Chapter 4: Cost-Volume-Profit Relationships

1) Explain how changes in activity affect contribution margin and operating income.

Cost-volume-profit (CFP) analysis helps managers understand the relationships among cost, volume, and profit and how profits are affected by five elements:

  1. Prices of products
  2. Volume or level of activity
  3. Per unit variable costs
  4. Total fixed costs
  5. Mix of products sold for multi-product companies.

The Basics of CVP Analysis

  • Examine changes in activity levels, selling prices, variable costs per units, or fixed costs, and how they affect profits.
  • Contribution Income Statement emphasizes the behaviour of costs and therefore helpful in judging the impact on profits of changes in selling price, cost, or volume.
  • Used by managers, and operating income is the measure of profit. Image result for contribution income statement

Contribution Margin

  • CM is used to cover fixed expenses, then the remaining is allocated for profits.
  • Total Contribution Margin = Sales – Variable cost
  • CM per unit = Price – Variable Cost per unit
  • Break-even Point – The level of sales at which profit is zero.
  • Could be defined as the point where total sales equals total expenses, or as the point where total contribution margin = total fixed expenses.
  • Equation Method - X * Sales per unit – X * VC per unit – FC = 0
  • Contribution Margin Method - BEP in units = fixed expense / contribution margin per unit.

2) Prepare and interpret a cost-volume-profit graph.

CVP Graph – Highlights relationships between revenues, costs, and level of activity presented in graphic form.

Preparing the CVP Graph

  • The CVP Graph (or Break-even chart) represents unit volume over the x axis and dollars on the vertical y-axis.
  • Steps
  • Plot the line parallel to the volume axis that represents fixed expenses.
  • Plot the line representing total expenses (fixed + variable) at various activity levels.
  • Plot the line representing total sales revenues in dollars at various activity levels.

  • Interpretation
  • Profit/Loss is measured by the vertical distance between the total revenue line (sales) and the total expenses line.
  • Break-even point at the intersection of total revenue and total expenses

  • Profit graph – simpler linear form of the CVP graph that plots:
  • Profit = Unit CM x Q – Fixed Expenses
  • Where Q = quantity of items sold.
  • Volume on x-axis and profit in dollars on y-axis.
  • Break even point at point in which profit = 0.

Image result for profit graph

3) Use the contribution margin ratio to compute changes in contribution margin and operating income resulting from changes in sales volume.

Contribution Margin Ratio

  • CM Ratio – The contribution margin as a percentage of total sales.
  • CM ratio = Contribution margin per unit / sales per unit
  • The effect on operating income of any dollar change in total sales can be computed by applying the CM ratio to the dollar change.
  • Change in CM = CM Ratio x Change in Sales
  • The CM ratio is useful when trade-offs must be made between more dollar sales of one product and more dollar sales of another.
  • Emphasize products that yield the greatest amount of CM per dollar of sales.

4) Show the effects on contribution margin of changes in variable costs, fixed costs, selling price, and volume.

Applications of CVP Concepts

  • Variable Expense Ratio – Ratio of variable expenses to sales.
  • VE Ratio = Variable Expenses / Sales
  • Relation of ratios – CM Ratio = 1 – Variable Expense Ratio

Important Formulas for Contribution Format Income Statements

  • Operating Income = Unit CM x Q – Fixed Expenses
  • CM = Sales – Variable Expenses
  • CM per unit = Per unit sales – Per unit variable expenses
  • CM Ratio = Total CM / Total Sales or CM Ratio = Per unit CM / Per Unit Sales
  • Variable Expense Ratio = Variable Expenses / Sales

Changes in Fixed Cost and Sales Volume

  • Analysis 1 - Expected Total Contribution Margin = New projected Sales x CM Ratio
  • Current Total Contribution Margin = Current Sales x CM Ratio
  • Incremental Contribution Margin = Subtract Current from expected
  • Change in Fixed Cost: Less incremental new expense (The amount of money needed to pay for this increase in sales)
  • Increased Operating Income = Incremental CM – change in fixed cost.
  • Analysis 2 - Incremental Contribution Margin = Increase in total sales x CM Ratio
  • Less Incremental Expense = change in fixed cost.
  • Increased Operating Income = Incremental CM – change in fixed cost.
  • Incremental Analysis – an analytical approach that focuses only on those items of revenue, cost, and volume that will change as a result of a decision.
  • Both methods use IA, which is simpler.

Change in Variable Costs and Sales Volume

  • For a variable cost change, incremental analysis may be used.
  • Analysis - Expected Total Contribution Margin with change in variable costs = (Increased sales x New CM)
  • Current Total Contribution Margin = (current sales x old unit CM)
  • Increase in Total Contribution Margin = Expected CM – Current CM.

Change in Fixed Costs, Selling Price, and Sales Volume

  • Analysis - Expected Total Contribution Margin with Lower Selling Price = New Sales number x New unit CM
  • Current Total Contribution Margin = Old sales number x old unit CM
  • Incremental Contribution Margin = Expected CM – Current CM
  • Change in Fixed Costs = Less the incremental expense that has to be paid.
  • Addition (or Reduction) in Operating Income = Incremental CM – incremental expense.
  • If the change in income is negative, the changes should not be made.

Change in Variable Costs, Fixed Cost, and Sales Volume

  • Variable Cost Analysis - Expected Total Contribution Margin with Changes = New sales x new CM
  • Current Total Contribution Margin = Old sales x old CM
  • Increase/Decrease in Total Contribution Margin = expected CM – old CM.
  • Fixed Cost Analysis
  • Change in fixed costs - Add cost avoided.
  • Increase/Decrease in Operating Income = Sum of increase/decrease in CM and change in fixed cost.

Importance of the Contribution Margin

  • CVP analysis seeks the most profitable combination of variable costs, fixed costs, selling price, and sales volume.
  • The best way to improve profits – increase total CM by reducing selling price (increases volume) and sometimes by trading off variable and fixed costs with changes in volume.

Cost-Volume-Profit Analysis

  • Changes affecting:
  • Selling price per unit
  • Variable Unit Costs
  • Fixed Cost
  • Volume
  • Decision Rule
  • Make Change if: Increase in CM > increase in fixed cost.
  • Decrease in CM < decrease in fixed costs.
  • Don’t Make Change if: Increase in CM < increased in fixed cost.
  • Decrease in CM > Decrease in fixed costs.

Unit CM and CM ratio are significant towards the actions a company is willing to take to improve profits.

5) Compute the break-even point in unit sales and sales dollars.

Break-Even Computations

  • Designed to answer questions of “safety nets,” how far sales could drop, etc.
  • Derivation - Contribution Format Income Statement in equation form
  • Profits = (Sales – Variable Expenses) – Fixed expenses
  • Or - Profits [P x Q] – [VC x Q] – Fixed Expenses
  • Where P = selling price per unit, Q = number of units sold; VC = variable costs per unit.
  • Further simplified: Profits = [CM x Q] – Fixed Expenses
  • At break-even, profits are zero. Therefore, total contribution margin must equal total fixed expenses.
  • Break-even point in units sold = Fixed Expenses / Unit Contribution Margin
  • Variation: Break-even point in total sales dollars = Fixed Expenses / CM Ratio
  • You can get break-even point in sales dollars by multiplying the break-even level of unit sales by selling price per unit.

6) Determine the level of sales needed to achieve a desired target profit.

Target Operating Profit Analysis

  • CVP formulas can be used to determine sales volume needed to achieve a target operating profit.
  • Derivation - Utilize the simplified profit equation: Profits [P x Q] – [VC x Q] – Fixed Expenses but include the target operating profit.
  • Rearranged - Units sold to attain target = (Fixed expenses + target operating profit) / Unit contribution margin
  • Check - Substitute the computed amount of units into the profit equation.
  • Dollar Sales to attain target profit = (Fixed Expenses + target Operating Profit) / CM Ratio
  • Alternative Solution: Number of products required to achieve target x selling price

After-Tax Analysis

  • Operating profit after taxes can be computed as a fixed percentage of income before taxes.
  • Income Taxes - tax rate (t) x operating profit before taxes (B)
  • Derivation of After-Tax Profit - Profit after taxes = Before-tax profit – Taxes
  • Profit after taxes = B – t(B)
  • Profit after taxes = B(1-t)
  • Income Before Taxes: B = Profit after taxes / (1-t)
  • For target profit with tax rate incorporated:(Fixed Expenses + [Target after-tax profit / 1 – Tax rate]) / Unit Contribution Margin

Single-Product CVP Analysis

  • Break-Even
  • Units - Break-even point in units sold = Fixed Expenses / Unit Contribution Margin
  • Sales Dollars - Break-even point in total sales dollars = Fixed Expenses / CM Ratio
  • Target Operating Profit
  • Units - Units sold to attain target profit = (Fixed expenses + [Target after-tax profit / 1 – Tax Rate]) / Unit Contribution Margin
  • Sales Dollars - Units sold to attain target profit = (Fixed expenses + [Target after-tax profit / 1 – Tax Rate]) / CM Ratio

7) Compute the margin of safety and explain its significance.

The Margin of Safety

  • Margin of Safety – Excess of budgeted (or actual) sales over the break-even volume of sales.
  • States the amount by which sales can drop before losses begin to occur.
  • Higher margin of safety = lower risk of not breaking even.
  • Margin of Safety - Total Budgeted (or Actual) sales – Break-even sales
  • Margin of safety percentage = Margin of Safety in dollars / total Budgeted (or actual) sales.

8) Explain Cost structure, compute the degree of operating leverage at a particular level of sales, and explain how operating leverage can be used to predict changes in operating income.

Cost Structure and Profit Stability

Cost Structure – Relative proportion of fixed and variable costs incurred by an organization.

  • It has latitude in trading off fixed and variable costs. Different cost structures have advantages and disadvantages.


  • High fixed costs and low variable costs = wider swings in operating income as sales change – greater profits in good years and greater losses in bad years.
  • Low fixed costs and higher variable costs = greater operating income stability and protected from losses during bad years at the cost of lower operating income in good years.

Operating Leverage

  • Operating Leverage – A measure of how sensitive operating income is to a given percentage change in sales.
  • Degree of Operating Leverage – A measure, at a given level of sales, of how a percentage change in sales volume will affect profits.
  • Degree of Operating Leverage = contribution margin / operating income
  • Operating income grows x times as fast as its sales, where x is degree of OL.
  • To find percentage change in operating income, take degree of operating leverage x change in sales.
  • Greatest at sales levels near break-even point and decreases as sales and profits rise.

Indifference Analysis

  • CVP Analysis aids in decision-making for comparative profitability of alternatives.
  • Relative profitability depends on activity levels.
  • Product with higher fixed costs requires a higher sales activity level than a product with low fixed costs and high variable costs.
  • To calculate the point at which a company is indifferent about two different cost systems, use an indifference analysis.
  • Determine the unit CM multiplied by the number of units (Q) minus the total fixed costs of each alternative.
  • Unit CM x Q – total cost = Unit CM (alternative) x Q – total cost (alternative)
  • Set up an equation with each alternative on opposite sides of the equal sign.
  • Solve for Q, the indifference point.
  • At sales below the indifference point, profitability is lower for the option with lower CM.
  • At sales above the indifference point, profitability is higher for the option with higher CM.

9) Compute the break-even point for a multi-product company and explain the effects of changes in the sales mix on the contribution margin and the break-even point.

Sales Mix

  • Sales Mix – The relative proportions in which a company’s products are sold.
  • Sales mix - Sales of each product as a percentage of total sales.
  • Managers try to achieve the combination or mix that yields the most profits.
  • Profits are greater if high-margin items make up a large proportion of sales.
  • Changes in sales mix causes variation in profits, independent of sales volume.

Sales Mix and Break-Even Analysis

  • With more than one product, break-even analysis depends on the mix in which the products are sold.
  • Analysis can be done in units or sales dollars.
  • Calculate Number of Units of product that must be sold to break even.
  • Total number of units to break even x Sales Mix percentage
  • Calculate Sales in dollars of each product that must be sold to break even.
  • Previous answer x cost per unit.
  • Sales mix changes - Break even point changes.

Assumptions of CVP Analysis

  • Assumptions underlie CVP analysis.
  • Selling price is constant throughout the entire relevant range. Price of a product/service doesn’t change as volume changes.
  • Costs are linear throughout the entire relevant range and can be accurately divided into variable and fixed elements.
  • Variable costs per unit are constant and fixed costs are constant in total over the entire relevant range.
  • In multi-product companies, the sales mix is constant.
  • In manufacturing companies, inventories do not change. The number of units produced = number of units sold.

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