5.5 Break-even analysis

Full video class on YouTube, summary and notes on Instagram, class extracts on TikTok, text below. Have fun!

The main point of this class is to learn how to construct break-even chart and calculate break-even quantity and other aspects of break-even analysis.

Contribution is the difference between the selling price and variable costs. In other words, contribution is what is remained to pay for fixed costs. There are two kinds of contribution: per unit and total. Contribution per unit = unit price (P) – unit (average) variable cost (AVC). Total contribution = total revenue (TR) – total variable costs (TVC). Another way to calculate total contribution is to multiply contribution per unit by quantity (Q). Remember all the three formulae below:

Contribution per unit = P – AVC

Total contribution = TR – TVC

Total contribution = contribution per unit ⨉ Q

It sometimes seems weird to my students why we actually need to learn what contribution is and why it is called this way. Well, the answer to both issues is fixed costs. Contribution, basically, helps to find out how different products contribute to paying for fixed costs and hence its name. Now, why else is it important to know how different products contribute towards fixed costs? See some of the reasons below.

Break-even analysis. Later in this class you will see that once you know contribution, you may find out break-even quantity really quickly.

Calculating profit. Usually, profit is calculated by breaking down the “TR – TC” equation (total revenue minus total costs), but once you know contribution and fixed costs, you may calculate profit in an alternative way, by subtracting total fixed costs from total contribution (total contribution – TFX).

Pricing strategy. There are many factors that impact pricing decisions, and contribution is one of them. If a product price is below contribution, then perhaps it’s worth raising (unless there are reasons to keep it low, for example as in loss leader pricing).

Product portfolio management. Knowing contribution can help marketing managers make decisions about eliminating products from the portfolio and it helps finance and accounts managers to plan the budgets.

IB requirement. Well, the ultimate reason why we learn things that we learn is because it is required by the IBO, haha.

Now imagine you are a manager at a school cafeteria and you have information about contribution of different items from the menu. Using the reasons above, think how useful (or not) the information provided is.

Assessment objectives are always important, but for this part of class they are super important! We’ll achieve them one by one in the sequence stated above. In this class, it will not be possible to isolate objectives or achieve them in a random order. So, here we go: let’s learn to construct and analyse break-even chart.

For starters, let’s make sense of this nonsense: BE, BEQ, BEC, BEP, BEA.

Break-even (BE) is a state at which organisation’s revenue equals its costs. If organisation is breaking even, it means that it does not have any profits, but also it does not experience loss. It means that revenue is just enough to cover the costs.

Break-even chart (BEC) is a graph that illustrates the value of costs and revenues against the volume of output. Basically, it is a picture that demonstrates what break-even is.

Break-even quantity (BEQ) is the number of items that have to be sold in order to break even, i.e. how many units of output need to be sold in order to cover the costs. Anything that is sold beyond BEQ would be profit, and anything below would be loss.

Break-even point (BEP) is the spot on the break-even chart (BEC) where total revenue (TR) and total costs (TC) intersect. Break-even point (BEP) is what represents break-even quantity (BEQ) on the break-even chart (BEC). It only exists on break-even chart.

Break-even analysis (BEA) is a business management tool that is used to determine the appropriate level of sales. It includes all of the definitions above and is a decision-making process that helps managers to set the right price, or reduce costs, or make other decisions that would help organisations to maximise profits.

Please mind the key words in italics in the definitions above. They really help to distinguish between all those concepts that basically refer to different aspects of the same thing: BE is a state, BEC is a graph, BEQ is a number, BEP is a spot, BEA is a tool.

Now let’s finally get to constructing the break-even chart. At this point, you might feel that you do not entirely understand everything, and it’s great if you feel this way. It means that everything’s going according to the plan. First, we’ll start with the big picture, and then we’ll narrow it down to smaller steps. So, please do not worry if something is unclear at the first step. You’ll get all the answers eventually. The basic rule for constructing a break-even chart is:

“Calculate — draw — label — indicate”

Now let’s see what exactly to calculate, draw, label and indicate:

1. Calculate BEQ, costs and revenues (if needed)
2. Draw the axes and total revenue (TR), total costs (TC), total fixed costs (TFC) lines
3. Label the chart, all lines and axes
4. Indicate BEP, BEQ, profit, loss

Congratulations! You have achieved the first assessment objective (AO) and you have a basic understanding of how to construct break-even charts. If you know how to construct it, then you can break it down into elements and explain what they mean, which means that you can also analyse break-even chart. Now let’s go more in-depth and figure out how exactly to calculate and analyse all the necessary aspects of break-even analysis: BEQ, BEP, profit or loss, margin of safety, target profit output, target profit and target price. This way, you will also achieve the second assessment objective (AO) for this part of class.

Some of the aspects of BEA are related, so for convenience, I have divided all the aspects into the following 5 groups:

1. BEQ & BEP
2. Profit or loss
3. Margin of safety
4. Target profit output
5. Target profit and target price

I hope that you remember that break-even quantity (BEQ) and break-even point (BEP) are essentially one thing, but BEQ is a number, and BEP is a spot on break-even chart (BEC) that represents BEQ. When you construct a BEC, make sure it includes both of them! BEP is at the intersection of TR and TC lines and BEQ is on the Output axis. See Figure 2 above and find both of them there.

How to calculate BEQ? There are three ways: break-even chart, “TR = TC” rule, and “Contribution per unit” rule.

The first way is the most time-consuming and inconvenient. What you do is draw the chart and find the BEP and BEQ there. You may take random quantity in order to draw TR and TC lines and then see where they intersect. That intersection would be BEP which, as we’ve already learnt, represents BEQ. That would work if break-even chart is really precise. If it’s just drawn on a napkin with a pencil, I’m afraid this is the least suitable way. Luckily, we have two more.

The second way is "TR = TC" rule. “TR = TC” rule basically refers to breaking down the TR = TC equation. In case you forgot, TR is total revenue and TC is total costs. Break-even quantity (BEQ) is the quantity when TR = TC, so just break down the equation until you can figure out Q:

TR = TC

P x Q = TFC + TVC

P x Q = TFC + (AVC x Q)

(P – AVC) x Q = TFC

Q = TFC ÷ (P – AVC)

(Q in this equation refers to BEQ)

If you get it, then you can skip this paragraph. If not, then total revenue (TR) is the same as price (P) multiplied by quantity (Q) of units sold. Total costs (TC) is a sum of total fixed costs (TFC) and total variable costs (TVC). Review class 3.3 if you have no clue about costs. Total variable costs (TVC) are calculated by multiplying the quantity (Q) of units sold and average variable costs (AVC), which are the same as variable costs per unit. Then, once you broke down TR and TC into P, Q, TFC and AVC, just do the math, move the variables until you get to Q, which represents break-even quantity (BEQ).

The last method is “Contribution per unit” rule. This one is my favourite. We have already learnt contribution in the first part of class and I’ve already mentioned that it can be used to calculate BEQ in an extremely convenient way:

BEQ = TFC ÷ contribution per unit

In the picture below, you may see how BEQ is calculated using all three ways. Keep in mind, that it never hurts to use two ways at the same time to make sure you did not make a mistake. And final reminder: BEP is the same as BEQ but it is a spot on the chart where TR and TC lines intersect.

Important warning! No decimals in BEQ, please. Always round numbers up. This is because in reality organisations cannot produce 3,872 of a chair, table, smartphone or anything else. It’s always whole numbers.

Profit and loss are the two sides of the same thing. Profit is the positive difference between revenues and costs. Loss is the negative difference between revenues and costs. There are two ways how to calculate profit/loss. One way is to subtract total costs (TC) from total revenues (TR):

Profit/loss = TR – TC

The second way is to subtract total fixed costs (TFC) from total contribution:

Profit/loss = total contribution – TFC

Just a reminder that total costs (TC) equal to a sum of total foxed costs (TFC) and total variable costs (TVC):

TC = TFC + TVC

Speaking of the break-even chart, both profit and loss are between TR and TC lines, but profit is to the right of the break-even point (BEP), and loss is to the left of the BEP, as indicated in the picture below.

Margin of safety (MOS) is the difference between the break-even quantity (BEQ) and actual level of output. In other words, the difference between break-even quantity and how many units organisation managed to sell in reality. The greater the margin of safety, the better, because it results in more profit. Additionally, the greater the margin of safety, the lower the risk is for the organisation in terms of demand fluctuations. If margin of safety is low, it means that actual/real sales are very close to BEQ, so a sudden change in demand will have a serious impact on organisational finance. However, if MOS is high, organisation is less sensitive to demand fluctuations.

Margin of safety can be expressed either as a number of units or as a percentage. With regards to percentage, it may be relative to either break-even quantity or actual sales. Let’s see the formula and figure out what it means.

MOS (units) = actual output – BEQ

MOS (%, relative to BEQ) = MOS (units) ÷ BEQ

MOS (%, relative to sales) = MOS (units) ÷ actual sales

For example, if XYZ Ltd needs to sell 100 chairs to break even, but it actually sells 140 chairs, then MOS is 40 chairs, or 40% greater than break-even quantity, or is about 29% of actual sales.

Target profit output is the level of output/quantity (Q) required to earn desired profit. Basically, it means “how many units should organisation produce to achieve desired profit”. There are 3 ways to calculate target profit output: finding it on break-even chart, using a formula, and “TR = TC” rule.

Using the break-even chart is not the most convenient method because it is really time-consuming and, if break-even chart is hand-drawn, it would not be precise enough to determine the exact target profit output. Regardless of that, if you just find the target (desired) profit on break-even chart (space between TR and TC lines) and see which level of output it refers to on the “output” axis, you will determine target profit output. See Figure 6 below for a worked example.

The second way to calculate target profit output is using the formula below:

Target profit output = (fixed costs + target profit) ÷ contribution per unit

For example, XYZ Ltd sells chairs for 5000₽ per item and has fixed costs of 300.000₽ and contribution of 3000₽ per chair. Let’s say, they want to make 100.000₽ profit, so how many chairs do they have to sell to achieve their dream profit? Or, in other words, what is their target profit output? Applying the formula above, we’ll get (300.000 + 100.000) ÷ 3.000 = 133.33 chairs. Do you remember that when it comes to break-even analysis, there are no decimal points? That’s because it is impossible to produce 133,33 of a chair. That is why, we’ll round the answer and record it as 133 chairs. So, XYZ Ltd has to sell 133 chairs to make a 100.000₽ profit.

The last way to calculate target profit output is to break down “Target profit = TR – TC” equation down to Q, whereby Q would be target profit output. This is very similar to breaking down “TR = TC” equation in calculating break-even quantity (BEQ). Eventually, after all the manipulations with the formula, you’ll get to this:

Target profit = P x Q – (TFC + TVC)

Target profit = P x Q – (TFC + AVC x Q)

Target profit + TFC = (P – AVC) x Q

Q = (target profit + TFC) ÷ (P – AVC)

Let’s continue with XYZ Ltd and its chairs and assume that target profit that XYZ Ltd dreams of is still 100.000₽ and that they want to keep charging 5.000₽ per chair. Another thing we need to know is average variable costs (AVC), which equals to price (P) minus contribution per unit: 5.000₽ – 3.000₽ = 2.000₽. Then, target profit output would be (100.000₽ + 300.000₽) ÷ (5.000₽ – 2.000₽) = 133 chairs. Again, XYZ Ltd has to sell 133 chairs to make a 100.000₽ profit.

All the examples how to calculate target profit output using three different ways are summarised in the picture below:

This is very similar to target profit output. “Target”, in this context, means “desired under certain conditions”. So, target profit is the required profit at the target profit output level and target price is price that is required to achieve target profit. There are no special formulae for target profit and target price, all you do is just alternate the variables with the desired target profit/price/output and calculate whatever you have to calculate. You may use the same three ways as in the the target profit output.

Let’s use XYZ Ltd and chairs again. Assuming that the fixed costs are still 300.000₽, contribution per unit is still 3.000₽ and target profit output is now 200 chairs (not 133 like in the previous example), then what should the target profit be? I’ll use my favourite formula for target profit output where a sum of fixed costs and target profit is divided by contribution per unit (see Aspect 4): 200 = (300.000 + target profit) ÷ 3.000, which means that target profit = 200 ⨉ 3.000 – 300.000 = 300.000₽. This means that if XYZ Ltd manages to sell 200 chairs at 5.000₽ per chair, it will make a profit of 300.000₽. You can play with this formula however you want. For example, assume that XYZ Ltd can sell 1000 chairs at 10.000₽ per chair and the rest of the variables are the same, what would the target profit be then?

Now, target price. Let’s say fixed costs are still 300.000₽, contribution is still 3.000₽ (which means that variable costs are still 2.000₽ per chair), target profit output is still 200 chairs, but XYZ Ltd now wants to make 500.000₽ under these conditions, then what should the price be? We’ll use the same formula for target profit output as in the previous paragraph. However, this time, we’ll have to replace “contribution per unit” with “target price – average variable costs”, because it is the same thing (we just broke down contribution into two elements), but we need target price, hence this extra step. So, after all the manipulations we get the following: 200 = (300.000 + 500.000) ÷ (target price – 2.000), which means that target price = (200 ⨉ 2.000 + 300.000 + 500.000) ÷ 200 = 6.000₽. So, in order to reach a target profit of 500.000₽ by selling 200 chairs, XYZ Ltd needs to charge 6.000₽ per chair.

If you understand all the formulae we’ve mentioned so far and understand how basic arithmetics work, you should not have a problem alternating the variables and figuring out target profit and target price under different circumstances.

Congratulations! If everything is clear so far, then you have achieved the first two assessment objectives (AO) for this part of class and now we will move on to the last one — to learn to explain and determine the effects of changes in price/cost on BEQ, profit and MOS, using graphical and quantitative methods.

Let’s break down the last assessment objective (AO) for this part of class. We have three elements: changes, effects, and some aspects of the break-even analysis. The goal here is to see how changes in price and costs (fixed and variable) impact break-even quantity (BEQ), profit and margin of safety (MOS). So we have three things that have impact on the other three things. Below there is a summary of all the effects of changes.

There are two ways how we can analyse the effects of changes in price and costs: graphical method and quantitative method. The former means that you use break-even chart (BEC), alter some of the lines that indicate changes in price, variable and fixed costs and see how BEQ, profit and MOS change. The latter means that you alter the variables in the formulae to see how it impacts BEQ, profit and MOS. This is not very difficult, as long as you are fine with basic math. Below are worked examples of analysing the effects of changes in price and costs (fixed and variable) on all three variables (BEQ, profit and MOS), using graphical method.

On the one hand, break-even analysis is a relatively simple tool, that is quite quick to apply. As long as you have data about costs and revenues, you may quickly figure out break-even quantity that will help greatly with decision-making in so many aspects! For example, it helps with determining the price, with production planning (identifying how much output to produce), with cost management. It also helps to anticipate different hypothetical scenarios (with different levels of target profit output, target price and target profit) and determine appropriate strategies.

On the other hand, reality is not linear, as in break-even chart. Fixed costs change, prices change, demand changes, etc. So, all these changes are not indicated in a break-even chart which is a static and simplistic indicator of reality with straight lines indicating all the costs and revenues. It’s still very helpful for a lot of decisions but it makes it really hard to apply break-even analysis for rapidly changing businesses and environments. If prices, demand or costs change often, then there is not much value in break-even analysis as it will outdate quickly.

Make sure you can define all of these:

1. Contribution
2. Contribution per unit
3. Total contribution
4. Break-even
5. Break-even chart
6. Break-even point
7. Break-even quantity
8. Break-even analysis
9. Profit
10. Loss
11. Margin of safety
12. Target profit output
13. Target profit
14. Target price