6.1 Break-even Analysis

A. Introduction

One of the primary goals of many businesses is to make a profit. Achieving this goal requires that total revenue surpasses the total costs incurred. As such, it becomes essential for business owners to carefully examine variables like the selling price per unit, the quantity of products sold, and the relationship between total revenue and costs. A powerful tool to aid this understanding is the Cost-Volume-Profit (CVP) analysis.

A part of CVP analysis looks at the ‘break-even point’. This represents the sales volume at which total revenue exactly equals total costs, resulting in a balance where neither profit nor loss is realized. This type of analysis is known as break-even analysis.

While the break-even point serves as a foundational benchmark, most businesses want to do better than just breaking even. They want to earn a profit. Therefore, understanding the impact of sales on earnings is important. Furthermore, to be competitive in the market, businesses try to set competitive selling prices and look into ways to lower their costs to increase profitability. This chapter will delve into these critical aspects of business decision-making, looking at how CVP and break-even analyses can help businesses make these decisions.

B. Compute Break-even Point Using Break-even Analysis

Imagine Sarah, who has started a small business selling hand-made candles. Each candle costs her $5 to make (materials), and she sells them for $25 each. Sarah also has fixed costs, which include her monthly rent for the workshop, totaling $2000 a month. Before she can make any profit, Sarah needs to know how many candles she must sell to cover her costs, or in other words, to break even. Table 5.1.1 summarizes the revenue, cost, and profit for Sarah’s business based on the different numbers of candles sold in a given month.

Table 5.1.1: Revenue, cost, and profit for different numbers of candles sold in a given month

Note that at the point where 100 candles are sold, the net income is zero. This is recognized as the break-even point, a point where total revenue precisely matches total costs, resulting in a net income of zero. Selling fewer than 100 units (e.g., 0 and 10 candle cases) results in a financial loss, whereas selling more than the break-even point of 100 units (such as the 200 candle case) ensures a profit.

C. CVP Assumptions

To streamline our discussion on CVP and break-even analyses, let us establish some foundational assumptions:

1. Selling Price: The price at which we sell each unit remains the same, so as we sell more, our total revenue increases at a consistent rate.
2. Costs categories and behavior: Costs are linear and can be classified as either fixed or variable cost
   • Fixed costs remain constant over a given time period. They do not depend on volume and remain unchanged throughout a specific time period. Examples include rent, mortgage, property taxes, and salaries.
   • Variable costs are constant per unit regardless of volume. Some of the examples are direct material, labor costs, and sales commissions.
3. Inventory: All units produced in a particular operating period are sold, i.e., there is no inventory.
4. Multiple products: When multiple products are produced within a business, the ratio of various products remains constant.

Note that in reality, fixed costs don’t always remain the same across different levels of production; they can change. Similarly, the unit variable cost can vary, often due to economies of scale. Furthermore, it’s not always clear-cut to label costs as either fixed or variable. Some costs have elements of both and are termed semi-fixed costs or mixed costs.

D. CVP Formulas

The CVP analysis focuses on profitability. In its simplest form, the income statement shows that profit is the result of subtracting total costs (TC) from total revenue (TR). This profit is commonly known as net income (NI) or operating income.

[asciimath]"Net Income" = "Total Revenue" - "Total Costs"[/asciimath]

Or

 [asciimath]NI=TR-TC[/asciimath]Formula 6.1

Assuming the business only sells one product, the total revenue is the product of the unit selling price and the number of units sold or volume. Given [asciimath]"SP"[/asciimath] represents the selling price and [asciimath]X[/asciimath] the volume (quantity), the total revenue can be written as

 [asciimath]TR=SP*X[/asciimath]Formula 6.2

Formula 6.2 is referred to as the total revenue function. 

The total costs are the sum of the total variable costs (TVC) and the fixed costs (FC).

 [asciimath]TC=TVC+FC[/asciimath]Formula 6.3

The total variable costs can be written in terms of the variable cost per unit (VC) and the volume (X).

 [asciimath]TVC=VC*X[/asciimath]Formula 6.4

Substituting Formula 6.4 into Formula 6.3 yields

 [asciimath]TC=VC*X+FC[/asciimath]Formula 6.5

Formula 6.5 is called the total cost function. Now plugging Formulas 6.2 and 6.5 into Formula 6.1 gives

 [asciimath]NI=(SP*X)-(VC*X+FC)[/asciimath]Formula 6.6a

Formula 6.6a is referred to as the net income function and it shows the relationship among net income (NI), the unit selling price (SP), unit variable cost (VC), fixed costs (FC), and volume (X).

E. Compute Break-even Point Using Break-even Analysis

The breakeven point (BP) is the point at which a business neither makes a profit nor incurs a loss. Therefore, to find out the quantity needed to break even, you can use Formula 6.6a and set the net income (NI) to zero.

 [asciimath]0= SP*X-(VC*X+FC)[/asciimath]Formula 6.6b

Alternatively, we can use the fact that at break-even total costs are equal to the total revenue.

[asciimath]TR=TC[/asciimath]

 [asciimath]SP*X=VC*X+FC[/asciimath]Formula 6.6c

Both Formulas 6.6b and 6.6c give the quantity at the breakeven point. The break-even point may be expressed in various ways such as

• number of units 
• total revenue at break-even units, or
• as a percent of capacity.

When Unit Selling Price And Variable Cost Are Known

In most cases, the unit selling price and the unit variable cost are known. Therefore, the values are directly input in Formula 6.6b and the volume is solved for.

When Unit Selling Price And Variable Cost Are Unknown

The relationship between total variable costs (TVC) and total revenue (TR) can be expressed as a constant ratio, equivalent to the ratio of unit variable costs (VC) to the selling price (SP). This is represented as:

[asciimath](TVC)/(TR)=(VC*X)/(SP*X)[/asciimath]

Simplifying this, we get:

 [asciimath](TVC)/(TR)=(VC)/(SP)[/asciimath]

In cases where we only have the total dollar values without specific unit selling prices or variable costs, we can still determine the break-even point in dollar terms (i.e., the total revenue at the break-even point) rather than in units. By assuming a unit selling price (SP) of $1, we can then calculate the unit variable cost (VC) as follows:

[asciimath](TVC)/(TR)=(VC)/1[/asciimath]

[asciimath]VC = (TVC)/(TR)[/asciimath]

Using the assumed SP of $1 and the calculated VC value, we can apply Formula 6.6c to find ‘X’, where ‘X’ now represents the total revenue at the break-even point, rather than the number of units sold.

Section 6.1 Exercises