Profit volume analysis, sometimes called cost-profit-volume analysis, is the process of calcu- lating the profits of a business at different volumes, or revenue levels. Break-even analysis, a component of profit volume analysis, is simply the calculation of the revenue level at which a business shows neither a profit nor a loss.

Generally, profit volume analysis involves five steps. First, set a range of business volumes for which you examine costs and profits. This step is probably one of the most critical be- cause all the information you input—unit sales price, variable costs, fixed costs, and costs varying with profits—is usually valid only over a limited range of volumes. By carefully considering the relationships between costs and changes in volume over a specific range, you can increase the accuracy of your analysis.

Second, calculate the unit sales price, the amount for which you sell your product or ser- vice. For example, if you build and sell single-family homes and your average sales price is $100,000, your unit sales price is $100,000.

Third, identify the costs that vary with revenue, the variable costs. Typically, it’s easiest to express and calculate these variable costs either as an amount determined per unit or as an amount determined as a percentage of revenues. For example, if you build houses, many of your costs are best described as an amount per house. For example, you land costs might average $15,000 per house and your material costs and your labor costs each might aver- age $40,000. Other costs, however, are better described as a percentage of revenues. For example, you might calculate sales commissions as 7% of the sales price and a state sales tax as 1 1/2% of the sales price. The key assumption for the purpose of profit volume analysis, however, is that within the range of business volumes you define, the variable costs change proportionally, based on revenue.

Fourth, determine your fixed costs. Fixed costs are those that stay constant, within the range of business volumes you define. You label these costs “fixed,” not because you cannot change them, but because small to moderate changes in revenue don’t change them. Examples of fixed costs are salaries of administrative personnel, office rent, and business insurance.

Fifth, calculate your profits and any costs that vary with profits. Examples of these costs are income taxes and profit-sharing plans. Obviously, the precise determination of income taxes and similar costs requires detailed tax accounting. But you might be able to estimate these income taxes and costs by applying an appropriate percentage to the profits before income taxes and other costs that vary with profits.

You can calculate the contribution margin (the revenues minus the variable costs) and profit at any volume within the range for which your inputs are valid. Although the analysis is only as good as your assumptions and is subject to the inevitable inaccuracies that creep into any projection of the future, profit volume analysis allows you to see roughly what happens to your profits over the likely range of business volumes.

One common profit volume analysis calculation is estimating the revenue level that pro- vides exactly enough contribution margin to cover fixed costs. In this calculation, because no profits exist, none of the costs that vary with profits exist. At the break-even point, rev- enues leave exactly enough contribution margin to cover fixed costs. The general formula used to calculate the break-even point is as follows:

*Break-even point in units=Fixed costs/Contribution margin per unit*

If you use more than one of the vary-with-profit cost categories presented in the profit volume and break-even analysis starter workbook, you need to recognize the correct relationships be- tween variables as you input them. Three types of relationships exist: independent-independent, independent-dependent, and dependent-dependent. Independent-independent is easiest to calculate because all of the costs that vary with profits are calculated independent of the other. The other two types of relationships can be more difficult. With the independent-dependent relationship, you need to calculate one cost so that you can calculate the next. As an example of this relationship and how you might recognize it in your inputs, suppose that the state income tax rate is 10% and is deducted from the Profit Before Vary-with-Profits Costs (PBVPC) and that after deducting the state income tax from the PBVPC, the federal income tax rate of 20% is applied to the PBVPC. The correct input percentage for the state income tax rate is 10%, because 10% of the PBVPC calculates the correct state income tax cost, as follows:

*State Income Tax=10%*PBVPC*

However, the federal income tax percentage must recognize the state income tax costs:

*Federal Income Tax=20%*(PBVPC-(10%*PBVPC))*

This formula can be further modified as follows to express the federal income tax rate as a percentage of the PBVPC and, therefore, your input to the profit volume and break-even analysis starter workbook:

*Federal Income Tax=18%*PBVPC*

A third type of relationship that might exist between the costs that vary with profits is a dependent-dependent, or circular, relationship. For example, suppose that you have an employee bonus cost equal to 10% of the after-tax profits. You need to know the amount of the bonus before you can calculate the federal income taxes because it’s a tax-deductible expense, and you need to know the federal income tax because it determines the after-tax profits, upon which the bonus is calculated, before you can calculate the bonus. Assuming that your only tax is a federal income tax rate of 20%, you calculate your federal income tax as follows:

*Federal Income Tax=20%*(PBVPC-bonus)*

Assuming that the employee bonus is 10% of the after-tax profits, your employee bonus cost equals:

*Bonus=10%*(PBVPC-Federal Income Tax)*

Given these definitions, you can define the federal income tax percentage by substituting the formula for the bonus in the federal income tax formula, as follows:

*Federal Income Tax=20%*(PBVPC-(10%*(PBVPC-Federal Income Tax)))*

You could state this formula algebraically as:

*Federal Income Tax=20%*(PBVPC-(10%*PBVPC)+(10%*Federal Income Tax))*

or:

*Federal Income Tax=20%*((90%*PBVPC)+(10%*Federal Income Tax))*

or:

*Federal Income Tax=(18%*PBVPC)+(2%*Federal Income Tax)*

or:

*98%*Federal Income Tax=18%*PBVPC*

or to show the federal income tax as a percentage of the PBVPC:

*Federal Income Tax=18.3673%*PBVPC*

Given this number, it’s easy to define the bonus as a percentage of the PBVPC by substi- tuting the following formula for federal income tax in the bonus formula:

*Bonus=10%*(PBVPC-(18.3673%*PBVPC))*

or, to show the bonus as a percentage of the PBVPC:

*Bonus=10%*(81.6327%*PBVPC)*

or, to show the bonus as a percentage of the PBVPC in another way:

*Bonus=8.16372%*PBVPC*

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