Formulae and calculations

Numbers run through every unit of business management: the cost of a workforce, the profit in a set of accounts, the share of a market, the break-even point of a product. Calculations are tested mostly in Paper 2, where most questions are quantitative, and occasionally in Paper 1. HL students meet extra formulas in Paper 2 and use them as evidence in Paper 3 plans.

Every formula and calculation in the course is collected here, unit by unit. Each entry gives the formula in the form examiners expect, a short worked example with invented figures, the unit the answer is written in and the error students most often make. HL-only formulas sit under headings marked "(HL only)". The arithmetic is rarely difficult, and most lost marks come from using the wrong figure, the wrong formula, the wrong unit or no working.

How calculation questions are marked

Calculation questions use the command term "calculate": obtain a numerical answer showing the relevant stages in the working. Because the definition asks for the stages, a typical [2] or [3] calculation gives one mark for the correct method (the right formula with the right figures substituted) and one for the correct answer with its unit. A bare answer that is correct usually earns full marks, but a bare answer that is wrong earns nothing, while a wrong answer with sound working can still earn the method mark.

Many calculations are followed by a second part: "Comment on your answer" [2] or "Explain what your result shows" [2]. In these parts the result is used as evidence. A comment compares the result with something (a previous year, a competitor, a target, a rule of thumb such as a current ratio of 1.5–2) and says what it means for the business in the stimulus material.

1. Write the formula in words, exactly as on the formula sheet or as learnt.

2. Substitute the figures from the stimulus, with their units ($000, units, days).

3. Calculate, keeping full accuracy until the last step.

4. State the answer with its unit and to the precision asked (2 decimal places for ratios unless told otherwise).

5. Label what the answer is ("Gross profit margin = 40.00%"), so the examiner can find it at a glance.

The own figure rule

Paper 2 questions often build on each other: a contribution per unit is used to find the break-even quantity, which is then used to find the margin of safety. If an early answer is wrong, the later parts are not automatically lost. Under the own figure rule, an examiner marks each later part using the student's own earlier figure, and awards the marks if the method applied to that figure is correct.

The rule only works if the examiner can see where the figure came from. A later answer of "margin of safety = 900 units" with no working cannot be checked against an earlier mistake, so it earns nothing if it is wrong. Writing "margin of safety = 3,500 − 2,600 = 900 units" earns the mark even though 2,600 was the wrong break-even quantity.

The formula sheet and the formulas to learn

A formula sheet is provided in the exam. It lists the profitability and liquidity ratios, the investment appraisal formulas and, for HL, the efficiency ratios and the capacity utilisation and productivity formulas. HL students are also given a table of discount factors when a question needs one. Every other formula must be learnt, and the table sorts the formulas into the two groups.

Rounding, units and presentation

Rounding and units cost more marks than difficult arithmetic. The conventions examiners use are consistent across the course.

Three habits prevent most rounding errors. Keep full accuracy in the calculator until the final step, and round once. Round a break-even quantity or target profit output up to the next whole unit, because 2,666 units would still leave a small loss when the exact answer is 2,666.67. State the unit in the data: if the stimulus is "in $000", an answer of "360" means $360,000, and should be written "$360,000" or "360 ($000)".

Negative figures and brackets

In financial statements, cash flow forecasts and budgets, a negative figure or a subtraction is written in brackets: a net cash flow of minus $6,000 is (6) in a forecast "in $000". In a calculation written out in a line, the minus sign is clearer: NPV = $197,155 − $200,000 = −$2,845. A negative net present value, a negative market growth rate or a negative PED is a meaningful answer and keeps its sign.

Using a calculator

A simple four-function calculator is enough for every calculation in the course, and a graphic display calculator is also allowed. Brackets need care, because a calculator follows the standard order of operations and divides before it subtracts.

• Use brackets for every formula with a sum or difference on top: (current assets − stock) ÷ current liabilities must be entered as (600 − 240) ÷ 300, not 600 − 240 ÷ 300.

• Store or write down intermediate results in full (average stock, capital employed, contribution per unit) and reuse them, rather than retyping rounded figures.

• Estimate first. A gross profit of 960 on sales of 2,400 is a little under half, so a margin near 40% is expected; an answer of 4% or 400% shows a slip.

• Check percentages. Divide, then multiply by 100; the % key on some calculators behaves differently and is best avoided.

Unit 1: Introduction to business management

Added value

Added value is the difference between the price of a product and the cost of the inputs bought in to make it. It measures how much the business itself has contributed through its work, skill and branding.

Added value is not profit. The $280 still has to pay the maker's wages, rent and other costs; only what is left after those is profit. The common error is to subtract every cost, which gives profit, or to subtract nothing and call the price the added value.

Unit 2: Human resource management

Time rates, piece rates and commission

Pay calculations follow directly from the definitions of each reward. A time rate pays for hours worked; a piece rate pays for units produced; commission pays a percentage of the value of sales, often on top of a basic salary.

Labour turnover (HL only)

Labour turnover measures how many employees leave a business in a period, as a percentage of the average size of the workforce.

Unit 3: Finance and accounts

Costs, revenue and profit

Every financial calculation starts from three building blocks. Total cost is the sum of fixed and variable costs; total revenue is price multiplied by quantity sold; profit is the difference between them. Average cost is the cost of each unit, and average revenue the revenue per unit, which equals the price when every unit sells at the same price.

The statement of profit or loss

The statement of profit or loss is a chain of subtractions. Each subtotal is the line above it minus the next cost, in the fixed IB order. Cost of sales is found from stock figures when it is not given directly.

The statement of financial position and working capital

The statement of financial position has its own arithmetic. Total assets minus total liabilities gives net assets, which always equals total equity. Working capital is the money available for day-to-day running.

Depreciation (HL only)

Depreciation spreads the cost of a non-current asset over its useful life. The straight line method charges the same amount every year; the units of production method charges according to how much the asset is used. The net book value is what remains of the cost after the depreciation charged so far.

Profitability ratios

The three profitability ratios are on the formula sheet and should be written exactly as it gives them. Capital employed is the long-term finance in the business: non-current liabilities plus equity.

Liquidity ratios

The two liquidity ratios compare what a business can turn into cash quickly with what it must pay soon. The acid test removes stock, the least liquid current asset.

Liquidity ratios are written as a number to 2 decimal places, not a percentage. The common error is to multiply by 100 out of habit from the profitability ratios.

Efficiency ratios (HL only)

Stock turnover measures how quickly stock is sold, either as a number of times a year or as a number of days. Debtor days and creditor days show how long customers take to pay and how long the business takes to pay its suppliers. The gearing ratio shows how much of the capital employed comes from long-term borrowing.

Cash flow forecasts

A cash flow forecast repeats two calculations for every month. The net cash flow is the difference between total cash inflows and total cash outflows; the closing balance is the opening balance plus the net cash flow, and becomes the next month's opening balance.

Payback period

The payback period is the time a project takes to repay its initial cost. It is found from the cumulative cash flow, adding each year's net cash flow to the cost (shown as a negative figure in year 0) until the total turns positive. The months within the payback year assume that the year's cash flow arrives evenly.

Average rate of return

The average rate of return (ARR) is the average annual profit of a project as a percentage of its cost. It is on the formula sheet.

Net present value (HL only)

Net present value (NPV) allows for the time value of money. Each future net cash flow is multiplied by the discount factor for its year and the chosen interest rate to give its present value; the original cost, paid now, is subtracted from the total.

The discount factors come from a table provided with the paper. The extract shows the rates that appear in it.

Budgets and variances (HL only)

A variance is the difference between a budgeted figure and the actual figure. Whether it is favourable or adverse depends on the line. For revenue, actual above budget is favourable; for costs, actual above budget is adverse.

The common error is to label a variance by its sign alone. A cost that came in $800 below budget is favourable even though actual is the smaller number; write the size of each variance and its label, and check the label against the type of line.

Unit 4: Marketing

Market share and market growth

Market share is a firm's sales as a percentage of total sales in the market. Market growth is the percentage change in the size of the market from one period to the next.

Mark-up pricing

Cost-plus pricing adds a percentage mark-up to the cost of each unit. The mark-up can also be found backwards from a price.

Mark-up is a percentage of cost, while the gross profit margin is a percentage of sales revenue. A 35% mark-up gives a gross margin of only 2.80 ÷ 10.80 × 100 = 25.93%, and mixing up the two is a frequent error.

Price elasticity of demand (HL only)

Price elasticity of demand (PED) compares the percentage change in quantity demanded with the percentage change in price. It is normally negative, and its size (ignoring the sign) shows whether demand is elastic (greater than 1) or inelastic (less than 1).

Contribution pricing sets the price at the variable cost per unit plus a chosen contribution towards fixed costs and profit. A product with a variable cost of $14 per unit and a target contribution of $6 per unit is priced at $14 + $6 = $20.

Moving averages (HL only)

A moving average smooths out short-term ups and downs to show the trend. Each three-period average is the mean of three consecutive values, placed against the middle one; the window then moves on by one period.

With quarterly data, four-quarter moving averages fall between two quarters, so each pair is averaged again to give a centred moving average that lines up with a quarter. The seasonal variation for a quarter is the actual value minus the centred trend value.

Unit 5: Operations management

Break-even analysis

Break-even calculations all start from the contribution per unit: the price minus the variable cost per unit, which is what each sale contributes towards fixed costs and then profit.

The break-even chart for the water filters shows the same results. Total revenue (TR) and total cost (TC) cross at 2,400 filters and $60,000. At 3,500 filters the vertical distance between TR ($87,500) and TC ($71,000) is the $16,500 profit, and the horizontal distance from the break-even quantity is the margin of safety of 1,100 filters.

Capacity utilisation and productivity (HL only)

Capacity utilisation compares actual output with the most that could be produced. The productivity rate compares total output with total input, both measured in comparable terms, often their value in money. Labour productivity and capital productivity measure output per unit of one input. The defect rate is the percentage of output that is faulty.

Operating leverage shows how sensitive profit is to a change in sales, and is greater when fixed costs are a large share of total costs. The degree of operating leverage compares the two percentage changes.

If sales revenue rises by 10% and profit before interest and tax rises by 25%, the degree of operating leverage is 25 ÷ 10 = 2.5: each 1% change in sales changes profit by 2.5%, in either direction.

Stock control (HL only)

The reorder level is the stock level at which a new order must be placed so that it arrives just as stock falls to the buffer stock. It depends on how fast stock is used and the lead time of the supplier. The maximum stock level is the buffer stock plus the reorder quantity.

The stock control chart shows why the formula works. Stock falls from 2,300 to the reorder level of 1,100 in four weeks; the order placed then takes two weeks, during which 600 more bags are sold, and arrives when stock reaches the buffer of 500. The reorder quantity of 1,800 lifts stock back to the maximum.

Cost to buy and cost to make (HL only)

The common error is to compare the supplier's price with the variable cost per unit alone ($4.50 against $2.80), which ignores the extra fixed costs of making the part and makes in-house production look far cheaper than it is.

Business management toolkit

Expected value in decision trees

A decision tree values each option by its expected value: the outcomes at a chance node multiplied by their probabilities and added. The cost of the option is then subtracted to give its net gain. The probabilities at each chance node must add up to 1.

Relative market share (BCG matrix)

The horizontal axis of the BCG matrix is relative market share: a product's market share compared with that of the largest competitor. A value above 1 means the product is the market leader.

A cereal with 18% of its market, where the largest rival holds 24%, has a relative market share of 18 ÷ 24 = 0.75. The vertical axis is the market growth rate, calculated like market growth.

Descriptive statistics

The mean is the total divided by the number of values; the median is the middle value of ordered data; the mode is the most frequent value. Quartiles split ordered data into quarters, and the interquartile range (IQR) is the spread of the middle half. The standard deviation measures how far values lie from the mean on average.

The mean (19) is above the median (18) because one high day (31) pulls it up; for skewed data like this the median and IQR describe a typical day better.

Float in critical path analysis (HL only)

In critical path analysis, each node shows its earliest start time (EST) and latest finish time (LFT). Total float is how long an activity can be delayed without delaying the whole project; free float is how long it can be delayed without delaying the next activity. Activities on the critical path have zero float.

Absorption costing (HL only)

Absorption costing shares indirect costs among departments or products in proportion to a chosen basis, such as floor space, labour hours or machine hours.

Rent of $90,000 shared by floor space between departments of 600 m², 300 m² and 100 m² gives $54,000, $27,000 and $9,000. In contribution costing, a special order is accepted on financial grounds if its price is above the variable cost per unit: 500 units at $18 with a variable cost of $14 add a total contribution of 500 × $4 = $2,000.

Simple linear regression (HL only)

A line of best fit through a scatter diagram can be written as y = a + bx, where a is the value of y when x is zero and b is the change in y for each one-unit rise in x. If weekly sales (y) and advertising spending in $000 (x) follow y = 120 + 4.5x, spending of $30,000 predicts sales of 120 + 4.5 × 30 = 255 units. Predictions outside the range of the data (extrapolation) are less reliable.

Checking an answer in ten seconds

Before moving on from any calculation, a short check catches most slips.