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CHAPTER 24

DISCUSSION QUESTIONS

24-1

Q24-1. Before making a decision under conditions ofuncertainty, a manager should try to assess theprobabilities associated with alternative possi-ble outcomes in order to determine the proba-ble result of each alternative action. Unless theprobabilities associated with possible outcomesare determined, the effect of uncertainty cannotbe accounted for adequately, which may resultin inconsistent and unreliable decisions.

Q24-2. Expected value is the weighted average valueof the events for a probability distribution, i.e.,it is the average value of the events that areexpected to occur.

Q24-3. The standard deviation of the expected value isa measure of the variability of events within aprobability distribution and, as such, is viewedas a measure of risk. The larger the standarddeviation, the greater the risk that the actualresult will differ from the expected value.

Q24-4. The coefficient of variation relates the stan-dard deviation for a probability distribution toits expected value, thus allowing for differ-ences in the relative size of different probabil-ity distributions. The coefficient of variation

provides a comparative measure of risk foralternatives with different expected values.Q24-5. A joint probability is the probability of the simul-

taneous occurrence of two or more events(e.g., the probability of the occurrence of bothevent A and event B, denoted as P(AB)),whereas a conditional probability is the proba-bility of the occurrence of one event given thatanother event has occurred (e.g., the probabili-ty of the occurrence of event A given that eventB has already occurred, denoted as P(AIB)).Aconditional probability implies that some rela-tionship exists between the events.

Q24-6. Management should be interested in revisingprobabilities as new information becomesavailable, because new information may alterthe expected outcomes (i.e., probabilities)enough to warrant making a different deci-sion. As a consequence, the revision of prob-abilities may be necessary in order to providea basis for making the best decision.

Q24-7. Decision trees graphically portray alternativesand their expected values and include asequential decision dimension in the analysisThey highlight decision points, alternativesestimated results, related probabilities, andexpected values. They are especially useful inevaluating alternatives requiring sequentiadecisions that depend upon uncertain out-comes.

Q24-8. In a discrete probability distribution, the possible outcomes are limited to certain finite values (e.g., 10, 11, 12, etc.). The number oshipments, orders, units of product, etc. areevents that could be described adequately bya discrete probability distribution. For convenience, the outcomes that occur in a discreteprobability distribution are often limited to afairly small number, but this need not necessarily be the case. In contrast, the possibleoutcomes that may occur in a continuousprobability distribution are infinite even withina limited range. Time, weight, volume, lengthtemperature, and economic value are exam-ples of continuous variables because they

can take on an infinite number of valueswithin a limited range (e.g., between 10 and11 seconds times of 10.1 seconds, 10.53 seconds, 10.926 seconds, etc. could occur)Although such items are measured in discreteunits, conceptually they can be subdividedinto infinitely small units of measure (e.g., $2$2.34, $2.627, $2.8935, etc.), and practicallythe number of different discrete values anitem may have without subdivision is large(e.g., the range of sales of $1 items between10,000 and 20,000 units).

Q24-9. The normal distribution has the following

attractive properties:(a) The normal distribution is symmetric and

it has only one mode. This means that theexpected value (which is the mean of thedistribution) is equal to the most likely single event (the mode). Consequently, thesingle best guess is also the expectedvalue.

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24-2 Chapter 24

24-2

(b) The relationship between the portion ofthe area under the curve for any giveninterval from the mean, as measured instandard deviations, is constant for all nor-mal distributions. This makes it possible to

determine the probability of the occur-rence of an event within any interval if themean and standard deviation are known.

Q24-10. Monte Carlo simulation is used to obtain aprobabilistic approximation of the outcomeof a business system or problem that con-tains numerous stochastic variables, but canbe modeled mathematically. Its procedureutilizes statistical sampling techniques andis computer oriented.

Q24-11. A normal distribution is a symmetrical distri-bution. The expected value (the mean) andthe most likely event (the mode) are equal.Since the most likely event would be usedeven when the distribution of probable out-comes is not considered specifically, andsince the most likely event and the expectedvalue are the same for a normal distribution,the expected net present value would be thesame whether probability analysis is incorpo-rated or not. Nevertheless, probability analy-sis should be incorporated into the capitalexpenditure evaluation because it provides away for management to evaluate risk.

Q24-12. A mutiperiod problem expands the analysis

from a single variable to multiple variables(i.e., the cash flows from each period aretreated as different random variables). As aconsequence, the expected net present valueof a capital expenditure proposal is treated asa random variable drawn from a multivariateprobability distribution. The variance for amultivariate distribution is computed by sum-ming the variances for each variable if thevariables are independent, or by summingthe standard deviations and squaring the totalif the variables are perfectly correlated(squaring the total incorporates the interac-

tion between the dependent variables). Toconsider the time value of money in amutiperiod capital expenditure proposal, theperiodic variances and the periodic standarddeviations should be discounted at the com-panys weighted average cost of capital.

Q24-13. Cash flows are independent if the magni-tude of cash flows in one period is not in anyway affected by the magnitude of cash flows

in another period. Independent cash flowsmight be expected to occur when a capitalexpenditure relates to the production of anestablished product or service; the demandfor which is expected to vary in response to

temporary changes in consumer tastes andpreferences or the capacity to purchase,which are uncorrelated between periods.

Q24-14. Cash flows are perfectly correlated if the mag-nitude of cash flows in a subsequent period isdependent upon the magnitude of cash flowsin a preceding period. Perfectly correlatedcash flows might be expected to occur if acapital expenditure relates to the production ofa new product or the entrance of a productinto a new market. In such a case, consumeracceptance of the product in one period mightbe expected to have a direct bearing an thelevel of sales in the following period.

Q24-15. If the periodic cash flows are neitherindependent nor perfectly correlated, thevariance of the net present value of a capitalexpenditure can be computed by (a) dividingthe period cash flows into independent anddependent components; (b) computing theperiodic variances for the independent cashflows and then discounting and summing toget the variance for the net present value ofthe independent cash flows; (c) computingthe periodic variances for the dependent

cash flows, taking the square root of eachvariance to get the periodic standard devia-tions, discounting and summing the periodicstandard deviations, and squaring the totalto get the variance for the net present valueof the dependent cash flows; and (d) addingthe variance for the net present value of theindependent cash flows to the variance ofthe net present value of the dependent cashflows.

Q24-16. MADM stands for multi-attribute decisionmodel, and it is an expenditure evaluationtool that explicitly incorporates both quanti-

tative and nonquantitative factors into thedecision analysis. Traditional economic eval-uation tools do not incorporate qualitativefactors into the decision model, yet most ofthe benefits to be derived from investmentsin new technologies are strategic and diffi-cult to quantify. MADM attempts to remedythis problem by giving weight to noneco-nomic variables.

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Chapter 24 24-3

EXERCISES

E24-1(1)

xi P(xi) E(x)Income or Income orMonthly (Loss) (Loss)

Sales Conditional ExpectedVolume Value Probability Value

3,000 $(35,000) .05 $(1,750)6,000 5,000 .15 7509,000 30,000 .40 12,000

12,000 50,000 .30 15,00015,000 70,000 .10 7,000

1.00 $33,000

(2) (1) (2) (3) (4) (5)xi (xi E(x)) (xi E(x))

2 P(xi) P(xi)(xi E(x))2

DifferenceIncome from

or (Loss) ExpectedConditional Value

Value ($33,000) (2) Squared Probability (3) (4)

$(35,000) $(68,000) $4,624,000,000 .05 $231,200,0005,000 (28,000) 784,000,000 .15 117,600,00

30,000 (3,000) 9,000,000 .40 3,600,00050,000 17,000 289,000,000 .30 86,700,00070,000 37,000 1,369,000,000 .10 136,900,000

Variance ........................................................................... $576,000,000

Coefficient

Standard deviation

Expected value= =

( )

( ( ))

$ ,

E x

24 0000

33 000727

$ ,.=of variation

Standard deviation $576,000,000( $ ,) = = 24 000

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E24-2(1) (1) (2) (3) (4) (5)

xi P(xi) E(x)Monthly Unit Conditional Frequency Expected

Sales Contribution Value Based ValueVolume Margin (1) (2) Probability (3) (4)

10,000 $10 $100,000 9/60 = .15 $ 15,00011,000 10 110,000 15/60 = .25 27,50012,000 10 120,000 18/60 = .30 36,00013,000 10 130,000 9/60 = .15 19,50014,000 10 140,000 6/60 = .10 14,00015,000 10 150,000 3/60 = .05 7,500

60/60 = 1.00 $119,500

(2) (1) (2) (3) (4) (5)xi (xi E(x)) (xi E(x))

2 P(xi) P(xi)(xi E(x))2

Deviation fromConditional $119,500

Value Expected Value (2) Squared Probability (3) (4)

$100,000 $(19,500) $380,250,000 .15 $ 57,037,500110,000 (9,500) 90,250,000 .25 22,562,500120,000 500 250,000 .30 75,000130,000 10,500 110,250,000 .15 16,537,500140,000 20,500 420,250,000 .10 42,025,000

150,000 30,500 930,250,000 .05 46,512,500Variance (2).............................................................................. $184,750,000

Coefficient

Standard deviation

Expected value= =

( )

( ( ))

$ ,

E x

13 5992

119 500114

$ ,.=

of variation

Standard deviation ) Vari $184,750,000( ( ) $ , = ance 2 13 592= =

24-4 Chapter 24

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Chapter 24 24-5

E24-3Cost to purchase thermocouplers:

Units needed annually (18,000 (1 .10)) ........................................ 20,000Unit cost................................................................................................ $15

Total estimated cost if thermocouplers purchased.......................... $300,000

Weighted average unit cost (expected value) to manufacture thermocouplers:

Estimated Weighted Averageper Unit Variable Unit Cost

Cost Probability (Expected Value)

$10 .1 $ 1.0012 .3 3.6014 .4 5.6016 .2 3.20

$13.40

Estimated variable manufacturing cost (18,000 units $13.40) .... $241,200Additional fixed manufacturing cost ................................................ 32,500

Total estimated cost if thermocouplers manufactured ................... $273,700

Manufacturing yields an estimated savings of $26,300 ($300,000 $273,700), subject tothe accuracy of estimated data. If data are accurate, manufacturing appears desirableassuming that the savings represents an acceptable rate of return on additionalinvested capital, there is no better alternative use of limited available facilities andequipment, and quality and production schedule demands can be met.

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24-6 Chapter 24

E24-4Table of expected values of possible strategies (000s omitted):

ExpectedPurchases/Sales 100 120 140 180 Value

100 $25 $25 $25 $25 $25.0120 15 40 40 40 37.5140 5 301 55 55 42.52

180 (15) 10 35 85 32.5Probability .1 .3 .4 .2

1Contribution margin for ordering 140,000 units and selling 120,000 units:Sales (120,000 $1.25)........................................................................ $150,000Cost of units ($50,000 + (140,000 $.50)) ......................................... 120,000

$ 30,000

2

Expected value for purchasing 140,000 units:$ 5 .1.................................................................................................. $ .530 .3.................................................................................................. 9.055 .4 ................................................................................................. 22.055 .2.................................................................................................. 11.0

$42.5

Jessica Company should purchase 140,000 units for December, according to the expectedvalue decision model, because this number of units produces the largest expectedvalue, $42,500.

E24-5(1) Payoff table of expected values of possible strategies

Sales ExpectedContribution

Order 10,000 20,000 30,000 40,000 Margin

10,000 $2,000 $2,000 $2,000 $2,000 $2,00020,000 (1,000) 4,000 4,000 4,000 3,50030,000 (4,000) 1,0001 6,000 6,000 4,0002

40,000 (7,000) (2,000) 3,000 8,000 2,500

Probability 5 50 = .1 10 50 = .2 20 50 = .4 15 50 = .3

1Contribution margin for ordering 30,000 hot dogs and selling 20,000 hot dogs:Sales (20,000 $.50)............................................................................ $10,000Cost of hot dogs (30,000 $.30) ........................................................ 9,000

Contribution margin............................................................................. $ 1,000

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Chapter 24 24-7

E24-5 (Concluded)

2Expected contribution margin for ordering 30,000 hot dogs:$(4,000) .1..................................................................................... $ (400)

$ 1,000 .2 .................................................................................... 200$ 6,000 .4 .................................................................................... 2,400$ 6,000 .3 .................................................................................... 1,800

Expected value..................................................................................... $4,000

(2) The expected value of perfect information is the difference between the averagecontribution margin using the best strategy (ordering 30,000 hot dogs) and theprobabilities and average contribution margin if Wurst knew in advance what thesales level would be each Saturday.

Average contribution margin if Wurst knew sales level:$2,000 .1........................................................ $ 200$4,000 .2........................................................ 800$6,000 .4........................................................ 2,400$8,000 .3........................................................ 2,400 $5,800

Average contribution margin using expected valuedecision rule to determine best strategy (from 1) 4,000

Contribution margin improved by............................... $1,800

Since the contribution margin would be improved by $1,800, Wurst could affordto pay up to $1,800 for perfect information.

E24-6(1) (2) (3) (4) (5)

PriorProbability Conditional Posterior

Prior Conditional Probability ProbabilityDemand Probability Probability (2) (3) (4) (4) Total

30,000 .10 .20 .02 .1040,000 .10 .50 .05 .2550,000 .50 .20 .10 .5060,000 .30 .10 .03 .15

1.00 1.00 .20 1.00

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24-8 Chapter 15

Since the expected value of not moving exceeds that of moving, the managershould not move the stereo store to the shopping mall ($42,000 > 40,000).

CGA-Canada (adapted). Reprint with permission.

Market demand increases (.3)

Market demand increases (.3)

Market demand remains same (.5)

Market demand remains same (.5)

Market demand declines (.2)

Market demand declines (.2)

Moving cost

$40,000

$42,000

Move toMall

Do notmove

Payoffs

$100,000

50,000

25,000

80,000

40,000

10,000

ExpectedValue

$ 30,000

25,000

5,000

$ 50,00010,000

$ 40,000

$ 24,000

20,000

2,000

$ 42,000

E24-7

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Chapter 24 24-9

The firm should make the sub-assembly rather than buy it because the expectedvalue of making the sub-assembly is $26,000, which is greater than the expectedvalue of buying ($24,500).

CGA-Canada (adapted). Reprint with permission

High demand (.4)

High demand (.4)

Medium demand (.3)

Medium demand (.3)

Low demand (.3)

Low demand (.3)

$26,000

$24,500

Make

Buy

Payoffs

$ 50,000

35,000

30,000 9,000

1,500

$ 24,500

5,000

ExpectedValue

$ 14,000

30,000

10,000

$ 20,000

9,000

3,000

$ 26,000

E24-8

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24-10 Chapter 24

1$300,000 expected profit $10,000 cost of applying for rezoning.2$100,000 expected profit $10,000 cost of applying for rezoning.

The land developer should bid on parcel B, and, if successful, apply for rezoningbecause the expected value of this alternative is greater than any other.CGA-Canada (adapted). Reprint with permission.

Successful (.6)

Successful (.5)

Unsuccessful (.4)

Unsuccessful (.5)

Unsuccessful (.2)

Apply

for R

ezonin

g

$120,000

$190,000

Bid

onParcelA

ExpectedPayoff

ExpectedValue

0

90,0002

$ 0

$ 120,000

$ 145,000

$ 100,000

$ 200,000

$ 290,0001

$ 100,000

0

$ 120,000

45,000

$ 190,000

$ 0

DoNotApp

lyforR

ezoning

Successful (.8)$190,000

Bid

onParcelB

$152,000

E24-9

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Chapter 24 24-11

E24-10

(1)

(2)

The probability of making a profit is equal to the area under the normal curveabove the breakeven point, which is approximately 93% (.43 for the area betweenthe breakeven point and the mean, which is the area for an interval of 1.5 stan-dard deviations from the mean found in Exhibit 24-8 of the textbook, plus .50 for

the area above the mean).

E24-11(1) (2) (3) (4)

Expected Value Present Valueof After-Tax Present of Expected

Net Cash Value After-Tax(Outflow) of $1 Net Cash Flow

Year Inflow @10% (2) (3)

0 $(20,000) 1.000 $(20,000)1 5,000 .909 4,545

2 5,000 .826 4,1303 5,000 .751 3,7554 5,000 .683 3,4155 5,000 .621 3,1056 5,000 .564 2,820

Expected net present value............................. $ 1,770

OR

(1) (2) (3) (4)Expected Value Present Present Value

of After-Tax Value of of ExpectedNet Cash Annuity After-Tax(Outflow) of $1 Net Cash Flow

Year Inflow @10% (2) (3)

0 $(20,000) 1.000 $(20,000)1 6 5,000 4.355 21,775

Expected net present value............................. $ 1,775*

*The difference in the results is due to rounding in the present value tables.

$ , cos

$,

,

193 750

538 750

50 000

fixed t

CM per unitunits to breakeven

u

=

nnits at mean units to breakeven

units in s dard devi

38 750

7 496

,

, tan aation=1 5.

=

55,000 units 45,000 units

(.667 2)

=

10 000

1 334

,

.

units

= 7 496, units

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24-12 Chapter 24

E24-12(1) (2) (3) (4) (5) (6)

PresentValue of Present

Periodic Present Periodic $1 at 12% Value ofStandard Value of Variance Squared Variance

Year Deviation $1@12% (2) (2) (3) (3) (4) (5)

0 0 1.000 0 1.000000 0.001 $500 .893 $250,000 .797449 $199,362.252 500 .797 250,000 .635209 158,802.253 500 .712 250,000 .506944 126,736.004 500 .636 250,000 .404496 101,124.005 500 .567 250,000 .321489 80,372.256 500 .507 250,000 .257049 64,262.257 500 .452 250,000 .204304 51,076.00

8 500 .404 250,000 .163216 40,804.00Variance of net present value ..................................................... $822,539.00

E24-13(1) (2) (3) (4)

Present ValuePeriodic Present of StandardStandard Value of Deviation

Year Deviation $1 @ 10% (2) (3)0 0 1.000 01 $1,000 .909 $9092 1,000 .826 8263 1,000 .751 7514 1,000 .683 6835 1,000 .621 621

Standard deviation of NPV .................... $3,790

OR

(1) (2) (3) (4)Present Present ValuePeriodic Value of of StandardStandard Annuity of Deviation

Year Deviation $1@ 10% (2) (3)

0 0 1.000 01-5 $1,000 3.791 $3,791

Standard deviation of NPV .................... $3,791*

*The difference in the results is due to rounding in the present value tables.

Standard deviation of net present value = =$ , $ .822 539 906 94

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Chapter 24 24-13

E24-14(1) (2) (3) (4) (5) (6)

Periodic PresentStandard Value of Present

Deviation of Present Periodic $1 at 12% Value ofIndependent Value of Variance Squared Variance

Year Cash Flow $1@12% (2) (2) (3) (3) (4) (5)

0 0 1.000 0 1.000000 0.001 $1,000 .893 $1,000,000 .797449 $797,4492 1,000 .797 1,000,000 .635209 635,2093 1,000 .712 1,000,000 .506944 506,9444 1,000 .636 1,000,000 .404496 404,4965 1,000 .567 1,000,000 .321489 321,4896 1,000 .507 1,000,000 .257049 257,0497 1,000 .452 1,000,000 .204304 204,304

Variance of NPV for independent cash flows.......................... $3,126,940

(1) (2) (3) (4)Present Value

Periodic Present of StandardStandard Value of Deviation

Year Deviation $1 @ 10% (2) (3)

0 0 1.000 0.001 $1,500 .893 $1,339.50

2 1,500 .797 1,195.503 1,500 .712 1,068.004 1,500 .636 954.005 1,500 .567 850.506 1,500 .507 760.507 1,500 .452 678.00

Standard Deviation of NPV fordependent cash flows...................... $6,846.00

Variance of NPV for dependent cash flows = ($6,846)2 = $46,867,716

Variance of NPV for independent cash flows ............ $ 3,126,940Variance of NPV for dependent cash flows................ 46,867,716

Variance of total NPV of investment ........................... $49,994,656

Standard deviation of total NPV of investment = =$ , , $ ,49 994 656 7 070..69

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24-14 Chapter 24

E24-15

(1) The 95% confidence interval for the net present value is a range between a lowof $20,000 ($30,000 expected NPV (2 $25,000 standard deviation)) and a high

of $80,000 ($30,000 expected NPV + (2 $25,000 standard deviation)).

(2) There is a .88493 probability that the NPV of the investment will be positive, i.e.,the .5 area above the mean plus the .38493 area below the mean (determinedfrom the table of Z values in Exhibit 24-8 of the text for ( x) = ($30,000expected NPV 0) $25,000 = 1.20).

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Chapter 24 24-15

PROBLEMS

P24-1(1) Deterministic approach:

Sales (60,000 units most likely sales volume $100) ................................................................. $6,000,000Variable costs:

Direct materials (60,000 units $25) ................. $1,500,000Direct labor (60,000 units $8.80 per

hour most likely rate 2 hours)................ 1,056,000Variable overhead (60,000 units

($.40 supplies + $.35 materialshandling + $1.25 heat, light, and power) 2 hours) .................................................... 240,000

Promotion fee (60,000 units $6) ...................... 360,000 3,156,000

Contribution margin ..................................................... $2,844,000Additional fixed costs:

Supervisor salary ................................................ $ 28,000Equipment lease rentals ..................................... 150,000 178,000

Annual pretax advantage of introducing newproduct ................................................................. $2,666,000

(2) Expected value approach:Sales in Units Probability Expected Value

50,000 .25 12,500

60,000 .45 27,00070,000 .20 14,00080,000 .10 8,000

61,500

Labor Hour Rate Probability Expected Value

$8.50 .30 $2.558.80 .50 4.409.00 .20 1.80

$8.75

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24-16 Chapter 24

P24-1 (Concluded)

Sales (61,500 expected value $100)......................... $6,150,000Variable costs:

Direct materials (61,500 units $25) ................. $1,537,500Direct labor (61,500 units $8.75expected value 2 hours) ........................ 1,076,250

Variable overhead (61,500 $2 per laborhour 2 hours) ........................................... 246,000

Promotion fee (61,500 units $6) ...................... 369,000 3,228,750

Contribution margin ..................................................... $2,921,250Additional fixed costs:

Supervisor salary ................................................ $ 28,000Equipment lease rentals ..................................... 150,000 178,000

Expected annual pretax advantage ............................ $2,743,250

(3) In this situation, Monte Carlo simulation could be used. A linear equation for thenet advantage would have to be developed that included the two variable items(sales volume and hourly direct labor costs) treated as independent stochasticvariables. The probability distributions for sales volume and hourly direct laborcost would be simulated and pairs of values would be selected for entry into theequation, using a random number generator.The net pretax advantage would becalculated and recorded, and then a new set of values for the stochastic vari-ables would be determined and reentered into the equation. A large number ofiterations would be calculated and recorded to determine the approximatedistribution of the net pretax advantage. The distribution would have a calculat-ed mean (which would be interpreted as the expected annual net pretax advan-tage) and a standard deviation (which could be interpreted as a measure of theproducts risk).

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Chapter 24 24-17

P24-2Video Recreation Inc. should adopt Plan 3 because it results in the least cost of thethree alternatives as demonstrated below:

ExpectedNumber of Number

Service Calls Probability = of Calls

400 .1 40700 .3 210900 .4 360

1,200 .2 240

1.0 850

Expected

Parts Cost Value ofPer Repair Frequency = Parts Cost

$30 .15 $ 4.5040 .15 6.0060 .45 27.0090 .25 22.50

1.00 $60.00

Plan 1Vendor fees (6 vendors $15,000 fee per vendor)........................... $ 90,000Service calls (850 calls $250 per call) ............................................ 212,500

Parts ($60 expected value per call 850 calls (1 + 10% markup)) 56,100Estimated total cost of Plan 1 ............................................................ $358,600

Plan 2Urban service calls (850 calls 60% urban $450 per call) .......... $229,500Rural service calls (850 calls 40% rural $350 per call) ............. 119,000Parts ($60 expected value per call 850 calls)................................. 51,000

Estimated total cost of Plan 2 ............................................................ $399,500

Plan 3Employee salaries (9 employees $24,000 average salary) ........... $216,000Employee fringe benefits ($216,000 employee wages 35%) ........ 75,600Preventive maintenance parts (200 calls per employee

9 employees $15 in parts per call) ........................................ 27,000Repair parts ((850 calls (1 30%))

($60 expected value per call (1 20%)))................................ 28,560

Estimated total cost of Plan 3 ............................................................ $347,160

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24-18 Chapter 24

P24-3Expected Value of Bearings Expected Value of BearingsRejected During Assembly Rejected During Performance Testing

Expected Expected

Quantity Probability Value Quantity Probability Value100 .50 50.0 20 .40 8.060 .25 15.0 15 .30 4.530 .15 4.5 10 .20 2.05 .10 0.5 5 .10 0.5

70.0 15.0

Hourly cost to = Direct labor + Variable overheadreplace bearing cost per hour cost per hour

= ($8) + (1.5)($8)

= $20

Cost of rejections Expected value of Replacement Cost toduring assembly = bearings rejected time per replace each

per lot during assembly unit bearing

= 70 6 / 60 hour $20 per hour

= $140

Cost of rejections Expected value ofduring performance

=bearings rejected

Replacement Cost to

testing per lot during performance time per replace each

testingunit bearing

= 15 1 hour $20 per hour

= $300

Maximum amount Cost of rejections Cost of rejectionsNumber

for quality = during assembly + during performance of lots

control program per lot testing per lot

= ($140 + $300) (1,000,000 units 1,000 units per lot)

= $440,000

(

(

(((

(( (

((

((

((

((

( ((

(((

((

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Chapter 24 24-19

P24-4(1) The payoff table of expected contribution margins for Kenton Clothiers shirt

order sizes follows:

Possible Actions Contribution Margin (Conditional Value) Contribution Margin(Quantities to for Possible Sales Quantities (Expected Value of

be Ordered) 100 200 300 400 Each Strategy)

100 $ 700* $ 700 $ 700 $ 700 $ 700200 100** 1,600 1,600 1,600 1,420300 (300) 1,200 2,700 2,700 1,620400 (500) 1,000 2,500 4,000 1,480***

Probability 3/25 = .12 12/25 = .48 9/25 = .36 1/25 = .04

* 100 shirts at the regular $30 sales price $7 CM per shirt = $700 CM** (100 shirts at the regular $30 sales price $8 CM per shirt) (100 shirts at the

$15 reduced price $7 loss per shirt) = $100 CM*** (.12 probability $(500)) + (.48 probability $1,000) + (.36 probability $2,500) +

(.04 probability $4,000) = $1,480 CM

(2) The best strategy for Kenton Clothiers would be to order 300 shirts each yearbecause it would result in the largest contribution margin over time. The coeffi-cient of variation for the best strategy (i.e., purchasing 300 shirts each year) is.615 determined as follows:

(1) (2) (3) (4) (5)x

i

(xi E(x)) (x

i E(x))2 P(x

i) P(x

i)(x

i E(x))2

Deviation fromConditional $1,620 Columns

Value Expected Value Col. (2)2 Probability (3) (4)

$ (300) $1,920 $3,686,400 .12 $442,3681,200 (420) 176,400 .48 84,6722,700 1,080 1,166,400 .36 419,9042,700 1,080 1,166,400 .04 46,656

Variance (2) ................................................................................. $993,600

Standard deviation Variance

Coeffic

( ) ( ) $ , $ . = = =

2

993 600 996 795

iient

Standard Deviation

Expected Value= =

( )

( ( ) )

$ .

$ ,

E

996 795

1 6620615= .

of variation

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24-20 Chapter 24

P24-4 (Concluded)

(3) The expected value of perfect information is $364 determined as follows:Average contribution if Kenton Clothiers knew sales in advance and ordered just

enough to meet sales demand:

CM Per ExpectedQuantity Unit Sold Total CM Probability Value

100 $ 7 $ 700 .12 $ 84200 8 1,600 .48 768300 9 2,700 .36 972400 10 4,000 .04 160

Expected value with perfect information .............................. $ 1,984Less expected value of best strategy under uncertainty

(ordering 300 shirts from requirement (1) above).......... 1,620

Expected value of perfect information................................... $ 364

P24-5

(1) (1) (2) (3) (4)Prior Conditional Posterior

Events Probability Probability (1) (2) Probability

1,600 .20 .25 .050 .12502,000 .50 .25 .125 .31252,400 .20 .75 .150 .3750

2,800 .10 .75 .075 .18751.00 .400 1.0000

(2)Actions Events (House Size Most in Demand)

Size To ExpectedBuild 1,600 2,000 2,400 2,800 Value

1,600 $200,000 $180,000 $160,000 $140,000 $167,5002,000 160,000 400,000 360,000 320,000 340,0002,400 120,000 320,000 600,000 540,000 441,2502,800 80,000 240,000 480,000 800,000 415,000

Posteriorprobability .1250 .3125 .3750 .1875

Gant should be advised to build the 2,400 square-foot houses on the tract ofland because that course of action has the highest expected value.

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Chapter 24 24-21

P24-6

(1) Expected value of outside printers offer:Expected

Enrollments Probability Value25,000 .05 1,25026,000 .15 3,90027,000 .40 10,80028,000 .25 7,00029,000 .15 4,350

1.00 27,300

Fees to be paid to the outside printer:Fixed fee.......................................................... $325,000Variable fee ((27,300 25,000) $15) ........... 34,500 $359,500

Savings available from closing Printing Department:Lease income from renting equipment............ $ 33,000Avoidable fixed costs:

Salaries and benefits ($160,000 110%)................................ $176,000Less cost of part-timeclerk ($16,000 110% 3/5 week) ............................. (10,560)

Less employee severance pay(($160,000 $16,000) 12 months). ......................... (12,000) 153,440

Telephone($4,000 ($80 12 months)).................. 3,040

Occupancy and administration($10,800 + $7,300) ................................... 18,100

Avoidable variable costs:Materials, supplies, and postage

((($165,100 26,000) $1) 27,300) 146,055 353,635

Increase in total costs from acceptance ofprinters offer...................................................... $ 5,865

In this case, the outside printers offer should not be accepted because the totacosts would increase by $5,865.

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24-22 Chapter 24

P24-6 (Concluded)

(2) Revised expected value of outside printers offer:(1) (2) (3) (4) (5)

RevisedPrior Conditional Posterior ExpectedEnrollments Probability Probability (1) (2) Probability Value

25,000 .05 .90 .045 .045 .260 = .173 4,32526,000 .15 .90 .135 .135 .260 = .519 13,49427,000 .40 .10 .040 .040 .260 = .154 4,15828,000 .25 .10 .025 .025 .260 = .096 2,68829,000 .15 .10 .015 .015 .260 = .058 1,682

1.00 .260 1.000 26,347

Fees to be paid to the outside printer:Fixed fee .............................................................................. $325,000Variable fee ((26,347 25,000) $15)................................ 20,205 $345,205

Savings available from closing Printing Department:Lease income from renting equipment ............................ $ 33,000Avoidable fixed costs:

Salaries and benefits ($160,000 110%)............................................... $176,000Less cost of part-time clerk

($16,000 110% 3/5 week) ..................................... (10,560)

Less employee severance pay(($160,000 $16,000) 12 months) . .............................. (12,000) 153,440

Telephone and telegraph($4,000 ($80 12 months)).................................. 3,040

Occupancy and administration($10,800 + $7,300) .................................................. 18,100

Avoidable variable costs:Materials, supplies, and postage

((($165,100 26,000) $1) 26,347) ..................... 140,956 348,536

Decrease in total costs from acceptance ofprinters offer....................................................................... $ (3,331)

Considering the new information, the outside printers offer should be acceptedbecause the total costs would decrease by $3,331.

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P24-7The tests should be administered because the expected value is $115 per appli-cant greater than the case where no test is administered ($1,015 $900).

Chapter 24 24-23

PayoffsExpectedValue

$ 1,750

$ 440

$ 1,200

150

640

300

$ 1,600

$ 200

$ 900

$ 2,5001

$ 2,2002

$ 2,4003

500

800

600

200

200

0

Satisfactory (.7)

Satisfactory (.2)

Satisfactory (.5)

Unsatisfactory (.3)

Unsatisfactory (.8)

Unsatisfactory (.5)

Abbreviated Training (.9)

Full training (.1)

Full training

Not hired (.1)

Not hired (.9)

Not hired

$1,600

$1,4204

UnacceptableScore

AcceptableScore

$ 900

$ 200

Test

NoTest

(.75)

(.75)

$ 200

$ 900

$1,0155

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24-24 Chapter 24

P24-7 (Concluded)

1Successful hire salary savings ................ $3,000Less costs:

Testing.................................................... $200Abbreviated training ............................ 300 500

Payoff ......................................................... $2,500

2Successful hire salary savings ................. $3,000Less costs:

Testing ................................................... $200Full training............................................ 600 800

Payoff ......................................................... $2,200

3

Successful hire salary savings ................. $3,000Less full training cost................................. 600

Payoff ......................................................... $2,400

4Expected value of abbreviated training ................................ $1,600 .9 = $1,440Expected value of not hiring ................................................... 200 .1 = 20

Expected value when test score acceptable ................................................... $1,420

5Expected value of acceptable test score ............................... $1,420 .75 = $1,065Expected value of unacceptable test score........................... 200 .25= 50

Expected value of administering test .............................................................. $1,015

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Chapter 24 24-25

P24-8Sales Material State of SalesPrice Lot Size Economy Demand Expected Payoff

$5.25 200,000 Weak 180,000 ($5.25 180,000) ($3 200,000) = $345,0005.25 200,000 Strong 200,000 ($5.25 200,000) ($3 200,000) = $450,000

5.25 240,000 Weak 180,000 ($5.25 180,000) ($2.90 240,000) = $249,0005.25 240,000 Strong 200,000 ($5.25 200,000) ($2.90 240,000) = $354,0005.00 200,000 Weak 200,000 ($5 200,000) ($3 200,000) = $400,0005.00 200,000 Strong 240,000 ($5 200,000) ($3 200,000) = $400,0005.00 240,000 Weak 200,000 ($5 200,000) ($2.90 240,000) = $304,0005.00 240,000 Strong 240,000 ($5 240,000) ($2.90 240,000) = $504,000

Slick Inc. should set the sales price at $5.00 per unit and order 200,000 units of mate-rial, because this course of action will result in the greatest expected value ($400,000contribution margin).

PayoffsExpectedValue

$ 207,000

$ 149,400

$ 240,000

$ 182,400

180,000

141,600

160,000

201,600

$ 387,000

$ 291,000

$ 400,000

$ 384,000

$345,000

249,000

400,000

304,000

450,000

354,000

400,000

504,000

Weak economy (.6)

Weak economy (.6)

Weak economy (.6)

Weak economy (.6)

Strong economy (.4)

Strong economy (.4)

Strong economy (.4)

Strong economy (.4)

$387,000

$291,000

$400,000

$384,000

Order 200,000

Order 200,000

Order 240,000

Order 240,000

$400,000

$387,000

Select$5.25SalesPrice

Select$5.00SalesPrice

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24-26 Chapter 24

P24-9(1)

ExpectedPayoff

ExpectedValue

$ 500,000

$ 2,800,000

$ 700,000

$ 2,000,000

500,000

2,000,000

1,000,000

$ 2,300,000

$1,300,000

$ 1,000,000

$ 500,000

$ 3,500,0002

3,500,000

$ 4,000,0004

$ 2,500,000

3

2,500,000

2,000,0005

$ 500,000

Do Not Introduce

Successful (.8)

Successful (.2)

Successful (.5)

Unsuccessful (.2)

Unsuccessful (.8)

Unsuccessful (.5)

Do Not Introduce

Intro

duce

New

Product

Intro

duce

New

Product

Test

Not

Succe

ssful(.5

)

$2,300,000

$2,300,000

$ 1,300,000

$ 1,000,000

$ 500,000

Test

Suc

cess

ful(

.5)

Test

Cam

paig

n

Strat

egy1

$1,000,000

$900,0001

NationwidePro

motionNoTestCam

paign

Strategy2

1$2,300,000 .5 = $1,150,000 500,000 .5 = 250,000

$ 900,000

2Successful with test = ($40 $30 $6 $.5) million = $3.5 million3Unsuccessful with test = ($16 $12 $6 $.5) million = $ 2.5 million4Successful without test = ($40 $30 $6) million = $4 million5Unsuccessful without test = ($16 $12 $6) million = $ 2 million

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P24-9 (Concluded)

(2) If the probability estimates can be relied upon, management should conduct thenationwide promotion and distribution without first performing a test campaign

because the expected value of Strategy 2 is $100,000 greater than the expectedvalue of Strategy 1.

(3) Criticism of the expected value decision criterion would include:(a) Selection of the probabilities associated with the possible outcomes for the

alternative strategies is a subjective process. If the probability estimates arebiased, the expected values will be biased.

(b) The values for the alternative courses of action are estimates that could beinaccurate.

(c) The decision model does not incorporate psychological factors. Forinstance, people are often risk averse, and personal evaluations will not nec-

essarily coincide with monetary evaluations.(d) A model is often overly simplified to make it manageable and may conse-quently leave out important considerations or assumptions.

Chapter 24 24-27

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P24-10(1) Expected value of periodic cash flows:

(1) (2) (3) (4) (5)Expected Expected

Expected Value of Value ofValue of Annual cash Annual AnnualAnnual Contribution Inflow Fixed Pretax NetSales Margin From Sales Cash Cash Inflow

in Units Per Unit (1) (2) Outflow (3) (4)

4,000 $14 $56,000 $8,100 $47,900

(1) (2) (3) (4)Tax Annual

Tax Basis Depreciation TaxYear (Cost) Rate Depreciation

1 $200,000 .143 $ 28,6002 200,000 .245 49,0003 200,000 .175 35,0004 200,000 .125 25,0005 200,000 .089 17,8006 200,000 .089 17,8007 200,000 .089 17,8008 200,000 .045 9,0009 200,000 .000 010 200,000 .000 0

$200,000

(1) (2) (3) (4) (5) (6)Expected Expected Expected

Expected Value of Value Value ofValue of Taxable of Tax After-Tax Net

Pretax Net Tax Income Liability Cash FlowYear Cash Flow Depreciation (2) (3) (4) 40% (2) (5)

0 $(200,000) 0 0 0 $(200,000)1 47,900 $28,600 $19,300 $ 7,720 40,1802 47,900 49,000 (1,100) (440) 48,340

3 47,900 35,000 12,900 5,160 42,7404 47,900 25,000 22,900 9,160 38,7405 47,900 17,800 30,100 12,040 35,8606 47,900 17,800 30,100 12,040 35,8607 47,900 17,800 30,100 12,040 35,8608 47,900 9,000 38,900 15,560 32,3409 47,900 0 47,900 19,160 28,740

10 47,900 0 47,900 19,160 28,740

$ 167,400

24-28 Chapter 24

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P24-10 (Continued)

Expected value of the periodic standard deviation:

(1) (2) (3) (4) (5)Pretax After-TaxCash Flow Cash Flow

Standard Value of Value ofDeviation Pretax Standard After-Tax Standardin Units Cash Flow Deviation Portion Deviationof Sales per Unit (1) (2) (1 40%) (3) (4)

1,750 $14 $24,500 60% $14,700

(2) Expected net present value of investment:

(1) (2) (3) (4)Expected PresentValue of Present Value of

After-Tax Net Value of After-Tax NetYear Cash Flow $1 at 12% Cash Flow

0 $(200,000) 1.000 $ (200,000)1 40,180 .893 35,8812 48,340 .797 38,5273 42,740 .712 30,4314 38,740 .636 24,6395 35,860 .567 20,333

6 35,860 .507 18,1817 35,860 .452 16,2098 32,340 .404 13,0659 28,740 .361 10,375

10 28,740 .322 9,254

Expected net present value ................................... $ 16,895

Chapter 24 24-29

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P24-10 (Concluded)

(3) Variance and standard deviation of expected net present value:(1) (2) (3) (4) (5) (6)

PresentValue of PresentPeriodic Periodic Present $1 at 12% Value ofStandard Variance Value of Squared Variance

Year Deviation Col. (2)2 $1 at 12% Col. (4)2 (3) (5)

0 0 0 1.000 1.000000 01 $14,700 $216,090,000 .893 .797449 $172,320,7542 14,700 216,090,000 .797 .635209 137,262,3133 14,700 216,090,000 .712 .506944 109,545,5294 14,700 216,090,000 .636 .404496 87,407,5415 14,700 216,090,000 .567 .321489 69,470,558

6 14,700 216,090,000 .507 .257049 55,545,7187 14,700 216,090,000 .452 .204304 44,148,0518 14,700 216,090,000 .404 .163216 35,269,3459 14,700 216,090,000 .361 .130321 28,161,065

10 14,700 216,090,000 .322 .103684 22,405,076

Variance of net present value............................................. $761,535,950

(4)

(5) The probability that the net present value will exceed zero is approximately 73%,i.e., the 50% area under the curve that is above the mean plus the approximately23% area under the curve that is below the mean but above zero (determinedfrom the table of Z values in Exhibit 24-8 of the text for ( X) = ($16,895 0) $27,596 = .61, which is about 23% of the total area under the normal curve).

Standard deviation

Variance of netpresent value= = $ ,761 535,, $ ,950 27 596=

=

of net present value

CoefficientStandard deviiation

Expected net present valueof vari = =$ ,

$ , .27 596

16 895 1 633aation

24-30 Chapter 24

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P24-11(1) Expected value of periodic cash flows:

(1) (2) (3) (4) (5)Contribution Expected Expected

Expected Margin per Value of Value ofValue of Unit (Cash Annual Cash Annual AnnualAnnual Inflow Net Inflow Fixed Pretax NetSales of Outflow From Sales Cash Cash Inflow

in Units per Unit) (1) (2) Outflow (3) (4)

5,000 $18 $90,000 $10,000 $80,000

(1) (2) (3) (4)Tax

Tax Basis Depreciation Tax

Year (Cost) Rate Depreciation1 $180,000 .143 $ 25,7402 180,000 .245 44,1003 180,000 .175 31,5004 180,000 .125 22,5005 180,000 .089 16,0206 180,000 .089 16,0207 180,000 .089 16,0208 180,000 .045 8,100

$180,000

(1) (2) (3) (4) (5) (6)Expected Expected Expected

Expected Value of Value of Value ofValue of Taxable Tax After-Tax Net

Pretax Net Tax Income Liability Cash FlowYear Cash Flow Depreciation (2) (3) (4) 40% (2) (5)

0 $(180,000) 0 0 0 $ (180,000)1 80,000 $25,740 $54,260 $21,704 58,2962 80,000 44,100 35,900 14,360 65,640

3 80,000 31,500 48,500 19,400 60,6004 80,000 22,500 57,500 23,000 57,0005 80,000 16,020 63,980 25,592 54,4086 80,000 16,020 63,980 25,592 54,4087 80,000 16,020 63,980 25,592 54,4088 80,000 8,100 71,900 28,760 51,240

$ 276,000

Chapter 24 24-31

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P24-11 (Continued)

Expected value of the periodic standard deviation:

(1) (2) (3) (4) (5)Pretax After-TaxCash Flow Cash Flow

Standard Value of Value ofDeviation Pretax Standard After-Tax Standardin Units Cash Flow Deviation Portion Deviationof Sales per Unit (1) (2) (1 40%) (3) (4)

2,000 $18 $36,000 .6 $21,600

(2) Expected net present value of investment:

(1) (2) (3) (4)Expected PresentValue of Present Value of

After-Tax Net Value of After-Tax NetYear Cash Flow $1 at 12% Cash Flow

0 $(180,000) 1.000 $ (180,000)1 58,296 .893 52,0582 65,640 .797 52,3153 60,600 .712 43,1474 57,000 .636 36,2525 54,408 .567 30,849

6 54,408 .507 27,5857 54,408 .452 24,5928 51,240 .404 20,701

Expected net present value................................. $ 107,499

24-32 Chapter 24

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P24-11 (Concluded)

(3) Standard deviation of expected net present value:

(1) (2) (3) (4)PresentValue of

Periodic Present StandardStandard Value of Deviation

Year Deviation $1 at 12% (2) (3)

0 0 1.000 01 $21,600 .893 $ 19,2892 21,600 .797 17,2153 21,600 .712 15,3794 21,600 .636 13,738

5 21,600 .567 12,2476 21,600 .507 10,9517 21,600 .452 9,7638 21,600 .404 8,726

Standard deviation of netpresent value ..................................................... $107,308

(4)

(5) The probability that the net present value will exceed zero is approximately 84%i.e., the 50% area under the curve that is above the mean plus the 34% area underthe curve that is below the mean but above zero (determined from the table of Zvalues in Exhibit 24-8 of the text for ( X) = ($107,499 0) $107,308 = 1.0which is about 34% of the total area under the normal curve.)

CoefficientStandard deviation

Expected net present value= =

$107,,

$ ,.

308

107 499998=

of variation

Chapter 24 24-33

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P24-12

(1) Expected net present value of mixed cash flows:

(1) (2) (3) (4) (5) (6)PresentExpected Expected Total Value of

Independent Dependent Expected ExpectedAfter-Tax After-Tax After-Tax Net After-Tax NetNet Cash Net Cash Cash Inflow Present Cash Inflow

Inflow Inflow (Outflow) Value of (Outflow)Year 70% 30% (2) + (3) $1 at 10% (4) (5)

0 $(30,000) 1.000 $ (30,000)1 $5,600 $2,400 8,000 .909 7,2722 7,700 3,300 11,000 .826 9,086

3 7,000 3,000 10,000 .751 7,5104 6,300 2,700 9,000 .683 6,1475 4,900 2,100 7,000 .621 4,347

Expected net present value ............................................................... $ 4,362

(2) Variance and standard deviation of expected net present value:(1) (2) (3) (4) (5) (6)

Independent Independent PresentCash Flow Cash Flow Value of PresentPeriodic Periodic Present $1 at 10% Value ofStandard Variance Value of Squared Variance

Year Deviation Col. (2)2 $1 at 10% Col. (4)2 (3) (5)0 0 0 1.000 1.000000 01 $1,000 $1,000,000 .909 .826281 $ 826,2812 1,000 1,000,000 .826 .682276 682,2763 1,000 1,000,000 .751 .564001 564,0014 1,000 1,000,000 .683 .466489 466,4895 1,000 1,000,000 .621 .385641 385,641

Variance of expected NPV for independent cash flows................... $2,924,688

24-34 Chapter 24

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P24-12 (Concluded)

(1) (2) (3) (4)Dependent Present

Cash Flow Value ofPeriodic Present StandardStandard Value of Deviation

Year Deviation $1 at 10% (2) (3)

0 0 1.000 01 $500 .909 $ 4552 500 .826 4133 500 .751 3764 500 .683 3425 500 .621 311

Standard deviation of NPV .................................. $1,897

Variance of net Standard deviation of 2

present value for = net present value fordependent cash flows dependent cash flows

= ($1,897)2 = $3,598,609

Variance of NPV for dependent cash flows .................................. $3,598,609Variance of NPV for independent cash flows............................... 2,924,688

Variance of total NPV of investment.............................................. $6,523,297

(3)

(4) The probability that the net present value will exceed zero is approximately 96%i.e., the 50% area under the curve that is above the mean plus the approximately

46% area under the curve that is below the mean but above zero (determinedfrom the table of Z values in Exhibit 24-8 of the text for ( X) = ($4,362 0) $2,554 = 1.71, which is about 46% of the total area under the normal curve.)

CoefficientStandard deviation

Expected net present value= =

$ ,2 5554

4 3620 586

$ ,.=

of variation

Standard deviation of

Variance of totalnet present valuetotal =net present value

= =$ , , $ ,6 523 297 2 554

Chapter 24 24-35

((

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24-36 Chapter 24

3ontheresultsof

theMADMworksheetbelow,Glotynemanagementsh

ouldchoosetheCIMsyste

m

because

mpositeweighted

scoreishigherthanthealternative.Basedonthisanalysis,theCIMsystem

isexpectedto

adequatelysatisfy

managementsmodernizationgoals.

GLO

TYNECORPORATION

Capit

alExpenditureProposal

MADMWorksheet

Relative

M

odernizeWithExistingTechnolo

gy

ModernizeWithNewT

echnology

Importance

Performance

Likelihood

Weighted

Performance

Likelihood

Weighted

Weighting

Rating

Estimate

Score

Rating

Estimate

Score

sentvalue....................................

30

2

.8

48.0

0

.5

0

setuptime...................................

20

0

.5

0

2

.9

36.0

throughputtime.........................

15

1

.5

7.5

2

.9

27.0

eproductquality..........................

15

1

.9

13.5

2

.5

15.0

inventorylevels.........................

10

0

.9

0

1

.6

6.0

eimagetooutsiders....................

10

1

.5

5.0

1

.6

6.0

......................................................

100

74.0

90.0