CHAPTER 24 DISCUSSION QUESTIONS Q24-1 Before making a decision under conditions of uncertainty, a manager should try to assess the probabilities associated with alternative possible outcomes in order to determine the probable result of each alternative action Unless the probabilities associated with possible outcomes are determined, the effect of uncertainty cannot be accounted for adequately, which may result in inconsistent and unreliable decisions Q24-2 Expected value is the weighted average value of the events for a probability distribution, i.e., it is the average value of the events that are expected to occur Q24-3 The standard deviation of the expected value is a measure of the variability of events within a probability distribution and, as such, is viewed as a measure of risk The larger the standard deviation, the greater the risk that the actual result will differ from the expected value Q24-4 The coefficient of variation relates the standard deviation for a probability distribution to its expected value, thus allowing for differences in the relative size of different probability distributions The coefficient of variation provides a comparative measure of risk for alternatives with different expected values Q24-5 A joint probability is the probability of the simultaneous occurrence of two or more events (e.g., the probability of the occurrence of both event A and event B, denoted as P(AB)), whereas a conditional probability is the probability of the occurrence of one event given that another event has occurred (e.g., the probability of the occurrence of event A given that event B has already occurred, denoted as P(AIB)) A conditional probability implies that some relationship exists between the events Q24-6 Management should be interested in revising probabilities as new information becomes available, because new information may alter the expected outcomes (i.e., probabilities) enough to warrant making a different decision As a consequence, the revision of probabilities may be necessary in order to provide a basis for making the best decision Q24-7 Decision trees graphically portray alternatives and their expected values and include a sequential decision dimension in the analysis They highlight decision points, alternatives, estimated results, related probabilities, and expected values They are especially useful in evaluating alternatives requiring sequential decisions that depend upon uncertain outcomes Q24-8 In a discrete probability distribution, the possible outcomes are limited to certain finite values (e.g., 10, 11, 12, etc.) The number of shipments, orders, units of product, etc are events that could be described adequately by a discrete probability distribution For convenience, the outcomes that occur in a discrete probability distribution are often limited to a fairly small number, but this need not necessarily be the case In contrast, the possible outcomes that may occur in a continuous probability distribution are infinite even within a limited range Time, weight, volume, length, temperature, and economic value are examples of continuous variables because they can take on an infinite number of values within a limited range (e.g., between 10 and 11 seconds times of 10.1 seconds, 10.53 seconds, 10.926 seconds, etc could occur) Although such items are measured in discrete units, conceptually they can be subdivided into infinitely small units of measure (e.g., $2, $2.34, $2.627, $2.8935, etc.), and practically, the number of different discrete values an item may have without subdivision is large (e.g., the range of sales of $1 items between 10,000 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 the expected value (which is the mean of the distribution) is equal to the most likely single event (the mode) Consequently, the single best guess is also the expected value 24-1 24-2 Chapter 24 (b) The relationship between the portion of the area under the curve for any given interval from the mean, as measured in standard deviations, is constant for all normal distributions This makes it possible to determine the probability of the occurrence of an event within any interval if the mean and standard deviation are known Q24-10 Monte Carlo simulation is used to obtain a probabilistic approximation of the outcome of a business system or problem that contains numerous stochastic variables, but can be modeled mathematically Its procedure utilizes statistical sampling techniques and is computer oriented Q24-11 A normal distribution is a symmetrical distribution The expected value (the mean) and the most likely event (the mode) are equal Since the most likely event would be used even when the distribution of probable outcomes is not considered specifically, and since the most likely event and the expected value are the same for a normal distribution, the expected net present value would be the same whether probability analysis is incorporated or not Nevertheless, probability analysis should be incorporated into the capital expenditure evaluation because it provides a way 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 are treated as different random variables) As a consequence, the expected net present value of a capital expenditure proposal is treated as a random variable drawn from a multivariate probability distribution The variance for a multivariate distribution is computed by summing the variances for each variable if the variables are independent, or by summing the standard deviations and squaring the total if the variables are perfectly correlated (squaring the total incorporates the interaction between the dependent variables) To consider the time value of money in a mutiperiod capital expenditure proposal, the periodic variances and the periodic standard deviations should be discounted at the company’s weighted average cost of capital Q24-13 Cash flows are independent if the magnitude of cash flows in one period is not in any way affected by the magnitude of cash flows in another period Independent cash flows might be expected to occur when a capital expenditure relates to the production of an established product or service; the demand for which is expected to vary in response to temporary changes in consumer tastes and preferences or the capacity to purchase, which are uncorrelated between periods Q24-14 Cash flows are perfectly correlated if the magnitude of cash flows in a subsequent period is dependent upon the magnitude of cash flows in a preceding period Perfectly correlated cash flows might be expected to occur if a capital expenditure relates to the production of a new product or the entrance of a product into a new market In such a case, consumer acceptance of the product in one period might be expected to have a direct bearing an the level of sales in the following period Q24-15 If the periodic cash flows are neither independent nor perfectly correlated, the variance of the net present value of a capital expenditure can be computed by (a) dividing the period cash flows into independent and dependent components; (b) computing the periodic variances for the independent cash flows and then discounting and summing to get the variance for the net present value of the independent cash flows; (c) computing the periodic variances for the dependent cash flows, taking the square root of each variance to get the periodic standard deviations, discounting and summing the periodic standard deviations, and squaring the total to get the variance for the net present value of the dependent cash flows; and (d) adding the variance for the net present value of the independent cash flows to the variance of the net present value of the dependent cash flows Q24-16 MADM stands for multi-attribute decision model, and it is an expenditure evaluation tool that explicitly incorporates both quantitative and nonquantitative factors into the decision analysis Traditional economic evaluation tools not incorporate qualitative factors into the decision model, yet most of the benefits to be derived from investments in new technologies are strategic and difficult to quantify MADM attempts to remedy this problem by giving weight to noneconomic variables 24-2 Chapter 24 24-3 EXERCISES E24-1 (1) Monthly Sales Volume 3,000 6,000 9,000 12,000 15,000 (2) xi Income or (Loss) Conditional Value $(35,000) 5,000 30,000 50,000 70,000 P(xi) Probability 05 15 40 30 10 1.00 (1) xi E(x) Income or (Loss) Expected Value $(1,750) 750 12,000 15,000 7,000 $33,000 (2) (3) (4) (xi – E(x)) (xi – E(x)) P(xi) Difference Income from or (Loss) Expected Conditional Value Value ($33,000) (2) Squared Probability $(35,000) $(68,000) $4,624,000,000 05 5,000 (28,000) 784,000,000 15 30,000 (3,000) 9,000,000 40 50,000 17,000 289,000,000 30 70,000 37,000 1,369,000,000 10 Variance Standard deviation ( σ) = $576,000,000 = $24, 000 00 Coefficient = Standard deviation( σ ) = $24, 00 of variation Expected value (E( x )) $33, 000 = 727 (5) P(xi)(xi – E(x))2 (3) × (4) $231,200,000 117,600,00 3,600,000 86,700,000 136,900,000 $576,000,000 24-4 E24-2 (1) (2) Chapter 24 (1) (2) Monthly Sales Volume 10,000 11,000 12,000 13,000 14,000 15,000 Unit Contribution Margin $10 10 10 10 10 10 (3) xi Conditional Value (1) × (2) $100,000 110,000 120,000 130,000 140,000 150,000 (4) P(xi) Frequency Based Probability 9/60 = 15 15/60 = 25 18/60 = 30 9/60 = 15 6/60 = 10 3/60 = 05 60/60 = 1.00 (1) xi (5) E(x) Expected Value (3) × (4) $ 15,000 27,500 36,000 19,500 14,000 7,500 $119,500 (2) (3) (4) (5) (xi – E(x)) (xi – E(x))2 P(xi) P(xi)(xi – E(x))2 Deviation from Conditional $119,500 Value Expected Value (2) Squared Probability (3) × (4) $100,000 $(19,500) $380,250,000 15 $ 57,037,500 110,000 (9,500) 90,250,000 25 22,562,500 120,000 500 250,000 30 75,000 130,000 10,500 110,250,000 15 16,537,500 140,000 20,500 420,250,000 10 42,025,000 150,000 30,500 930,250,000 05 46,512,500 σ ) $184,750,000 Variance (σ Standard deviation ( σ ) = Variance ( σ ) = $184,750,000 = $13, 592 92 Standard deviation ( σ ) $13, 59 Coefficient = 114 = = of variation Expected value (E( x )) $119, 500 Chapter 24 24-5 E24-3 Cost to purchase thermocouplers: Units needed annually (18,000 ÷ (1 – 10)) Unit cost Total estimated cost if thermocouplers purchased 20,000 × $15 $300,000 Weighted average unit cost (expected value) to manufacture thermocouplers: Estimated per Unit Variable Cost $10 12 14 16 Probability Weighted Average Unit Cost (Expected Value) $ 1.00 3.60 5.60 3.20 $13.40 Estimated variable manufacturing cost (18,000 units × $13.40) Additional fixed manufacturing cost Total estimated cost if thermocouplers manufactured $241,200 32,500 $273,700 Manufacturing yields an estimated savings of $26,300 ($300,000 – $273,700), subject to the accuracy of estimated data If data are accurate, manufacturing appears desirable; assuming that the savings represents an acceptable rate of return on additional invested capital, there is no better alternative use of limited available facilities and equipment, and quality and production schedule demands can be met 24-6 Chapter 24 E24-4 Table of expected values of possible strategies (000s omitted): Purchases/Sales 100 120 140 180 Probability 100 $25 15 (15) 120 $25 40 301 10 140 $25 40 55 35 180 $25 40 55 85 Expected Value $25.0 37.5 42.52 32.5 1Contribution margin for ordering 140,000 units and selling 120,000 units: Sales (120,000 × $1.25) Cost of units ($50,000 + (140,000 × $.50)) $150,000 120,000 $ 30,000 2Expected value for purchasing 140,000 units: $ × 30 × 55 × 55 × $ 9.0 22.0 11.0 $42.5 Jessica Company should purchase 140,000 units for December, according to the expected value decision model, because this number of units produces the largest expected value, $42,500 E24-5 (1) Payoff table of expected values of possible strategies Sales Order 10,000 20,000 30,000 40,000 Probability 1Contribution 10,000 $2,000 (1,000) (4,000) (7,000) 20,000 $2,000 4,000 1,0001 (2,000) ÷ 50 = 10 ÷ 50 = 30,000 $2,000 4,000 6,000 3,000 40,000 $2,000 4,000 6,000 8,000 Expected Contribution Margin $2,000 3,500 4,0002 2,500 20 ÷ 50 = 15 ÷ 50 = margin for ordering 30,000 hot dogs and selling 20,000 hot dogs: Sales (20,000 × $.50) $10,000 Cost of hot dogs (30,000 × $.30) 9,000 Contribution margin $ 1,000 Chapter 24 24-7 E24-5 (Concluded) 2Expected contribution margin for ordering 30,000 hot dogs: $(4,000) × $ 1,000 × $ 6,000 × $ 6,000 × Expected value (2) $ (400) 200 2,400 1,800 $4,000 The expected value of perfect information is the difference between the average contribution margin using the best strategy (ordering 30,000 hot dogs) and the probabilities and average contribution margin if Wurst knew in advance what the sales level would be each Saturday Average contribution margin if Wurst knew sales level: $2,000 × $4,000 × $6,000 × $8,000 × Average contribution margin using expected value decision rule to determine best strategy (from 1) Contribution margin improved by $ 200 800 2,400 2,400 $5,800 4,000 $1,800 Since the contribution margin would be improved by $1,800, Wurst could afford to pay up to $1,800 for “perfect” information E24-6 (1) Demand 30,000 40,000 50,000 60,000 (2) Prior Probability 10 10 50 30 1.00 (3) Conditional Probability 20 50 20 10 1.00 (4) Prior Probability × Conditional Probability (2) × (3) 02 05 10 03 20 (5) Posterior Probability (4) ÷ (4) Total 10 25 50 15 1.00 Chapter 15 24-8 E24-7 Payoffs Expected Value $100,000 $ 30,000 Market demand remains same (.5) 50,000 25,000 Market demand declines (.2) –25,000 Market demand increases (.3) $40,000 Moving cost Move to Mall –5,000 $ 50,000 –10,000 $ 40,000 Do not move Market demand increases (.3) 80,000 $ 24,000 Market demand remains same (.5) 40,000 20,000 Market demand declines (.2) –10,000 $42,000 –2,000 $ 42,000 Since the expected value of not moving exceeds that of moving, the manager should not move the stereo store to the shopping mall ($42,000 > 40,000) CGA-Canada (adapted) Reprint with permission Chapter 24 24-9 E24-8 Payoffs Expected Value $ 50,000 $ 20,000 Medium demand (.3) 30,000 9,000 Low demand (.3) –10,000 –3,000 High demand (.4) $26,000 $ 26,000 Make Buy High demand (.4) 35,000 $ 14,000 Medium demand (.3) 30,000 9,000 Low demand (.3) 5,000 $24,500 1,500 $ 24,500 The firm should make the sub-assembly rather than buy it because the expected value of making the sub-assembly is $26,000, which is greater than the expected value of buying ($24,500) CGA-Canada (adapted) Reprint with permission 24-10 Chapter 24 E24-9 Expected Payoff Successful (.6) Expected Value $ 200,000 $ 120,000 –0– –0– $120,000 el A Unsuccessful (.4) Bi d on Pa rc $ 120,000 Successful (.5) $ 290,0001 $ 145,000 lB 90,000 45,000 pl y fo r Unsuccessful (.5) $190,000 Successful (.8) $ 190,000 Ap arce on P Re Bid zo n in g $190,000 Do No or ly f pp tA $152,000 ing zon Re Unsuccessful (.2) 1$300,000 2$100,000 $ 100,000 $ 100,000 $ $ –0– –0– expected profit – $10,000 cost of applying for rezoning expected profit – $10,000 cost of applying for rezoning The land developer should bid on parcel B, and, if successful, apply for rezoning because the expected value of this alternative is greater than any other CGA-Canada (adapted) Reprint with permission 24-22 Chapter 24 P24-6 (Concluded) (2) Revised expected value of outside printer’s offer: (1) (2) (3) Enrollments 25,000 26,000 27,000 28,000 29,000 Prior Probability 05 15 40 25 15 1.00 Conditional Probability 90 90 10 10 10 (1) × (2) 045 135 040 025 015 260 (4) (5) Revised Posterior Expected Probability Value 045 ÷ 260 = 173 4,325 135 ÷ 260 = 519 13,494 040 ÷ 260 = 154 4,158 025 ÷ 260 = 096 2,688 015 ÷ 260 = 058 1,682 1.000 26,347 Fees to be paid to the outside printer: Fixed fee Variable fee ((26,347 – 25,000) × $15) Savings available from closing Printing Department: Lease income from renting equipment Avoidable fixed costs: Salaries and benefits ($160,000 × 110%) $176,000 Less 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) Telephone and telegraph ($4,000 – ($80 × 12 months)) Occupancy and administration ($10,800 + $7,300) Avoidable variable costs: Materials, supplies, and postage ((($165,100 ữ 26,000) $1) ì 26,347) Decrease in total costs from acceptance of printer’s offer $325,000 20,205 $345,205 $ 33,000 153,440 3,040 18,100 140,956 348,536 $ (3,331) Considering the new information, the outside printer’s offer should be accepted because the total costs would decrease by $3,331 Chapter 24 24-23 P24-7 The tests should be administered because the expected value is $115 per applicant greater than the case where no test is administered ($1,015 – $900) Payoffs Satisfactory (.7) $ 2,500 Expected Value $ 1,750 $1,600 Abbreviated Training (.9) $1,420 Unsatisfactory (.3) Not hired (.1) Acceptable Score –500 –150 –200 $ 1,600 (.75) $1,015 Test (.75) Unacceptable Score Satisfactory (.2) Full training (.1) Not hired (.9) Satisfactory (.5) No Test Full training $ 900 $ 440 –800 –640 –200 $ –200 $ 2,400 $ 1,200 $ 900 Unsatisfactory (.5) Not hired $ –200 Unsatisfactory (.8) $ –200 $ 2,200 –600 –300 $ 900 24-24 Chapter 24 P24-7 (Concluded) 1Successful hire salary savings Less costs: Testing Abbreviated training Payoff 2Successful $200 300 hire salary savings Less costs: Testing Full training Payoff 3Successful $3,000 hire salary savings Less full training cost Payoff 500 $2,500 $3,000 $200 600 800 $2,200 $3,000 600 $2,400 4Expected $1,440 –20 $1,420 5Expected $1,065 –50 $1,015 value of abbreviated training $1,600 × = Expected value of not hiring –200 × = Expected value when test score acceptable value of acceptable test score $1,420 × 75 = Expected value of unacceptable test score –200 × 25= Expected value of administering test Chapter 24 24-25 P24-8 Sales Price $5.25 5.25 5.25 5.25 5.00 5.00 5.00 5.00 Material Lot Size 200,000 200,000 240,000 240,000 200,000 200,000 240,000 240,000 State of Economy Weak Strong Weak Strong Weak Strong Weak Strong Sales Demand 180,000 200,000 180,000 200,000 200,000 240,000 200,000 240,000 Order 200,000 Expected Payoff ($5.25 × 180,000) – ($3 × 200,000) = $345,000 ($5.25 × 200,000) – ($3 × 200,000) = $450,000 ($5.25 × 180,000) – ($2.90 × 240,000) = $249,000 ($5.25 × 200,000) – ($2.90 × 240,000) = $354,000 ($5 × 200,000) – ($3 × 200,000) = $400,000 ($5 × 200,000) – ($3 × 200,000) = $400,000 ($5 × 200,000) – ($2.90 × 240,000) = $304,000 ($5 × 240,000) – ($2.90 × 240,000) = $504,000 Payoffs Expected Value Weak economy (.6) $345,000 $ 207,000 Strong economy (.4) 450,000 180,000 $ 387,000 Weak economy (.6) 249,000 $ 149,400 Strong economy (.4) 354,000 141,600 $ 291,000 Weak economy (.6) 400,000 $ 240,000 Strong economy (.4) 400,000 160,000 $ 400,000 Weak economy (.6) 304,000 $ 182,400 Strong economy (.4) 504,000 201,600 $ 384,000 $387,000 $387,000 Select $5.25 Sales Price Select $5.00 Sales Price Order 240,000 Order 200,000 $291,000 $400,000 $400,000 Order 240,000 $384,000 Slick Inc should set the sales price at $5.00 per unit and order 200,000 units of material, because this course of action will result in the greatest expected value ($400,000 contribution margin) 24-26 Chapter 24 P24-9 (1) Do Not Introduce Expected Payoff Expected Value $ –500,000 $ –500,000 $2,300,000 $900,000 y1 teg St — ign pa am $ 2,800,000 $ 2,500,000 –500,000 $ 2,300,000 $2,300,000 ) (.5 ful ss ce uc tS No st Te Unsuccessful (.2) Do Not Introduce $ –500,000 Te s tC $ 3,500,000 t uc Te st od Pr Su cc es s ew eN ful uc (.5 ) rod Int Successful (.8) $ –500,000 $1,000,000 ew eN uc rod Int $ 3,500,000 700,000 t e id uc nw od tio Pr Na Successful (.2) Pr ot om $ –1,300,000 n io Unsuccessful (.8) –2,000,000 $ –1,300,000 –2,500,000 No n ig pa am tC s Te — gy te St $ 1,000,000 Successful (.5) Unsuccessful (.5) $2,300,000 – 500,000 × × 2Successful $ 4,000,000 $ 2,000,000 –2,000,000 –1,000,000 $ 1,000,000 = $1,150,000 = –250,000 $ 900,000 with test = ($40 – $30 – $6 – $.5) million = $3.5 million with test = ($16 – $12 – $6 – $.5) million = $ –2.5 million 4Successful without test = ($40 – $30 – $6) million = $4 million 5Unsuccessful without test = ($16 – $12 – $6) million = $ –2 million 3Unsuccessful Chapter 24 24-27 P24-9 (Concluded) (2) If the probability estimates can be relied upon, management should conduct the nationwide promotion and distribution without first performing a test campaign because the expected value of Strategy is $100,000 greater than the expected value of Strategy (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 are biased, the expected values will be biased (b) The values for the alternative courses of action are estimates that could be inaccurate (c) The decision model does not incorporate psychological factors For instance, people are often risk averse, and personal evaluations will not necessarily coincide with monetary evaluations (d) A model is often overly simplified to make it manageable and may consequently leave out important considerations or assumptions 24-28 Chapter 24 P24-10 (1) Expected value of periodic cash flows: (1) (2) (3) Expected Expected Value of Value of Annual cash Annual Contribution Inflow Sales Margin From Sales in Units Per Unit (1) × (2) 4,000 $14 $56,000 (1) (2) Year 10 Tax Basis (Cost) $200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000 (1) (2) (3) Year 10 Expected Value of Pretax Net Cash Flow $(200,000) 47,900 47,900 47,900 47,900 47,900 47,900 47,900 47,900 47,900 47,900 Tax Depreciation $28,600 49,000 35,000 25,000 17,800 17,800 17,800 9,000 0 (3) Tax Depreciation Rate 143 245 175 125 089 089 089 045 000 000 (4) Expected Value of Taxable Income (2) – (3) $19,300 (1,100) 12,900 22,900 30,100 30,100 30,100 38,900 47,900 47,900 (4) Annual Fixed Cash Outflow $8,100 (5) Expected Value of Annual Pretax Net Cash Inflow (3) – (4) $47,900 (4) Annual Tax Depreciation $ 28,600 49,000 35,000 25,000 17,800 17,800 17,800 9,000 0 $200,000 (5) Expected Value of Tax Liability (4) × 40% $ 7,720 (440) 5,160 9,160 12,040 12,040 12,040 15,560 19,160 19,160 (6) Expected Value of After-Tax Net Cash Flow (2) – (5) $(200,000) 40,180 48,340 42,740 38,740 35,860 35,860 35,860 32,340 28,740 28,740 $ 167,400 Chapter 24 24-29 P24-10 (Continued) Expected value of the periodic standard deviation: (2) (1) (2) Standard Deviation in Units of Sales 1,750 Pretax Cash Flow per Unit $14 (3) Pretax Cash Flow Value of Standard Deviation (1) × (2) $24,500 (4) After-Tax Portion (1 – 40%) 60% Expected net present value of investment: (1) (2) (3) (4) Expected Present Value of Present Value of After-Tax Net Value of After-Tax Net Year Cash Flow $1 at 12% Cash Flow $(200,000) 1.000 $ (200,000) 40,180 893 35,881 48,340 797 38,527 42,740 712 30,431 38,740 636 24,639 35,860 567 20,333 35,860 507 18,181 35,860 452 16,209 32,340 404 13,065 28,740 361 10,375 10 28,740 322 9,254 Expected net present value $ 16,895 (5) After-Tax Cash Flow Value of Standard Deviation (3) × (4) $14,700 24-30 Chapter 24 P24-10 (Concluded) (3) Variance and standard deviation of expected net present value: (1) (2) (3) (4) (5) (6) Present Value of Present Periodic Periodic Present $1 at 12% Value of Standard Variance Value of Squared Variance $1 at 12% Col (4)2 (3) × (5) Year Deviation Col (2)2 0 1.000 1.000000 $14,700 $216,090,000 893 797449 $172,320,754 14,700 216,090,000 797 635209 137,262,313 14,700 216,090,000 712 506944 109,545,529 14,700 216,090,000 636 404496 87,407,541 14,700 216,090,000 567 321489 69,470,558 14,700 216,090,000 507 257049 55,545,718 14,700 216,090,000 452 204304 44,148,051 14,700 216,090,000 404 163216 35,269,345 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 Standard deviation Variance of net = present value = $761, 535,, 950 = $27, 596 of net present value (4) $27, 596 Standard deviiation Coefficient = = = 1.633 of variation Expected net present value $16, 895 (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 approximately 23% area under the curve that is below the mean but above zero (determined µ – X) ÷ σ = ($16,895 – 0) from the table of Z values in Exhibit 24-8 of the text for (µ σ, which is about 23% of the total area under the normal curve) ÷ $27,596 = 61σ Chapter 24 24-31 P24-11 (1) Expected value of periodic cash flows: (1) (2) (3) Contribution Expected Expected Margin per Value of Value of Unit (Cash Annual Cash Annual Inflow Net Inflow Sales of Outflow From Sales in Units per Unit) (1) × (2) 5,000 $18 $90,000 (1) (2) Year Tax Basis (Cost) $180,000 180,000 180,000 180,000 180,000 180,000 180,000 180,000 (3) Tax Depreciation Rate 143 245 175 125 089 089 089 045 (1) (2) (3) Year Expected Value of Pretax Net Cash Flow $(180,000) 80,000 80,000 80,000 80,000 80,000 80,000 80,000 80,000 Tax Depreciation $25,740 44,100 31,500 22,500 16,020 16,020 16,020 8,100 (4) Expected Value of Taxable Income (2) – (3) $54,260 35,900 48,500 57,500 63,980 63,980 63,980 71,900 (4) Annual Fixed Cash Outflow $10,000 (5) Expected Value of Annual Pretax Net Cash Inflow (3) – (4) $80,000 (4) Tax Depreciation $ 25,740 44,100 31,500 22,500 16,020 16,020 16,020 8,100 $180,000 (5) Expected Value of Tax Liability (4) × 40% $21,704 14,360 19,400 23,000 25,592 25,592 25,592 28,760 (6) Expected Value of After-Tax Net Cash Flow (2) – (5) $ (180,000) 58,296 65,640 60,600 57,000 54,408 54,408 54,408 51,240 $ 276,000 24-32 Chapter 24 P24-11 (Continued) Expected value of the periodic standard deviation: (2) (1) (2) Standard Deviation in Units of Sales 2,000 Pretax Cash Flow per Unit $18 (3) Pretax Cash Flow Value of Standard Deviation (1) × (2) $36,000 (4) After-Tax Portion (1 – 40%) Expected net present value of investment: (1) (2) (3) Expected Value of Present After-Tax Net Value of Year Cash Flow $1 at 12% $(180,000) 1.000 58,296 893 65,640 797 60,600 712 57,000 636 54,408 567 54,408 507 54,408 452 51,240 404 Expected net present value (4) Present Value of After-Tax Net Cash Flow $ (180,000) 52,058 52,315 43,147 36,252 30,849 27,585 24,592 20,701 $ 107,499 (5) After-Tax Cash Flow Value of Standard Deviation (3) × (4) $21,600 Chapter 24 24-33 P24-11 (Concluded) (3) Standard deviation of expected net present value: (1) (2) (3) Periodic Present Standard Value of Year Deviation $1 at 12% 0 1.000 $21,600 893 21,600 797 21,600 712 21,600 636 21,600 567 21,600 507 21,600 452 21,600 404 Standard deviation of net present value (4) Present Value of Standard Deviation (2) × (3) $ 19,289 17,215 15,379 13,738 12,247 10,951 9,763 8,726 $107,308 (4) Standard deviation $107,, 308 Coefficient = = = 998 of variation Expected net present value $107, 499 (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 under the curve that is below the mean but above zero (determined from the table of Z µ – X) ÷ σ = ($107,499 – 0) ÷ $107,308 = 1.0σ σ, values in Exhibit 24-8 of the text for (µ which is about 34% of the total area under the normal curve.) 24-34 Chapter 24 P24-12 (1) Expected net present value of mixed cash flows: (1) (2) (3) (4) (5) (6) Present Expected Expected Total Value of Independent Dependent Expected Expected After-Tax After-Tax After-Tax Net After-Tax Net Net Cash Net Cash Cash Inflow Present Cash Inflow Inflow Inflow (Outflow) Value of (Outflow) Year 70% 30% (2) + (3) $1 at 10% (4) × (5) $(30,000) 1.000 $ (30,000) $5,600 $2,400 8,000 909 7,272 7,700 3,300 11,000 826 9,086 7,000 3,000 10,000 751 7,510 6,300 2,700 9,000 683 6,147 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) Independent Independent Present Cash Flow Cash Flow Value of Periodic Periodic Present $1 at 10% Standard Variance Value of Squared $1 at 10% Col (4)2 Year Deviation Col (2) 0 1.000 1.000000 $1,000 $1,000,000 909 826281 1,000 1,000,000 826 682276 1,000 1,000,000 751 564001 1,000 1,000,000 683 466489 1,000 1,000,000 621 385641 Variance of expected NPV for independent cash flows (6) Present Value of Variance (3) × (5) $ 826,281 682,276 564,001 466,489 385,641 $2,924,688 Chapter 24 24-35 P24-12 (Concluded) (2) (3) Dependent Cash Flow Periodic Present Standard Value of Year Deviation $1 at 10% 0 1.000 $500 909 500 826 500 751 500 683 500 621 Standard deviation of NPV Variance of net present value for dependent cash flows = ( (4) Present Value of Standard Deviation (2) × (3) $ 455 413 376 342 311 $1,897 Standard deviation of net present value for dependent cash flows = ($1,897)2 ( (1) = Variance of NPV for dependent cash flows Variance of NPV for independent cash flows Variance of total NPV of investment Standard deviation of total net present value $3,598,609 $3,598,609 2,924,688 $6,523,297 Variance of total = net present value = $6, 523, 297 = $2, 554 (3) Standard deviation $2, 554 Coefficient = = = 0.586 of variation Expected net present value $4, 362 (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 (determined X) ữ = ($4,362 – 0) from the table of Z values in Exhibit 24-8 of the text for (µ σ, which is about 46% of the total area under the normal curve.) ÷ $2,554 = 1.71σ Factors Net present value Reduce setup time Reduce throughput time Improve product quality Reduce inventory levels Improve image to outsiders Total Relative Importance Weighting 30 20 15 15 10 10 100 Modernize With Existing Technology Performance Likelihood Weighted Rating Estimate Score 48.0 5 7.5 13.5 5.0 74.0 GLOTYNE CORPORATION Capital Expenditure Proposal MADM Worksheet Modernize With New Technology Performance Likelihood Weighted Rating Estimate Score 36.0 27.0 15.0 6.0 6.0 90.0 Based on the results of the MADM worksheet below, Glotyne management should choose the CIM system because its composite weighted score is higher than the alternative Based on this analysis, the CIM system is expected to more adequately satisfy management’s modernization goals P24-13 24-36 Chapter 24 ... during assembly 70 overhead (Variable cost per hour ( ($8) ( = + ( Direct labor ( cost per hour ( Cost of rejections during assembly per lot = ( Hourly cost to replace bearing Expected Value... $300,000 Weighted average unit cost (expected value) to manufacture thermocouplers: Estimated per Unit Variable Cost $10 12 14 16 Probability Weighted Average Unit Cost (Expected Value) $ 1.00... 3.60 5.60 3.20 $13.40 Estimated variable manufacturing cost (18,000 units × $13.40) Additional fixed manufacturing cost Total estimated cost if thermocouplers manufactured $241,200 32,500