
5. Estimate for 4,000 loaves.
   ₦27,000 + (4,000 × ₦8) = ₦27,000 + ₦32,000 = ₦59,000.
6. Check on an unused month.
   At 4,800 loaves the function gives ₦27,000 + (4,800 × ₦8) = ₦65,400, against an actual ₦65,000.
   The small gap of ₦400 shows the estimate is reasonable, though it also shows that the two end points alone do not capture every month exactly.

The scattergraph method

The high-low method uses two points only. The scattergraph method uses all of them, by eye. Plot output on the horizontal axis and total cost on the vertical axis, mark every observation, and draw a straight line through the middle of the points so that about as many lie above as below.

Where the line cuts the vertical axis is the estimate of fixed cost, and the slope is the variable cost per unit.

Data. Output (units) and total cost (₦): 2,000 → 31,000; 3,000 → 36,000; 4,000 → 43,000; 5,000 → 47,000; 6,000 → 55,000.

• High-low:
  variable cost = (55,000 − 31,000) ÷ (6,000 − 2,000) = ₦6 a unit.
  Fixed cost = 31,000 − (2,000 × 6) = ₦19,000.
• Scattergraph: a line drawn through the five points would cut the vertical axis at about ₦19,000 to ₦20,000, and rise by about ₦6 for each extra unit.
  The middle three points lie close to the line y = 19,000 + 6x (which gives 37,000, 43,000 and 49,000 at 3,000, 4,000 and 5,000 units, against actual 36,000, 43,000 and 47,000).

The scattergraph shows whether the relationship is really linear and whether any observation is an outlier. Its weakness is that a line drawn by eye is subjective, so two people can draw slightly different lines. Regression removes this subjectivity.

The three methods compared

When costs are recorded at different price levels

If inflation has pushed prices up between the high and low months, the cost difference includes a price effect as well as a volume effect. The variable cost will be overstated. Adjust all costs to the same price level first. Example. At 4,000 units, cost was ₦46,000 when the price index was 100. At 6,000 units, cost was ₦66,000 when the index was 110.

1. Restate the later cost at the earlier price level: ₦66,000 × 100 ÷ 110 = ₦60,000.
2. Variable cost.
   (₦60,000 − ₦46,000) ÷ (6,000 − 4,000) = ₦7 a unit (at index 100).
3. Fixed cost.
   ₦46,000 − 4,000 × ₦7 = ₦18,000 (at index 100).

To forecast cost at a future price level, multiply the answer by the expected index. At an index of 120 and 5,000 units, cost = (₦18,000 + 5,000 × ₦7) × 1.20 = ₦53,000 × 1.20 = ₦63,600. Without the adjustment, the high-low method would have given ₦10 a unit (₦20,000 ÷ 2,000), a big overstatement.

Splitting a semi-variable overhead for a budget

Budgets often need the fixed and variable parts of a cost such as electricity. Suppose the electricity bill was ₦92,000 in a month of 10,000 machine hours and ₦116,000 in a month of 14,000 machine hours.

• Variable cost.
  (₦116,000 − ₦92,000) ÷ (14,000 − 10,000) = ₦6 per machine hour.
• Fixed cost.
  ₦92,000 − 10,000 × ₦6 = ₦32,000 a month.
• Budget for 12,000 machine hours = ₦32,000 + 12,000 × ₦6 = ₦104,000.

The same figures feed a flexible budget: fixed ₦32,000 plus ₦6 for each machine hour, whatever the activity. This is how the budget cost allowance in the budgeting topic is found in practice.

Where the cost equation is used

• Flexible budgets: to find the allowed cost at the actual activity level.
• Cost-volume-profit (CVP) analysis: the variable cost per unit and fixed cost are the inputs for break-even and target profit.
• Pricing and special orders: the variable cost per unit shows the minimum price.
• Forecasting: total cost at a future activity level.
• Standard costs: the variable overhead rate per hour.

Common mistakes in the exam

• Picking the highest and lowest costs instead of the highest and lowest activity levels.
• Dividing the cost difference by the wrong number of units, or by the total units instead of the difference.
• Forgetting to find the fixed cost after finding the variable cost.
• Using the variable cost for only one of the two points and getting different fixed costs. Check both.
• Not adjusting for a known step in fixed cost, or adjusting the wrong total (add to the low or deduct from the high, not both).
• Mixing up the two sides of a step when forecasting. Use the function that fits the activity level asked for.
• Writing the final equation with a and b the wrong way round.

Quick revision

• y = a + bx. a = fixed cost. b = variable cost per unit (the gradient).
• b = (high cost − low cost) ÷ (high units − low units). a = total cost − b × units.
• Step in fixed costs given in naira: add to the low total or deduct from the high total. Find b, find a on the unadjusted pair, then reverse the adjustment for the other side.
• Step given as a percentage, or a percentage change in variable cost: use a third level so that two levels share the same fixed cost (or same unit variable cost).
• Check with the other point. Use the function to estimate cost at any activity level.
• High-low uses two points only. Regression uses all points and measures reliability.

Practice questions

These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer.

Section A style: multiple choice

Section B style: written questions

A further written question

Model answers

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