# 2.4 Contribution Rate

## LEARNING OBJECTIVES

• Solve cost-volume-profit analysis problems using the contribution rate approach.

It is difficult to compare different products and their respective dollar amount contribution margins if their selling prices and costs vary widely. For example, how do you compare a unit contribution margin of $1,390 (selling price of$2,599.99) on a big screen television to a unit contribution margin of $0.33 on a chocolate bar (selling price of$0.67)? On a per-unit basis, which contributes relatively more to fixed costs? To facilitate these comparisons, the products must be placed on equal terms, requiring you to convert all dollar amount contribution margins into percentages, called contribution rates. A contribution rate is a contribution margin expressed as a percentage of the selling price.

## Contribution Rate

The contribution rate ($CR$) is the contribution margin expressed as a percentage of the selling price.  The contribution margin is the percentage of the selling price that is available to first pay for the fixed costs and then to produce profit.

$\displaystyle{CR=\frac{CM}{S} \times 100\%}$

where

• $CR$ is the contribution rate. This represents the percentage of the unit selling price that is available to pay for all of the fixed costs of the business. When all fixed costs are paid for, this percentage is the portion of the selling price that will remain as profit.
• $CM$ is the contribution margin. This is the amount of money that remains available to pay for fixed costs once the unit variable cost is removed from the selling price of the product.
• $S$ is the selling price per unit. The unit selling price of the product.

## EXAMPLE

Suppose the selling price of a product is $5 and the variable cost is$3.50.  Calculate the contribution rate.

Solution:

Step 1:  The given information is

$\begin{eqnarray*} S & = & \5 \\ VC & = & \3.50 \end{eqnarray*}$

Step 2:  Calculate the contribution margin.

$\begin{eqnarray*} CM & = & S-VC \\ & = & 5-3.50 \\ & = & \1.50 \end{eqnarray*}$

Step 3:  Calculate the contribution rate.

$\begin{eqnarray*} CR & = & \frac{CM}{S} \times 100\% \\ & = & \frac{1.50}{5} \times 100\% \\ & = & 30\%\end{eqnarray*}$

The contribution rate is 30%.  This means that 30% of the $5 selling price is available to pay for the fixed costs. The remaining 70% of the$5 selling price goes to the variable costs.

### NOTES

1. The contribution rate remains the same regardless of the number of units sold and the fixed costs.  Changes to the selling price or variable costs (or both) will change the contribution rate.
2. What does the contribution rate really mean?  Suppose the contribution rate is 60%.  This means that 60% of the selling price goes to pay the fixed costs below the break even point or becomes profit above the break-even point.  For example, if the selling price is $1, a 60% contribution rate means$0.60 of that $1 selling price goes to pay the fixed costs. The remaining 40% of the selling price goes to pay the variable costs. For the$1 selling price, a 60% contribution rate means $0.40 of that$1 selling price goes to pay the variable costs.

There may be instances when the unit selling price and unit variable costs are unknown or unavailable.  In such circumstances, it would be impossible to calculate the contribution rate using the above formula.  Sometime, the only information you have available is aggregate information such as total revenue and total variable costs. In such cases it is still possible to calculate the contribution rate.

$\displaystyle{CR=\frac{TR-TVC}{TR} \times 100\%}$

where

• $CR$ is the contribution rate. This represents the percentage of the unit selling price that is available to pay for all of the fixed costs of the business. When all fixed costs are paid for, this percentage is the portion of the selling price that will remain as profit.
• $TR$ is the total revenue. This is the total amount of money that the company has received from the sale of the product.
• $TVC$ is the total variable cost. This is the total cost associated with the level of output. When total variable costs are subtracted from the total revenue, the remainder represents the portion of money left over to pay the fixed costs.

## EXAMPLE

Suppose a chocolate bar manufacturer has total monthly revenue of $3,886 and total variable costs of$1,972.  Calculate the contribution rate.

Solution:

Step 1:  The given information is

$\begin{eqnarray*} TR & = & \3,886 \\ TVC & = & \1,972 \end{eqnarray*}$

Step 2:  Calculate the contribution rate.

$\begin{eqnarray*} CR & = & \frac{TR-TVC}{TR} \times 100\% \\ & = & \frac{3,886-1,972}{3,886} \times 100\% \\ & = & 49.25\%\end{eqnarray*}$

The contribution rate is 49.25%.  This means that 49.25% of the selling price of each chocolate bar is available to pay for the fixed costs.  The remaining 50.75% of the selling price of each chocolate bar goes to the variable costs.

## Net Income using Contribution Rate

Recall that net income is the difference between total revenue ($TR$) and total costs ($TVC+FC$).  So, net income can be expressed as

$\displaystyle{NI=TR-TVC-FC}$

Rearranging this formula, we get

$\displaystyle{TR-TVC=FC+NI}$

From the contribution rate formula, $TR-TVC=CR \times TR$.  Substituting into the above formula and solving for $TR$, we get

$\displaystyle{TR = \frac{FC+NI}{CR}}$

where

• $CR$ is the contribution rate. This represents the percentage of the unit selling price that is available to pay for all of the fixed costs of the business. When all fixed costs are paid for, this percentage is the portion of the selling price that will remain as profit.
• $TR$ is the total revenue. This is the total amount of money that the company has received from the sale of the product.
• $TVC$ is the total variable cost. This is the total cost associated with the level of output. When total variable costs are subtracted from the total revenue, the remainder represents the portion of money left over to pay the fixed costs.
• $NI$ is the net income.  The amount of money left over after all costs have been paid is the net income. If the number is positive, then the business is profitable. If the number is negative, then the business suffers a loss.
• $FC$ is the fixed costs. These are the costs that do not change regardless of the level of production.

## EXAMPLE

The manufacturer of a product has fixed costs of $200,000 per year. The variable costs are 65% of the selling price. 1. What is the contribution rate? 2. What is the total revenue to earn a profit of$25,000?
3. What is the total revenue to incur a loss of $10,000? 4. What is the net income if the total revenue is$500,000?
5. What is the net income if the total revenue is $750,000? Solution: Step 1: The given information is $\begin{eqnarray*} FC & = & \200,000 \end{eqnarray*}$ Step 2: The contribution rate is 35%. Because 65% of the selling price goes to the variable costs, the remaining 35% of the selling price is available for the fixed costs, which is the contribution rate. Step 3: Calculate the total revenue for $NI=25,000$. Because this is a profit, the net income is positive. $\begin{eqnarray*} TR & = & \frac{FC+NI}{CR} \\ & = & \frac{200,000+25,000}{0.35} \\ & = & \642,857.14 \end{eqnarray*}$ The total revenue to earn a profit of$25,000 is $642,857.14. Step 4: Calculate the total revenue for $NI=-10,000$. Because this is a loss, the net income is negative. $\begin{eqnarray*} TR & = & \frac{FC+NI}{CR} \\ & = & \frac{200,000-10,000}{0.35} \\ & = & \542,857.14 \end{eqnarray*}$ The total revenue to incur a loss of$10,000 is $542,857.14. Step 5: Calculate the net income for $TR=500,000$. $\begin{eqnarray*} TR & = & \frac{FC+NI}{CR} \\ 500,000 & = & \frac{200,000+NI}{0.35} \\ 500,000 \times 0.35 & = & 200,000+NI \\ 175,000 & = & 200,000+NI \\ 175,000-200,000 & = & NI \\ -\25,000 & = & NI \end{eqnarray*}$ A loss of$25,000 occurs when the total revenue is $500,000 Step 6: Calculate the net income for $TR=750,000$. $\begin{eqnarray*} TR & = & \frac{FC+NI}{CR} \\ 750,000 & = & \frac{200,000+NI}{0.35} \\ 750,000 \times 0.35 & = & 200,000+NI \\ 262,500 & = & 200,000+NI \\ 262,500-200,000 & = & NI \\ \62,500 & = & NI \end{eqnarray*}$ A profit of$62,500 occurs when the total revenue is $750,000 ### NOTE When you calculate the contribution rate, you will probably end up with a number that has lots of decimals. When you use the contribution rate in another calculation, such as finding the total revenue, net income, or break-even revenue, keep all of the decimals in the contribution rate. This way you will avoid any round-off error in your next calculation. ## TRY IT Your company has fixed costs$195,000.  Your variable costs are 72% of your sales.

1. What is the contribution rate?
2. What is the total revenue to incur a loss of $20,000? 3. What is the net income for a total revenue of$600,000?
Click to see Solution

1. The contribution rate is 28%.

2. Total revenue for loss of $20,000. $\begin{eqnarray*} TR & = & \frac{FC+NI}{CR} \\ & = & \frac{195,000-20,000}{0.28} \\ & = & \625,000 \end{eqnarray*}$ 3. Net income for revenue of$600,000

$\begin{eqnarray*} TR & = & \frac{FC+NI}{CR} \\ 600,000 & = & \frac{195,000+NI}{0.28} \\ 168,000 & = & 195,000+NI \\ -\27,000 & = & NI \end{eqnarray*}$

## Break-Even Analysis using Contribution Rate

The break-even revenue is the revenue so that the net income is zero.  To find the break-even revenue, we set $NI=0$ in the above formula.  Then,

$\displaystyle{\mbox{Break-Even Revenue}=\frac{FC}{CR}}$

where

• $CR$ is the contribution rate. This represents the percentage of the unit selling price that is available to pay for all of the fixed costs of the business. When all fixed costs are paid for, this percentage is the portion of the selling price that will remain as profit.
• $FC$ is the fixed costs.  These are the costs that do not change regardless of the level of production.

### NOTE

Using the contribution rate, we can only find the revenue at the break-even point, and not the actual number of units sold.  This is especially useful when we only have the total revenue and total costs, and not the unit selling price and unit variable costs.  If the unit selling price and unit variable costs are known, it is possible to find the number of units to break-even using one of the other methods (revenue/cost functions or contribution margin).

## EXAMPLE

Suppose you are starting your own business.  You determine that your total fixed costs are $420,000. After analyzing the financial statements of your competitors, you estimate a contribution rate of 35%. What is your break-even revenue? Solution: Step 1: The given information is $\begin{eqnarray*} FC & = & \420,000 \\ CR & = & 0.35 \end{eqnarray*}$ Step 2: Calculate the break-even revenue. $\begin{eqnarray*} \mbox{Break-Even Revenue} & = & \frac{FC}{CR} \\ & = & \frac{420,000}{0.35} \\ & = & \1,200,000 \end{eqnarray*}$ The break-even revenue is$1,200,000.

Your company has fixed costs $195,000. Your variable costs are 72% of your sales. What is the break-even revenue? Click to see Solution $\begin{eqnarray*} \mbox{Break-Even Revenue} & = & \frac{FC}{CR} \\ & = & \frac{195,000}{0.28} \\ & = & \696,428.57 \end{eqnarray*}$ ## EXAMPLE Based on last year’s production, your company recorded the following information.  Total Revenue$650,000 Total Fixed Costs $125,000 Total Variable Costs$409,500

You expect the same level of production next year.

1. What is the contribution rate?
2. What is the break-even revenue?
3. Suppose the fixed costs decrease by $25,000 with no change in the total revenue and total variable costs. What is the new break-even revenue? 4. Suppose the total variable costs increase by$10,500 dollar with no change in the total revenue and total fixed costs.  What is the new break-even revenue?

Solution:

Step 1:  The given information is

$\begin{eqnarray*} TR & = & \650,000 \\ TVC & = & 409,500 \\ FC & = & \125,000 \end{eqnarray*}$

Step 2:  Calculate the contribution rate.

$\begin{eqnarray*} CR & = & \frac{TR-TVC}{TR} \times 100\% \\ & = & \frac{650,000-409,500}{650,000} \times 100\% \\ & = & 37\%\end{eqnarray*}$

Step 3:  Calculate the break-even revenue.

$\begin{eqnarray*} \mbox{Break-Even Revenue} & = & \frac{FC}{CR} \\ & = & \frac{125,000}{0.37} \\ & = & \337,837.84 \end{eqnarray*}$

The break-even revenue is $337,837.84. Step 4: Calculate the break-even revenue with $FC=125,000-25,000=100,000$. Because there is no change in the total revenue and total variable costs, the contribution rate is unchanged. $\begin{eqnarray*} \mbox{Break-Even Revenue} & = & \frac{FC}{CR} \\ & = & \frac{100,000}{0.37} \\ & = & \270,270.27 \end{eqnarray*}$ If the fixed costs decrease by$25,000, the break-even revenue is $270,270.27. Step 5: Calculate the break-even revenue with $TVC=409,500+10,500=420,00$. Because the total variable costs changed but the total revenue did not, the contribution rate will change. The contribution rate must be recalculated to find the new break-even revenue. $\begin{eqnarray*} CR & = & \frac{TR-TVC}{TR} \times 100\% \\ & = & \frac{650,000-420,000}{650,000} \times 100\% \\ & = & 35.384... \%\\ \\ \mbox{Break-Even Revenue} & = & \frac{FC}{CR} \\ & = & \frac{125,000}{0.353854...} \\ & = & \353,260.87\end{eqnarray*}$ If the total variable costs increase by$10,500, the break-even revenue is $353,260.87. ## TRY IT In the annual report to shareholders, Borland Manufacturing reported total gross sales of$7,200,000, total variable costs of $4,320,000, and total fixed costs of$2,500,000.

1. Determine Borland’s break-even revenue.
2. Suppose the fixed costs increase by 10% with no changes to the total revenue and total variable costs.  What is the new break-even revenue?
3. Suppose the total sales decreases by $200,000 with no changes to the fixed costs and total variable costs. What is the new break-even revenue? Click to see Solution 1. Break-even revenue. $\begin{eqnarray*} CR & = & \frac{TR-TVC}{TR} \times 100\% \\ & = & \frac{7,200,000-4,320,000}{7,200,000} \times 100\% \\ & = & 40\% \\ \\ \mbox{Break-Even Revenue} & = & \frac{FC}{CR} \\ & = & \frac{2,500,000}{0.4} \\ & = & \6,250,000 \end{eqnarray*}$ 2. Break-even revenue for $FC=2,750,000$. $\begin{eqnarray*} \mbox{Break-Even Revenue} & = & \frac{FC}{CR} \\ & = & \frac{2,750,000}{0.4} \\ & = & \6,875,000 \end{eqnarray*}$ 3. Break-even revenue for $TR=7,000,000$. $\begin{eqnarray*} CR & = & \frac{TR-TVC}{TR} \times 100\% \\ & = & \frac{7,000,000-4,320,000}{7,000,000} \times 100\% \\ & = & 38.285...\% \\ \\ \mbox{Break-Even Revenue} & = & \frac{FC}{CR} \\ & = & \frac{2,500,000}{0.38285...} \\ & = & \6,529,850.75 \end{eqnarray*}$ ## Exercises 1. Last year, A Child’s Place franchise had total sales of$743,000. If its total fixed costs were $322,000 and net income was $\81,000$, what was its contribution rate? Click to see Answer 54.24% 2. In the current year, a small Holiday Inn franchise had sales of$1,800,000, fixed costs of $550,000, and total variable costs of$750,000. Next year, sales are forecast to increase by 25% but costs will remain the same. How much will net income change (in dollars)?

$262,000 3. Monsanto Canada reported the following on its income statement for one of its divisions: • Sales=$6,000,000
• Total Fixed Costs=$2,000,000 • Total Variable Costs=$3,200,000
1. What is the contribution rate?
2. What is the break-even revenue?
3. What is the net income for a revenue of $5,500,000? 4. What is the total revenue for a profit of$200,000?
5. What is the total revenue for a loss of $300,000? Click to see Answer a. 46.67%; b.$4,285,714.29; c. $566,666.67; d.$4,714,285.71; e. $3,642,857.14 4. Procter and Gamble is budgeting for next year. For one of its brands, P&G projects it will operate at 80% production capacity next year and forecasts the following: • Sales=$80,000,000
• Total Fixed Costs=$20,000,000 • Total Variable Costs=$50,000,000
• Net Income=$10,000,000 Determine the net income if sales are higher than expected and P&G realizes 90% production capacity. Click to see Answer$13,750,000

5. Suppose at a production level of 131,000 units, the total fixed costs are $3,200,000, the total variable costs are$5,009,440, and the selling price per unit is $99.97. 1. What is the contribution rate? 2. What is the break-even revenue? 3. Suppose fixed costs rise by 10% with no changes to total revenue or total variable costs. At 131,000 units, what is the new net income? 4. Suppose the selling price decreased by 25% during a sale and the resulting number of units increased by 50% from the current 131,000. What is the new net income? 5. Suppose fixed costs are lowered by 5%, total variable costs rise by 3%, the selling price lowered by 5% and the level of output rises by 10%. What is the new net income? Click to see Answer a. 61.75%; b.$5182,310.06; c. $4,556,650; d.$4,109,410; e. $4,969,078 6. You are thinking of starting your own business and want to get some measure of feasibility. You have determined that your total fixed costs would be$79,300. From annual business reports and competitive studies, you estimate your contribution rate to be 65%. What is your break-even in dollars?

$122,000 7. If a business has total revenue of$100,000, total variable costs of $60,000, and total fixed costs of$20,000, determine its break-even point in dollars.

$50,000 8. If your organization has a contribution rate of 45% and knows the break-even point is$202,500, what are your organization’s total fixed costs?

$91,125 9. Burton Snowboards reported the following figures last week for its Custom V Rocker Snowboard: • Sales=$70,000
• Total Fixed Costs=$19,000 • Total Variable Costs=$35,000
• Net Income=$16,000 If the above numbers represent 70% operational capacity, express the weekly break-even point in dollars as a percentage of maximum capacity. Click to see Answer 38% 10. Boston Beer Company, the brewer of Samuel Adams, reported the following financial information to its shareholders: • Total Revenue=$488,600,000
• Total Variable Costs=$203,080,000 • Total Fixed Costs=$182,372,000

What is the break-even revenue?

$312,086,576.10 11. In the beverage industry, PepsiCo and The Coca-Cola Company are the two big players. The following financial information, in millions of dollars, was reported to its shareholders:  Pepsi Co. Coca Cola Company Total Revenue$14.296 $21.807 Total Variable Costs$7.683 $12.663 Total Fixed Costs$6.218 $8.838 Compare the break-even points in total dollars between the two companies based on these reports. Click to see Answer Pepsi=$13,442,088,008.47, Coca-Cola=$21,077,238,188.98, Pepsi is 36.22% lower. 12. The Puzzle Company had total revenue of$4,750,000, total fixed costs of $1,500,000, and total variable costs of$2,750,000. If the company desires to earn a net income of $1,000,000, what total sales volume is needed to achieve the goal? Click to see Answer$5,937,500

13. Whirlpool Corporation had annual sales of $18.907 billion with a net income of$0.549 billion. Total variable costs amounted to $16.383 billion. 1. What are the fixed costs? 2. Determine Whirlpool Corporation’s break-even revenue. 3. Suppose the fixed costs increase by 25%. What is the new break-even revenue? 4. If Whirlpool Corporation managed to increase revenues by 5% the following year while implementing cost-cutting measures that trimmed variable costs by 2%, determine the percent change in the break-even dollars. Click to see Answer a.$1.975 billion; b. $14.975 billion; c.$18.493 billion; d. -30.2%