# 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. #### 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. ### 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 / 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.

Generally:

• 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.