3.1.2: Expressing CVP Relationships

Expressing CVP Relationships

How was the Excel spreadsheet in Exhibit 3-1 constructed? Underlying the exhibit are equations that express the CVP relationships and influence the structure of the contribution income statement in Exhibit 3-1. There are three related ways (we will call them “methods”) to model CVP relationships:

The equation method
The contribution margin method
The graph method

Different methods are useful for different decisions. The equation method and the contribution margin method are most useful when managers want to determine operating income at a few specific sales levels (for example, 5, 15, 25, and 40 units sold). The graph method helps managers visualize the relationship between units sold and operating income over a wide range of quantities.

Equation Method

Each column in Exhibit 3-1 is expressed as an equation.

$Revenues - Variable costs - Fixed costs = Operating income$

How are revenues in each column calculated?

$text Revenues end text space of 0.2em ?=? space of 0.2em text Selling?price end text thin space left parenthesis S P right parenthesis straight thin space space of 0.2em straight cross times back space space of 0.2em space of 0.2em text Quantity?of?units?sold end text thin space left parenthesis Q right parenthesis$

How are variable costs in each column calculated?

$Variable costs = Variable cost per unit (V C U) \times Quantity of units sold (Q)$

So,

(Equation 1)

$[(\begin{matrix} Selling \\ price \end{matrix}) \times (\begin{matrix} Quantity of \\ units sold \end{matrix}) - (\begin{matrix} Variable cost \\ per unit \end{matrix}) \times (\begin{matrix} Quantity of \\ units sold \end{matrix})] - \begin{matrix} Fixed \\ costs \end{matrix} = \begin{matrix} Operating \\ income \end{matrix}$

Equation 1 becomes the basis for calculating operating income for different quantities of units sold. For example, if you go to cell F7 in Exhibit 3-1, the calculation of operating income when Emma sells 5 packages is

$($ 200 \times 5) - ($ 120 \times 5) - $ 2,000 = $ 1,000 - $ 600 - $ 2,000 = −$1,600$

Contribution Margin Method

Rearranging equation 1,

(Equation 2)

$\begin{array}{r} [(\begin{matrix} Selling \\ price \end{matrix} - \begin{matrix} Variable cost \\ per unit \end{matrix}) \times (\begin{matrix} Quantity of \\ units sold \end{matrix})] - \begin{matrix} Fixed \\ costs \end{matrix} & = & \begin{matrix} Operating \\ income \end{matrix} \\ (\begin{matrix} Contribution margin \\ per unit \end{matrix} \times \begin{matrix} Quantity of \\ units sold \end{matrix}) - \begin{matrix} Fixed \\ costs \end{matrix} & = & \begin{matrix} Operating \\ income \end{matrix} \end{array}$

The contribution margin per unit is $inline style straight $ 80 straight thin space left parenthesis straight $ 200 minus straight $ 120 right parenthesis back space straight comma$ so when Emma sells 5 packages,

$Operating income = ($ 80 \times 5) - $ 2, 000 = - $ 1, 600$

Equation 2 expresses the basic idea we described earlier—each unit sold helps Emma recover $80 (in contribution margin) of the $2,000 in fixed costs.

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Graph Method

The graph method helps managers visualize the relationships between total revenues and total costs. Exhibit 3-2 illustrates the graph method for GMAT Success. Because we have assumed that total costs and total revenues change linearly with units sold, the graph shows each relationship as a line. We need only two points to plot each line.

TOTAL COSTS LINE. The total costs line is the sum of fixed costs and variable costs. Fixed costs are $2,000 for all quantities of units sold within the relevant range. To plot the total costs line, use as one point the $2,000 fixed costs at zero units sold (point A) because variable costs are $0 when no units are sold. Select a second point by choosing any other output level (say, 40 units sold) and determine the corresponding total costs. Total variable costs at this output level are $$ 4,800 (40 units \times $ 120 per unit) .$ Remember, fixed costs are $2,000 for all quantities of units sold within the relevant range, so total costs at 40 units sold equal $$ 6,800 ($ 2,000 + $ 4,800),$ which is point B in Exhibit 3-2. The total costs line is the straight line from point A through point B.
TOTAL REVENUES LINE. One convenient starting point is $0 revenues at 0 units sold, which is point C in Exhibit 3-2. Select a second point by choosing any other convenient output level and determining the corresponding total revenues. At 40 units sold, total revenues are $$ 8,000 ($ 200 per unit \times 40 units),$ which is point D in Exhibit 3-2. The total revenues line is the straight line from point C through point D.

The profit or loss at any sales level can be determined by the vertical distance between the two lines at that level in Exhibit 3-2. For quantities fewer than 25 units sold, total costs exceed total revenues, and the purple area indicates operating losses. For quantities greater than 25 units sold, total revenues exceed total costs, and the blue-green area indicates operating incomes. At 25 units sold, total revenues equal total costs. Emma will break even by selling 25 packages.

The profit or loss at any sales level can be determined by the vertical distance between the two lines at that level in Exhibit 3-2. For quantities fewer than 25 units sold, total costs exceed total revenues, and the purple area indicates operating losses. For quantities greater than 25 units sold, total revenues exceed total costs, and the blue-green area indicates operating incomes. At 25 units sold, total revenues equal total costs. Emma will break even by selling 25 packages.

Cost-volume graph depicts the operating loss area, total costs line, breakeven point, operating income area, and the total revenues line.

Y-axis shows the cost in dollars ranging from 0 to 10,000 with increment of 2,000 and X-axis shows the number of units sold ranging from 0 to 50 with increment of 10.

Graph shows two lines
- Total costs line: Slope of the total costs line is the variable cost per unit equals 120 dollars
- Total revenues line: Slope of the total revenues line is the selling price equals 200 dollars
The point A with coordinates 0 units and 2,000 dollars; and point B with coordinates 40 units and 6,800 dollars; lie on the total costs line, whereas the point C with coordinates 0 units and 0 dollars; and point D with coordinates 40 units and 8,000 dollars; lie on the total revenues line.
The two lines intersect at the breakeven point with coordinates 25 units and 5,000 dollars.
The area between the lines and the Y-axis that lies below the breakeven point shows the operating loss area.
The area between the lines that lies above the breakeven point depicts the operating income area.