After a public relations debacle over a possible outbreak of pulluscular pig disorder (PPD) at one of Bowman-Lyons-Centerville's hog farms, Paul Segal, head of the hog farming division, is considering vaccinating another herd at a hog farm located near the original outbreak. Immunity to PPD is fairly common among pigs; the probability that the entire herd in question is immune is fairly high: 60%.
However, if any pigs in the herd are not immune, the expected cost to Bowman-Lyons- Centerville is $150,000. This cost has been calculated based on the probability of a PPD outbreak and the ensuing costs to contain the disease and manage the expected PR fallout. The cost of vaccinating the herd is $40,000. What is the EMV of the cost of not vaccinating the herd?
Enter the EMV in $thousands as a decimal number with one digit to the right of the decimal point (e.g., enter "$5,500" as "5.5"). Round if necessary.
The EMV of the cost of not vaccinating the herd is $60,000. Since the cost of vaccinating the herd — $40,000 — is lower than the expected cost of not vaccinating the herd, Paul should have the herd vaccinated, barring the emergence of any further information.
There is a test that Paul could perform on the herd to determine if the herd is immune to PPD. This test delivers two results: "Positive" for immunity, or "Negative" for immunity. The test is not perfectly accurate: if the entire herd is immune, the test will report "Positive" with probability 85% and "Negative" with probability 15%. Similarly, if any of the pigs in the herd are not immune, the test will report "Positive" with probability 30% and
"Negative" with probability 70%.
Using Paul's prior probability for the immunity of the herd, calculate P(Positive), the marginal probability that the test will report a "Positive" result.
Enter the probability as a decimal number with three digits to the right of the decimal point (e.g., enter "50%" as "0.500"). Round if necessary.
First, calculate the joint probabilities P(Positive & Immune) and P(Positive and Not Immune), using the marginal probabilities for immunity and non-immunity — 60% and
40%, respectively — and the conditional probabilities that quantify the reliability of the test - P(Positive | Immune) and P(Positive | Not Immune).
Then, sum the joint probabilities to find the marginal probability of a "Positive" test result: 63%. Since the test results "Positive" and "Negative" are mutually exclusive and collectively exhaustive events, the probability of a "Negative" is simply 100% minus the probability of a "Positive," i.e., P(Negative) is 37%.
The tree below will help Paul determine whether or not ordering the immunity test will be worthwhile. The calculated probabilities of the test results are entered in the appropriate branches. However, to reach a decision, Paul needs the conditional probabilities of immunity and non-immunity conditioned on a "Positive" test result. What is P(Immune | Positive)? Enter the probability as a decimal number with three digits to the right of the decimal point (e.g., enter "50%" as "0.500"). Round if necessary.
To calculate P(Immune | Positive), use the definition of conditional probability and divide the joint probability P(Positive & Immune) by the marginal probability P(Positive).
P(Immune | Positive) is 81%. Since immunity and non-immunity are mutually exclusive and collectively exhaustive events, P(Not Immune | Positive) is simply 100% minus P(Immune | Positive) i.e., 19%.
Calculate the remaining probabilities on the decision tree. What is the highest amount Paul should be willing to spend on the immunity test?
Enter the expected value of sample information in $thousands as a decimal number with one digit to the right of the decimal point (e.g., enter "$5,500" as "5.5"). Round if necessary. The expected value of not vaccinating given a "Positive" test result is $28,500. Since this is lower than the cost of vaccination, Paul should choose not to vaccinate on EMV grounds. The probability that the herd is immune given that the test reports "Negative" is 24.3%. Since the EMV of the cost of vaccinating the herd ($40,000) is lower than the EMV of the cost of not vaccinating ($113,550) Paul should choose to vaccinate if the test result is "Negative".
The EMV of conducting the immunity test is $32,755, i.e., the sum of the EMVs of "Negative" and "Positive" test results, weighted by their respective probabilities.
The expected value of sample information, EVSI, is the difference between the EMVs of testing and not testing for immunity, i.e., $7,245. The most Paul should be willing to pay for the immunity test is $7,245.
Risk Analysis
Just as an e-learning course must have a bittersweet last section, so too must your internship on Hawaii have a final day. A parting Pacific frolic is refreshing preparation for your meeting with Leo.
Introducing Risk
As you dry in the morning sun, Alice invites you to consider Leo's position should he wager the Kahana on the Tethys' success. "Even if the potential profits are substantial, losing the Kahana if the Tethys tanks would be a severe blow. I wonder how seriously he's considered the
potential downside."
Imagine: upon your arrival at business school you buy yourself a nice new car. It's a sleek, powerful, Burgundy Ben Hur by Chariot — with both a Sweetone stereo and a Theft
Discouragement System — worth over $25,000.
City driving can be rough: the probability that you'll be involved in an accident that "totals" your car — reduces its value from $25,000 to zero — in your first year is 0.01%. Collision insurance to cover such a loss in your new city is optional: it will cost you $1,600 per year above and beyond the premium for required liability insurance. Will you buy the collision
insurance?
a. Yes, I will buy collision insurance.
There is no single correct answer to this question. Your answer depends on your risk tolerance. Continue on with the text to explore ways to think about risk.
b. No, collision insurance is for the weak.
There is no single correct answer to this question. Your answer depends on your risk tolerance. Continue on with the text to explore ways to think about risk.
c. A friend of mine in town can get me much cheaper insurance than that!
Good for you! Please pass your friend's contact information on through this application's discussion board.
d. Could I please have the Bennie in Champagne instead?
There is no single correct answer to this question, but there are an infinite number of wrong ways to answer, responding with an unrelated question being one of them.
Most people would pay the premium to mitigate the risks associated with a car accident leading to such large losses. But let's look at the decision tree for buying collision insurance. For simplicity, we'll ignore accidents that do not total your car.
After a year in school, your Bennie's market value will have a present value of $25,000. If you don't buy collision insurance and the car is totaled, you'll lose the full value of the car — $25,000. If — as we all hope — you and your car survive the city's roads unscathed, you won't lose anything.
Opting for collision insurance entails a $1,600 insurance premium. If you are spared the
calamity of a wreck, you'll have incurred just the premium cost at the end of the year. If traffic tragedy befalls you, your insurance company will pay out the value of the car minus a $1,000 deductible: so at the end of the year you will be out $2,600: the premium plus the $1,000 deductible.
Which option is better in terms of the EMV? a. Buying collision insurance.
This is not the best answer. Think about what a "better" outcome is in this situation: a higher EMV or a lower EMV?
b. Not buying collision insurance. This is the best answer.
At $2.50, the EMV of not buying collision insurance is much better than the EMV of buying the insurance. Essentially, if you buy the collision insurance, you are paying $1,600 to avoid an expected loss of $2.50! We shouldn't be surprised — insurance companies are not charities, and from their perspective, selling you insurance has a very favorable EMV.
Still, a majority of drivers choose collision insurance, even though it has a significantly lower EMV. Why might this be the case?
Let's consider a different situation. Suppose you owned a bright and shiny miniature windup toy Bennie for $25. Would you take out a $1.60 insurance policy on it, even if the probability were upwards of 50% that it would be stepped on or lost in the coming year?
Probably not. A $25 loss is not even a minor catastrophe. If you lost or accidentally destroyed your miniature Bennie, you'd either replace it quickly, or accept its departure without fuss. If instead of a nice, new, $25,000 Bennie you owned a 20-year old beat-up and rusty
Oldsmobile Delta-88, worth about $2,500, you'd also think twice about paying $160 to insure it against collision. A loss of $2,500 — though stinging — is unlikely to be a major setback, especially when weighed against a $160 premium.
But, imagine that you — assisted by your MBA degree — amass a multi-million dollar fortune over the next decades. Would you still insure a $25,000 car against collision? Perhaps not. Clearly, the value of the potential loss relative to your net worth has an important influence on your willingness to take on the risk associated with uncertainty. Suppose you are invited to play a gambling game in which you could win $10 million with 50% probability or lose $2 million with 50% probability. Even though the EMV of playing this game is $4 million, you'd probably decline participation unless you can afford a $2 million loss.
For most of us, the pain of the $2 million loss would weigh more heavily than the joy of winning $10 million, so we would not accept this gamble. For someone who can afford a $2 million loss, this game might be very profitable, especially if played multiple times.
There are formal ways to measure such assessments by assigning a personal utility value to each outcome. Then we can show that maintaining the status quo — our current assets — provides greater utility than playing the game does. Rigorous methods of quantifying utility are beyond this course's scope. For now, let's look at how we might gain insight into the personal utility we associate with different monetary outcomes.
Returning to collision insurance, let's summarize the scenarios and their probabilities and outcome values. You have two options: "Buy collision insurance" and "Don't buy collision insurance.
If you insure against collision, there are two possible outcomes: either you avoid a wreck for one year and lose the insurance premium of $1,600, or misfortune strikes and your Bennie is totaled. In the latter case, you lose the premium and the deductible. The probabilities
associated with these two outcomes are 99.99% and 0.01%, respectively. The possible outcomes and their respective probabilities are listed in the table below.
If you don't insure against collision and park your car safely at year's end, you won't sustain any loss at all. If the the unfortunate alternative occurs and your Bennie is totaled, you'll have lost its value of $25,000. Again, the probabilities of these outcomes are 99.99% and 0.01%, respectively.
We now have two tables that completely summarize the possible scenarios of our decision. The monetary values of these scenarios are arranged from top to bottom, from the best outcomes to the worst. These tables — one for each option — are together referred to as the risk profiles for the decision.
From the risk profiles, it easy to recognize that the option of not buying insurance will deliver the most preferred outcome if you don't have an accident but will deliver the least desirable outcome if you do. This insight provides the basis for you to prefer to buy the collision insurance: it exposes you to lower risk, even though it has a lower EMV.
Summary
The EMV criterion is not the only criterion that informs decisions. Besides wanting to
maximize our average outcomes in the long run, we want to minimize our exposure to risk, especially when potential losses are high relative to our own net worth.
Risk Attitudes
Recall Seth Chaplin and S&C Films' production of the movie Cloven. Seth has a choice between two options. In a deal with Pony Pictures, S&C produces the film, and Pony acquires the complete rights to Cloven for a flat production fee. In a second deal with K2 Classics, S&C retains part ownership, and the financial outcomes for S&C depend on Cloven's performance at the box office.
Seth knows from previous analysis that the K2 deal has a higher EMV: $2.04 million vs. $1.48 million for the Pony deal. However, he also knows the K2 deal is riskier. Before making a final decision, Seth wants to understand the full implications of his two options. Let's build risk profiles for each deal.
There are seven possible scenarios that can occur. The outcome values of these scenarios depend on Seth's choice of a production partner, on whether or not superstar actor Shawn Connelly lends his gravelly baritone to the film's lead character, and on the audience's reception of the film.
Let's summarize the possible scenarios. If Seth chooses the Pony option, two scenarios could occur. In one, Connelly participates in the movie and Seth's profits are $2.2 million. In the other, Connelly does not participate in the movie and Seth's profits are only $1 million. If Seth opts for the Pony deal, the probabilities of these two scenarios occurring are 40% and 60%, respectively.
If Seth takes the K2 option, there are five possible scenarios, but only three possible outcomes: "Blockbuster" success, a "Lackluster" performance, and a "Flop.
A "Blockbuster" success can occur whether or not Connelly participates in the production. The probability of a "Blockbuster" is 38%: this probability is calculated by adding the joint probability that Cloven stars Connelly and is a "Blockbuster" to the joint probability that Cloven is a "Blockbuster" without Connelly's participation.
A "Lackluster" performance can also occur whether or not Connelly participates in the production. The probability of a "Lackluster" performance is 50%: this probability is
calculated by adding the joint probability that Cloven stars Connelly and has a "Lackluster" performance to the joint probability that Cloven has a "Lackluster" performance without Connelly's participation.
A "Flop" occurs only when Connelly doesn't take part in the production of Cloven. The total probability of a "Flop" is the joint probability of Cloven being a "Flop" and Connelly not taking part: 12%.
Seth now has complete risk profiles for the two options. The risk profiles give Seth a quick overview of the possible outcomes — and their respective probabilities — for each choice. Note that we've combined scenarios so that we now have probabilities for each distinct outcome value. How does a manager like Seth use risk profiles to inform a decision? Risk profiles allow Seth to compare options not just in terms of their EMVs but in terms of the risk they expose him to. From Seth's risk profile we can see that, although the K2 option has the higher EMV, it is also associated with greater risk: a $2 million loss is possible, whereas the Pony option doesn't include any possible loss.
Which option Seth should choose ultimately depends on how comfortable he is with the risk associated with the K2 option. The Pony option is certain to deliver a profit. The K2 option might generate a substantial loss.
If a $2 million loss is more than Seth believes his company can — or should — sustain, he may want to choose the Pony option despite its lower EMV. If Seth chooses the Pony option, he will be demonstrating that he is risk averse: he is choosing an option with a lower EMV in return for lower exposure to risk.
Most people are at least slightly risk averse, as shown by their inclinations to buy insurance. Some people tend to be risk seeking, that is, they are willing to forgo options with higher EMVs and accept higher levels of risk in return for the possibility of extremely high returns. Although risk-seeking behavior is generally rare in the population when significant down- side risk is involved, many people tend to be somewhat risk seeking when the value of the loss risked is low. For example, the EMV for playing the lottery is lower than for not playing the lottery. However, since the amount risked by playing the lottery is so low — the price of a lottery ticket — many people choose to pay it in return for the unlikely outcome that they will win a multi-million dollar prize.
When we use the EMV as the basis for decision making, we are acting in a risk neutral manner. Most people are risk neutral when they make routine "day in, day out" types of decisions — those for which potential losses are not terribly harmful to the decision-maker or her organization.
People feel comfortable using the EMV for routine decisions because they feel comfortable "playing the averages": some decisions will result in losses and some in gains, but over the long run, the outcomes will average out. We expect that, over time, choosing options with the highest EMVs will lead to highest total value.
Most people feel comfortable using EMV as the basis for decisions that are not made repeatedly — provided the decisions have outcome values similar to those of other, more routine decisions that they make on a regular basis.
Why would Pony Pictures agree to purchase the rights to Cloven and take on all the risk associated with marketing and distributing the film? For Pony, the deal with Seth is a routine decision. Pony is a large enough company to sustain a potential multi-million dollar loss if Cloven flops. Pony also distributes 10 to 12 movies a year, so overall, the positive EMV of a film release promises that in the long run, Pony's operations will be profitable. Seth knows that the K2 deal is very risky: if he chooses the K2 deal, he will tie his
company's financial success to box office performance and to Shawn Connelly's whims. If Cloven "Flops," S&C Films might well go bankrupt.
But Seth likes a gamble... Summary
Risk profiles allow us to assess the utility different outcomes bring us, as opposed to their monetary value. The concise summary risk profiles provided helps us compare and
contrast our different decision options, allowing us to choose the option we prefer based on our attitude to risk: risk averse, risk seeking, or risk neutral.
Solving the Market Research Problem (II)
"Although Leo's market research event doesn't pay off in terms of its expected value," Alice explains, "it could help him manage his risk."
Let's assemble the risk profile for Leo's decision, including the decision about whether or not to run his market research event. We have to keep in mind that the cost of running his event is $240,000.
Which of the following tables correctly summarizes the outcomes and probabilities if Leo chooses to run his market research event?
a. a
This is not the best answer. Think about the costs associated with the market research event. b. b
This is not the best answer. Think about the costs associated with the market research event. c. c
This is not the best answer. Think about what the probabilities of any set of mutually exclusive events must add up to.