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# MANAGERIAL ECONOMICS ECON 4343

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This 74 page Class Notes was uploaded by Moshe Fritsch on Friday October 2, 2015. The Class Notes belongs to ECON 4343 at Arkansas State University taught by Daniel Marburger in Fall. Since its upload, it has received 34 views. For similar materials see /class/217734/econ-4343-arkansas-state-university in Economcs at Arkansas State University.

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Date Created: 10/02/15

CHAPTER 12 DECISIONMAKING UNDER RISK AND UNCERTAINTY LEARNING OBJECTIVES In this chapter students Will be able to 1 Set up a decision tree 2 Calculate and interpret expected value 3 Calculate standard deviation and the coefficient of variation and use them to infer the level of risk 4 Analyze alternatives When subsequent decisions must be made 5 Use decision theory to determine the value of information 6 Explain the limitations inherent in the maximin and minimax criteria Overview of Freemark Abbey Winery Freemark Abbey was a winery located in California s Napa Valley It produced only premium wines One of the partners that owned Freemark William Jaeger was confronted with an important decision Recent weather reports suggested that a storm might hit the Napa Valley If the storm hit the valley the rainwater could conceivably swell the berries and reduce their concentration If so the selling price of the wine would decrease by 85bottle The possibility also existed however that the storm would cause the botrytis mold to form on the grape skins If the mold formed the wine would be highly valued by connoisseurs and could be sold at more than double the normal price The higher price however would be partially offset by a decrease in the quantity sold The alternative would be to harvest the grapes in advance of the storm If the grapes were harvested immediately the wine could be sold at a price of 285bottle In making his decision Jaeger also had to determine the rami cations if he waited for the storm and it did not hit the region Ultimately the price depended on the sugar concentration In general the higher the percentage of sugar concentration the higher the price In determining whether to harvest the grapes or wait for the storm Jaeger had to estimate the likelihood the storm would hit the odds that the mold would form if the storm hit and the probabilities of various sugar concentration levels if the storm did not hit the region Relevant Revenue Cost Analysis and Uncertainty Your rm is trying to decide where to locate its new retail outlet You determine that if you locate the outlet on the west side of town it will generate operating pro ts of 5 millionyear If you position the outlet on the east side of town it will produce an annual pro t of 8 million Where should you build the outlet If only decisions were this easy In the real world a rm s annual operating pro ts are uncertain What if other rms locate an outlet near yours What if they launch an aggressive marketingpricing strategy What if property taxes rise What if the minimum wage increases Until now we ve implemented relevant revenuerelevant cost analysis under the assumption that both the cost and revenue gures were known In fact that will rarely be the case In the real world a rm s revenues and costs depend on a variety of factors These factors are often referred to as states of nature because they are beyond the control of the rm If the rm could control these factors it would manipulate them to its bene t Instead the states of nature represent constraints the company must deal with As the states of nature vary the level of uncertainty associated with a strategic decision increases and uncertainty breeds risk Hence in this chapter we move from discussing relevant revenues and costs to relevant expected revenues and costs Unfortunately as the number of factors mount the number of potential outcomes increases sometimes exponentially This can make it increasingly difficult for the firm to digest the information in such a way as to make an objective accurate decision In this chapter we will discuss the tools to allow a firm lay out its potential outcomes and to utilize simple summary statistics that make it relatively easy for firms to compare alternatives and make decisions Decision Trees To help lay out the various outcomes associated with the firm s initiative we can piece together a decision tree The purpose of a decision tree is to create a roadmap of sorts that lays out the various cost and revenue figures associated with a given decision The decision tree includes an assortment of branches each one dictated by the states of nature Some will affect the firm s revenues whereas others affect the firm s costs At the end of each branch is an outcome and the probability the outcome is likely to occur The first fork in the decision tree refers to the decision the manager wishes to make In the Freemark Abbey Winery case William Jaeger needed to decide whether to harvest the grapes immediately or leave them on the vine and await a possible storm If he decides not to harvest the grapes his revenues depend on whether the storm hits If the storm hits his revenues will depend on whether the mold forms If the storm does not hit the selling price of a bottle of wine will depend on the level of sugar concentration The decision tree allows the decisionmaker to lay out the potential outcomes and the probabilities that each outcome will occur Let s create our own example to illustrate a decision tree Suppose a rm is trying to determine whether to expand into the Detroit market or the Twin Cities Minneapolis St Paul Because the rm presumably will locate at the most pro table location the manager should compile a list of the states of nature that could affect the revenues and costs at each location The manager can use basic economic theory to compile a list of states of nature On the demand side the quantity demanded depends on the price consumer tastes and preferences the price of complementary goods the price of substitute goods consumer incomes price expectations and the number of potential buyers Although the rm has control over a few of these factors such as the product s price the price the rm charges depends on factors beyond its control such as the price charged by competing rms On the cost side the rm s expenses are equal to the some of its xed and variable costs which are often in uenced by states of nature beyond the rm s control For example if market wages are rising the rm will be forced to offer higher wages to attract and retain quali ed workers Let s walk through a simple example The rst step in building a decision tree is to create a fork in the road that identi es management s alternatives In our example the rm wishes to determine whether to expand into Detroit or Minneapolis St Paul Next the manager seeks to determine the relevant costs associated with each location An investigation reveals that if the rm locates in Detroit it will face the following costs Fixed costs 300000 Variable costs 12unit The Twin Cities location has the following cost structure Fixed costs 600000 Variable costs 10unit The rm believes there is a 20 chance the raw materials needed for production will be in short supply If so the variable costs are expected to rise to 14 in Detroit and 13 in the Twin Cities This is incorporated into the decision tree through a pair of branches one branch shows the existing cost structure and the other exhibits the higher costs Next the manager seeks to determine the relevant revenues associated with each location He she determines that unit sales are sensitive to the state of the economy If the economy is normal the rm will likely sell 350000 units per year in Detroit and 375000 in the Twin Cities Ifthe economy is in a recession sales fall to 200000 units per year in Detroit and 180000 in the Twin cities The economy averages one recession every four years Accordingly then the decision tree illustrates the uncertainty in unit sales by creating a pair of branches at each location If the rm locates in Detroit and the economy is normal the rm will sell 350000 units If the economy falls into a recession unit sales will fall to 200000 units On the other hand if the rm locates in the Twin Cities it will sell 375000 units ifthe economy is normal and 180000 during a recession Based on past history the likelihood of a recession is 25 which of course implies the probability of a normal economy is 75 We will note the probabilities at their corresponding branches Relevant revenues are also affected by the price The price depends on whether one of the rm s competitors triggers a price war Price wars tend to occur once every three years Without a price war the rm can expect to charge 20unit at either location During a price war the price is expected to fall to 15unit in Detroit and 13 in the Twin Cities The market demand for the good is fairly inelastic If a price war breaks out unit sales will tend to rise by 10 at either location We accommodate the uncertainty in prices in the decision tree through a subsequent set of branches In the absence of a price war the rm expects to charge 20unit in Detroit Should a price war occur the price will drop to 15 The rm also expects to charge 20 in the Twin Cities in the absence of a price war However if a price war were to take place the rm s price will fall to 13 These branches also incorporate the change in unit sales that may arise from a price war The rm anticipates the likelihood of a price war to be 33 thus the branches post the respective probabilities Figure 1 here Figure 1 shows the decision tree Note how the decision tree allows the manager to determine the various pro t outcomes arising from the relevant revenues and costs at each location In this case the branches of the tree lead to eight different outcomes each one exhibiting the rm s pro ts under various scenarios Assuming the states of nature are independent of each other the probability of any three states of nature occurring simultaneously is the product of the individual probabilities For example if the probability of a lower cost structure is 080 the probability of a normal economy is 075 and the probability of a price war is 033 then the probability of all three occurring simultaneously is 080 X 075 X 033 4020 We can now combine the outcomes and probabilities for each plant as shown in Table 1 Scenario Low Cost Normal Economy No Price War Low Cost Recession No Price War Low Cost Normal Economy Price War Low Cost Recession Price War High Cost Normal Economy No Price War High Cost Recession No Price War High Cost Normal Economy Price War High Cost Recession Price War Detroit Outcome 2500000 1300000 855000 360000 1800000 900000 85000 80000 Table 1 Probability 4020 1340 1980 0660 1005 0335 0495 0165 Twin Cities Outcome 3150000 1200000 63 7500 6000 2025000 660000 600000 600000 Probability 4020 1340 1980 0660 1005 0335 0495 0165 Although the decision tree lists the various outcomes at each location and their corresponding probabilities the manager is left to digest the information In the next section of this chapter we will discuss several summary statistics that provide concise yet accurate information from which an objective decision can be made IMPORTANT IMPLICATIONS FOR THE MANAGER The decision tree is a tool managers can use to anticipate the various cash ows that might occur if a given decision is implemented Each branch of a decision tree details an external factor aka state of nature that could affect the firm s price output andor cost structure Expected Value Let s ignore the probabilities for the time being The outcomes associated with each location are shown in Table 2 Scenario Low Cost Normal Economy No Price War Low Cost Recession No Price War Low Cost Normal Economy Price War Low Cost Recession Price War High Cost Normal Economy No Price War High Cost Recession No Price War High Cost Normal Economy Price War High Cost Recession Price War Detroit Outcome 2500000 1300000 855000 360000 1800000 900000 85000 80000 Table 2 Twin Cities Outcome 3150000 1200000 63 7500 6000 2025000 660000 600000 600000 Which location is the most pro table The easiest way to compare alternatives is to calculate the average cash ow at each location As you know the mean is calculated as In this case the average annual cash ow in Detroit is 965000year as compared to 80831250year in the Twin Cities Before choosing Detroit we need to be careful When one calculates the mean heshe implicitly assumes each outcome is equally likely In realistic business scenarios this is highly unlikely Some outcomes are almost always more likely to occur than others If so then a straight comparison of mean outcomes may be biased and could mislead the decisionmaker When the probabilities of various outcomes differ as they do in Table 1 the appropriate way to determine the average is to calculate the expected value The expected value u is a weighted average It weights each outcome by the probability it will occur The expected value is calculated as y n 13X 11 where P is the probability event 139 will occur and X is the outcome of event 139 Let s calculate the expected value for each location Based on the figures in Table 1 if we locate in Detroit our expected annual cash ow is 2500000 x 4020 1300000 x 1340 855000 x 1980 360000 x 0660 1800000 x 1005 900000 x 0335 85000 x 0495 7 80000 x 0165 158618750 The expected annual cash ow at the Twin Cities location is 3150000 X 4020 1200000 X 1340 637500 X 1980 6000 X 0660 2025000 X 1005 660000 X 0335 600000 X 0495 7 600000 X 0165 173895150 Note how incorporating the likelihood each event will occur alters the gures Calculating the average annual outcome in Detroit using a straight mean was 965000 By building in the probabilities of each outcome the gure rises to 158618750 Similarly the average annual pro t at the Twin Cities without incorporating probabilities was 80831250 The weighted average on the other hand was 173895150 The large difference between the straight mean and the eXpected value stems from the fact that the more positive scenarios are signi cantly more likely to occur than the least favorable scenarios For eXample the probability of a low cost structure combined with a normal economy and no price war is 402 In contrast the likelihood of a high cost structure combined with a recession and a price war is only 165 The straight average attached equal weight to each of these outcomes resulting in a mean that was biased downward IMPORTANT IMPLICATIONS FOR THE MANAGER 1 When the likelihood of various outcomes differs the manager can determine the mean outcome by calculating the eXpected value 2 The eXpected value u is a weighted average It is calculated by multiplying each outcome by its corresponding probability and then summing them up across all outcomes or u PIXI 11 Standard Deviation Let s alter the cash ows for the Twin Cities The cash ows at the two locations and their respective probabilities are shown in Table 3 Table 3 Detroit Twin Cities Scenario Outcome Probability Outcome Probability Low Cost Normal Economy No Price War 2500000 4020 3000000 4020 Low Cost Recession No Price War 1300000 1340 1100000 1340 Low Cost Normal Economy Price War 855000 1980 561000 1980 Low Cost Recession Price War 360000 0660 51750 0660 High Cost Normal Economy No Price War 1800000 1005 1650000 1005 High Cost Recession No Price War 900000 0335 462500 0335 High Cost Normal Economy Price War 85000 0495 890250 0495 High Cost Recession Price War 80000 0165 73493150 0165 The gures for Detroit are the same hence the expected value at Detroit is still 158618750 For the Twin Cities however the expected value is 3000000 X 4020 1100000 X 1340 561000 X 1980 7 51750 X 0660 1650000 X 1005 462500 X 0335 7 890250 X 0335 7 73493150 X 0165 158618750 According to the eXpected value gures the average annual cash ows are the same at each location Over time therefore the locations should be equally pro table This begs the question based on the numbers in Table 3 do you have a preference Some students may prefer Detroit If the rm locates at Detroit the worst case scenario is a negative cash ow of 80000 as compared with a potential loss of 890250 at the Twin Cities location Other students may prefer the Twin Cities based on the fact that cash ows could potentially reach 3000000 as compared with a best case scenario of only 2500000 Detroit In general students who prefer Detroit notice that the outcomes are fairly close together whereas the outcomes at the Twin Cities are more dispersed When the outcomes are fairly close together the decisionmaker is willing to forego the potential for highly desirable outcomes to avoid the less desirable outcomes These individuals have risk averse preferences Those who prefer the Twin Cities have risk seeking preferences They are willing to risk less favorable outcomes to have a shot at the most desirable result Of course other students may decide the location doesn t matter because the average eXpected cash ow is the same in each region Such persons are eXhibiting risk neutral preferences Ultimately those with risk averse or riskseeking preferences are using the standard deviation of the outcomes to help them make a decision As you may recall from your basic statistics class standard deviation 6 is a measure of the individual outcomes around the mean The basic calculation of standard deviation is where X is the value of observation 139 and 7 is the mean outcome However as with the calculation of the mean the basic calculation of standard deviation implies each observation is equally likely to occur To adjust for differing probabilities we multiply each squared difference by its corresponding probability instead of summing the squared differences and dividing by n J The proper calculation of standard deviation when the outcomes are not equally likely to occur is therefore aPX z where P is the probability outcome 139 will occur Notice also that the expected value u replaces the mean Yfrom the standard calculation Figure 2 shows the distribution of two sets of outcomes Assuming some students have encountered a dense buildup of brain cobwebs from the time they first took their basic statistics class let s review the concept of the normal curve The outcomes appear on the horizontal axis and the frequency of the outcomes ie the number of times the outcome occurred The taller the curve the more frequently the outcome occurred Notice that the frequency is greatest at the expected value As we move farther away from the expected value in either direction the number of times the outcome occurred decreases The bellshaped normal distribution assumes the distribution of outcomes is symmetric around the mean Figure 2 here In interpreting the concept of standard deviation visually note that in distribution a the outcomes are relatively close to the mean Because the range of potential outcomes is relatively small the standard deviation is relatively small This implies less uncertainty regarding the outcomes and therefore less risk In contrast the outcomes on distribution b are more spread out As a result the standard deviation will be higher implying more uncertainty and risk Let s calculate the standard deviation of the cash ow gures for the Detroit and Twin Cities locations For Detroit the standard deviation is 2500000 1586187502 X 4020 1300000 1586187502 X 1340 855000 1586187502 X 1980 360000 1586187502 X 0660 1800000 1586187502X 1005 900000 1586187502X 0335 85000 1586187502 X 0495 7 80000 1586187502 X 0165 854098 The standard deviation for the Twin Cities is 3000000 1586187502 X 4020 1100000 1586187502 X 1340 561000 1586187502 X 1980 51750 1586187502 X 0660 1650000 1586187502X 1005 462500 1586187502X 0335 890250 1586187502 X 0495 7 734932 1586187502 X 0165 1286686 What do these numbers mean If you examine the equation you ll see that the standard deviation is the square root of the average squared deviation between each outcome and the mean outcome If that explanation leaves your head spinning let s simply note that your first operation in calculating standard deviation is to take the difference between each outcome and the mean outcome These differences are then squared to assure they do not cancel each other out when summed Thus the more spread out the outcomes the larger the difference between each outcome and the mean Consequently the more dispersed the outcomes the larger the standard deviation Because the standard deviation for the Twin Cities is higher than for Detroit we know the range of outcomes for the Twin Cities is greater Statisticians apply Chebyshev s Theorem to make inferences from standard deviation Chebyshev proved the proportion of any data set lying within k standard deviations of the mean is at least 1 7 lkz where k is any positive number greater than 1 For example we can infer that at least 1 7 122 or 75 of the outcomes will lie within two standard deviations of the mean When the distribution of outcomes is normally distributed we can make better inferences than Chebyshev s Theorem permits Sometimes called the Gaussian distribution named after German mathematician Friedrich Gauss we can infer the following empirical rules which will become extremely useful when we get to the chapter on regression analysis In general when the number of observations is very large a 90 of the observations lie within 1645 standard deviations of the mean b 95 of the observations lie within 196 standard deviations of the mean and c 99 of the observations lie within 258 standard deviations of the mean Assuming the distribution of cash ows are normally distributed we can infer that 90 of the time the annual cash ows in Detroit will range between 181196 and 2991179 At the riskier Twin Cities location annual cash ows will range between 530411 and 3702786 90 of the time Clearly coupling expected value with standard deviation allows the decisionmaker to assess not only the average outcome but also the level of risk associated with the alternatives IMPORTANT IMPLICATIONS FOR THE MANAGER 1 The level of risk associated with a decision can be inferred from the standard deviation 5 The standard deviation is a measure of the dispersion of outcomes around the mean It is calculated as 03213 Z 2 The larger the standard deviation the more dispersed the outcomes Therefore in general the higher the standard deviation the greater the risk associated with the decision If the number of observations is relatively large managers can use the Gaussian distribution to infer the following a 90 of the observations lie within 1645 standard deviations of the mean b 95 of the observations lie within 196 standard deviations of the mean and c 99 of the observations lie within 258 standard deviations of the mean 0 Coef cient of Variation Although standard deviation is a good tool to assess risk it can be misleading To illustrate let s take a look at the following alternatives A rm is considering launching one of two initiatives The outcomes associated with each initiative along with the corresponding probabilities appear in Table 4 The expected value for Alternatives A and B are 70 and 1000040 respectively Table 4 Alternative A Alternative B Cash Flow Probability Cash Flow Probability 0 10 1000100 10 100 10 999800 10 350 20 1000500 20 50 20 999950 20 0 40 999900 40 Which alternative is riskier The standard deviation for Alternative A is 0 702 x 10 100 702 x 10 350 702 x 20 50 702 x 20 0 702 x 40 145 The standard deviation for Alternative B is 1000100 10000402 X 10 999800 10000402 X 10 1000500 10000402 X 20 999950 10000402 X 20 999900 10000402 X 40 240 At first glance B appears to be the riskier of the two because the standard deviation is nearly twice as large for B as it is for A But let s employ the empirical rule of thumb used earlier HA is initiated 90 ofthe cash ows will lie between 168 and 309 If B is implemented 90 ofthe cash ows will lie between 999645 and 1000435 So which alternative is riskier In comparing Alternatives A and B we need to distinguish between absolute risk and relative risk Stande deviation allows one to infer absolute risk On an absolute dollar level B does imply a greater spread among likely outcomes after all 1000435 999645 790 which is greater than 309 7 168 477 However on a relative basis when one considers the expected value for each alternative 70 for A and 1000040 for B a swing between 168 and 309 is a lot larger than the 790 difference between 1000435 and 999645 On a relative basis Alternative B is practically a certain outcome Although absolute risk is hardly a worthless concept using standard deviation to assess risk when the difference between the expected values of two or more alternatives is fairly large is inherently an applesandoranges comparison Because relative risk is frequently more important to the decisionmaker we calculate the coef cient of variation y to allow for an applestoapples comparison of risk Calculating the coefficient of variation is fairly intuitive We inferred that B was less risky than A because its standard deviation was small relative to its expected value Not surprisingly then the coefficient of variation is simply the standard deviation divided by the expected value or 611 The corresponding coefficients of variation for A and B are A 14570 207 and B 2401000040 000024 By dividing the standard deviations by the expected value we are measuring the dispersion of outcomes per expected dollar in cash ow For each expected dollar in A s cash ow there is 207 in dispersion among outcomes For each expected dollar in B s cash ow there is 000024 in dispersion among outcomes Note how it s easier to compare the levels of risk in A and B by measuring the dispersion per expected dollar IMPORTANT IMPLICATIONS FOR THE MANAGER 3 Standard deviation measures the absolute risk associated with a decision 4 When comparing alternatives managers may wish to examine relative risk This can be done by calculating the coefficient of variation V which is the standard deviation divided by the expected value or y Gu The coefficient of variation determines the dispersion of outcomes per unit of expected value For example if expected value measures the expected profit in dollars the coefficient of variation measures the dispersion of outcomes per dollar of expected profit This allows the relative risk from differing alternatives to be easily compared U Let s return to our original DetroitTwin Cities example Table l The summary statistics for each location are 20 Detroit Expected Value 158618750 Standard Deviation 854098 Coef cient of Variation 54 Twin Cities Expected Value 173895150 Standard Deviation 1299534 Coef cient of Variation 75 As the analysis indicates the expected pro t in the Twin Cities 173895150 exceeds that of Detroit 158618750 However the Twin Cities market is more risky than the Detroit market as evidenced by the coef cient of variation 75 per expected dollar in the Twin Cities as opposed to 54 per expected dollar in Detroit Although accountants can readily piece together the pro ts associated with each outcome where do the probabilities come from Unfortunately in business situations determining the probability of a normal economy coupled with a price war is not the same as ipping a coin an in nite number of times and projecting the percentage of times the coin ip comes up heads Although the probabilities in decision tree analysis may re ect historical odds they are inherently subjective even historical odds may not predict future odds Is it possible a manager s decision may be swayed by assessing probabilities incorrectly Indeed that s a possibility However through sensitivity analysis the manager can toy with the numbers to determine the impact of different probabilities on expected value Let s return to the DetroitTwin Cities example to illustrate Given the 21 assumed probabilities of a higher cost structure a normal economy and a price war we determined the expected values of the Detroit and Twin Cities markets to be 158618750 and 173895150 respectively How would changing the probabilities affect the summary statistics Is it possible that a different set of probabilities could cause the manager to change hisher decision We can examine the probability associated with each state of nature For example if we assume a low cost structure the expected values for Detroit and the Twin Cities are 1715313 and 1941161 respectively Ifwe assume a high cost structure the expected values are 1069688 and 93011250 respectively Assume the probability of a lower cost structure P is unknown If so the expected pro t in Detroit and the Twin Cities would be Detroit 1715313P 106968817 P and Twin Cities 1941161P 9301125017 P where P is the probability of a lower cost structure To determine the probability of a lower cost structure that would yield identical expected values for the two locations we set the expected values equal to each other and solve for P or 1715313P 106968817 P 1941161P 9301125017 P or P 382 This provides useful information to the decisionmaker The Detroit location yields a higher expected value than the Twin Cities only if a high cost structure exists If we hold the assumptions regarding the state of the economy and the price war constant we determined that Detroit would yield the higher expected value only if the probability of a low cost structure was less than 382 Because we originally assumed the probability of 22 a low cost structure was 80 this strengthens our conclusion the possibility of future cost increases are unlikely to sway our location decision We could do the same with the other two states of nature In each case we hold the assumptions for the other states of nature constant and then solve for the probability of the remaining state of nature For example let s hold constant the cost structure and price war assumptions In Detroit the expected value if a normal economy exists is 1812530 With a recession the expected value is 907160 In the Twin Cities the expected values for a normal economy and recession are 2088450 and 690456 respectively If the probability of a normal economy is P the two locations will have the same expected value if 1812530P 9071601 7 P 2088450P 6904561 7 P or ifP 44 Thus holding the other assumptions constant the Detroit location yields the higher expected pro t only if the probability of a normal economy is less than 44 This is well below our original projection of 75 so the manager can feel fairly secure about the Twin Cities location Finally we can examine the impact of the price war on the expected values in the two locations Holding constant the assumptions regarding the cost structures and the state of the economy the expected value in Detroit without a price war is 2075000 With a price war the expected value in Detroit is 593750 In the Twin Cities the expected value without a price war is 2466750 Ifa price war occurs the expected value is 261300 Labeling the probability that a price war does not occur as P the two locations have the same expected value if 23 2075000P 5937501 7 P 2466750P 2613001 7 P or P 459 This indicates that holding the other assumptions constant Detroit will exhibit a higher expected pro t than the Twin Cities only if the probability of competition without a price war is less than 459 Again we assumed a 33 chance ofa price war in any given time period so the Twin Cities choice seems more secure The Value of Information This discussion segues into the value of information Although the probabilities of various states of nature are rarely known and must be estimated we can sometimes use decision tree analysis to infer the value of information The value of information is equal to the expected value with the information minus the expected value without the information It determines the maximum a rm should be willing to spend to acquire additional information For example suppose a rm is considering launching a new product into the tri state area of western Pennsylvania eastern Ohio and northern West Virginia If launched throughout its retail outlets it anticipates a 50 chance its profit contribution will total 2 millionyear a 25 chance its profit contribution will be 05 millionyear a 10 chance that the profit contribution will total 01 million annually a 10 chance its product will have a negative contribution of 02 million and a 5 chance its good will lose 05 million annually Based on these projections the expected profit contribution is 2 million x 50 05 million x 25 01 million x 10 02 million x 10 05 million x 05 or 109 millionyear The product manager anticipates the good will have a life of five years after which it will become obsolete Based on a cost of capital of 8 the present value of the income stream is 435 million 24 One alternative is to conduct a test market in its retail outlets in Akron Ohio If the product generates a positive pro t contribution in Akron it will be launched throughout the region The 2 million pro t contribution for the entire region will be inferred if the test market yields a pro t contribution of 200000 Similarly the region wide pro t contributions of 05 million and 01 million will be projected if the test market yields pro t contributions of 005 million and 001 million respectively The product will not be launched if the pro t contribution in Akron is less than 001 million By test marketing the product suppose the revised expected value for a region wide launch is 2 million X 50 05 million X 25 01 million X 10 0 X 15 or 1135 million Over the anticipated veyear life of the product the discounted present value is 453 million Because the discounted present value of the eXpected pro t contribution with the test market is 453 million and the eXpected pro t contribution without the test market study is 435 million management should be willing to spend no more than 453 million 435 million or 180000 on the test market study We can also determine the value of information in the Freemark Abbey Winery case Jaeger must determine whether to harvest the grapes immediately or wait for the upcoming storm How much money would Jaeger be willing to pay to nd out if the storm will hit the region Ironically the answer is nothing If Jaeger knows with certainty that the storm will hit the eXpected revenues total 34584 which is more than what he would earn if he harvested the grapes before the storm This assumes he determined in advance that he would not bottle an inferior wine if the mold did not form Even if he knew with certainty the storm would not hit his eXpected revenues would be equal to 37200 which again eXceeds his earnings from an immediate harvest 25 What does matter to Jaeger is whether a storm is conducive to the formation of the botrytis mold If he knew the storm would not likely result in the mold he would harvest the grapes immediately and earn 34200 If he knew the storm was conducive to the mold but might not hit the region his expected earnings from waiting for the storm would be 41400 For this reason Jaeger would be willing to spend up to 41400 35892 the expected revenues from delaying the harvest or 5508 to learn more details about the storm IMPORTANT IMPLICATIONS FOR THE MANAGER Sometimes a manager can pay to acquire information to assist in decisionmaking The manager can determine the value of the information in the following manner 1 First based on the decision tree analysis determine the expected value of the alternatives with existing information This is the expected value without information Second determine what decision would be made with better information This is the expected value with information The difference between the expected values with and without information represents the maximum the firm should be willing to pay to acquire additional information J U Decision Trees and Capital Budgeting In the chapters on capital budgeting and estimating cash ows we noted that largescale decisions over an intermediate or longterm time frame entail a great deal of uncertainty The general notion that each year s aftertax cash ow can be discounted to its present value implicitly assumes the cash ow figures are certain In fact as we know economic 26 conditions may change competitors may enter or exit the market exchange rates may change government regulations may increase costs or restrict production etc As these states of nature change prices output and unit costs may rise or fall Decision tree analysis can be an invaluable tool for estimating a proj ect s future cash ow By compiling a list of scenarios the manager can project a series of cash ows based on a variety of circumstances The expected value of each year s cash ow can be discounted to its present value Moreover to provide a sensitivity analysis the expected cash ows can be in ated or de ated by one to two standard deviations One tool for developing a sensitivity analysis is through a Monte Carlo simulation This is typically available through simulation so ware packages or as an addon to Microsoft Excel In simulation the manager selects an expected value and standard deviation for each relevant variable unit sales price average variable cost etc The distribution of values can either follow a normal triangular uniform or lognormal distribution see Figure 3 The computer then selects a random number from the distribution and calculates the NPV Next a second set of values for each variable is selected at random and the NPV is calculated again This process is repeated over and over again resulting in numerous NPV calculations The mean NPV standard deviation and coefficient of variation are then reported Figure 3 here 27 IMPORTANT IMPLICATIONS FOR THE MANAGER 1 Decision trees are especially useful for determining the cash ows associated with capital budgeting decisions 2 Monte Carlo simulation available in many spreadsheet packages allows the firm to simulate the impact of various combinations of factors on cash ows Decision Trees and Sequential Decisions Thus far we have introduced decision trees to determine a single decision by management The rst fork in the decision tree de nes the choices available to the manager Each subsequent fork refers to factors that in uence the company s pro ts but are beyond the rm s control Quite often however a decision tree may require the manager to make more than one decision For example suppose a rm is trying to decide whether to launch a new product Clearly the decision analysis must incorporate the pro ts generated by the good Of course if the rm goes ahead and launches the product it will have to make a pricing decision Naturally the decision to launch the product hinges on the assumption the manager will choose the optimal price as conditions change How can the manager determine the expected value of an initiative when some of the outcomes require additional decisions The key to making sequential decisions in decision tree analysis is to make the decisions from right to left In other words the manager needs to examine the last decision mandated by the tree By calculating the summary statistics the manager can determine what heshe would do if the situation were to arise From this point the manager can backtrack to the next decision and repeat the process Once all subsequent 28 decisions have been determined the manager can utilize decision analysis to evaluate the overall project We can see examples of the need for sequential decisionmaking in the Freemark Abbey Winery case If a storm hits the region and the mold does not form the winery could either sell the wine in bulk sell the grapes directly or bottle and sell the wine anyway Although the latter option would generate twice as much revenue in the short run selling an inferior wine could damage the winery s reputation and inhibit future sales Hence in piecing together the decision tree Jaeger needs to determine at the front end what he would do if a storm hit the region but the mold did not form Let s work through our own example to illustrate how sequential decisions are made Suppose a movie studio is trying to decide whether to nance a 150 million big budget lm or two smaller lms costing 70 million each Quite often the director requests additional funding to make changes andor additions to the lm Sometimes the additional funds improve the box of ce appeal of the movie and increase the lm s gross On other occasions the movie does quite well if the director is held to the original budget The studio executives anticipate a request for an additional 50 million for the bigbudget lm and a combined 25 million for the two smaller lms If the additional funds for the bigbudget lm are approved the studio execs expect an 80 chance it will gross 300 million a 10 chance it will gross 200 million and a 10 chance it will gross 80 million If the budget increase is not approved the probabilities of a 300 million gross fall to 60 with the likelihood of a 200 million rising to 25 and the probability of an 80 million gross increasing to 15 29 If the two smaller lms are nanced and budget increases of a combined 25 million are approved the executives believe there is a 60 chance the lms will collectively gross 300 million This probability falls to 30 if the budget increases are not approved Similarly the likelihood the smaller lms will gross a combined 200 million is expected to be 30 if the budget increase is approved and 40 if it is not approved Finally if the additional funds are approved the likelihood the lms will gross 80 million is 10 and 30 if not approved What makes this case different from the others discussed so far is that the studio executives cannot decide whether to lnd the bigbudget lm or the two smaller lms without rst anticipating the effect of the inevitable request for a budget increase Figure 4 sets this up as a decision tree to illustrate the studio executives dilemma Figure 4 here This is the essence of decision trees with sequential decisions to answer the broad question of whether to nance the bigbudget lm or the two smaller lms the studio executives must decide whether they would approve the request for additional funding The decisionmaker must ultimately make decisions by solving the decision tree from right to left In this case the studio executives must rst determine whether they would approve a budget increase before they examine the larger issue of which movies to nance Let s work with the bigbudget movie rst If the director requests an additional 50 million the expected gross is 300 million x 80 200 million x 10 80 million x 10 50 million or 218 million if the budget increase is approved and 300 million x 60 200 million x 25 80 million x 15 or 242 million if it is not approved 30 Hence if the studio executives based their decision exclusively on expected value they would not approve the additional 50 million budget increase If the smaller lms are nanced and the directors request an additional 25 million in combined budgets the expected combined gross is 300 million x 60 200 million x 30 80 million x 10 25 million or 223 million ifthe increase is approved and 300 million x 30 200 million x 40 80 million x 30 or 194 million if the additional nancing is not approved Therefore should the studio nance the smaller lms the studio executives would approve the budget increase Now that this decision has been made in advance the studio executives can work backward to determine which lms to nance Because the budget increase for the big budget movie would not be approved its expected pro ts are 242 million 150 million 92 million Assuming the request for additional funds for the smaller lms are approved their expected combined pro ts are equal to 223 million 140 million 83 million Absent risk considerations therefore the executives would approve the big budget movie IMPORTANT IMPLICATIONS FOR THE MANAGER When the primary decision a manager must make involves one or more sequential decisions the manager should use backward induction Specifically he she should determine what heshe would do if confronted with the last decision in the chain of events and work backward to the primary decision 3l Maximin and Minimax Two other simple tools for making decisions under risk and uncertainty are the maximin and minimax methods Unfortunately as you ll soon see whereas managers are attracted to these tools because of their simplicity they suffer because of their simplicity and are not recommended Under maximin the manager lays out the possible outcomes associated with various alternatives Heshe identi es the worst possible outcome associated with each alternative and selects the alternative with the best worstcase scenario hence the manager maximizes the minimum outcome Let s work through a quick example A manager is choosing between three projects A B and C By working with accounting the manager identi es the annual aftertax cash ows associated with each project The states of nature that drive that cash ows are ranked from least favorable l to most favorable 4 These appear in Table 5 Table 5 State of Nature 1 2 3 4 Project A 2 million 1 million 1 million 3 million Project B 5 million 05 million 3 million 6 million Project C 4 million 0 million 2 million 4 million Using maXimin the manager bases hisher decision on the worst case scenario Here project A results in a potential loss of 2 million whereas projects B and C could potentially result in losses of 5 million and 4 million respectively Hence the manager decides to go with project A because it has the most favorable worst case scenario 32 In addition to its simplicity maXimin does not require the decisionmaker to know the probability of each outcome which adds to its appeal However this is one of its weaknesses As we critique maXimin we must note that only the worst case scenarios matter It ignores not only other potential cash ows but also the likelihood that the worst case scenario is going to occur Let s use an exaggerated example to illustrate the problems with maXimin Suppose the manager is choosing between two alternatives The cash ows associated with each alternative appear in Table 6 And to make matters worse let s assume the probability state of nature 1 is going to occur is l in 1 million Table 6 State of Nature 1 2 Alternative A 01 02 Alternative B 01 100 million Under the maXimin criterion Alternative A would be selected because its worst case scenario is better than B s The fact that state of nature 2 generates 100 million as compared to 02 for A is completely ignored Moreover the fact that state of nature 1 is highly unlikely to occur is never factored into the decision Another simple tool is minimaX Here the manager seeks to minimize the opportunity cost of making the wrong decision Let s return to the scenario in Table 6 to show how minimaX works We will show the original table and then use it to construct the minimaX table Table 7 If state of nature 1 were to appear the best solution would be to go with project A because it results in the smallest loss If the manager had chosen project A therefore the opportunity cost of selecting A is 0 ie he she could not have bene ted by selecting another project However had the manager selected B heshe 33 would have lost 3 million more than if he she had chosen A Likewise had heshe chosen C the manager would lose 2 million more than if heshe had selected A Table 7 State of Nature 1 2 3 4 Project A 2 million 1 million 1 million 3 million Project B 5 million 05 million 3 million 6 million Project C 4 million 0 million 2 million 4 million State of Nature 1 2 3 4 Project A 0 1 million 2 million 3 million Project B 3 million 05 million 0 0 Project C 2 million 0 1 million 2 million Using minimaX the objective of the manager is to minimize the maXimum opportunity cost associated with each project For example by choosing projects A or B the manager could miss out on as much as 3 million However by making the wrong decision the manager could never miss out by more than 2 million by choosing C Hence using minimaX the manager would choose project C Again as with maximin the weakness of minimaX lies in the fact that it ignores the probabilities associated with each state of nature and focuses only on a single outcome In this case C would be the recommended alternative even if the odds that states of nature 1 and 4 were ten million to one and the likelihood of state of nature 3 occurring was 9999 The lesson to be learned is that like the payback method described in the capital budgeting chapter 34 managers should not be seduced by a method s simplicityif a decisionmaking tool seems too easy it probably is IMPORTANT IMPLICATIONS FOR THE MANAGER 1 Using a maximin criterion the manager sorts out the various outcomes associated with alternatives He she then chooses the alternative with the most favorable worst case scenario 2 Using a minimax criterion the manager sorts out the various outcomes associated with alternatives For each state of nature he she determines the opportunity cost associated with making the wrong decision ie the difference in profits between choosing the right alternative versus the wrong alternative The manager identifies the maximum opportunity cost associated with each decision and chooses the alternative with the lowest maximum opportunity cost The primary limitation of the maximin and minimax decision criteria is that they focus only on one outcome to the exclusion of all others and do not incorporate the likelihood the outcome will occur 0 SUMMARY 1 A manager can use a decision tree to lay out the outcomes associated with a decision To build a decision tree the manager must determine the factors that in uence the cash ows associated with a decision infer the probability the event will occur and estimate the cash ows that will exist if the event occurs N When the probabilities of the outcomes associated with an initiative vary the proper method to assess the mean outcome is to calculate the expected value The 35 E 4 V39 0 gt1 expected value of a list of outcomes is a weighted average calculated by summing the outcomes multiplied by their expected probabilities The level of absolute risk associated with a decision can be inferred from the standard deviation The larger the standard deviation the more risk associated with the decision In choosing between alternatives the level of relative risk can be inferred from the coefficient of variation This is calculated as the standard deviation divided by the expected value It reports the dispersion among outcomes for each unit of expected value The value of information is the difference between the expected value of the alternative with information less the expected value of the alternative without information When managers must make sequential decisions they should do so using backward induction beginning with the last decision on the decision tree and working backward until they reach the primary decision Using the maximin criterion the manager chooses the alternative that provides the most favorable of the worst case scenarios Using the minimax criterion the manager determines the opportunity cost associated with making the wrong decision in each possible scenario identifies the highest opportunity cost for each alternative and selects the alternative that yields the lowest maximum opportunity cost The weakness in either of these decision criteria is that they ignore all but one outcome and do not factor in the probability it will occur 36 MiniCase Luggage Delivery Service As a veteran business traveler Samuel Clayton had an idea for a business enterprise Tired of having the airlines lose his luggage in transit Clayton considered creating a luggage delivery service For a price of 30bag his service would pick up the luggage at the traveler s home and deliver it to the individual s destination His preliminary investigation suggested variable costs of 20bag and annual xed costs of 50000 The volume of business would depend on the level of interest in this type of service and the state of the economy If the level of interest was high Samuel believed he could generate a volume of 25000 bagsyear in a booming economy 20000 bagsyear in a normal economy and 17500 bagsyear in a recession If the level of interest was moderate he anticipated an annual volume of 15000 bags in a booming economy 10000 bags in a normal economy and 6250 bags in a recession Finally ifthe interest level was low Samuel expected to generate 2500 bagsyear in a booming economy 2000 bagsyear in a normal economy and 500 bagsyear in a recession Clayton thought there was a 60 likelihood that interest in his service would be high a 30 chance of moderate interest and a 10 probability that interest would be low Based on an examination of historical economic data Clayton thought the likelihood ofa booming economy was 20 and the likelihood of a recession was also 20 Starting up his business would be a fulltime venture Clayton would not start up the luggage delivery service unless it could increase his income beyond the 75000 annual salary he currently received Because he is uneasy about quitting his job for a potentially risky business venture Samuel spoke with a market research rm about 37 conducting a study to determine the level of interest in his business The market research study would cost 35000 Should Clayton start up the luggage delivery business Should he pay for the market research study PROBLEMS 1 A computer software company has to decide which of two advertising strategies to adopt TV commercials or newspaper ads Sales depend on the total viewership when the commercials are run and the total readership when the newspaper ads appear Experience dictates the corresponding level of sales when viewershipreadership is high medium or low TV Commercials Newspaper Ads Viewership Sales Readership Sales High 16000 High 12000 Medium 12000 Medium 10000 Low 8000 Low 8000 Media reports show that viewership is high 30 of the time medium 40 of the time and low 30 of the time Newspaper readership is high 40 of the time medium 40 of the time and low 20 of the time 38 The cost of the television commercials is 5000 whereas the cost of the newspaper ads is 1500 Calculate the expected pro t standard deviation and coef cient of variation Which advertising strategy would you recommend 2 A manufacturer of digital cameras is trying to decide whether to adopt a highprice strategy or a lowprice strategy The rm39s pro ts depend on the competitor39s reaction to the rm s strategy your competitor must also decide whether to use a high or lowprice strategy and the state of the economy either a booming economy a normal economy or a recession If the rm and its competitor adopt a highprice strategy then the rm s pro ts will equal 60000 in a booming economy 40000 in a normal economy and 20000 in a recession If the rm adopts a highprice strategy and its competitor goes with the lower price the rm s pro ts will equal 50000 in an economic boom 30000 in a normal economy and 20000 in a recession The likelihood that the competitor will go with a highprice strategy if the rm does is 60 Assuming the rm goes with a lowprice strategy and its competitor adopts a highprice strategy the rm s pro ts will be 50000 in a booming economy 40000 in a normal economy and 25000 in a recession 39 If its competitor uses the lowprice strategy at the same time the rm does the rm s pro ts will be 35000 in a boom 30000 in a normal economy and 25000 in a recession The likelihood that its competitor will go with a lowprice strategy if the rm does is 80 Historical data shows that the probability of an economic boom in any given year is 30 and the probability ofa recession is 20 Calculate the expected value standard deviation and coef cient of variation associated with each strategy Which strategy would you recommend Explain 3 DigiMusic is a manufacturer of MP3 players It has developed a player that can hold up to three times the video and audio images of competitors brands The company is considering applying for a patent The estimated cost of applying for a patent is 20000 If the patent is approved DigiMusic believes it can earn a pro t contribution of l50unit and sell two million unitsyear over the 20year life of the patent Convincing the Us Patent and Trademark Of ce that their product is patentable is not a given Following consultations with patent lawyers DigiMusic believes there is a 30 chance it will obtain a patent Management expects the pro t contribution will fall to 50unit after the patent expires and unit sales will fall to 500000 Similarly if DigiMusic goes into production without 40 seeking a patent it expects to earn 150unit and sell 2 million units for three years after which competition will reduce DigiMusic s pro t contribution and unit sales to 50unit and 500000 units respectively The USPTO publishes all patent ling applications 18 months after ling DigiMusic is concerned this will give competitors advance notice of its innovation If DigiMusic s patent application is not approved competitors will manufacture their own brands one year earlier than would normally be expected DigiMusic also worries about the possibility of patent infringements Competitors may decide to manufacture their own brands and ght a legal battle over the infringement charge A prolonged legal battle could cost DigiMusic 250 million If DigiMusic wins the legal battle it can obtain an injunction against the rm violating the patent and obtain legal damages that essentially restore its pro t contribution to its previous level throughout the infringement period In other words it will not recover the legal expenses but can retain its monopoly position for the remainder of the patent The company believes there is a 25 likelihood a competitor will violate the patent If this should occur legal experts believe DigiMusic s probability of having its patent upheld in court to be 80 a Create a spreadsheet to determine whether DigiMusic should ght a legal battle over infringements in each year of its patent ie is it worthwhile to ght an infringement that occurs in the patent s last year nexttolast year etc Assume a cost of capital of 8 Fquot Should DigiMusic seek a patent or go directly into production 41 4 Baby Rest Inc a manufacturer of car seats for babies is a defendant in a product liability case The plaintiff and defendant are preparing for trial but a jury has not yet been selected Based on extensive communications with their attorney the CEO believes there is a 10 chance a jury will award the plaintiff 5 million a 30 chance it will award 1 million a 30 chance it will award 500000 and a 30 chance it will award 0 Sustokovich Jury Consulting Associates has offered its services In preliminary meetings the consults claimed their services would reduce the likelihood of the 5 million award to zero percent and increase the likelihood of no award to 40 What is the maximum Baby Rest should be willing to pay for this service 5 All Things Christmas is preparing for the upcoming holiday season Management is trying to decide if it should offer gift wrapping as a free service during checkout Historically the store had not giftwrapped its items but offered giftwrapping at an additional charge Roughly 15 of purchasers paid extra for the giftwrapping which generated a profit contribution of 2item the cost of giftwrapping was 50unit The average pro t contribution excluding giftwrapping was 15 item If it offers complimentary giftwrapping management believes there is a 25 probability it will sell 3000 items a 25 chance it will sell 2500 items a 30 chance it will sell 2000 items and a 20 chance it will sell 1000 items If it does not offer complimentary giftwrapping it expects a 10 chance it will sell 3000 items a 20 chance it will sell 2500 items a 40 chance it will sell 2000 items and a 30 chance it will sell 1000 items 42 Should All Things Christmas offer complimentary giftwrapping 6 Bill is a seasonal farmer and grows vegetables during the spring and summer to sell at the local farmers market According to various sources Punxsutawney Phil and the Farmer s Almanac there is a chance of a late spring frost during the first week of April Bill could either plant his vegetables now midMarch after the predicted frost mid April or May 151 If Bill planted now there would be a 30 chance of a spring frost If the frost hit Bill would have to replant He could either replant immediately after the frost or wait two weeks to ensure the cold weather had passed If he replanted immediately after the frost there would still be a 20 chance of a frost If he waited two weeks after the frost there would only be a 5 chance of another frost If he waited until May 1st to plant he could avoid a frost altogether The problem Bill faces lies in the fact that he does not have a greenhouse while most of the vendors in the farmers market grow their vegetables in a greenhouse and do not have to worry about the possibility of a frost If Bill plants now and the frost does not hit his vegetables will be ready for the opening week at the farmers market If the frost does hit Bill s vegetables will not be ready until a month after the market opens There is some possibility he could be ready for the opening week of the farmer s market if he immediately replants around April 1 If Bill waits and plants in midApril his vegetables still will not be ready until a month after the market opens If he plants in April and suffers a frost in midApril his vegetables will not be ready until siX weeks after the market opens The same is true if he avoids the frost altogether his crops will not be ready until siX weeks after the market opens In either case most of the other vendors 43 will have established customers and Bill will have to reduce his price by 20 to gain market share Moreover if Bill misses a month at the market he will sell 30 less volume than he has predicted If he misses siX weeks he will sell 40 less Bill will also incur replanting costs of 100 each time he has to replant If Bill plants early and misses the frost he will sell 1500 pounds at 2lb When should Bill plant his vegetables 7 BioJoe is a company known for its ability to produce rice Its operation averages 10 higher yields than the industry average BioJoe wants to know if it should plant a rice that is resistant to sheath blight Sheath blight Rhizoctonia solani is a fungus that feeds off rice Once humidity reaches 95 the sheath blight infection begins to grow rapidly and will decrease yields depending on the severity of the infection However the rice has to be exposed to the fungus before the infection can occur BioJoe believes that there is a 40 chance the rice will be exposed to the fungus If the rice is infected with sheath blight BioJoe is estimating 110 bushelsacre compared to 190 bushelsacre if it isn t affected in a 100acre plot Resistant rice costs 775acre to produce compared to 520acre for rice that isn t resistant Both breeds will generate 475bushel BioJoe is trying to decide which breed to produce for this year Should BioJoe plant a breed that is resistant to sheath blight 44 8 Office Partners is considering extending its market into developing countries that show the highest percentage growth in personal computers Unfortunately because it lacks historical data it can only make sales projections based on various assumptions At the present time Of ce Partners has narrowed down its market to either Elizea or East Katia The net present values from market expansion in either country appear below Sales Projections Very Poor Average Good Very in millions of dollars Poor Good Country Elizea 35 10 l 25 60 East Katia 12 1 1 16 25 a If you were to make your decision via maximin which country would you choose b Suppose the probabilities of each sales level was as follows Very Poor 2 Poor 15 Average 5 5 Good 22 Very Good 5 How would this affect your recommendation 45 HANDS0N EXERCISES Decision Trees Decision trees represent a convenient way to lay out the outcomes associated with a decision The first branch de nes the decision the manager wishes to make Subsequent branches establish the factors that affect the outcomes of the decision Read over the below scenario We ll build the decision tree based on the information A firm is trying to decide whether to build a manufacturing plant in Detroit Michigan or the Twin Cities in Minnesota If it locates in Detroit its fixed costs will equal 300000 and it will incur variable costs of 12unit If the plant is located in Minnesota it will incur fixed costs of 600000 and variable costs of 10unit However there is a 20 chance a decrease in supply will drive up variable costs If so the variable costs will rise to 14 in Detroit and 13 in the Twin Cities Unit sales will depend on the state of the economy If economic conditions are normal the firm expects to sell 350000 units in Detroit and 375000 in the Twin Cities Ifthe economy is in a recession it anticipates unit sales of 200000 units in Detroit and 180000 in the Twin Cities Historically recessions have been known to occur once every four years The unit price will be 20 in either location unless a price war occurs in which case the price will fall to 15 in Detroit and 13 in the Twin Cities When price wars occur unit sales tend to rise by 10 Price wars tend to occur once every three years a What decision is the firm trying to make Indicate each alternative on one of the branches 46 b The rst factor that could in uence the rm s pro ts is the cost structure List each cost structure in the respective branch and the probability it will occur Detroit Twin Cities c The next factor that may in uence pro ts is the state of the economy List each possibility and the probability it will occur on one of the branches Low Cost Fixed 300000 Variable 12 39 P 80 High Cost Fixed 300000 Variable 1 4 unit P 20 Twin Cities Low Cost Fixed 600000 aria e un P 80 High Cost Fixed 600000 Variable 1 3unit P 20 48 c The last factor that affects pro tability is the price war List each possibility and its corresponding probability in each branch Norm al 332000 0lt Recession Low Cost Q 200000 Fixed 300000 0 P 25 Variable 12unit P 80 0 Normal Detroit 0 Q 350000 P 75 H hc Recession 1g 0 Q 200000 Fixed 300000 P 25 Variable 1 4unit P 20 Normal Low Cost Q 375000 Fixed 600000 P 75 O Variable I Twm 10unit O Recessron Cities P 80 Q 180000 39 P 25 o Normal Q 375000 High Cost P 75 0 Fixed 600000 Variable l3unit P 20 I Recessron Q 180000 P 25 49 a Calculate the pro t and probability associated with each branch Recall that the probability of independent events is the product of the individual probabilities Pro t Prob Norm al o 133533000 i Recession Low Cost Q 200900 Fixed 300000 0 p 25 Variable 12unit p 80 O 7 Normal Detroit 0 QP35000 Recession High Cost 0 Q 200000 Fixed 300000 P 25 Variable 7 7 1 4unit P 20 Normal Low Cost Q 375000 Fixed 600000 P 75 0 Twin Variable O Recessmn 10unit Cities P 80 Q 180000 P 25 o 7 Normal Q 375000 High Cost 75 Fixed P O Var1able O l3unit 7 P 20 Recess1on Q 180000 5 Expected Value a Fill in the pro ts for each of the below scenarios Calculate the average pro t at each location Detroit Twin Cities Scenario Outcome Outcome Low Cost Normal Economy No Price War Low Cost Recession No Price War Low Cost Normal Economy Price War Low Cost Recession Price War High Cost Normal Economy No Price War High Cost Recession No Price War High Cost Normal Economy Price War High Cost Recession Price War Mean Pro t b The below table shows both the outcomes and the probability that each outcome will occur Detroit Twin Cities Scenario Outcome Probability Outcome Probability Low Cost Normal Economy No Price War 2500000 4020 3150000 4020 Low Cost Recession No Price War 1300000 1340 1200000 1340 Low Cost Normal Economy Price War 855000 1980 637500 1980 Low Cost Recession Price War 360000 0660 6000 0660 High Cost Normal Economy No Price War 1800000 1005 2025000 1005 High Cost Recession No Price War 900000 0335 660000 0335 High Cost Normal Economy Price War 85000 0495 600000 0495 High Cost Recession Price War 80000 0165 600000 0165 Which outcome is least likely to occur Which outcome is most likely to occur c Based on your answer to b do you think the mean outcome is an accurate assessment of the average pro t at each location When the mean of a set of outcomes is calculated it implicitly assumes the probability each outcome will occur is the same When the probabilities differ as they usually do for business decisions the appropriate method for determining the average outcome is to calculate the expected value u The expected value is a weighted average It is calculated by multiplying each outcome by the probability it will occur and then summing them up or u ZPIXI 11 d Calculate the expected value for each location Scenario Low Cost Normal Economy No Price War Low Cost Recession No Price War Low Cost Normal Economy Price War Low Cost Recession Price War High Cost Normal Economy No Price War High Cost Recession No Price War High Cost Normal Economy Price War High Cost Recession Price War Expected Value Detroit Outcome 2500000 1300000 855000 360000 1800000 900000 85000 80000 Probability 4020 1340 1980 0660 1005 0335 0495 0165 Twin Cities Outcome 3150000 1200000 63 7500 6000 2025000 660000 600000 600000 Probability 4020 1340 1980 0660 1005 0335 0495 0165 Standard Deviation a Let s alter the cash ows for the Twin Cities The cash ows at the two locations and their respective probabilities are shown below Calculate the respective expected values Detroit Twin Cities Scenario Outcome Probability Outcome Probability Low Cost Normal Economy No Price War 2500000 4020 3000000 4020 Low Cost Recession No Price War 1300000 1340 1100000 1340 Low Cost Normal Economy Price War 855000 1980 561000 1980 Low Cost Recession Price War 360000 0660 51750 0660 High Cost Normal Economy No Price War 1800000 1005 1650000 1005 High Cost Recession No Price War 900000 0335 462500 0335 High Cost Normal Economy Price War 85000 0495 890250 0495 High Cost Recession Price War 80000 0165 73493150 0165 Expected Value b If your recommendation were based exclusively on the average annual pro t at each location which location would you recommend c Considering all of the information before you which location would you recommend Explain d What do your answers to questions b and c suggest about using expected value exclusively to make your decision e Given all of the infomation why might someone recommend Detroit The Twin Cities If you have a preference in this scenario you are considering risk along with expected value The more widely dispersed the outcomes the greater the risk Stande deviation 6 calculates the dispersion of outcomes around the mean It is calculated as As the equation implies the greater the difference between the outcomes and the mean the greater X I u2 Therefore as the standard deviation increases the dispersion of outcomes increases which implies more risk f Calculate the standard deviations for Detroit and the Twin Cities Decisionmakers may not always seek to avoid risk In this scenario whereas one decisionmaker may choose Detroit because the outcomes are closer together another decisionmaker may prefer the Twin Cities because it has the potential to pay off better than Detroit In other words the person who prefers the Twin Cities is willing to risk one of the lower payoffs in the Twin Cities for a chance at one of the higher payoffs An individual who prefers to avoid risk and therefore prefers a smaller standard deviation has risk averse preferences An individual who prefers risk and therefore prefers a larger standard deviation has risk seeking preferences Finally someone who is indifferent between standard deviations when the alternatives have the same expected value is risk neutral Coef cient of Variation Location A Outcomes Probability of Occurring 1 0 10 2 100 10 3 350 20 4 50 20 5 0 40 Location B Outcomes Probability of Occurring 1 1000100 10 2 999800 10 3 1000500 20 4 999950 20 5 999900 40 a Calculate the expected value and standard deviation for each of the locations Expected Value Standard Deviation b If you were to assess risk exclusively on the size of the standard deviation which location would you have indicated as the most risky c As you glance over the above information would you in fact identify this location as the more risky of the two d Is a simple comparison of standard deviation gures suf cient to assess risk Why or why not Standard deviation measures the actual dispersion of outcomes around the mean In this manner it is a measure of absolute risk In this scenario however a standard deviation of 145 when the expected value is 70 is on a relatively basis is fairly risky On the other hand a standard deviation of 240 when the expected value is 1000040 is pretty close to a sure bet When the expected values of alternatives differ managers may prefer to measure the level of relative risk This can be assessed by calculating the coef cient of variation y The coef cient of variation is determined by dividing the standard deviation by the expected value or y ou e Calculate the coef cient of variation for each location Interpret each number Decision Trees with Sequential Decisions A movie studio is trying to decide whether to nance a 150 million bigbudget lm or two smaller lms costing 70 million each The studio executives anticipate a subsequent request for an additional 50 million for the bigbudget lm and a combined 25 million for the two smaller lms well after the original nancing has been approved and production is under way If the additional funds for the bigbudget lm are approved the studio execs expect an 80 chance it will gross 300 million a 10 chance it will gross 200 million and a 10 chance it will gross 80 million If the budget increase is not approved the probabilities of a 300 million gross fall to 60 with the likelihood of 200 million rising to 25 and the probability of an 80 million gross increasing to 15 If the two smaller lms are nanced and budget increases of a combined 25 million are approved the executives believe there is a 60 chance the lms will collectively gross 300 million This probability falls to 30 if the budget increases are not approved Similarly the likelihood the smaller lms will gross a combined 200 million is expected to be 30 if the budget increase is approved and 40 if it is not approved Finally if the additional funds are approved the likelihood the lms will gross 80 million is 10 and 30 if not approved Again let s build the decision tree from scratch What decision is the movie studio trying to make List each alternative on one of the branches lt After the lm is approved the studio anticipates requests for additional funding 50 million if the bigbudget lm is approved and 25 million combined if the two smaller budget lms are approved List the sequential decisions on the next set of branches Bigbudget Film 15 0 Two million lms 0 0 60 List the anticipated gross and the probabilities those grosses will be realized on the corresponding branches Gross Probability Approve Additional 50 million lt Big Budget Film 150 million Do not approve Additional budget Approve 0 Additional 25 39 lon Do not Approv e Additional V f Notice that you can t determine the expected value standard deviation and coef cient of variation for the bigbudget and two smaller budget lms until you rst decide how to deal with the request for additional nancing List the outcomes and probabilities associated with the request for additional funding for the bigbudget lm Gross Probability 50 million 0 Determine the expected value standard deviation and coef cient of variation associated with the budget increase request don t forget to subtract the additional budget cost if approved Expected Value Standard Deviation Coef cient of Variation Would you approve the budget increase Explain List the outcomes and probabilities associated with the request for additional funding for the smallerbudget lm Gross Probability 25 million Approved Determine the expected value standard deviation and coef cient of variation associated with the budget increase request don t forget to subtract the additional budget cost if approved Expected Value Standard Deviation Coefficient of Variation Would you approve the budget increase Explain Redo the decision tree with the subsequent decisions predetermined Gross 150 million BigBudget Additional 50 million Not appro Two 70 mi 39 lms 25 million additio Budget approved Probability Determine the expected value standard deviation and coef cient of variation associated with the bigbudget and two smallerbudget lms Expected Value bigbudget Standard Deviation bigbudget Coef cient of Variation bigbudget Expected Value two smaller lms Standard Deviation two smaller lms Coef cient of Variation two smaller lms The Value of Information A rm is considering launching a new product into the region If launched throughout its retail outlets it anticipates the following annual outcomes and probabilities Pro t contribution Probability 2 million 50 05 million 25 01 million 10 02 million 10 05 million 05 a Calculate the expected value b The product manager anticipates the good will have a life of ve years after which it will become obsolete Based on a cost of capital of 8 estimate the present value of the income stream 65 c An alternative to launching the product throughout the region is to conduct a test market in a single market The pro t contribution for the region will be inferred from the following test market results Test Market Result Inferred Regional Pro t Contribution 200000 2 million 50000 05 million 10000 01 million lt 10000 0 product will not be launched Probability 50 25 d Calculate the revised expected value based on the results of the test market e Compare the expected value with the test market results with the expected value without the test market How much should the rm be willing to pay for the test market 66 Maximin Criterion Part I A rm is trying to decide between three projects A B and C It has identi ed four outcomes associated with each project The probabilities of each outcome are not known State of Nature 1 2 3 4 Project A 2 million 1 million 1 million 3 million Project B 5 million 05 million 3 million 6 million Project C 4 million 0 million 2 million 4 million a Circle the least favorable outcome associated with each project b Which of the circled outcomes is the most favorable Under the maximin criterion the project that has the most favorable worst case outcome is selected 67 Part 11 Suppose a rm is trying to choose between alternatives A and B The outcomes associated with each alternative are shown below State of Nature 1 2 Alternative A 01 02 Alternative B 01 100 million a Circle the least favorable outcome associated with each project b Which of the circled outcomes is the most favorable c Which alternative would you choose based on maXimin d Would your decision remain the same if the probability of state 1 was 10 1 01 e Based on the above implications what weakness is associated with the maXimin criterion 68 Minimax Criterion a Based on the below table circle the most favorable outcomes corresponding to each state of nature State of Nature 1 2 3 4 Project A 2 million 1 million 1 million 3 million Project B 5 million 05 million 3 million 6 million Project C 4 million 0 million 2 million 4 million b In the below table calculate the opportunity cost associated with choosing the wrong alternative corresponding to each state of nature For example if state of nature 1 took place the best alternative is A which loses 2 million Had the rm chosen A it could not have done better so its opportunity cost is 0 Had it chosen B it would lose 5 million which is 3 million worse than if it had chosen A Had it chosen C it would lose 4 million which is 2 million worse than if it had chosen A Complete the table State of Nature 1 2 3 4 Project A 0 7 7 7 Project B 3 million 7 7 Project C 2 million 7 7 c Circle the maximum opportunity cost associated with each project d Which project has the lowest maximum opportunity cost 69 Using the minimaX criterion the project that has the lowest maximum opportunity cost will be selected e The below table shows the opportunity cost table According to the table Project C has the smallest maximum opportunity cost State of Nature 1 2 3 Project A 0 1 million 2 million Project B 3 million 05 million 0 Project C 2 million 0 1 million 4 3 million 0 2 million How might your decision change if the probabilities of the states of nature were as follows State of Nature Probability l 01 2 97 3 01 4 01 f What is the primary limitation of the minimaX criterion 70 Figure 1 Outcome Probability Cost Detroit Twin Cities 2500000 4020 LoW 300000 600000 12unit 10unit 855000 1980 High 300000 600000 13unit 14unit 1300000 1340 Economy Normal 350000 units 375000 units Recession 200000 units 180000 units 360000 0660 Price War No 20 20 1800000 1005 Yes 15 13 Q 110 QT10 85000 0495 900000 0335 80000 0165 NW 3150000 4020 Normal 0 Low Cost 637500 1980 No Price War Rece 39 n 1200000 1340 39 War 6000 0660 High W 2025000 1005 0W0 Price War 600000 0495 Recession O NO Price War 660000 0335 Pmiwar 600000 0165 71 Figure 2 a Frequency 1 Outcome b Frequency 1 Outcome Figure 3 a Normal distribution Frequency Outcome b Triangular distribution Frequency Outcome c Uniform distribution Frequency Outcome d Lognormal distribution Frequency Outcome 75000 Figure 4 Gross 300 million Approve Additional 200 million 50 million lt Big Budget Film 80 million 150 million 300 million Do not approve 200 million Additional budget 80 million 300 million APprove 0 Additional 200 million 25 39 ion 80 million Do not Approve 300 million Additional Budget Y 200 million 80 million Probability 80 30 40

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