Review Questions
5.1 Profit before taxes = [(Price – Unit variable cost) × Sales volume in units] – Fixed costs = Unit contribution margin × Sales volume in units – Fixed costs.
5.2 The contribution margin statement.
5.3 The sales volume at which profit equals zero.
5.4 The sales dollars at which profit equals zero.
5.5 The unit contribution margin divided by price.
5.6 Taxes reduce profit by a certain percentage beyond the breakeven point. Above the breakeven point, the slope of the profit line decreases by taxes paid.
5.7 We can use the CVP relation to estimate profit at each price, quantity combination.
5.8 The amount by which sales exceed breakeven sales. It equals (Sales in units – Breakeven volume)/Sales in Units or, equivalently, (Revenues – Breakeven revenues)/Revenues.
5.9 The percentage change in profit = the percentage change in sales volume (or revenues) × (1/Margin of safety).
5.10 Operating leverage is a measure of risk from having more fixed costs. It equals
Fixed costs/Total costs.
5.11 The relative proportion in which a company expects to sell products – e.g., two units of product A for every unit of product B.
5.12 The contribution margin per average unit.
5.13 The contribution margin per average sales dollar.
5.14 It is easier to work with revenues directly and comparing contribution margin ratios across products makes more sense than comparing unit contribution margins.
5.15 (1) Revenues increase proportionally with sales volume, (2) variable costs increase proportionally with sales volume, (3) selling prices, unit variable costs, and fixed costs are known with certainty, (4) a single-period analysis, (5) a known and constant product mix, (6) CVP analysis does not always provide the “best” solution to a short-term decision, and (7) the availability of capacity.
Discussion Questions
5.16 Unit contribution margin equals unit selling price less unit variable cost.
Assuming that unit contribution margin is positive, unit selling price is a bigger number than unit variable cost, and therefore a 10% increase in unit selling price will increase the unit contribution margin more than a 10% decrease in unit variable cost. To illustrate, let the unit selling price be $50 and the unit variable cost be $30. The unit contribution margin will be $20 (=$50 -$30). A 10% increase in unit selling price will increase the unit contribution margin to $25 (=$55-$20), but a 10% reduction in unit variable cost will increase the unit contribution margin to only $23 (=$50-$27).
5.17 Profit before taxes = .15 * Revenues (fact 1)
Profit before taxes = .40*Revenues – $200,000 (fact 2)
Setting these equations equal to each other… Revenues = $200,000/.25 = $800,000.
5.18 It is generally advisable to conduct CVP analysis on a cash basis. Non-cash items
such as depreciation are not relevant. However, it is not uncommon to see CVP analysis being used in conjunction with accounting profits---which would include depreciation as an expense---rather than net cash flow. Such an analysis can be particularly erroneous for start-ups and growth firms because the magnitude of non-cash items or accruals is likely to be large.
5.19 Yes. Let us consider an example. Suppose the unit contribution margin is $5 and
the fixed costs are $200,000. The breakeven quantity then is 40,000 units
(=$200,000/$5). Let us say that the fixed costs increase by $100,000 to $300,000 but the unit contribution margin stays at $5. The new breakeven quantity is 60,000 (=$300,000/$5). That is we need an additional volume of 20,000 units to
breakeven. We can also calculate this additional volume needed to breakeven by dividing the change in fixed costs by the unit contribution margin (=$100,000/$5).
5.20 Many countries use a progressive tax structure. That is, the tax rate increases for
higher income brackets. However, the CVP analysis is fundamentally the same except that the profit equation is more elaborate. Consider an example where the tax rate is 30% up to $300,000, but increases to 40% beyond $300,000. In this case, we have to modify the computation of profit after taxes as:
If profit before taxes is less than or equal to $300,000 then [EQ]Profit after taxes = Profit before taxes × (1 – 30%) If profit before taxes is greater than $300,000 then [EQ]Profit after taxes = $300,000 × (1 – 30%) +
(Profit before taxes - $300,000) × (1 – 40%) Thus, for the first $300,000 of profit, the company would pay tax at the rate of 30%; beyond $300,000 the applicable tax rate would be 40%. Keep in mind, however, that having multiple tax brackets has no consequence for the calculation of the breakeven point because, at the breakeven point, the profit is zero and there are no taxes.
5.21 Yes, we can modify the CVP relation to include step costs. With step costs, fixed
costs do not stay fixed for all volumes. It stays fixed for a volume range beyond which it increases to the next level. Consider, for example, a company that leases a copier for its needs and pays a monthly rent. The copier has a certain fixed capacity to make copies over a certain time. Till this capacity is reached, the rent does not vary with the volume of copies made. However, once the company’s copying needs exceeds its capacity, another copier may have to be rented and the rent payment increases by a step to include the rent of the next copier.
When fixed cost increases in steps, the CVP analysis may have to be repeated a few times to converge to the answer. Think about computing the breakeven point. First, assume that the breakeven point would fall within the first step. With this assumption, we can calculate the breakeven point in the usual manner described in the text. If this calculation yields a breakeven point that is within the volume range over which the fixed cost does not increase to the next step, we are done. Otherwise, we change the fixed cost to the next step value and repeat our breakeven calculation. We repeat this process until we reach a point where the breakeven volume falls within the range of the assumed step fixed cost!
5.22 Software companies typically have a high proportion of fixed costs in their cost
structure (i.e., high operating leverage) because their primary resource is trained software professionals. Most of these professionals are paid fixed salaries and wages that do not vary with the volume of software programs they generate, or the number of software programs a company like Microsoft sells. Relatively speaking, an automobile company such as Ford would have a greater proportion of variable costs in its cost structure, although over time this proportion has decreased because of increased automation.
5.23 The practice of selling the same good at different prices to different customers is
called discriminatory pricing. In general the Robinson-Patman Act of 1936 prohibits discriminatory pricing in certain situations (such as, for example, a wholesaler selling the same product to two retailers at different prices with the purpose of influencing their competitive standings). However, in other situations, such differential pricing may well be necessary depending on the nature of the customer or the specific market (because of customer-specific or market-specific costs). We will discuss this aspect later in Chapter 10, when we discuss Customer Profitability Analysis.
5.24 Margin of safety is a “cushion” that the existing level of operations allows
managers in dealing with operating risk. The smaller this cushion, the closer is the manager to making a loss. Thus, when demand uncertainty increases
unexpectedly, this cushion “protects” managers from incurring losses. In such situations, it gives them some room to offer discounts or promotions to keep up the volume in the short-term in order to preserve profitability.
5.25 As the sales volume increases, the total variable costs increase but fixed costs stay
the same. Recall that operating leverage is the ratio of fixed costs to total costs. Because fixed costs stay the same and the total costs increase (because of the increase in variable costs), the operating leverage decreases.
5.26 As demand conditions fluctuate, short-term profits are more sensitive to
consequent changes in sales volume for firms with higher operating leverage. This is why operating leverage is viewed as measure of risk. Referring to Exhibit 5.14, the operating leverage of Sierra Plastics increases with the new technology. Notice that its profits (profit before taxes) fluctuate more with the sales volume with the new technology than without the technology.
5.27 In general, divisions of large firms often have very different cost structures and
serve different markets. Because the CVP analysis is essentially a tool for short-term decision making that helps managers in deciding the level of operations, it makes more sense for individual divisional heads to perform CVP analysis at their respective divisions. At the firm level, the effects of these short-term decisions can then be aggregated to determine the overall state of the firm in the short run, and which divisions are contributing in what measure in this respect.
5.28 Yes. In general the unit of one product is not necessarily comparable to a unit of
another dissimilar product because they require different amounts of resources---such as raw material, labor, machining time, finishing time---to produce.
Therefore, we cannot say that the sports car is more profitable to produce because its unit contribution margin is higher than the contribution margin of an entry-level vehicle. However, with the contribution margin ratio, we can express relative profitability in terms of sales dollars. We can say for instance that for every sales dollar, the sports car contributes twenty cents toward profit (i.e., CMR of 20%), and the entry-level vehicle contributes ten cents toward profit (i.e., CMR of 10%).
5.29 CVP analysis should be considered as a convenient tool to understand the
relations between cost, volume and profit. It makes many assumptions as
discussed in the text. Let us consider a few assumptions assumption and identify a setting in which it would be violated.
Assumption 1: Revenues increase proportionally with sales volume. This assumption essentially means that price per unit is constant and does not vary with volume. However, it is well known that as you decrease price per unit, you can sell more and vice versa. In other words, price per unit and sales volume are inversely related. When we allow for this possibility, this assumption is violated. Assumption 2: Variable costs increase proportionally with sales volume. In other
words, unit variable cost stays the same over the relevant range of operations so that variable costs increase linearly with volume. This assumption will be violated whenever the sales volume goes beyond relevant range (e.g., when firms stretch existing capacity to meet demand). In such cases, variable costs can increase more than proportionately.
Assumption 3: Selling prices, unit variable costs, and fixed costs are known with certainty. In the real world, we have to deal with uncertainty all the time. The assumed selling price, and variable/fixed costs may turn out be different from the actual price and costs because of changes in demand conditions or resource availability.
Assumption 4: Single-period analysis. Most business relationships extend beyond a single period, and most short-term decisions have longer-term implications. Please refer to a discussion of such implications in Chapter 2. Such implications would result in a violation of this assumption.
Assumption 5: Product-mix assumption. With many products, CVP analysis assumes a known and constant product mix. However, in most instance, the product-mix itself has to be decided. Changing the product-mix may be best the way to react to changes in demand for the different products in the mix.
Exercises 5.30
a. Recall that:
Profit before taxes = (unit contribution margin sales volume in units) – fixed costs.
Additionally,
Unit contribution margin = Unit selling price – Unit variable cost. = $3.00 – $1.00 = $2.00 per package. The problem also informs us that Ajay’s fixed costs for the month = $600. Thus, Ajay’s profit is:
Profit before taxes = ($2.00 number of packages sold) – $600.
b. Breaking even implies a profit of zero. Thus, we have: $0 = ($2.00 Breakeven volume) – $600,
Or, breakeven volume in packages =
00 . 2 $ 600 $ = 300 packages. c.
Substituting Ajay’s target profit of $1,400 into the expression for profit we developed in part [a], we have:
$1,400 = $2.00 Required number of packages – $600.
OR, Required sales = $600$2.00$1,400.
5.31 a.
A 50% increase changes Ajay’s variable costs from $1.00 per package to $1.00 (1 + .50), or $1.50 per package.
Consequently, the revised unit contribution margin = $3.00 – $1.50 = $1.50 per package.
Setting target profit to zero in the expression for profit, we obtain: 0 = $1.50 Breakeven volume – $600.
OR, Breakeven volume = $$1600.50= 400 packages.
Note: If Exercise 5.30 had been assigned, you may notice that the breakeven volume has increased by 100 packages – this occurs because Ajay now makes a lower contribution margin per package wrapped.
b. Writing-out Ajay’s profit in detail, we have:
Profit before taxes = (Price – Unit variable cost) number of packages – Fixed costs.
Substituting using the given data, we have: $2,400 = (Price – $1.00) 3,000 – $600. OR, Price = $2.00.
c. If revenue were $4,500, Ajay would have wrapped
00 . 3 $ 500 , 4 $ = 1,500 packages. This sales volume would yield a total contribution margin of: 1,500 packages $2 per package = $3,000. Netting out the fixed costs of $600 yields profit of $2,400. Alternatively, we can substitute 1,500 packages into the profit equation to yield:
5.32 CVP Relation and Profit Planning, Contribution Margin Ratio Approach (LO1, LO2).
a. As discussed in the text, we can re-write the profit as:
Profit before taxes = (Contribution margin ratio Revenue) – Fixed costs.
For Gina, her contribution margin ratio is
(
)
$1.00
$.25
-$1.00
= 0.75.
Additionally, her fixed costs amount to $6,000 per month. Thus, we have:
Profit before taxes = (.75 Revenue) – $6,000.
b. Using the profit equation from part [a] and setting profit equal to $0, we have: $0 = (.75 Breakeven revenue) – $6,000.
Breakeven revenue = $8,000.
c. Substituting sales of $10,000 into Gina’s profit calculation yields:
Profit before taxes = (0.75 $10,000) – $6,000 = $1,500.
5.33 CVP and Profit Planning, Hercules (LO1, LO2, LO3
a. We know that the profit equation is:
Profit before taxes = (unit contribution margin sales volume in units) – fixed costs.
At breakeven profit before taxes = 0. For Hercules, unit contribution margin is revenue – variable costs = $100 - $35 = $65 per member per month. Fixed costs are $40,950 per month. Substituting, we have:
0 = $65 × Breakeven volume - $40,950
Thus, breakeven volume = $40,950/$65 per member = 630 members.
b. First, let us gross up the after-tax target to a required pre-tax amount. We know:
$11,375 / 0.65 = $17,500 = required pre-tax profit. Using this estimate in the pre-tax profit equation, we have: $17,500 = $65× required volume - $40,950
Required volume = $58,450/$65 per member = 899 members (rounded). c. We know from part [b] that Hercules needs to earn $17,500 before taxes to
reach its goal. We calculate the contribution margin ratio as $65/$100 = 0.65 or 65%. We know:
Profit before taxes = contribution margin ratio × required revenue – Fixed costs. $17,500 = 0.65 × required revenue - $40,950
Required revenue = $89,900 (rounding down). Of course, we also can calculate this answer from the answer to part [b]: 899 members × $100 per member = $89,900.
d. The Margin of safety = [current sales – breakeven sales] / current sales. Using the answer from part [a], we have: [950 - 630] /950 = 33.68 % e. Operating leverage = Fixed cost / Total cost
= $40,950 / [40,950 + 950 × $35] = 55.19%
5.34 Contribution Margin, Unit level costs (LO1).
First, we need to calculate unit contribution and fixed costs. We have unit
contribution = (price – variable costs) = ($800 - $440 - $40) = $320. At a volume of 15,000 units, fixed costs are $110 + $50 = $160 per unit, or $2,400,000 in total. Then, from the profit equation:
0 = unit contribution × breakeven volume – fixed costs 0 = $320 per unit × breakeven volume - $2,400,000
5.35 CVP Relation and solving for unknowns, Contribution Margin Ratio Approach (LO1, LO2).
a. Substituting a target profit of $3,600 into the monthly profit equation, we have:
$3,600 = (0.75 Required revenue) – $6,000.
OR, Required revenue =
(
)
0.75
$3,600
$6,000
= $12,800 per
month.
b. Substituting the data into Gina’s profit equation, we have: $4,000 = (0.75 $15,000) – Fixed costs.
Maximum expenditure on fixed costs = $11,250 – $4,000 = $7,250.
c. Gina’s new variable cost is $0.25 (1 + 0.5) = $0.375 per $1.00 of sales.
Thus, Gina’s new contribution margin ratio is:
(
)
$1.00
$.375
-$1.00
= 0.625.
Substituting this information into the profit calculation and setting profit equal to $0, we have:
$0 = (0.625 Breakeven revenue)– $6,000.
Breakeven revenue = $9,600.
Note: If Exercise 5.32 has also been assigned, notice that Gina’s breakeven
revenue increases by $1,600. From a pedagogical standpoint, instructors may wish to point out that exercises 5.30-5.34 are quite similar. The only difference is whether the problem is cast in terms of sales in units or sales in dollars. This framing, though, very much affects how we proceed.
5.36 CVP Relation and Profit Planning, Unit Contribution Margin Approach, Taxes (LO1, LO2).
a. When taxes are proportional to pre-tax profit, we can write the profit as: Profit after taxes = Profit before taxes (1 – Tax rate).
= [(Unit contribution margin × Quantity) – Fixed costs] (1 – Tax rate).
For SpringFresh,
Unit contribution margin = $1.50 – $0.50 = $1.00. Annual Fixed cost = $50,000 × 12 = $600,000. The tax rate = .25
Consequently, SpringFresh’s annual after-tax profit is:
Profit after taxes = [($1.00 Pounds laundered) – $600,000] × .75.
Note: Please be careful to convert monthly fixed costs to an annual amount. b. SpringFresh’s profit before taxes = ($1.00 750,000 pounds) – $600,000 =
$150,000.
Taxes paid = $150,000 .25 = $37,500.
Profit after taxes = $150,000 .75 = $112,500.
c. Since profit and taxes = $0 at the breakeven point, we know that the profit expression in part [a] reduces to:
$0 = $1.00 Breakeven volume – $600,000. Thus, Breakeven volume = 600,000 pounds.
5.37
Profit after taxes = [(Unit contribution margin × Quantity) – Fixed costs] (1 – Tax rate).
$120,000 = [($1.00 Required volume in pounds) – $600,000] × .75. Thus, Required volume = 760,000 pounds.
5.38 CVP Relation in Non-Profits, Contribution Margin Ratio Approach (LO1, LO2).
a. In this case, “profit” is the amount left over after paying for fixed costs. We have:
$21,000 = (# of attendees × $50) - $15,000 Or, we need 720 attendees.
b. Let us first write out the profit equation:
$21,000 = (# of attendees × $50) + (# of attendees × 0.50 × $20) - $15,000 Solving, we find # of attendees = 600
Effectively, the cash bar raises the contribution per member to $60, which lowers the required volume.
5.39 CVP and Profit Planning, Contribution Margin Ratio Approach, Taxes (LO1, LO2).
a. Recall that using the contribution margin ratio, we can express profit after taxes as:
Profit after taxes = [(Contribution margin ratio Billings) – Fixed costs] × (1 – Tax rate).
For Arena, the contribution margin ratio is 1 – 0.30 = 0.70 (70% of billings). Further, Arena’s monthly fixed costs = $14,000 and the tax rate is 35%. Consequently, Arena’s monthly profit is:
Profit after taxes = [(.70 Billings) – $14,000] × .65.
b. At the breakeven point, profit after taxes = profit before taxes = $0 (i.e., no tax is due because there is no profit). Consequently, we have (since the tax rate is irrelevant at breakeven):
$0 = (0.70 Breakeven revenue) – $14,000. Solving, we find breakeven billings = $20,000. c. Using the profit expression in [a], we have:
Profit before taxes = (0.70 $50,000) – $14,000 = $21,000.
Profit after taxes = $21,000 .65 = $13,650.
d. Again, using the profit expression in (a), we have: $7,280 = [(0.70 Required billings) – $14,000] × .65.
Required billings = $36,000.
5.40 CVP Relation, Inferring Cost Structure, Extension to Decision Making (LO2, LO3).
a. The contribution margin ratio =
)
(
Price
cost
able
Unit vari
-Price
.With Zap’s information, we have the contribution margin ratio =
)
(
$22.00
cost
able
Unit vari
-$22.00
= 0.60.Unit variable cost = $8.80 for each “ZAP” kit.
Alternatively, from the formula for contribution margin ratio, notice that
Contribution margin ratio = 1 – (Unit variable cost/Price). The latter term is the “variable cost ratio.” Then,
Contribution margin ratio = 1 – Variable cost ratio. Applying the data from the problem, we have:
0.60 = 1 – Variable cost ratio, or Variable cost ratio = 40% of sales price That is, unit variable cost = 0.40 × $22 = $8.80 per unit.
b. Let us use the profit expression:
Profit before taxes = (Contribution margin ratio Revenue) – Fixed costs. We know that Zap expects to break even at 17,500 “ZAP” kits – thus, Breakeven revenue = 17,500 × $22.00 = $385,000. Additionally, we know that profit = $0 at the breakeven point. Thus, we have:
$0 = (0.60 $385,000) – Fixed costs. Solving, we find that fixed costs = $231,000.
Alternatively, we could use the unit contribution margin formulation. We can calculate the breakeven point as:
$0 = (Unit contribution margin Breakeven volume) – Fixed costs. From part [a], we know that the unit variable cost = $8.80. Because the selling price = $22.00, we know that the unit contribution margin = $22.00 – $8.80 = $13.20. Thus, we have:
$0 = ($13.20 17,500) – Fixed costs. Again, we find that fixed costs = $231,000.
c. The free shipping and handling offer reduces Zap’s revenue per “ZAP” kit to $20.00. With the knowledge acquired in parts [a] and [b] (i.e., the variable cost per “ZAP” kit and Zap’s monthly fixed costs, respectively), we can calculate Zap’s breakeven volume as:
$0 = (Unit contribution margin Breakeven volume) – Fixed costs. or, $0 = [($20.00 – $8.80) Breakeven volume] – $231,000
Breakeven number of kits ( Breakeven volume ) = 20,625.
Consequently, Zap must sell an additional 20,625 – 17,500 = 3,125 kits to break even if the company decides to offer “free” shipping.
Note: Instructors may wish to use this problem to emphasize the importance of
knowing both the unit contribution margin approach and the contribution margin ratio approach. In part [a], it was necessary to use the contribution margin ratio to arrive at the variable cost per unit. In part [c], however, the unit contribution margin approach more readily accommodates a reduction in the sales price – i.e., it is relatively straightforward to calculate the unit contribution margin. Calculating the new contribution margin ratio is somewhat more involved, although it will lead to
contribution margin ratio is (20 – 8.80)/20 = 0.56 or 56%.
5.41 CVP Relation and Decision Making, Pricing based on a Demand Schedule (LO3).
Employing the CVP relation, we can compute the profit at alternative prices to determine the price that yields the maximum profit. The following table contains the detailed computations.
Price Revenue Variable
Costs
Fixed Costs Profit
$32.50 $9,750 $1,800 $3,000 $4,950
$30.00 $10,500 $2,100 $3,000 $5,400
$27.50 $11,000 $2,400 $3,000 $5,600
$25.00 $11,250 $2,700 $3,000 $5,550
$22.50 $11,250 $3,000 $3,000 $5,250
By inspection, we find $27.50 to be the profit-maximizing price. Greg earns $5,600 in profit at this price.
Note: Instructors may wish to point out that firms may use a demand function
instead of using a demand schedule like the table above. A demand function gives the revenue for every possible price. (For example, based on the data provided, Greg’s demand function is: Quantity = 950 – 20 × Price). In this case, we use calculus or numerical approximation techniques (via a spreadsheet such as Excel’s solver function) to determine the profit-maximizing price.
5.42 CVP relation and Decision Making, Choosing a Cost Structure, Operating Leverage (LO3, LO4).
a. One’s first inclination is to compare the profit across the various popcorn machines. However, for the same number of customers, revenue is equal across the three machines. Thus, we can rank order popcorn machines according to their total costs. In other words, the problem can be formulated as a cost minimization problem as the machine that minimizes cost also maximizes profit.
We start by assessing the number of patrons at which the small popcorn machine will cost the same as the medium popcorn machine. We have:
The number of patrons at which the cost is the same = 15 . 000 , 6 $ = 40,000.
Thus, when a theater expects less than 40,000 moviegoers a year (or
approximately 110 per day), it is optimal to rent the small popcorn machine. Comparing the medium and the large popcorn machines, we have:
$12,000 + (0.35 number of patrons) = $18,500 + (0.25 number of patrons).
The number of patrons at which the cost is the same =
10 . 500 , 6 $ = 65,000.
Thus, when a theater expects more than 65,000 moviegoers a year (or approximately 178 per day), it is optimal to rent the large popcorn machine. Additionally, the above analysis informs us that when a theater expects between 40,000 and 65,000 moviegoers a year, it is optimal to rent the medium popcorn machine.
Thus, we have the following decision rule:
Midwest Cinema Theaters Popcorn Machine Rental Model
Annual # of Moviegoers ( M ) Popcorn Machine Size
M < 40,000 Small
40,000 < M < 65,000 Medium
M > 65,000 Large
Note: Instructors also may wish to graphically represent the tradeoff – this is perhaps best accomplished by asking students to graph, for each size popcorn machine, how popcorn costs (y-axis) varies as a function of the number of patrons (x-axis). This allows students to see where the lines cross – The instructor can then shade the low cost frontier to see how the preferred machine depends on expected volume.
b. Operating leverage = Fixed costs/Total costs
For each popcorn machine, we have:
Fixed Costs Variable Costs Operating Leverage Small $6,000 $32,500 (= 65,000 × $0.50) .1558 Medium $12,000 $22,750 (= 65,000 × $0.35) .3453 Large $18,500 $16,250 (= 65,000 × $0.25) .5323
As discussed in the text, operating leverage frequently is used as a measure of risk – ceteris paribus, the higher the operating leverage, the higher the risk. Thus, while we see that the large popcorn machine is preferred for volumes of 65,000 moviegoers and higher, it also carries the highest risk, a factor that Leticia may wish to consider in her decision.
5.43 CVP Relation and Decision Making, Margin of Safety, Operating Leverage, Cash-Basis Breakeven Analysis (LO3, LO4).
a. Margin of safety =
)
sales
Current
sales
Breakeven
sales
Current
(
We first need to determine Cottage Bakery’s breakeven sales. Using the contribution margin ratio to write the profit equation, we have:
Profit before taxes = (Contribution margin ratio Revenue) – Fixed costs. Because we do not know Cottage Bakery’s fixed costs, we first have to use the model to “back out” fixed costs and then derive breakeven sales. Accordingly,
$7,500 = (0.4 $150,000) – Fixed costs Fixed costs = $52,500.
Next, setting target profit equal to $0 will allow us to calculate breakeven revenue:
$0 = (0.4 Breakeven revenue) – $52,500 Breakeven revenue = $131,250.
Cottage Bakery’s Margin of safety =
)
(
$150,000
$131,250
-$150,000
= 12.5%.In dollars, the margin of safety equals $150,000 – $131,250 = $18,750. b. Operating leverage = Fixed costs/Total costs.
We know that the contribution margin (in dollars) = Contribution margin ratio Revenue. Thus, we have:
Contribution margin = 0.4 150,000 = $60,000.
Because contribution margin = revenues – variable costs, variable costs can be calculated as $90,000. Given that fixed costs are $52,500, the total costs are $142,500.
Consequently, Operating leverage = fixed costs / total costs
= $52,500/$142,500 = 0.368 (rounded)
c. If 30% of the fixed costs represent non-cash expenses, the cash fixed expenses equal: 0.70 $52,500 = $36,750.
We are now in a position to write the cash profit as:
Cash profit = (Contribution margin ratio Revenue) – Cash fixed costs. To calculate the breakeven point, we set cash profit = $0.
$0 = (0.40 Cash breakeven revenue) – $36,750.
Cash breakeven revenue = $91,875.
Notice the large difference between the revenue required to breakeven on a cash basis ($91,875) and the revenue required to breakeven on a non-cash (accrual) basis ($131,250). Moreover, this problem presents a nice opportunity to talk with students about which profit model is more germane to the firm. While cash basis accounting may paint an unusual picture of the organization’s health (due to the slippage between cash-basis accounting and “economic reality”), many
organizations (and people) need to ensure that they remain solvent/liquid in the short-term and that cash expenditures do not exceed cash revenues.
5.44 Multi-Product CVP Analysis, Unit Contribution Margin Approach (LO5).
a. Let us employ a weighted unit contribution margin approach to solve the problem. For Mountain Maples, we have:
2,400 total trees sold – 800, or 1/3 are Butterfly, and 1,600, or 2/3, are Moonfire. Thus, we have:
Weighted unit contribution margin = 1/3 $100 + 2/3 $50. = $66.67.
In turn, Mountain Maples’ profit becomes:
Profit before taxes = ($66.67 total number of trees sold) – $75,000.
At the breakeven point, we have: $0 = ($66.67 Breakeven number of trees) – $75,000.
Solving, we find that the total number of trees sold to breakeven = 1,125. Of these, 1,125 1/3 = 375 Butterfly; 1,125 2/3 = 750 Moonfire. b. Using the weighted unit contribution margin approach, we have:
$50,000 = ($66.67 total number of trees) – $75,000. The total number of trees = 1,875.
Of these, 1,875 1/3 = 625 are Butterfly, and 1,875 2/3 = 1,250 are Moonfire
c. The change in the product mix affects Mountain Maple’s weighted contribution.
With the new information, we have:
Weighted contribution margin= (.50 $100) + (.50 $50) = $75.00
The weighted contribution margin is higher than in part [a] because the product mix has shifted toward Butterfly, which has the highest contribution margin per tree.
0 = ($75.00 Breakeven number of trees) – $75,000. Breakeven number of trees = 1,000.
Of these, 1,000 .50 = 500 are Butterfly, and 1,000 .50 = 500 are Moonfire
5.45 Multi-Product CVP Analysis, Contribution Margin Ratio Approach (LO5).
a. In this setting, we must use the weighted contribution margin ratio approach given the absence of unit-level data. Accordingly,:
Profit before taxes = (RevenueN Contribution margin rationN) +
(RevenueU Contribution margin ratioU) – Fixed costs,
The subscripts ‘N’ and ‘U’ stand for new and used. Additionally, we know that $1,500,000/$2,000,000, or 75% of the revenue is from new cars, and
$500,000/$2,000,000, or 25% of the revenue is from used cars.
Further, we can calculate the contribution margin ratio for each product using the product-level financial data. We have:
(Contribution margin ratio)N =
000 , 500 , 1 000 , 750 000 , 500 , 1 = .50.
(Contribution margin ratio)U =
000 , 500 000 , 200 000 , 500 = .60.
Thus, the weighted contribution margin ratio = (.50 × .75) + (.60 × .25) = .525. We can now write Select’s profit in terms of the weighted contribution margin ratio and total revenues:
Profit before taxes = (.525 × Total revenue) – $840,000. Setting profit equal to $0, we find:
Breakeven total revenue = $1,600,000.
This translates into $1,600,000 .75 = $1,200,000 in new auto sales and $1,600,000 .25 = $400,000 in used auto sales.
b. To answer this question, we plug in our desired profit in the equation for profit developed in part [a]. We now have:
$1,050,000 = (.525 Total revenue) – $840,000. Solving, we find:
Total revenue = $3,600,000.
This translates into $3,600,000 .75 = $2,700,000 in new auto sales and $3,600,000 .25 = $900,000 in used auto sales.
5.46 Multi-Product Analysis, weighted contribution margin & weighted contribution margin ratio approach, Hercules (LO5).
a. We know that individual and family memberships have a 3:1 ratio (900 individual to 300 families). Thus, the weighted contribution margin is: [3 × ($100-$35) + 1 × ($150 -$60)] / 4 = $71.25.
Then, plugging into the profit equation, we have: 0 = $71.25 × total memberships - $42,750,
Total memberships = $42,750/$71.25 per average membership= 600.
At this volume, Hercules has (3/4) × 600 = 450 individual and (1/4) × 600 = 150
family memberships.
b. We calculate the contribution margin for individual and family memberships at 0.65 (= [($100-$35)/$100] and 0.60 (= [($150-$60)/$150] respectively. We know that individual and family memberships have a 2:1 ratio in terms of total revenue (Individual revenue is 900 members × $100 per month = $90,000 and family revenue is 300 memberships × $150 per month = $45,000.) Thus, the weighted contribution margin ratio is:
[(2/3) × 0.65 + (1/3) × 0.60] = 0.6333 = 63.33%.
Please note that we weight the individual contribution margin ratios by their revenue shares. In contrast, we used the share of memberships to weight individual contribution margins in part [a].
Total memberships = $42,750/0.63333 = $67,500.
Note: You can verify this answer by using the answer for part [a]. Total revenue
realized at the answer to part [a] is 450 × $100 + 150 × $150 = $67,500!
PROBLEMS
5.47 CVP relation, Profit Planning, Unit Contribution Margin Approach, Extensions to Decision Making (LO1, LO2, LO3).
a. The general CVP model is:
Profit before taxes = [(Price – Unit variable cost) Quantity] – Fixed cost. = (Unit contribution margin Quantity) – Fixed costs. Plugging in both the membership fee and the fixed and variable cost information for Garnet’s Gym, we have:
Profit = [($500 – $200) Number of members] – $1,200,000. = ($300 Number of members) – $1,200,000.
b. To calculate the breakeven point in members, we set profit equal to $0 in the expression for profit in [a]:
$0 = 300 Breakeven volume – $1,200,000.
Number of members required to break even = 4,000.
c. Here, we plug in the number of members from the previous year into the profit equation developed in part [a]. Doing so yields:
Profit before taxes = ($300 5,000) – $1,200,000 = $300,000.
d. This strategy changes the annual membership fee to $500 .90 = $450. In turn, this changes the per-member contribution margin to $450 – $200 = $250. If membership increases to 6,500 because of the discount, then expected profit is:
Profit before Taxes = ($250 6,500) – $1,200,000 = $425,000.
This action would increase profit by $125,000 (i.e., $425,000 – $300,000) compared to the previous year.
This seems like a good option to increase profit unless there are significant “congestion costs.” The owners probably should think through the soundness of increasing membership by 1,500 persons, or 1,500/5,000 = 30%. It is possible that this will translate to increased waiting time for machines and equipment, or difficulty finding a parking spot. The owners may wish to survey current members to assess their preferences. Such a strategy could backfire if people believe that their “cozy” gym has become too crowded and, perhaps, filled with the “wrong” type of people (e.g., those who are not serious about training). On the other hand, current members may desire more members as this increases the potential for finding dates, friendships, and so on.
e. We start with our finding from part [d] that profit is expected to be $425,000 if the owners reduce the membership fee. We set this amount equal to our target profit and solve for the advertising expense. Accordingly, we have:
$425,000 = ($300 6,500) – $1,200,000 – Advertising.
We find that the maximum advertising expenditure = $325,000.
In terms of comparing the options, the owners should assess which option is likely to be more costly, in terms of foregone contribution margin or out-of-pocket expense, and which option is likely to lead to the greatest increase in membership (if this is the desired outcome).
f. The two approaches are mathematically equivalent. To see this, we start with the unit contribution margin approach:
Profit before taxes = Unit contribution margin Quantity – Fixed costs.
To arrive at the contribution margin ratio approach, we first multiply UCM Q by P/P, giving us:
(UCM Q) PP .
In turn, this expression can be re-written as:
P UCM
(Q P).
This should look familiar – it is equivalent to: CMR Revenue.
5.48 CVP Relation, Profit Planning, Contribution Margin Ratio Approach, Extensions to Decision Making (LO1, LO2, LO3).
a. Given the absence of unit-level data, we need to employ the contribution margin ratio approach. Additionally, since we are asked to find the breakeven point, and profit = $0 at the breakeven point, taxes are not relevant. Thus, we have:
$0 = (Contribution margin ratio Breakeven revenue) – Fixed costs. For Precious Stone Jewelry, the contribution margin ratio is:
)
(
$1,000,000
$1,000,000
-
600,000
= 40%.
Additionally, Precious Stone Jewelry’s fixed costs = $260,000. Thus, we have: $0 = (0.40 Breakeven revenue) – $260,000.
Solving, we find breakeven revenue = $650,000.
b. Increasing the selling price by 20% will increase revenues by 20% (because quantity stays the same) and, in turn, increase the contribution margin ratio. First, we have:
Revised revenues = 1.20 1,000,000 = $1,200,000. The new contribution margin ratio is:
)
(
$1,200,000
600,000
-$1,200,000
= 50%. Thus, we have: $0 = (0.50 Breakeven revenue) – $260,000.Solving, we find breakeven revenue = $520,000. Thus, breakeven revenue would decrease by $650,000 – $520,000 = $130,000.
c. If variable costs decrease by 20%, new variable costs will be: Variable costs = .80 600,000 = $480,000.
)
$1,000,000
480,000
-$1,000,000
(
= 52% Thus, we have: $0 = (0.52 Breakeven revenue) – $260,000.Solving, we find breakeven revenue = $500,000. Thus, breakeven revenue would decrease by $650,000 – $500,000 = $150,000.
d. As discussed in part [a], taxes have no effect on the breakeven point. Since taxes are a percentage of profit, any percentage of $0 is always $0. Thus, the tax rate is not relevant. As illustrated in part [e], however, taxes are relevant when the firm earns positive profit.
e. If all changes do take place, we have:
Revenues = $1,000,000 1.20 $1,200,000
Variable costs = $600,000 (1 – .20) 480,000
Contribution margin $720,000
Fixed costs (stay the same) 260,000
Profit before taxes $460,000
Taxes .30 $460,000 138,000
Profit after taxes $322,000
Thus, profit is expected to increase by $217,000, from $105,000 to $322,000. Both the change in price and unit variable cost increase Precious Stone’s
contribution margin and, in turn, profit. The increase in the tax rate diminishes the amount of Profit before taxes that Precious Stone retains – that is, the increase in the tax rate reduces the slope of the profit after taxes line.
5.49 CVP Relation and Profit Planning, Solving for Unknowns (LO1, LO2).
a. The general model for the city’s snow removal costs looks like (notice that there is no need for a revenue, or profit, component as the problem deals strictly with cost):
Total costs = Fixed costs + (Variable cost per snowfall Number of snowfalls). We are provided with total costs and snowfall data for two separate years: $300,000 = Fixed costs + Variable cost per snowfall 20.
$228,000 = Fixed costs + Variable cost per snowfall 12.
We now have two equations with two unknowns – accordingly, we can subtract the second equation from the first equation. Doing so yields:
$72,000 = Variable cost per snowfall 8. OR, variable cost per snowfall = $9,000.
Plugging this back into either year we find that fixed cost = $120,000. Thus, the city’s snow removal costs can be calculated as:
Total Snow Removal Costs = $120,000 + ($9,000 Number of major snowfalls).
b. For this question, it is simply a matter of plugging in the anticipated number of snowfalls into the expression for snow removal costs developed in part [a]. We have:
Expected Snow Removal Costs = $120,000 + ($9,000 26) = $354,000. Thus, the city should request $54,000 more than it spent this year! This problem, which links nicely back to Chapter 3, is useful for showing students how CVP can be used for planning/budgeting purposes. This problem, though, deals only with the C(ost) and the V(olume) portions of the model.
5.50 Building a CVP Relation that Incorporates Taxes and Bonus Payments using a Contribution Margin Ratio Approach (LO1, LO2).
The first step is to write Diamond Jubilee’s after-tax profit as follows:
Profit after taxes = (Total wagers – Winnings – Variable costs – Bonus – Fixed costs)
we will need to work with the contribution margin ratio approach. The model is more involved because the bonus is a function of pre-tax income. We begin by modeling pre-tax and pre-bonus profit:
Pre-tax and pre-bonus profit = (Contribution margin ratio Total wagers) – Fixed costs.
Here, contribution margin ratio is the contribution margin ratio without the bonus; it is $1.00 – $0.82 – $0.08 = $0.10 or 10%. Further, fixed costs equal $27,500. Thus, we have:
Pre-tax and pre-bonus profit = (.10 Total wagers) – $27,500.
We are now in a position to add the bonus payment to the model as the manager’s bonus equals 5% of the pre-tax and pre-bonus profit. After subtracting the bonus, which is 5% of the pre-tax and pre-bonus profit, we have pre-tax (but after bonus) profit as:
Pre-tax profit = [(.10 Total wagers) – $27,500] (1 – .05).
We can next add taxes to the model. With a tax rate of 25%, we have: After-tax profit = {[(.10 Total wagers) – $27,500] (1 – .05)} (1 – .25). At this point, we need only substitute the desired monthly after-tax, after-bonus profit of $28,500. Thus,
$28,500 = {[(.10 Total wagers) – $27,500] (1 – .05)} (1 – .25).
Solving, we find that the required monthly level of total wagers (gross gambling revenue) = $675,000.
This problem demonstrates how the “basic” CVP relation can be expanded to incorporate additional short-term profit considerations such as bonuses and taxes.
5.51 CVP Relation and Profit Planning, Choosing a Cost Structure (LO1, LO2, LO3).
a. To calculate breakeven revenue, we start with the profit calculation using the contribution-margin ratio:
Profit = (Contribution margin ratio Revenue) – Fixed costs. Setting profit = $0,
Breakeven revenue = Fixed costs/Contribution margin ratio. Given the information in the problem for each cost structure, we have:
Monthly breakeven revenue under current cost structure:
Fixed costs $36,000 per month
Contribution margin ratio 40%
Breakeven Revenue $36,000/0.40 = $90,000
Monthly breakeven revenue under new cost structure:
Fixed costs $60,000 per month
Contribution margin ratio 60%
Breakeven revenue $60,000/0.60 = $100,000
Thus, Cecelia’s breakeven revenue will increase by $10,000 if she acquires the new machines. Even though her contribution margin ratio increases (which, ceteris paribus, pushes the breakeven point down) by acquiring the new machines, the substantial increase in fixed costs drives the breakeven point up. b. As in part [a], the profit is:
Profit = (Contribution margin ratio Revenue) – Fixed costs.
We now plug in the various parameters to determine profit under each cost structure.
Current cost structure:
Profit at $95,000 in revenue = (0.40 $95,000) – $36,000 = $2,000. Profit at $150,000 in revenue = (0.40 $150,000) – $36,000 = $24,000.
New cost structure:
Profit at $95,000 in revenue = (0.60 $95,000) – $60,000 = ($3,000). Profit at $150,000 in revenue = (0.60 $150,000) – $60,000 = $30,000.
Thus, Cecelia prefers her current cost structure if monthly revenues are expected to be $95,000, and she prefers to acquire the new machines if revenues are expected to be $150,000.
c. We can find this point of indifference, or crossover point, by equating the profit equation under the two cost structures and solving for revenue. Let the required revenue level be $R. The profit at $R is:
New cost structure: (0.60 $R) – $60,000. Equating the two profit equations, we have:
(0.40 $R) – $36,000 = (0.60 $R) – $60,000. $R = ($60,000 – $36,000)/(0.60 – 0.40) = $120,000.
With $120,000 of revenue, Cecelia makes the same profit of $12,000 with either cost structure. Also notice that for sales greater than $120,000, Cecelia
prefers to acquire the new machines whereas for sales less than $120,000 Cecelia prefers her current cost structure. That is, the slope of the profit line is higher under the new cost structure than the old cost structure. Instructors may wish to graph the two cost structures to illustrate this point to students.
5.52 CVP Relation and Decision Making, Pricing Based on a Demand Schedule (LO3).
a. The following table provides the profit computations (and comparisons) if fixed costs were $1,500,000 per year and variable costs were $1 per copy.
Introductory Price $25/copy $15/copy $5/copy
Users (copies sold) 75,000 150,000 300,000
Year 1
Price per copy $25 $15 $5
Variable cost per copy $1 $1 $1
Contribution margin per copy $24 $14 $4
Total contribution margin $1,800,000 $2,100,000 $1,200,000
Fixed costs $1,500,000 $1,500,000 $1,500,000
Profit before taxes $300,000 $600,000 $(300,000)
Year 2
Price per copy $25 $25 $25
Variable cost per copy $1 $1 $1
Contribution margin per copy $24 $24 $24
Total contribution margin $1,800,000 $3,600,000 $7,200,000
Fixed costs $1,500,000 $1,500,000 $1,500,000
Profit before taxes $300,000 $2,100,000 $5,700,000 Year 1 + Year 2 Profit $600,000 $2,700,000 $5,400,000
The table clearly shows that Innova Solutions maximizes two-year profit by setting an introductory price of $5 per copy (this would be true for just about any discount rate).
b. The following table provides the profit computations (and comparisons) if fixed costs were $200,000 per year and variable costs were $15 per copy.
Introductory Price $25/copy $15/copy $5/copy
Users (copies sold) 75,000 150,000 300,000
Year 1
Price per copy $25 $15 $5
Variable cost per copy $15 $15 $15
Contribution margin per copy $10 $0 ($10)
Total contribution margin $750,000 $0 ($3,000,000)
Fixed costs $200,000 $200,000 $200,000
Profit before Taxes $550,000 ($200,000) ($3,200,000)
Year 2
Price per copy $25 $25 $25
Variable cost per copy $15 $15 $15
Contribution margin per copy $10 $10 $10
Total contribution margin $750,000 $1,500,000 $3,000,000
Fixed costs $200,000 $200,000 $200,000
Profit before Taxes $550,000 $1,300,000 $2,800,000 Year 1 + Year 2 Profit $1,100,000 $1,100,000 ($400,000)
Here, we see that Innova Solutions’ optimal pricing strategy changes – the table shows that Innova Solutions probably is best off by setting an introductory price of $25 per copy (inter temporal considerations also would lead Innova Solutions to go with $25 rather than $15).
c. In an industry where an “installed base” of customers is important (as in software or video games), firms often sacrifice today’s profit to build market share and tomorrow’s profit. The key idea is that that the firm can increase profitability in future years by taking advantage of consumers’ switching (transaction) costs.
The efficacy of this strategy depends on the current period tradeoff between demand and price. If demand increases sufficiently with a price drop, then the strategy of going for a low introductory price can generate significantly more profit in the long-run. However, as the tables show, the required increase in demand increases as variable costs increase. For Innova, “low-balling”
maximized profit when the variable cost was low but not when the variable cost was high. Thus, we often find such a pricing strategy being followed only by those firms with low variable costs (equivalently, high contribution margin ratios) as a percentage of price. While such a strategy may work for software or video games, it is unlikely to work for auto manufacturers.
between price and resulting demand. Moreover, firms often spend considerable effort and resources in constructing sophisticated models that capture the economic forces of supply and demand.
5.53 CVP Relation and Margin of Safety (LO4).
a. This is a non-standard problem in the sense that there is no variable cost. That said, we can view the commission as Brenda’s contribution margin ratio. With this interpretation, the profit is:
Profit = (Contribution margin ratio Transactions in dollars) – Fixed costs.
To calculate monthly breakeven transactions, we set profit = $0 and fixed costs equal to $18,000. This yields:
$0 = (0.03 Breakeven Transactions) – $18,000.
Thus, the volume of monthly transactions required to breakeven = $600,000.
b. Margin of safety =
)
sales
current
sales
breakeven
sales
current
(
. =(
)
$1,000,000
$600,000
-$1,000,000
. = 40%.Thus, Brenda’s transaction volume could decrease by 40%, or $400,000 before she incurs a loss in a month.
c. Using the setup from part [b], we find Brenda’s margin of safety for a transaction volume of $1,200,000 to be:
Margin of safety =
(
)
$1,200,000
$600,000
-$1,200,000
= 50%.Similarly, for a transaction volume of $1,600,000, we find: Margin of safety =
(
)
$1,600,000
$600,000
-$1,600,000
= 62.50%.d. First, we notice that margin of safety only makes sense at a given sales
volume. As sales change, the margin of safety also changes. Second, we notice that that margin of safety increases as transaction volume increases, reflecting the additional sales over the breakeven volume of sales (and, hence, a larger “cushion”). Finally, we notice that the relationship between sales and margin of safety is non-linear. In Brenda’s case, 20% and 60% increases in transaction volume lead to 25% and 56.25% increases in her margin of safety — i.e., 25% = (50-40)/40; 56.25 = (62.50-40)/40.
5.54 CVP Relation and Decision Making, Operating Leverage, Margin of Safety (LO3, LO4).
a. We have enough information to construct condensed income statements for each proposal: Proposal 1 Proposal 2 Revenues $2,750,000 $2,750,000 Variable Costs* 1,100,000 1,925,000 Contribution Margin** $1,650,000 $825,000 Fixed Costs 1,500,000 675,000
Profit before Taxes $150,000 $150,000
* = (1 – .60) $2,750,000; (1 – .30) $2,750,000.
** can also be calculated as: .60 $2,750,000; .30 $2,750,000.
Thus, expected profit is equal under both proposals. Turning to operating leverage, we have:
Operating leverage = Fixed costs/Total costs.
The condensed income statements provide us with all of the information necessary to compute operating leverage:
Operating leverage (proposal 1) = $1,500,000/$2,600,000 = 0.577 (rounded).
Operating leverage (proposal 2) = $675,000/$2,600,000 = 0.260 (rounded).
The margin of safety = (current sales – breakeven sales)/current sales. Thus, we need to calculate the breakeven revenue under each proposal. To do so, we calculate profit using the contribution margin ratio. We have:
$0 = Contribution margin ratio Breakeven revenue – Fixed costs; Breakeven revenue = Fixed costs/Contribution margin ratio.
Breakeven revenue (proposal 1) = $1,500,000/.6 = $2,500,000. Breakeven revenue (proposal 2) = $675,000/.3 = $2,250,000. We are now in a position to calculate the margin of safety:
Margin of safety (proposal 1) = ($2,750,000 – $2,500,000)/$2,750,000 = .0909
(rounded).
Margin of safety (proposal 2) = ($2,750,000 – $2,250,000)/$2,750,000 = .1818 (rounded).
Now that we have performed all of the requisite calculations, we are in a position to think about which proposal the bank is likely to support. This is a difficult question. First, notice that expected profit is identical under each alternative. Thus, we need to consider the proposals on some other dimension – the upside potential and/or the downside risk. In terms of the upside, each additional dollar of revenue generated under proposal #1 contributes $.60 to profit, whereas under proposal 2, each additional dollar of revenue generated contributes only $.30 to profit. Thus, proposal 1 has higher upside potential than proposal 2.
Both operating leverage and margin of safety provide us with some measure of business risk and, hence, the downside. Operating leverage is a measure of the extent of fixed costs in the business – notice that the first proposal has much higher operating leverage. This is because the first proposal has significantly higher fixed costs than the second proposal. The margin of safety provides us the amount by which sales revenue could decrease before Dan is in the red. Notice that the second proposal has a higher margin of safety than the first proposal – thus, there is more of a cushion in the second proposal (as reflected by the lower break-even point). Thus, proposal 1 has more downside risk than proposal 2. We really need to get inside the banker’s head and think about his/her objective, which likely is “will I get my money back?” Given this objective, a prudent (and probably risk-averse) banker is likely to push the less risky proposal #2. This conclusion is not, however, a fait accompli and this question can generate interesting discussions about risk preferences and decision-making under uncertainty.
b. To calculate profit, operating leverage, and margin of safety, we can repeat our approach from part [a]. We have:
Proposal 1 Proposal 2
Revenues $4,500,000 $4,500,000
Variable Costs* 1,800,000 3,150,000
Contribution Margin** $2,700,000 $1,350,000
Fixed Costs 1,500,000 675,000
Profit before Taxes $1,200,000 $675,000
* = (1 – .60) $4,500,000; (1 – .30) $4,500,000.
** = can also be calculated as: .60 $4,500,000; .30 $4,500,000. Turning to operating leverage, we have:
Operating leverage (proposal 1) = $1,500,000/$3,300,000 = .455 (rounded).
Operating leverage (proposal 2) = $675,000/$3,825,000 = .176 (rounded).
For margin of safety, we have:
Margin of safety (proposal 1) = ($4,500,000 – $2,500,000)/$4,500,000 = .4444
(rounded).
Margin of safety (proposal 2) = ($4,500,000 – $2,250,000)/$4,500,000 = .50.
Notice that as sales increase, proposal 1 becomes more attractive relative to proposal 2. First, expected profit is significantly higher ($525,000) under proposal 1 than proposal 2. Second, the upside potential of proposal 1 is higher than proposal 2 – as discussed in part [a], each additional $ of revenue contributes $0.60 to profit under proposal 1, but only $0.30 under proposal 2. Finally, compared to part [a], the differences in operating leverage and, particularly, the margin of safety between the proposals are smaller – proposal 1 is only slightly riskier than proposal 2. Given the profit difference and minimal risk, a prudent lender would likely push proposal 1 in this setting.
5.55 Multi-Product CVP Analysis (LO5).
a. The profit is:
Profit before Taxes = [(Price – unit variable cost) Quantity] – Fixed costs. Substituting for Campus Bagels’ specific information, we have:
Profit = [($1.00 – $0.40) number of bagels sold] – $100,000.
Given the 250,000 bagels sold in the previous year, profit was: Profit before taxes = $0.60 250,000 – $100,000.
Profit before taxes = $50,000.
b. Campus Bagels will now have two different products, bagels and bagel sandwiches. Accordingly, the profit is:
Profit = [Quantitybagels (1.00 – .40)]
+ [Quantitybagel sandwiches (Pricebagel sandwiches – 1.25)] – $125,000.
Notice that fixed costs have increased by $25,000 and, in order to determine net income, Campus Bagels needs to know the price of bagel sandwiches as well as the quantity of bagel sandwiches to be sold. One might also argue that the quantity of bagels that will now be sold is ambiguous.
More generally, whether Campus Bagels will continue to sell 250,000 bagels depends on whether bagels and bagel sandwiches are complements or substitutes. One could argue the case either way – on the one hand, some customers might switch from regular bagels to the “more refined” bagel sandwich. Additionally, some current customers might already convert the bagels they buy into bagel sandwiches. On the other hand, the products may complement each other. Strengthening the product line by adding bagel sandwiches may attract new customers to Campus Bagels who, in turn, not only buy bagel sandwiches but also are drawn to regular bagels.
c. It is important to note that the problem setup is somewhat atypical in that it is problematic to specify a quantity without knowing a price – it probably is useful to discuss this issue in class.
$125,000.
Solving, we find Price= $4.25.
d. This scenario is more typical as the demand for one product affects the demand for the other and vice-versa. Now, in addition to the incremental variable and fixed costs associated with introducing bagel sandwiches, there is an opportunity cost associated with selling bagel sandwiches – the lost
contribution margin on bagels.
We can calculate this opportunity cost directly and add it to the cost of bagel sandwiches or, perhaps more intuitively, we can use the model from part [b], which has this opportunity cost “built in.” In other words, notice that we can directly plug the new quantities into the model, or:
$100,000 = [225,000 (1.00 – .40)] + [25,000 (Pricebagel sandwiches – 1.25)] –
$125,000.
Solving, we find Price = $4.85.
Here, we see that relationships among products can be captured in our CVP formulation. At a more general level, we could explicitly model this relationship (e.g., Quantitybagels = 250,000 – Quantitybagel sandwiches). Moreover, when multiple
products exist we need to consider both complementary and substitute
relationships. Such relationships yield positive or negative externalities and, in turn, reduce or increase profitability. CVP models can help sort-out these issues.
5.56 Multi-Product CVP and Fixed Cost Allocations (LO5).
a.
The following table provides the required computations.
Retail Institutional
Traceable fixed costs 175,000 80,000
Allocated fixed cost 100,000 100,000
Total 275,000 180,000
Contribution margin ratio 66.67% 40%
Breakeven revenue $412,500 $450,000
At this volume, Jan breaks even for the entire company as well. After all, his total fixed costs are $175,000 + $80,000 + $200,000 = $455,000. Then, at the
computed volumes, he generates a contribution of $412,500 × 0.6667 + $450,000 × 0.4 = $455,000. The firm also breaks even at the total level of $862,500.
b. Jan’s weighted contribution margin ratio is $480,000/$900,000 = 53.33%. With this estimate, we can calculate breakeven revenue as
$455,000 / 0.53333 = $853,125
Or, $426,562.50 each in retail and institutional sales.
c. The answers in parts a and b differ because we “fix” different items. In part (b), we fixed the sales mix to be 50% from each segment. With this assumption, the breakeven sales are $853,125. However, sales mix is not fixed in part (a). Rather, we have “fixed” the allocation to be $100K to each segment. Thus, the final answer has a sales mix that is NOT 50-50 across the segments. Moreover, the fixed cost also is not allocated in proportion to the sales mix at breakeven! In general, it makes more sense to fix the sales mix as in part (b) in multi-product CVP analysis.
d. In general, we perform multi-product CVP for the entire firm. This is particularly appropriate when the products are similar (e.g., car dealership), are substitutes and share considerable fixed cost. However, we also encounter situations
in which products are distinct with few common fixed costs. In this case (e.g.,
divisions of General Electric or John Deere), it makes sense to compute a division level break even.
5.57 Multi-Product CVP Analysis (LO5).
a. Determining the product mix is the first step in computing the breakeven point in a multi-product setting. In Kim’s case, we have the following per 10
customers:
Sandwich Soup Salad Drink
Price $4.00 $3.00 $3.00 $1.00
Unit variable cost $1.25 $1.00 $0.75 $0.25 Unit contribution margin $2.75 $2.00 $2.25 $0.75 # of orders 5 7 4 6 Total contribution margin $13.75 $14.0 0 $9.00 $4.50 Total Price: $20.00 $21.0 0 $12.00 $6.00
Thus, we can calculate Kim’s weighted contribution margin ratio as $41.25/$59 = 69.91%.
Profit = (0.6991 × revenue) – $4,950. Setting profit = $0, we find:
Breakeven revenue = $4,950/0.6991 = $7,080.
Because 10 customers represent $59 in revenue, Kim needs to serve $7,080/$59 = 120 sets of customers = 1,200 customers per month to break even. At this volume,
Kim will serve 600 sandwiches (120 5), 840 bowls of soup (120 7), 480 salads (120 4), and 720 bottles of water or cans of soda (120 6).
b. The “free” drink offer creates inter-relations among the products. Perhaps the easiest way to deal with such complications is to modify the profit equation to include both paid and free drinks. We can then re-compute the contribution margin and selling price for a ‘bundled’ product. The following table provides the detailed computations:
Sandwich Soup Salad DrinkPaid DrinkFree
Price $4.00 $3.00 $3.00 $1.00 $0.00
Unit variable cost $1.25 $1.00 $0.75 $0.25 $0.25 Unit contribution margin $2.75 $2.00 $2.25 $0.75 -$0.25 # of orders 5 7 7 3 3 Total contribution margin $13.75 $14.00 $15.75 $2.25 -$.75 Total Price: $20.00 $21.0 0 $21.00 $3.00 $0.00
Thus, the price for the product mix is $65.00, and its contribution margin is $45.00.
In turn, Kim’s profit and breakeven sales are: Profit = ($45 Quantity) – $4,950.
Setting profit = $0, we find:
Breakeven number of units = $4,950/$45 = 110.
Because each bundle represents 10 customers, Kim needs to serve 10 110 = 1,100 customers per month to breakeven. At this volume, Kim will serve 550
sandwiches (110 5), 770 bowls of soup (110 7), 770 salads (110 7), and 660 bottles of water or cans of soda (110 6). Kim also will generate 110 $65 =
$7,150 in revenue (i.e., her breakeven point in sales dollars is the breakeven number of bundles multiplied by the revenue per bundle).
Thus, Kim needs to serve fewer customers) because the contribution margin for every ten customers has increased. Notice that Kim’s required revenue, however, has changed very little because her bundle selling price has increased by $6 compared to part [a] – this can be calculated as (3 $3 per salad) – (3 $1 per water/soda) = $6.
5.58 Multi-Product CVP Analysis, Weighted Contribution Margin Ratio Approach (LO5).
a. We start by setting a price for a new textbook – this will allow us to calculate the price of a used textbook and all of the associated variable costs. Let’s say Pricenew = $100.
If Pricenew = $100, then Unit variable costnew = (.75 $100) + (.05 $100) = $80.
In turn, the contribution margin ratio on a new book =
)
(
$100
$100
-
$80
= 20%.
For Priceused we have .75 $100 = $75. The unit variable costused is: (.25 $100)
+ (.05 $100)* = $30.
* Note that the variable selling costs (in $ per book, not %) are the same between new and used textbooks, or $5.
Accordingly, the contribution margin ratio on a used book =
)
(
$75
$75
-
$30
= 60%.
If 40% of University Bookstore’s revenues come from used books and 60% come from new books, we can write University Bookstore’s profit model as:
Profit = (.40 × Total revenues × .60) + (.60 × Total revenues × .20) – $360,000. That is, the weighted contribution margin ratio = (.40 × .60) + (.60 × .20) = .36. Setting profit equal to $0, we solve for the breakeven revenue as:
$0 = (.36 × Breakeven revenue) – $360,000. Thus, Breakeven revenue = $1,000,000.
