Cost-Volume-Profit Analysis

Before-tax income

After-tax income/(1 - Tax rate)

Operating income

(Price × Units) - (Unit variable cost × Units) - Fixed cost

Units to breakeven

(set Oi = 0) Q =F/Cm

Summary of Important Equations

1. Sales revenue 2. Operating income 3. Break-even point in units 4. Contribution margin ratio 5. Variable cost ratio 6. Break-even point in sales revenue 7. Margin of safety 8. Degree of operating leverage 9. Percentage change in operating income

Assumptions of Cost-Volume-Profit Analysis

1. The analysis assumes a linear revenue function and a linear cost function. 2. The analysis assumes that price, total fixed costs, and unit variable costs can be accurately identified and remain constant over the relevant range. 3. The analysis assumes that what is produced is sold. 4. For multiple-product analysis, the sales mix is assumed to be known. 5. The selling prices and costs are assumed to be known with certainty.

From Income Statement Total Sales Less: Total Variable Costs = Contribution Margin Less: Total Fixed Costs = Operating Income

Basic Formula Selling price/unit x number of units - Variable cost/unit x number of units = CM/ unit x number of units - Fixed costs = Operating income SQ - VQ = CmQ - F = Oi

Break-Even Point in Units for the Multiple-Product Setting

Break-even sales = Fixed Costs/Weighted Average Contribution Margin Ratio

Profit (Operating Income)

CmQ ‒ F

Sales revenue

Price × Units sold

Find the BEP in sales $ for Peggy's Kitchen Wares. The total revenue and total variable cost information below is based on the expected sales mix. Small Medium Large Total Expected 2,000 5,000 3,000 10,000 sales in units Total $20,000 $75,000 $54,000 $149,000 Revenue Total VC $8,000 $40,000 $33,000 $81,000 Total CM $12,000 $35,000 $21,000 $68,000 CMR 60.0% 46.7% 38.9% 45.6%

Compute the weighted average contribution margin ratio: WACmr = [(20/149) x 60%] + [(75/149) x 46.7%] + [(54/149) x 38.9%] = 45.6% or WACmr = Total contribution margin/Total revenue = $68,000/$149,000. Compute the BEP in sales $: F + 0/waCmr or $40,800/.456 = $89,474

Suppose that Bill's Briefcases has budgeted next year's sales at 5,000 units. Compute Bill's degree of operating leverage. Recall that S = $200, V = $80, F = $360,000, and the margin of safety percentage at 5,000 units is 40%.

Contribution margin and profit at 5,000 units: CM = = ($200 - $80) x 5,000 = $600,000 Profit = $600,000 - $360,000 = $240,000 Degree of operating leverage = $600,000/$240,000 = 2.5 or, degree of operating leverage ($360,000/$240,000) + 1 = 2.5 or degree of operating leverage = 1/40% = 2.5

Contribution margin ratio

Contribution margin/Sales or = (Price - Unit variable cost)/Price Total Sales - Total variable cost/Total sales SQ ‒ VQ/SQ

Percentage change in operating income

Degree of operating leverage × Percent change in sales

Currently Bill's salespersons have salaries totaling $80,000 (included in fixed costs of $360,000) and earn a 5% commission on each unit ($10 per briefcase). He is considering an alternative compensation arrangement where the salaries are decreased to $35,000 and the commission is increased to 20% ($40 per briefcase). Compute the BEP in units under the proposed alternative. Recall that S= $200 and V= $80 currently.

First compute F and V under the proposed plan: F= $360,000 - $45,000 decrease in salaries = $315,000 V= $80 + $30 increase in commission = $110 Then compute Q under the proposed plan: Units needed to break-even = F/Cm BEP = $315,000/CM (Price - VC) or $315,000/200 - 110 = 3,500 units

Break-even point in units

Fixed cost/(Price - Unit variable cost)

Break-even point in sales revenue

Fixed cost/Contribution margin ratio or = Fixed cost=(1 -Variable cost ratio)

how risk differs from uncertainty

Formally, risk differs from uncertainty in that with risk, the probability distributions of the variables are known. With uncertainty, the probability distributions are not known.

Previously Bill's degree of operating leverage was 2.5 & profits were $240,000 on 5,000 units sold.

If expected sales were to increase to 6,000 units (a 20% increase), then profits would increase by 2.5 x 20%, or 50%, to $360,000. $240,000 x 1.5 = $360,000; Proof: $240,000 current OI + (1,000 units increase x $120 CM) = $360,000 20% increase in sales → 50% (20% x 2.5) increase in profits If expected sales were to decreaseto 4,500 units (a 10% decrease), then profits would decrease by 2.5 x 10%, or 25%, to $180,000. $240,000 x .75 = $180,000; Proof: $240,000 current OI — (500 units decrease x $120 CM) = $60,000 10% decrease in sales → 25% (10% x 2.5) decrease in profits

Manual System vs. Automated System Price (P) Variable costs (VC) Fixed costs (FC) Contribution margin (CM) Break-even point (BEP) Margin of safety (MofS) Degree of operating leverage (D of OL) Downside risk (DR) Upside potential (UP)

MS vs. AS P Same Same VC Relatively higher Relatively lower FC Relatively lower Relatively higher CM Relatively lower Relatively higher BEP Relatively lower Relatively higher MofS Relatively higher Relatively lower D of OL Relatively lower Relatively higher DR Relatively lower Relatively higher UP Relatively lower Relatively higher

Margin of Safety percentage

Margin of safety in units/ Actual or estimated units Margin of safety in $/ Actual or estimated sales $

Operating Leverage

Measures exposure to earnings volatility risk Extent to which a company's net income reacts to a given change in sales. With a higher operating leverage profits increase rapidly when sales increase and plunge drastically when sales decrease. Contribution margin/Profit (Fixed costs/Profit) + 1 1/Margin of safety percentage

Net income

Operating income - Income taxes Operating income - (Tax rate × Operating income) Operating income (1 - Tax rate)

How many briefcases does Bill need to sell to reach a target after-tax profit of $319,200 if the tax rate is 30%? What level of sales revenue is this? Recall that S= $200, V= $80, and F= $360,000.

Pretax profit =After-tax profit/(1 -Tax rate) $319,200/(1 - 0.3) = $456,000 Units needed to reach target after-tax profit F + Oi/Cm $360,000 + $456,000/$120 =6,800 units Sales $ needed to reach target after-profit F + Oi/Cmr $360,000 + $456,000/(200 - 80/200) or $816,000/.6 = $1,360,000

Determining the Indifference Point Compute the volume of sales, in units, for which Bill is indifferent between the two alternatives.

Profit (current plan) = $120Q - $360,000 Profit (proposed plan) = $90Q - $315,000 remembering that OI (profit) = CmQ ‒ F $120Q - $360,000 = $90Q -$315,000 So, subtract 90Q from 120Q to get all q's on the left side of the equation and add 360,000 to -315,000 to get everything else on the right side of the equation. $30Q = $45,000, then Q = 1,500 units

What is the pretax profit if Bill sells 4,100 briefcases? If he sells 2,200 briefcases? Recall that S= $200, V= $80, and F= $360,000.

Profit at 4,100 units = CmQ ‒ F = $120 x 4,100 -$360,000 = $132,000. Remember that CmQ = Price - Variable Cost Profit at 2,200 units = CmQ ‒ F = $120 x 2,200 -$360,000 = ($96,000).

Units to achieve a target pretax profit

Q = F + Oi/Cm

Peggy's Kitchen Wares sells three sizes of frying pans. Next year she hopes to sell a total of 10,000 pans. Peggy's total fixed costs are $40,800. Each product's selling price and variable costs is given below. Find the BEP in units for this company.

Q3: Multiple Product Breakeven Point Small Medium Large Total Expected sales 2,000 5,000 3,000 10,000 in units Selling price $10.00 $15.00 $18.00 per unit Variable costs $4.00 $8.00 $11.00 per unit CM $6.00 $7.00 $7.00 per unit First note the sales mix in units is 20%:50%:30%, respectively; then compute the weighted average contribution margin: WACM = 20%x$6 + 50%x$7 + 30%x$7 = $6.80 Next, compute the BEP in terms of total units: F + 0/ WACm or $40,800/ ($6.80/unit) = 6,000 units But 6,000 units is the BEP in units only for the given sales mix: 20%: 50%: 30% The BEP should be stated in terms of how many of each unit must be sold: Units required to break even: Small pans 20% 1,200 Medium pans 50% 3,000 Large pans 30% 1,800 6,000 Units needed to reach breakeven

Sales = Variable + Fixed + Operating Costs Costs Income How do we get from this to Contribution Margin formula?

SQ = VQ + F + Oi Subtract VQ from both sides SQ -VQ = F + Oi (S -V)Q = F + Oi But Cm = S -V; substitute in equation CmQ = F + Oi (Contribution Margin Formula)

Margin of safety

Sales - Break-even sales or Units sold - Break-even units the units sold or expected to be sold or the revenue earned or expected to be earned above the break-even volume. a measure of how far past the breakeven point a company is operating, or plans to operate.

Operating Income

Sales - variable cost gives the Contribution Margin, and CM - fixed expenses = OI (or profit)

Where: S = V = Cm = Cmr = F = Q = SQ = VQ = CmQ = Oi =

Selling price per unit Variable cost per unit Contribution margin per unit Contribution margin ratio Fixed costs Number of units Total sales Total variable costs Total contribution margin Operating income (before taxes)

Degree of operating leverage

Total contribution margin/Operating income

Variable cost ratio

Total variable cost/Sales or = Unit variable cost/Price

How many briefcases does Bill need to sell to reach a target pretax profit of $240,000? What level of sales revenue is this? Recall that P = $200, V = $80, and F = $360,000.

Units needed to reach target pretax profit F + Profit/ P - V $360,000 + $240,000/ $120 per unit = 5,000 units Sales $ required to reach target pretax profit F + Profit/CMR or F+Profit/(P - VC/P) $600,000/(200 - 80/200) or $600,000/.6 (or 60%) = $1,000,000

Suppose Bill's marketing department says that he can sell 6,000 briefcases if the selling price is reduced to $170. Bill's target pretax profit is $210,000. Determine the highest level that his variable costs can be to make his target. Recall that F= $360,000.

Use the CVP formula for units, but solve for V: If Bill can reduce his variable costs to $75/unit, he can meet his goal. Q= 6,000 = F + Oi/Cm = $360,000 + $210,000/$170/unit -V $170/unit‒V = $360,000 + $210,000/6,000 unit = $95/unit $170/unit‒V = $95/unit V = $75/unit

Margin of Safety in $

actual or estimated sales $ - BEP in sales $

Margin of Safety in units

actual or estimated units of activity - BEP in units

Convert after tax profit to before tax profit

after tax profit/(1 - tax rate)

Common fixed expenses

fixed costs that are not traceable to the segments and that would remain even if one of the segments was eliminated

Direct fixed expenses

fixed costs that can be traced to each segment and would be avoided if the segment did not exist

The indifference point between alternatives

is the level of sales (in units or sales $) where the profits of the alternatives are equal.

Two measures of risk

margin of safety and operating leverage

Suppose that Bill's Briefcases has budgeted next year's sales at 5,000 units. Compute all three measures of the margin of safety for Bill. Recall that P = $200, V = $80, F = $360,000, the BEP in units = 3,000, and the BEP in sales $ = $600,000.

margin of safety in units = 5,000 units - 3,000 units = 2,000 units margin of safety in $ = $200 x 5,000 - $600,000 = $400,000 margin of safety percentage 2,000 units/5,000 units = $400,000/ $200 x 5,000 = 40% The margin of safety tells Bill how far sales can decrease before profits go to zero.

contribution margin (per unit)

price - variable cost Used extensively in CVP analysis. The amount of revenue remaining after deducting all variable costs (including variable production, selling & administrative costs) The amount left to cover fixed costs & "contribute" to net income.

Sales mix

relative combination of products being sold by a firm Defining a particular sales mix allows us to convert a multiple-product problem to a single-product CVP format

sales revenue for target profit

total fixed cost + target profit/contribution margin ratio F + Oi/Cmr

break-even sales revenue

total fixed cost/contribution margin ratio (set Oi= 0) SQ =F/Cmr

variable cost per unit

total variable costs/units

The sales mix should be stated as a proportion of total units sold

when performing CVP calculations for in units.

The sales mix should be stated as a proportion of total revenues

when performing CVP calculations in sales $.