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Calculating the internal rate of return
The expected value of the consultant’s report is determined by comparing the EMV for the consultant with the EMV for the best action determined based on prior probabilities. Based upon prior probabilities used in the previous example, Global Airlines charged the same fares with an expected monetary value of $48.65 million. If Global Airlines hires the consultant and acts on the consultant’s report, the expected monetary value increases to $48.8601. The difference between the expected monetary value given the consultant’s report and the expected monetary value based on prior probabilities is the maximum amount Global Airlines is willing to pay the consultant. This difference is $0.2101 million or $210,100 — $48.8601 minus $48.65.
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Taking Chances with Risk Preferences
Individuals have different risk preferences. Some people buy lottery tickets all the time, while others never buy them. Some people invest millions in the newest innovation, while others stay with the tried and true. Different risk preferences result from differences in individual satisfaction or dissatisfaction arising from risk.
As I note in Chapter 5, utility is a subjective measure of satisfaction that’s unique to an individual. A utility function is an index or scale that measures the relative utility of various outcomes. Economists use utils to measure the amount of an individual’s satisfaction. The concept of utility enables individual risk preferences to be incorporated in decision-making criteria.
Constructing a utility function
Because utility is subjective, one individual’s utility function can’t be compared to another individual’s utility function. You must construct a utility function by determining the individual’s utility associated with each possible outcome.
To construct a utility function, you need to compare two alternatives. One alternative is called the standard lottery. The standard lottery has two possible outcomes, such as a payoff of $A occurring with probability P and a payoff of $B occurring with probability 1 – P. The second alternative is called the certainty equivalent. This alternative has a certain payoff of $C.
You’re indifferent when the expected utility from the certainty equivalent (first alternative) equals the expected utility from the standard lottery (second alternative). Mathematically, an individual is indifferent if
Chapter 15: Risk Analysis: Walking Through the Fog
where U($) represents the utility associated with that payoff and P and (1 – P) represent probabilities.
In order to construct a utility function, you take the following steps:
1. Assign utility values to two outcomes.
For example, receiving $0 may have a utility value of 0, while receiving $100 has a utility value of 200.
2. Define the monetary certainty equivalent.
Assume the two outcomes specified in Step 1 represent a standard lottery, where each outcome has a 50 percent chance of occurring. After you specify the standard lottery, you must determine what amount of money received with certainty would make you indifferent to the standard lottery.
For example, you may decide that you’d be indifferent between receiving $45 for certain and taking the gamble represented by the standard lottery.
3. Determine the utility of the certainty equivalent.
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Thus, you get 100 utils of satisfaction from $45.
4. Repeat these calculations for other standard lottery and certainty equivalent combinations to lead to other utilities.
For example, the new standard lottery has two possible outcomes: receive $0 or $45. Each outcome has a 50 percent chance of occurring.
You decide that you’d be indifferent between receiving $25 for certain and taking the gamble represented by the standard lottery.
You get 50 utils of satisfaction from $25.
Risking attitude
Attitudes toward risk vary among individuals. Some individuals gladly take on additional risk, while other individuals willingly pay a substantial premium, such as with health insurance, to reduce their risk of loss. Individual attitudes toward risk are grouped into the following three categories:
✓ You’re risk adverse if you have two alternatives with the same expected monetary value, and you choose the alternative with less variation in outcomes.
✓ You’re risk neutral if you have two alternatives with different variations in outcomes and the same expected monetary value and you’re indifferent between those alternatives. Risk neutral individuals aren’t influenced positively or negatively by risk. Risk neutral individuals will always choose the alternative with the highest expected value. ✓ You’re a risk taker or risk lover if two alternatives have the same expected monetary value and you choose the one with the highest variability. Lottery players are risk takers.
Figure 15-3 illustrates the relationship between the expected payoff and utility for individuals with different risk preferences. Note that the additional satisfaction risk adverse individuals get from higher payoffs is decreasing while the additional satisfaction risk takers get from higher payoffs is increasing.
Figure 15-3:
Attitudes toward risk.
Using the expected utility criterion
After you determine the utilities, you maximize a decision’s expected utility. Expected utility equals the sum of each possible outcome’s utility multiplied by the probability of the outcome occurring.
