CHAPTER 13
COST-VOLUME-PROFIT RELATIONSHIPS
I. Questions1. The total “contribution margin” is the excess of total revenue over total variable costs. The unit contribution margin is the excess of the unit price over the unit variable costs.
2. Total contribution margin:
Selling price - manufacturing variable costs expensed - nonmanufacturing variable costs expensed = Total contribution margin.
Gross margin:
Selling price - variable manufacturing costs expensed - fixed manufacturing costs expensed = Gross margin.
3. A company operating at “break-even” is probably not covering costs which are not recorded in the accounting records. An example of such a cost is the opportunity cost of owner-invested capital. In some small businesses, owner-managers may not take a salary as large as the opportunity cost of forgone alternative employment. Hence, the opportunity cost of owner labor may be excluded.
4. In the short-run, without considering asset replacement, net operating cash flows would be expected to exceed net income, because the latter includes depreciation expense, while the former does not. Thus, the cash basis break-even would be lower than the accrual break-even if asset replacement is ignored. However, if asset replacement costs are taken into account, (i.e., on a “cradle to grave” basis), the long-run net cash flows equal long-run accrual net income, and the long-run break-even points are the same.
5. Both unit price and unit variable costs are expressed on a per product basis, as:
= (P1 - V1) X1 + (P2 - V2) X2 + + (Pn - Vn) Xn - F, for all products 1 to n where:
= operating profit,
P = average unit selling price, 13-1
V = average unit variable cost, X = quantity of units,
F = total fixed costs for the period.
6. If the relative proportions of products (i.e., the product “mix”) is not held constant, products may be substituted for each other. Thus, there may be almost an infinite number of ways to achieve a target operating profit. As shown from the multiple product profit equation, there are several unknowns for one equation:
= (P1 - V1) X1 + (P2 - V2) X2 + + (Pn - Vn) Xn - F, for all products 1 to n.
7. A constant product mix is assumed to simplify the analysis. Otherwise, there may be no unique solution.
8. Operating leverage measures the impact on net operating income of a given percentage change in sales. The degree of operating leverage at a given level of sales is computed by dividing the contribution margin at that level of sales by the net operating income.
9. Three approaches to break-even analysis are (a) the equation method, (b) the contribution margin method, and (c) the graphical method. In the equation method, the equation is: Sales = Variable expenses + Fixed expenses + Profits, where profits are zero at the break-even point. The equation is solved to determine the break-even point in units or peso sales. 10. The margin of safety is the excess of budgeted (or actual) sales over the
break-even volume of sales. It states the amount by which sales can drop before losses begin to be incurred.
11. The sales mix is the relative proportions in which a company’s products are sold. The usual assumption in cost-volume-profit analysis is that the sales mix will not change.
12. A higher break-even point and a lower net operating income could result if the sales mix shifted from high contribution margin products to low contribution margin products. Such a shift would cause the average contribution margin ratio in the company to decline, resulting in less total contribution margin for a given amount of sales. Thus, net operating income would decline. With a lower contribution margin ratio, the break-even point would be higher since it would require more sales to cover the same amount of fixed costs.
Exercise 1 (Using a Contribution Format Income Statement) Requirement 1
Total Per Unit
Sales (30,000 units × 1.15 = 34,500 units)...P172,500 P5.00 Less variable expenses... 103,500 3.00 Contribution margin...69,000 P2.00 Less fixed expenses... 50,000 Net operating income...P 19,000
Requirement 2
Sales (30,000 units × 1.20 = 36,000 units)...P162,000 P4.50 Less variable expenses... 108,000 3.00 Contribution margin...54,000 P1.50 Less fixed expenses... 50,000 Net operating income...P 4,000
Requirement 3
Sales (30,000 units × 0.95 = 28,500 units)...P156,750 P5.50 Less variable expenses... 85,500 3.00 Contribution margin...71,250 P2.50 Less fixed expenses (P50,000 + P10,000)... 60,000 Net operating income...P 11,250
Requirement 4
Sales (30,000 units × 0.90 = 27,000 units)...P151,200 P5.60 Less variable expenses... 86,400 3.20 Contribution margin...64,800 P2.40 Less fixed expenses... 50,000 Net operating income...P 14,800
Exercise 2 (Break-even Analysis and CVP Graphing) Requirement 1
The contribution margin per person would be:
Price per ticket...P30 Less variable expenses:
Dinner...P7 Favors and program... 3 10 Contribution margin per person...P20 The fixed expenses of the Extravaganza total P8,000; therefore, the break-even point would be computed as follows:
Sales = Variable expenses + Fixed expense + Profits P30Q = P10Q + P8,000 + P0
P20Q = P8,000
Q = P8,000 ÷ P20 per person
Q = 400 persons; or, at P30 per person, P12,000 Alternative solution:
or, at P30 per person, P12,000.
Requirement 2
Variable cost per person (P7 + P3)...P10 Fixed cost per person (P8,000 ÷ 250 persons)... 32 Ticket price per person to break even...P42
Requirement 3
Cost-volume-profit graph: Break-even point
in unit sales = Unit contribution marginFixed expenses = P20 per personP8,000
P0 P2,000 P4,000 P6,000 P8,000 P10,000 P12,000 P14,000 P16,000 P18,000 P20,000 P22,000 0 100 200 300 400 500 600 Number of Persons P es os
Exercise 3 (Break-even and Target Profit Analysis) Requirement 1
Sales = Variable expenses + Fixed expenses + Profits P900Q = P630Q + P1,350,000 + P0
P270Q = P1,350,000
Q = P1,350,000 ÷ P270 per lantern
Q = 5,000 lanterns, or at P900 per lantern, P4,500,000 in sales Alternative solution:
13-5
Fixed Expenses Total Expenses Total Sales Break-even point: 400 persons,
or P12,000 in sales
Break-even point
in unit sales = Unit contribution marginFixed expenses = P270 per lanternP1,350,000
or at P900 per lantern, P4,500,000 in sales
Requirement 2
An increase in the variable expenses as a percentage of the selling price would result in a higher break-even point. The reason is that if variable expenses increase as a percentage of sales, then the contribution margin will decrease as a percentage of sales. A lower CM ratio would mean that more lanterns would have to be sold to generate enough contribution margin to cover the fixed costs. Requirement 3 Present: 8,000 Lanterns Proposed: 10,000 Lanterns*
Total Per Unit Total Per Unit
Sales P7,200,000 P900 P8,100,000 P810 **
Less variable expenses 5,040,000 630 6,300,000 630 Contribution margin 2,160,000 P270 1,800,000 P180 Less fixed expenses 1,350,000 1,350,000 Net operating income P 810,000 P 450,000 * 8,000 lanterns × 1.25 = 10,000 lanterns
** P900 per lantern × 0.9 = P810 per lantern
As shown above, a 25% increase in volume is not enough to offset a 10% reduction in the selling price; thus, net operating income decreases.
Requirement 4
Sales = Variable expenses + Fixed expenses + Profits P810Q = P630Q + P1,350,000 + P720,000
P180Q = P2,070,000
Q = P2,070,000 ÷ P180 per lantern Q = 11,500 lanterns
Alternative solution:
Exercise 4 (Operating Leverage) Requirement 1
Sales (30,000 doors)...P18,000,000 P600 Less variable expenses... 12,600,000 420 Contribution margin...5,400,000 P180 Less fixed expenses... 4,500,000 Net operating income... 900,000P
Requirement 2
a. Sales of 37,500 doors represents an increase of 7,500 doors, or 25%, over present sales of 30,000 doors. Since the degree of operating leverage is 6, net operating income should increase by 6 times as much, or by 150% (6 × 25%).
b. Expected total peso net operating income for the next year is:
Present net operating income...P 900,000 Expected increase in net operating income next year
(150% × P900,000)... 1,350,000 Total expected net operating income...P2,250,000
Exercise 5 (Multiproduct Break-even Analysis) Requirement 1
13-7 Unit sales to attain
target profit =
Fixed expenses + Target profit Unit contribution margin = P1,350,000 + P720,000P180 per lantern = 11,500 lanterns
Degree of
operating leverage = Net operating incomeContribution margin = P5,400,000P900,000
Model E700 Model J1500 Total Company
Amount % Amount % Amount %
Sales P700,000 100 P300,000 100 P1,000,000 100
Less variable expenses
280,000 40 90,000 30 370,000 37 Contribution margin P420,000 60 P210,000 70 630,000 63 *
Less fixed expenses 598,500
Net operating income P 31,500 * 630,000 ÷ P1,000,000 = 63%.
Requirement 2
The break-even point for the company as a whole would be:
Requirement 3
The additional contribution margin from the additional sales can be computed as follows:
P50,000 × 63% CM ratio = P31,500
Assuming no change in fixed expenses, all of this additional contribution margin should drop to the bottom line as increased net operating income. This answer assumes no change in selling prices, variable costs per unit, fixed expenses, or sales mix.
Exercise 6 (Break-even Analysis; Target Profit; Margin of Safety) Requirement 1
Sales = Variable expenses + Fixed expenses + Profits P40Q = P28Q + P150,000 + P0
P12Q = P150,000
Q = P150,000 ÷ P12 per unit
Q = 12,500 units, or at P40 per unit, P500,000 Break-even point
in total peso sales = Overall CM ratioFixed expenses = P598,5000.63 = P950,000 in sales
Alternatively:
or, at P40 per unit, P500,000.
Requirement 2
The contribution margin at the break-even point is P150,000 since at that point it must equal the fixed expenses.
Requirement 3
Total Unit
Sales (14,000 units × P40 per unit)...P560,000 P40 Less variable expenses
(14,000 units × P28 per unit)... 392,000 28 Contribution margin
(14,000 units × P12 per unit)...168,000 P12 Less fixed expenses... 150,000 Net operating income...P 18,000
Requirement 4
Margin of safety in peso terms:
Margin of safety in pesos = Total sales – Break-even sales 13-9
Break-even point in unit sales =
Fixed expenses Unit contribution margin = P12 per unitP150,000
= 12,500 units
Unit sales to attain target profit =
Fixed expenses + Target profit Unit contribution margin = P150,000 + P18,000P12 per unit = 14,000 units
= P600,000 – P500,000 = P100,000 Margin of safety in percentage terms:
Requirement 5
The CM ratio is 30%.
Expected total contribution margin: P680,000 × 30%...P204,000 Present total contribution margin: P600,000 × 30%... 180,000 Increased contribution margin...P 24,000 Alternative solution:
P80,000 incremental sales × 30% CM ratio = P24,000
Since in this case the company’s fixed expenses will not change, monthly net operating income will increase by the amount of the increased contribution margin, P24,000. III. Problems Problem 1 (CVP Relationships) Requirement 1 Requirement 2 Margin of safety
percentage = Margin of safety in pesosTotal sales = P100,000P600,000
= 16.7% (rounded)
CM ratio = Contribution marginSelling price = P15P60 = 25% Variable expense ratio = Variable expenseSelling price = P45P60 = 75%
Sales = Variable expenses + Fixed expenses + Profits P60Q = P45Q + P240,000 + P0
P15Q = P240,000
Q = P240,000 ÷ P15 per unit
Q = 16,000 units, or at P60 per unit, P960,000 Alternative solution:
X = 0.75X + P240,000 + P0 0.25X = P240,000
X = P240,000 ÷ 0.25
X = P960,000; or at P60 per unit, 16,000 units
Requirement 3
Increase in sales... P400,000 Multiply by the CM ratio... x 25% Expected increase in contribution margin... P100,000
Since the fixed expenses are not expected to change, net operating income will increase by the entire P100,000 increase in contribution margin computed above.
Requirement 4
Sales = Variable expenses + Fixed expenses + Profits P60Q = P45Q + P240,000 + P90,000
P15Q = P330,000
Q = P330,000 ÷ P15 per unit Q = 22,000 units
Contribution margin method:
Requirement 5
Margin of safety in pesos = Total sales – Break-even sales
= P1,200,000 – P960,000 = P240,000
13-11 Fixed expenses + Target profit
Contribution margin per unit =
P240,000 + P90,000
Requirement 6
a.
b. Expected increase in sales... 8% Degree of operating leverage... x 5 Expected increase in net operating income... 40% c. If sales increase by 8%, then 21,600 units (20,000 x 1.08 = 21,600) will
be sold next year. The new income statement will be as follows:
Total Per Unit Percent ofSales
Sales (21,600 units)... P1,296,000 P60 100% Less variable expenses... 972,000 45 75% Contribution margin... 324,000 P15 25% Less fixed expenses... 240,000
Net operating income... P 84,000
Thus, the P84,000 expected net operating income for next year represents a 40% increase over the P60,000 net operating income earned during the current year:
Note from the income statement above that the increase in sales from 20,000 to 21,600 units has resulted in increases in both total sales and total variable expenses. It is a common error to overlook the increase in variable expense when preparing a projected income statement.
Requirement 7
a. A 20% increase in sales would result in 24,000 units being sold next year: 20,000 units x 1.20 = 24,000 units.
Total Per Unit Percent ofSales
Sales (24,000 units)... P1,440,000 P60 100% Less variable expenses... 1,152,000 48* 80% Contribution margin... 288,000 P12 20% Less fixed expenses... 210,000†
Net operating income... P 78,000 Margin of safety
percentage = Margin of safety in pesosTotal sales =
P240,000
P1,200,000 = 20%
Degree of operating leverage = Contribution margin Net operating income =
P300,000 P60,000 = 5 P84,000 – P60,000 P60,000 = P24,000 P60,000 = 40% increase
* P45 + P3 = P48; P48 P60 = 80%.
† P240,000 – P30,000 = P210,000.
Note that the change in per unit variable expenses results in a change in both the per unit contribution margin and the CM ratio.
b.
c. Yes, based on these data the changes should be made. The changes will increase the company’s net operating income from the present P60,000 to P78,000 per year. Although the changes will also result in a higher break-even point (17,500 units as compared to the present 16,000 units), the company’s margin of safety will actually be wider than before:
Margin of safety in pesos = Total sales – Break-even sales
= P1,440,000 – P1,050,000 = P390,000
As shown in requirement (5) above, the company’s present margin of safety is only P240,000. Thus, several benefits will result from the proposed changes.
Problem 2 (Basics of CVP Analysis; Cost Structure) Requirement 1
The CM ratio is 30%.
Total Per Unit Percentage
Sales (13,500 units) P270,000 P20 100 %
Less variable expenses 189,000 14 70 Contribution margin P 81,000 P 6 30 %
13-13 Break-even point
in unit sales = Contribution margin per unitFixed expenses = P12 per unitP210,000
= 17,500 units Break-even point
in peso sales = Fixed expensesCM ratio = P210,0000.20 = P1,050,000
The break-even point is:
Sales = Variable expenses + Fixed expenses + Profits P20Q = P14Q + P90,000 + P0
P 6Q = P90,000
Q = P90,000 ÷ P6 per unit Q = 15,000 units
15,000 units × P20 per unit = P300,000 in sales
Alternative solution:
Requirement 2
Incremental contribution margin:
P70,000 increased sales × 30% CM ratio...P21,000 Less increased fixed costs:
Increased advertising cost... 8,000 Increase in monthly net operating income...P13,000 Since the company presently has a loss of P9,000 per month, if the changes are adopted, the loss will turn into a profit of P4,000 per month.
Break-even point
in unit sales = Contribution margin per unitFixed expenses = P6 per unitP90,000
= 15,000 units Break-even point
in sales pesos = Fixed expensesCM ratio = P90,0000.30
Requirement 3
Sales (27,000 units × P18 per unit*)...P486,000 Less variable expenses
(27,000 units × P14 per unit)... 378,000 Contribution margin...108,000 Less fixed expenses (P90,000 + P35,000)... 125,000 Net operating loss...P(17,000) *P20 – (P20 × 0.10) = P18
Requirement 4
Sales = Variable expenses + Fixed expenses + Profits P 20Q = P14.60Q* + P90,000 + P4,500 P5.40Q = P94,500 Q = P94,500 ÷ P5.40 per unit Q = 17,500 units * P14.00 + P0.60 = P14.60. Alternative solution: ** P6.00 – P0.60 = P5.40. Requirement 5
a. The new CM ratio would be:
Per Unit Percentage
Sales P20 100 %
Less variable expenses 7 35
Contribution margin P13 65 %
The new break-even point would be: 13-15 Unit sales to attain
target profit = Fixed expenses + Target profitCM per unit = P90,000 + P4,500P5.40 per unit**
b. Comparative income statements follow:
Not Automated Automated
Total Per Unit % Total Per Unit %
Sales (20,000 units) P400,000 P20 100 P400,000 P20 100 Less variable expenses 280,000 14 70 140,000 7 35 Contribution margin 120,000 6P 30 260,000 P13 65 Less fixed expenses 90,000 208,000
Net operating income P 30,000 P 52,000
c. Whether or not one would recommend that the company automate its operations depends on how much risk he or she is willing to take, and depends heavily on prospects for future sales. The proposed changes would increase the company’s fixed costs and its break-even point. However, the changes would also increase the company’s CM ratio (from 30% to 65%). The higher CM ratio means that once the break-even point is reached, profits will increase more rapidly than at present. If 20,000 units are sold next month, for example, the higher CM ratio will generate P22,000 more in profits than if no changes are made.
The greatest risk of automating is that future sales may drop back down to present levels (only 13,500 units per month), and as a result, losses will be even larger than at present due to the company’s greater fixed costs. (Note the problem states that sales are erratic from month to month.) In sum, the proposed changes will help the company if sales continue to trend upward in future months; the changes will hurt the company if sales
Break-even point
in unit sales = Contribution margin per unitFixed expenses = P13 per unitP208,000
= 16,000 units
Break-even point
in sales pesos = Fixed expensesCM ratio = P208,0000.65 = P320,000 in sales
drop back down to or near present levels.
Note to the Instructor: Although it is not asked for in the problem, if time
permits you may want to compute the point of indifference between the two alternatives in terms of units sold; i.e., the point where profits will be the same under either alternative. At this point, total revenue will be the same; hence, we include only costs in our equation:
Let Q = Point of indifference in units sold P14Q + P90,000 = P7Q + P208,000
P7Q = P118,000
Q = P118,000 ÷ P7 per unit Q = 16,857 units (rounded)
If more than 16,857 units are sold, the proposed plan will yield the greatest profit; if less than 16,857 units are sold, the present plan will yield the greatest profit (or the least loss).
Problem 3 (Sales Mix; Multiproduct Break-even Analysis) Requirement 1
Products
Sinks Mirrors Vanities Total
Percentage of total sales 32% 40% 28% 100%
Sales P160,000 100 % P200,000 100 % P140,000 100 % P500,000 100%
Less variable expenses 48,000 30 160,000 80 77,000 55 285,000 57 Contribution margin P112,000 70 % P 40,000 20 % P 63,000 45 % 215,000 43%*
Less fixed expenses 223,600
Net operating income (loss)
P ( 8,600) * P215,000 ÷ P500,000 = 43%. Requirement 2 Break-even sales: 13-17 Break-even point
in total peso sales = Fixed expensesCM ratio = P223,6000.43 = P520,000 in sales
Requirement 3
Memo to the president:
Although the company met its sales budget of P500,000 for the month, the mix of products sold changed substantially from that budgeted. This is the reason the budgeted net operating income was not met, and the reason the break-even sales were greater than budgeted. The company’s sales mix was planned at 48% Sinks, 20% Mirrors, and 32% Vanities. The actual sales mix was 32% Sinks, 40% Mirrors, and 28% Vanities.
As shown by these data, sales shifted away from Sinks, which provides our greatest contribution per peso of sales, and shifted strongly toward Mirrors, which provides our least contribution per peso of sales. Consequently, although the company met its budgeted level of sales, these sales provided considerably less contribution margin than we had planned, with a resulting decrease in net operating income. Notice from the attached statements that the company’s overall CM ratio was only 43%, as compared to a planned CM ratio of 52%. This also explains why the break-even point was higher than planned. With less average contribution margin per peso of sales, a greater level of sales had to be achieved to provide sufficient contribution margin to cover fixed costs.
Problem 4 (Basic CVP Analysis) Requirement 1
The CM ratio is 60%:
Selling price P150 100%
Less variable expenses 60 40
Contribution margin 90P 60%
Requirement 2
Break-even point in total sales pesos =
Fixed expenses CM ratio = P1,800,0000.60
Requirement 3
P450,000 increased sales × 60% CM ratio = P270,000 increased contribution margin. Since fixed costs will not change, net operating income should also increase by P270,000.
Requirement 4
a.
b. 6 × 15% = 90% increase in net operating income.
Requirement 5
Last Year:
28,000 units 42,000 units*Proposed:
Total Per Unit Total Per Unit
Sales P4,200,000 P150.00 P5,670,000 P135.00**
Less variable expenses
1,680,000 60.00 2,520,000 60.00 Contribution margin 2,520,000 P 90.00 3,150,000 P 75.00 Less fixed expenses 1,800,000 2,500,000
Net operating income P 720,000 P 650,000 * 28,000 units × 1.5 = 42,000 units
** P150 per unit × 0.90 = P135.00 per unit No, the changes should not be made.
Requirement 6
Expected total contribution margin:
28,000 units × 200% × P70 per unit*...P3,920,000 Present total contribution margin:
28,000 units × P90 per unit... 2,520,000 Incremental contribution margin, and the amount by which
advertising can be increased with net operating income
remaining unchanged...P1,400,000 * P150 – (P60 + P20) = P70
Problem 5 (Break-Even and Target Profit Analysis)
13-19
Degree of operating leverage = Contribution margin
Requirement 1
The contribution margin per patch would be:
Selling price...P30 Less variable expenses:
Purchase cost of the patches...P15 Commissions to the student salespersons... 6 21 Contribution margin...P 9 Since there are no fixed costs, the number of unit sales needed to yield the desired P7,200 in profits can be obtained by dividing the target profit by the unit contribution margin:
Requirement 2
Since an order has been placed, there is now a “fixed” cost associated with the purchase price of the patches (i.e., the patches can’t be returned). For example, an order of 200 patches requires a “fixed” cost (investment) of P3,000 (200 patches × P15 per patch = P3,000). The variable costs drop to only P6 per patch, and the new contribution margin per patch becomes:
Selling price...P30 Less variable expenses (commissions only)... 6 Contribution margin...P24 Since the “fixed” cost of P3,000 must be recovered before Ms. Morales shows any profit, the break-even computation would be:
Target profit
Unit contribution margin =
P7,200
P9 per patch = 800 patches =
800 patches x P30 per patch P24,000 in total sales
= Unit contribution marginFixed expenses = P24 per patch = 125 patchesP3,000 Break-even point
125 patches x P30 per patch = P3,750 in total sales
If a quantity other than 200 patches were ordered, the answer would change accordingly.
Problem 6
Requirement 1: Break-even chart
Requirement 2: Profit-volume graph
13-21 600,000 500,000 400,000 300,000 200,000 100,000 (P) 5,000 10,000 15,000 20,000 25,000 30,000 (units) Break-even point TR TC FC
IV. Multiple Choice Questions 1. B 6. B 11. B 16. D 21. A 26. A 2. B 7. D 12. A 17. D 22. D 27. B 3. B 8. B 13. A 18. D 23. C 28. C 4. C 9. A 14. C 19. C 24. B 29. B 5. C 10. D 15. D 20. D 25. C 30. A 5,000 10,000 15,000 20,000 25,000 30,000 50,000 100,000 150,000 200,000 250,000 200,000 150,000 100,000 50,000 0 P R O F I T L O S S Break-even point
