Nancy is indifferent between a gamble that pays $625 with a probability of 20% and $100 with a probability of 80%, versus a sure payment of $269. She is also indifferent between another gamble that pays $269 with a probability of 50% and $10 with a probability of 50%, versus a sure payment of $49. Let (p1, p2, p3, p4, p5: 625, 269, 100, 49, 10) represent a prospect (gamble) where p1, p2, p3, p4, and p5 represent the probabilities of the prizes $625, 269, 100, 49, and 10, respectively.
Determine whether Nancy is risk-averse, risk-loving, or risk-neutral, based on her preferences.
Rank the following four prospects according to Nancy’s preferences.
A: (0.2, 0.2, 0.2, 0.2, 0.2: 625, 269, 100, 49, 10)
B: (0.4, 0, 0, 0, 0.6: 625, 269, 100, 49, 10)
C: (0, 0.1, 0.8, 0, 0.1: 625, 269, 100, 49, 10)
D: (0, 0, 1.0, 0, 0: 625, 269, 100, 49, 10)
To determine whether Nancy is risk-averse, risk-loving, or risk-neutral, we need to examine her choices and preferences towards risky prospects.
In the first scenario, Nancy is indifferent between a gamble that pays $625 with a probability of 20% and $100 with a probability of 80%, versus a sure payment of $269. This implies that she finds the expected value of the two gambles to be equal. To determine the expected value, we multiply each outcome by its respective probability and sum them up:
Gamble A: (0.2 * $625) + (0.8 * $100) = $125 + $80 = $205
Sure Payment: $269
Since $205 is less than $269, Nancy is risk-averse in this scenario since she preferred the sure payment over the gamble.
In the second scenario, Nancy is indifferent between a gamble that pays $269 with a probability of 50% and $10 with a probability of 50%, versus a sure payment of $49. Again, we calculate the expected value:
Gamble B: (0.5 * $269) + (0.5 * $10) = $134.5 + $5 = $139.5
Sure Payment: $49
Since $139.5 is greater than $49, Nancy is risk-loving in this scenario since she preferred the gamble over the sure payment.
Based on Nancy's preferences in the two scenarios, we can conclude that she is neither strictly risk-averse nor risk-loving but exhibits risk-neutral behavior. This means that Nancy views the risk in the first scenario as unfavorable and prefers the sure payment. However, she views the risk in the second scenario as favorable and chooses the gamble.
Now, let's rank the four prospects (A, B, C, D) according to Nancy's preferences:
1. Prospect B: (0.4, 0, 0, 0, 0.6: 625, 269, 100, 49, 10)
Nancy prefers this prospect over the others because she is risk-loving in this scenario and values the potentially higher payoff.
2. Prospect A: (0.2, 0.2, 0.2, 0.2, 0.2: 625, 269, 100, 49, 10)
Nancy is indifferent between Prospect A and Prospect B since the expected values of the two prospects are equal. Both prospects offer the same level of risk, and she values them equally.
3. Prospect C: (0, 0.1, 0.8, 0, 0.1: 625, 269, 100, 49, 10)
Nancy prefers Prospect C over Prospect D because it offers a higher expected value due to the 0.8 probability of the $100 payoff. However, it is ranked lower than Prospect A and B because she values the sure payment in both of those scenarios.
4. Prospect D: (0, 0, 1.0, 0, 0: 625, 269, 100, 49, 10)
Nancy prefers this prospect the least because it only offers the sure payment of $100, which is lower than the other prospects that have potentially higher payoffs.
Therefore, the ranking of the prospects according to Nancy's preferences is: B > A > C > D.