Cost-Volume-Profit Relationships Notes

Cost-Volume-Profit Relationships

Cost-Volume-Profit Analysis: Key Assumptions

• To simplify Cost-Volume-Profit (CVP) calculations, managers typically make the following assumptions:
  • Selling Price is Constant: The price of a product or service does not change as volume changes.
  • Costs are Linear: Costs can be accurately divided into variable and fixed components.
    • Variable costs are constant per unit.
    • Fixed costs are constant in total over the relevant range.
  • Constant Sales Mix: In multi-product companies, the mix of products sold remains constant.

Basics of Cost-Volume-Profit Analysis

• Contribution Margin (CM): The amount remaining from sales revenue after variable expenses have been deducted.
• The contribution income statement helps managers judge the impact on profits of changes in selling price, cost, or volume.
• The emphasis is on cost behavior.
• CM is used first to cover fixed expenses; any remaining CM contributes to net operating income.
• If Racing Bicycle Company (RBC) sells 400 units in a month, it operates at the break-even point.
• If RBC sells one more bike (401 bikes), net operating income increases by $200$.

Contribution Income Statement Example (RBC)

• For the Month of June (401 bicycles):
  • Sales: $200,500$ ($500$ per unit)
  • Less: Variable expenses: $120,300$ ($300$ per unit)
  • Contribution Margin: $80,200$ ($200$ per unit)
  • Less: Fixed expenses: $80,000$
  • Net Operating Income: $200$

Estimating Profits at a Particular Sales Volume

• We do not need to prepare an income statement to estimate profits.
• Simply multiply the number of units sold above break-even by the contribution margin per unit.
• If Racing sells 430 bikes, its net operating income will be $6,000$ (30 units x $200$ per unit).

CVP Relationships in Equation Form

• The contribution format income statement can be expressed as:
  • $Profit = (Sales – Variable\; expenses) – Fixed \;expenses$
• For RBC selling 401 units:
• $Profit = (Sales – Variable\; expenses) – Fixed \;expenses$
  • $Profit = ($200,500$–$120,300$) –$80,000
  • Profit = 200

Refining the Equation for Single-Product Companies

• When a company has only one product, the equation can be refined as:
  • Profit = (P × Q – V × Q) – Fixed\; expenses where P = Selling Price per unit, Q = Quantity of units sold, V = Variable expense per unit
• This equation can be used to show the 200 profit RBC earns if it sells 401 bikes.
  • Profit = ($500 × 401 –$300 × 401) – $80,000$
  • $200 = ($500 × 401 – $300 × 401) –$80,000

Unit Contribution Margin

• Unit CM = Selling price per unit – Variable expenses per unit
• It is often useful to express the simple profit equation in terms of the unit contribution margin (Unit CM):
  • Profit = (P – V) × Q – Fixed \;expenses
  • Profit = Unit\; CM × Q – Fixed\; expenses
• Unit\;CM = P – V
• For RBC selling 401 bikes:
  • Profit = ($500 –$300) × 401 – $80,000$
  • $Profit =$200 × 401 – $80,000$
  • $Profit =$80,200 – $80,000$
  • $Profit =$200

Contribution Margin Ratio (CM Ratio)

• The CM ratio is calculated by dividing the total contribution margin by total sales.

  • Example: $100,000 ÷$250,000 = 40

• Each 1 increase in sales results in a total contribution margin increase of 40¢.

• The CM ratio can also be calculated by dividing the contribution margin per unit by the selling price per unit.

  • CM\;Ratio = \frac{CM\;per\;unit}{SP\;per\;unit}
  • For RBC: CM\;Ratio = \frac{$200}{$500} = 40\%

• A 50,000$increase in sales revenue results in a$20,000$increase in CM ($50,000 × 40\% = $20,000$).

• If Racing Bicycle increases sales from 400 to 500 bikes ($50,000$), contribution margin will increase by $20,000$ (50,000 × 40\%$).

• The relationship between profit and the CM ratio can be expressed using the following equation:

  • Profit = (CM\;ratio × Sales) – Fixed\; expenses
  • Profit = (40\% × $250,000) –$80,000
  • Profit = $100,000 –$80,000
  • Profit = $20,000$

Break-even Analysis

• The equation and formula methods can be used to determine the unit sales and dollar sales needed to achieve a target profit of zero.

Break-even in Unit Sales: Equation Method

• Suppose RBC wants to know how many bikes must be sold to break-even (earn a target profit of $0$).
• Profits are zero at the break-even point.
  • $Profits = Unit\;CM × Q – Fixed\; expenses$
  • $0 =$200 × Q – $80,000$
  • $80,000 =$200 × Q
  • $Q = 400\;bikes$

Break-even in Unit Sales: Formula Method

• Unit sales to break even $= \frac{Fixed\; expenses}{CM\;per\;unit}$
  • Unit\;sales = \frac{$80,000}{$200} = 400

Break-even in Dollar Sales: Equation Method

• Suppose Racing Bicycle wants to compute the sales dollars required to break-even (earn a target profit of $0$).
  • $Profit = CM\;ratio × Sales – Fixed\; expenses$
  • $0 = 40\% × Sales –$80,000
  • 80,000 = 40\% × Sales
  • Sales = $80,000 ÷ 40\%$
  • $Sales =$200,000

Break-even in Dollar Sales: Formula Method

• Dollar sales to break even = \frac{Fixed\;expenses}{CM\;ratio}
  • Dollar\;sales = \frac{$80,000}{40\%} = $200,000$

Target Profit Analysis

• We can compute the number of units that must be sold to attain a target profit using either:
  • Equation method, or
  • Formula method.

Equation Method

• $Profit = Unit\;CM × Q – Fixed\; expenses$

• Our goal is to solve for the unknown “Q” which represents the quantity of units that must be sold to attain the target profit.

• Suppose RBC’s management wants to know how many bikes must be sold to earn a target profit of $100,000$.

  • $100,000 =$200 × Q – $80,000$
  • $200 × Q =$100,000 + $80,000$
  • $Q = ($100,000 + $80,000) ÷$200
  • Q = 900

The Formula Method

• Unit sales to attain the target profit = \frac{Target\;profit + Fixed\; expenses}{CM\;per\;unit}

• Suppose Racing Bicycle Company wants to know how many bikes must be sold to earn a profit of 100,000.

  • Unit\;sales = \frac{$100,000 + $80,000}{$200} = 900

Target Profit Analysis in Terms of Sales Dollars

• We can also compute the target profit in terms of sales dollars using either the equation method or the formula method.

Equation Method

• Profit = CM\;ratio × Sales – Fixed\; expenses
• Our goal is to solve for the unknown “Sales,” which represents the dollar amount of sales that must be sold to attain the target profit.
• Suppose RBC management wants to know the sales revenue that must be generated to earn a target profit of 100,000.
  • 100,000 = 40\% × Sales – $80,000$
  • $40\% × Sales =$100,000 + $80,000$
  • $Sales = ($100,000 + $80,000) ÷ 40\%$
  • $Sales =$450,000

Formula Method

• Dollar sales to attain the target profit = \frac{Target\;profit + Fixed\; expenses}{CM\;ratio}

• We can calculate the dollar sales needed to attain a target profit (net operating profit) of 100,000 at Racing Bicycle.

  • Dollar\;sales = \frac{$100,000 + $80,000}{40\%} = $450,000$

The Margin of Safety in Dollars

• The margin of safety is the excess of budgeted or actual sales dollars over the break-even volume of sales dollars.
• It is the amount by which sales can drop before losses are incurred.
• The higher the margin of safety, the lower the risk of not breaking even and incurring a loss.
• $Margin\;of\;safety\;in\;dollars = Total\; sales - Break-even\; sales$
• If we assume that RBC has actual sales of $250,000$, given that we have already determined the break-even sales to be $200,000$, the margin of safety is $50,000$.

The Margin of Safety Percentage

• RBC’s margin of safety can be expressed as 20% of sales. ($50,000 ÷$250,000)

The Margin of Safety in Units

• The margin of safety can also be expressed in terms of the number of units sold.
• The margin of safety at RBC is $50,000$, and each bike sells for $500$; hence, RBC’s margin of safety is 100 bikes.
• Margin\;of\;Safety\;in\;units = \frac{$50,000}{$500} = 100\;bikes

Cost Structure and Profit Stability

• Cost structure refers to the relative proportion of fixed and variable costs in an organization.
• Managers often have some latitude in determining their organization’s cost structure.
• There are advantages and disadvantages to high fixed cost / low variable cost and low fixed cost / high variable cost cost structures.
  • An advantage of a high fixed cost structure is that income will be higher in good years compared to companies with a lower proportion of fixed costs.
  • A disadvantage of a high fixed cost structure is that income will be lower in bad years compared to companies with a lower proportion of fixed costs.
  • Companies with low fixed cost structures enjoy greater stability in income across good and bad years.

Operating Leverage

• Operating leverage is a measure of how sensitive net operating income is to percentage changes in sales.
• It is a measure, at any given level of sales, of how a percentage change in sales volume will affect profits.
• $Degree\;of\;operating\;leverage = \frac{Contribution\;margin}{Net\;operating\;income}$
• Degree\;of\;Operating\;Leverage = \frac{$100,000}{$20,000} = 5
• With an operating leverage of 5, if RBC increases its sales by 10%, net operating income would increase by 50%.

The Concept of Sales Mix

• Sales mix is the relative proportion in which a company’s products are sold.
• Different products have different selling prices, cost structures, and contribution margins.
• When a company sells more than one product, break-even analysis becomes more complex.
• Let’s assume Racing Bicycle Company sells bikes and carts and that the sales mix between the two products remains the same.

Multi-Product Break-Even Analysis

• Bikes comprise 45% of RBC’s total sales revenue and the carts comprise the remaining 55%.
• \frac{$265,000}{$550,000} = 48.2\% (rounded)
• Dollar sales to break even$= \frac{Fixed\;expenses}{CM\;ratio}$
  • Dollar\;sales\;to\;break\;even = \frac{$170,000}{48.2\%} = $352,697