Robin Bea is the CEO of a small shipping company. She needs to decide whether or not to lease another truck to add to her current fleet.
If she leases the truck, she may be able to generate new business that would increase her company's profits. However, if additional business fails to materialize, she may not be able to cover the incremental leasing costs. Based on her assessment of future trends in the transportation sector, Robin identifies three scenarios that might transpire: "Boom," "Moderate Growth," and "Slowdown," depending upon the performance of the economy and its impact on the transportation sector. She associates with each scenario an estimate of the total profits an expanded fleet will generate for her firm.
If she doesn't lease the new truck, Robin probably won't generate as much revenue, but her leasing costs will be lower. Again, she breaks down the possible outcomes if she does not lease the additional truck into three scenarios corresponding to the performance of the economy. Then she associates an estimate of total firm profits with each scenario.
a. a.
This is not the best answer. Think about what kinds of nodes are present in this decision. b. b.
This is the best answer. c. c.
This is not the best answer. What order do the important events of this decision occur in? d. d.
This is not the best answer. What order do the important events of this decision occur in?
The first branching of the tree occurs at the point when Robin makes her decision: to lease or not to lease a new truck. A square decision node should represent this decision.
Each of Robin's options splits into three possible outcomes depending on the performance of the economy. Robin will learn which economic scenario transpires only after making her decision. The second branching represents this uncertainty. The nodes of these branches are chance nodes and should be drawn as circles. Exercise 4: Wheel of Fortune
You and a friend have a wheel of fortune that has a 75% chance of a "green" outcome and a 25% chance of a "red" outcome. Before you spin, the friend offers you $100 if you correctly predict the outcome. If you choose the wrong color, you receive nothing. The good news: you don't have to choose until the spin is complete. The bad news: you will have to keep your eyes closed until the wheel has stopped and you have made your prediction.
You close your eyes and keep them shut while the wheel spins. When the wheel stops, your eyes are still closed. You now must decide whether to choose "red" or "green.
Which of the following trees best represents your decision? a. a.
This is the best answer. b. b.
This is not the best answer. Think about the order in which you will discover the results of each node. c. c.
This is not the best answer. Think about the circumstances under which you win this game. d. d.
This is not the best answer. Think about the order in which you will discover the results of each node. The nodes of a decision tree are arranged from left to right in the order in which we discover their results, not in the order in which the events actually occur. Thus, even though the wheel stops and the result is finalized before you make your decision, because you don't know the result until after your decision is made, the chance node for the spin result should appear after — to the right of — the decision node.
Comparing the Outcomes
Alice's decision trees are neat devices to help organize and structure a decision problem. But how are they going to help Leo evaluate his options and choose the best one?
Introducing the Expected Monetary Value
Seth Chaplin must decide how to produce the movie Cloven. He has mapped out the logical structure of his decision in a decision tree and has incorporated the appropriate data,
evaluating each scenario in terms of its expected profits and its probability of occurring. Now, how should he use the tree to inform his decision?
If S&C Films works in partnership with K2 and retains part ownership of Cloven, S&C could earn a profit of $6 million, a highly desirable outcome. However, that scenario is relatively unlikely: 30%. How do we balance the high value of that outcome against its low probability? The answer is elegant and simple: we multiply the outcome value by its probability: $6 million * 0.3 = $1.8 million. Essentially, we "credit" the "Blockbuster" outcome with only 30% of its value.
The magnitude of the loss incurred if the film "Flops" is mitigated by its low likelihood of 20%.
Then we add these weighted values together, for a total of $1.4 million. This total is called the expected monetary value (EMV) of the K2 option. The EMV is a weighted average of the
expected outcomes of the scenarios in which S&C retains part ownership of Cloven as stipulated in the K2 deal.
Let's return to our wheel of fortune game to build our intuition for the concept of the EMV. This wheel consists of three areas: "blue," "green," and "red." The blue area is 30% of the total area. If a spin results in "blue," you win $6.
"Green" covers 50% and "red" covers 20% of the wheel's area. If a spin results in "green," you gain nothing at all, if the result is "red," you lose $2.
Suppose you play the game 100 times. How much money do you think you will have gained or lost over 100 spins? What do you estimate will be your average yield per game?
About 30% of the time, a spin will result in "blue." In other words, you can expect 30 of the 100 spins to yield $6.
About 50% of the time a spin will result in "green." You expect 50 of the 100 spins to yield $0. And about 20% of the time a spin will result in "red," — that is, you expect 20 of the 100 spins to cost you $2.
The total amount you can expect to win after playing the game 100 times is $140, and the average yield per spin is $1.40. That average is the expected monetary value (EMV) for a single spin.
The expected monetary value for a single spin is $1.40. If you spin the wheel once, which of the following results is least likely to occur as the outcome?
a. A loss of $2.
That is not the correct answer. Compute the probabilities of the four outcomes. Which is the least likely to occur?
b. No gain or loss (i.e. $0).
That is not the correct answer. Compute the probabilities of the four outcomes. Which is the least likely to occur?
c. A gain of $1.40.
This is the best answer. The probability of gaining $1.40 is 0. d. A gain of $6.
That is not the correct answer. Compute the probabilities of the four outcomes. Which is the least likely to occur?
The probability of each outcome is shown below. With a probability of 0%, $1.40 is the least likely outcome. It is important to understand the nature of the expected monetary value. We do not actually "expect" an outcome of $1.40. In fact, $1.40 is not even a possible outcome. The EMV of $1.40 is the long-term average value of the outcomes of a large number spins.
In the Cloven case, the EMV of $1.4 million is the average amount of profits Seth Chaplin can expect to make when he produces similar films in similar circumstances.
We can use the EMV as a measure with which to compare alternative options. First, we calculate the EMV for each chance node, beginning at the right of the tree. For the chance node associated with the K2 deal, the EMV is $1.4 million.
Now that we have calculated the EMV of the K2 chance node, we can "collapse" the branches emanating from the chance node to a single point. Going forward, we can treat the EMV of $1.4 million as the endpoint value of the K2 option.
The EMV for the Pony Option is simply $1.0 million: the outcome value multiplied by its probability of 100%. At a decision node, we choose the best EMV of all the branches emanating from that decision node. In our Cloven example, the best EMV is the one with the highest expected profits. The EMV of the K2 deal, $1.4 million, exceeds $1.0 million, the EMV of the Pony deal. Selecting the option with the best EMV and removing all other options from consideration is known as "pruning" the tree.
Any decision tree — no matter how large or complex — can be analyzed using two simple procedures. At each chance node, calculate the EMV, collapse the branches to a point, and
replace the chance node with its EMV. At each decision node, compare the EMVs and prune the branches with less favorable EMVs. This entire process is known as folding back the decision tree.
Summary
We often use expected monetary value (EMV) to quantify the value of uncertain outcomes. The EMV is the sum of the values of the possible outcomes of an uncertain event after each has been weighted by its probability of occurring. The EMV can be interpreted as the expected average outcome value of the uncertain event, if that uncertain event were repeated a large number of times. To analyze a tree, we "fold it back:" we move from right (the future) to left, finding the EMV for each node. For chance nodes we calculate the EMV as described below. For decision nodes we simply choose the option with the best EMV ! lower costs or higher profits ! among the choices represented by a decision node's branches and prune the others.
Relevant Costs
Burning to use decision trees and EMVs, you start to draw up the square and circular nodes that make up Leo's Chez Tethys decision. Alice, however, urges caution: "Before you get too trigger-happy with those decision trees, you should be aware of a few common pitfalls." Jen Amato has been driving "Millie," an old jalopy of a car, for the past five years. Millie — an Oldsmobile Delta-88 — is twenty years old. A week ago, the air conditioner broke down, and Jen had it replaced at a cost of $500.
Now, the drive train is worn out. Replacing it will cost $1,200. If she doesn't repair it, Jen can sell Millie "as is" for about $300. Jen must decide: should she sell her car or should she have the drive train replaced?
If Jen sells her car now, she will not even recoup her $500 investment in the air conditioner. If she has the drive train replaced, her beloved Millie will last a little longer, and Jen will be able to enjoy the cool air she spent so much money on. How should the fact that Jen hasn't had the opportunity to benefit from her air conditioner investment affect her decision?
With a broken drive train, Millie is worth about $300. According to her mechanic, if Jen pays $1,200 to replace the drive train, Millie will be worth approximately $1,100 in terms of resale value. In the analysis of Jen's decision, what role should her $500 investment in the air conditioner play?
In any scenario in which Jen sells her car, the $500 air conditioner cost has already been incurred, so it could be included in the total cost.
Likewise, the air conditioner repair costs are part of the prehistory of any scenario in which Jen has Millie's drive train replaced. Here, too, the $500 investment in the air conditioner could be included as a cost.
However, when we compare the total costs of the two options, we recognize that the $500 does not contribute to a difference in the outcome values. Because the $500 cost is incurred in both cases, it plays no role in the comparison of the two options, and thus is irrelevant to Jen's decision — she will have spent the $500 no matter what she decides to do now.
Costs that were incurred or committed to in the past, before a decision is made, contribute to the total costs of all scenarios that could possibly unfold after the decision is made. As such these costs — called sunk costs — should not have any bearing on the decision, because we cannot devise a scenario in which they are avoided.
It isn't wrong to include a sunk cost in the analysis as long as it is included in the value of every outcome. However, including sunk costs distracts from the differences between
scenarios — the relevant costs. Imagine the complexity of Jen's tree if she included every sunk cost she's incurred from owning Millie — from the original purchase price of the car to the cost of all the gasoline she's pumped into Millie over the years, to her expenditure on a dashboard hula dancer.
Misinterpreting a sunk cost as a cost that weighs on only some of the scenarios is a common error. After sinking $500 of repairs into her car, selling it for $300 will be quite painful for Jen. Nonetheless, good decisions are made based on possible future outcomes, not on the desire to correct or justify past decisions or mistakes.
Another common decision-making error is to omit from the analysis relevant costs that should have a bearing on the decision. If Jen has Millie's drive train replaced, the repair costs are just one of many costs involved with that option. Given Millie's age, there is a high likelihood of another repair cost soon. Similarly, if Jen sells Millie now, she will have costs associated with buying a new car or arranging other transportation services.
Opportunity costs are an important cost category that decision makers often neglect to include in their analyses. For example, selling the car will require Jen to devote 10 hours of her time that she would otherwise devote to her part-time job paying $12 per hour. Thus, Jen should add $120 in opportunity costs to the outcomes of the "sell Millie" decision.
We should also take non-monetary costs into account. Jen will feel sad at leaving her trusted vehicle and companion on many a road trip behind. Although such costs can be difficult to quantify, they should not be neglected. We will see shortly how to use sensitivity analysis to incorporate non-monetary costs into a decision analysis.
Summary
Among the most common errors in decision analysis is the failure to properly account for the costs involved in different possible scenarios. On the one hand, relevant costs such as opportunity costs or non-monetary consequences are often omitted. On the other hand, irrelevant costs such as sunk costs are incorrectly included in the analysis. Sunk costs are costs that were incurred or committed to prior to making the decision and cannot be recovered at the time the decision is being made. Since these costs factor into any possible future outcome, they can be safely omitted from the analysis; sunk costs must never be included in only selected branches of a decision tree.
Time Horizons
Jen has decided to replace her old car, Millie the Oldsmobile. Her friend, Sven, is leaving the country for two years and doesn't want to pay to store his Mazda Miata. He offers Jen two options.
In the first option, Jen leases the car, paying Sven $700 each year. When Sven returns from abroad, he reclaims his car. In the second option, Jen buys the car outright for $4,000. How should Jen compare these two options?
What is the appropriate cost difference Jen should use to compare the two options Sven has offered? a. $3,300
This is not the best answer. Read on to learn more. b. $2,600
This is not the best answer. Read on to learn more. c. $1,400
This is not the best answer. Read on to learn more. d. None of the above.
This is the best answer.
To begin, note that in the first option, the costs are spread across two years, whereas in the second option, Jen makes a single payment when she closes the deal with Sven. To compare the options meaningfully, we must define a time horizon for this problem, that is, we must compare their costs over a common time period. In this case, two years is a convenient time horizon.
Next, note that we cannot directly compare $1,400, the simple sum of the two lease payments made at two different times, to the $4,000 one-time cost of the purchase option paid entirely at the beginning of the first year. We can only compare costs when they are valued at the same point in time.
We'll compare the present value of the costs associated with each option — that is, the value of the costs at the time Jen makes her decision. In order to compare the costs, we first need to convert the second installment of the leasing payment into its present value.
Jen currently has sufficient cash for either alternative in an investment account with a 5% rate of return. What is the present value of the second installment of $700 paid under the leasing option?
a. $700.00
This is not the best answer. Remember that future cash flows must be discounted. b. $666.67
This is the best answer. c. $735.00
This is not the best answer. Think about how the value of money changes over time. d. None of the above.
This is not the best answer. Think about how the value of money changes over time.
The present value of the second installment of $700 is $666.67. At the end of one year, $666.67 residing in her investment account today will have increased in value to $666.67*(1.05) = $700, the amount she must pay for the second lease installment.
The present value of the total cost of the leasing option is $1,366.67. Can we use this number to compare the two options?
Although $1,366.67 and $4,000 are now comparable costs because they are given at their present value, we have not yet considered the fact that under the purchase option, Jen will own the Miata at the end of the two-year period. In two years, Jen expects that the Miata's market value will be around $3,000. The value of an asset at the end of the time horizon is typically called its terminal value.
What's the net present value of the cost of the purchase option - that is, the present value of future cash flows after subtracting, or "netting out," the initial payment to Sven? Recall that Jen currently has her money invested in accounts that earn a 5% average annual return.
a. $4,000.00
This is not the best answer. Think about the value of Jen's assets in two years. b. $2,721.09
This is not the best answer. Think about the value of Jen's assets in two years. c. $1,278.91
This is the best answer. d. $1,142.86
This is not the best answer. Think about the value of Jen's assets in two years.