You have available five different investment strategies and their respective payoffs for various states-of-nature as shown in the chart below. Which investment would you make under the different decision criteria?
States of Nature
Sever
Decline Moderate Decline Stable Moderate Advance Strong Advance
T-Bills 3.50 3.50 3.50 3.50 3.50
Paragon (22.50) (2.00) 20.00 35.00 50.00
Luster 28.00 14.70 0.00 (10.00) (20.00)
Apex 10.00 5.00 7.00 45.00 30.00
Market (13.00) 5.00 22.00 38.00 Portfolio 47.00
Probability 0.05 0.35 0.30 0.20 0.10
Note: For the following questions, ¡§Investment¡¨ is the name of the strategy not the number, i.e., Luster, Apex, etc.
a)Maximin criteria
Answer: Investment
b)Maximax criteria
Answer:Investment
c)Equally Likely criteria
Answer:Investment
d)Criterion of realism (assume Ą = 0.6)
Answer:Investment
e)Minimax regret criteria
Answer:Investment
f)Criterion of maximum expected value
Answer:Investment
a) Maxmin criteria: In this criteria, we select the investment strategy that maximizes the minimum payoff for each state of nature.
Looking at the chart, the minimum payoffs for each investment strategy are as follows:
- T-Bills: 3.50
- Paragon: -22.50
- Luster: -20.00
- Apex: 5.00
- Market: -13.00
The maximum of these minimum payoffs is 5.00, which corresponds to the Apex investment strategy. Therefore, under the Maxmin criteria, the investment to make would be "Apex".
b) Maximax criteria: In this criteria, we select the investment strategy that maximizes the maximum payoff for each state of nature.
Looking at the chart again, the maximum payoffs for each investment strategy are as follows:
- T-Bills: 3.50
- Paragon: 50.00
- Luster: 28.00
- Apex: 45.00
- Market: 47.00
The maximum of these maximum payoffs is 50.00, which corresponds to the Paragon investment strategy. Therefore, under the Maximax criteria, the investment to make would be "Paragon".
c) Equally Likely criteria: In this criteria, we assume that each state of nature has an equal probability of occurring. We calculate the expected payoff for each investment strategy and select the one with the highest expected payoff.
The expected payoffs for each strategy can be calculated by multiplying the payoffs for each state of nature by their respective probabilities and summing them up.
For example, the expected payoff for T-Bills would be:
(3.50 * 0.05) + (3.50 * 0.35) + (3.50 * 0.30) + (3.50 * 0.20) + (3.50 * 0.10) = 3.50
Similarly, calculating the expected payoffs for other strategies, we find:
- T-Bills: 3.50
- Paragon: 4.45
- Luster: 4.74
- Apex: 16.35
- Market: 17.70
The strategy with the highest expected payoff is "Market" with an expected payoff of 17.70. Therefore, under the Equally Likely criteria, the investment to make would be "Market".
d) Criterion of realism (assume θ = 0.6): In this criteria, we assume a risk tolerance factor θ and multiply each payoff by (1-θ). We calculate the adjusted payoffs for each strategy, and select the one with the highest adjusted payoff.
For example, the adjusted payoff for T-Bills with θ = 0.6 would be:
3.50 * (1-0.6) = 1.40
Similarly, calculating the adjusted payoffs for other strategies, we find:
- T-Bills: 1.40
- Paragon: -8.60
- Luster: -8.00
- Apex: 8.80
- Market: -2.80
The strategy with the highest adjusted payoff is "Apex" with an adjusted payoff of 8.80. Therefore, under the Criterion of Realism with θ = 0.6, the investment to make would be "Apex".
e) Minimax regret criteria: In this criteria, we calculate the regret for each strategy by subtracting the payoff of each strategy in each state of nature from the maximum payoff for that state of nature. We then select the strategy with the minimum maximum regret.
Calculating the regret for each strategy, we find:
- T-Bills: 46.50
- Paragon: 72.50
- Luster: 48.70
- Apex: 40.00
- Market: 54.00
The strategy with the minimum maximum regret is "Apex" with a maximum regret of 40.00. Therefore, under the Minimax Regret criteria, the investment to make would be "Apex".
f) Criterion of maximum expected value: In this criteria, we calculate the expected value for each strategy by multiplying the payoffs for each state of nature by their respective probabilities and summing them up. We then select the strategy with the highest expected value.
The expected values for each strategy can be calculated as described in the "Equally Likely criteria" (c).
The expected values for each strategy, previously calculated, are:
- T-Bills: 3.50
- Paragon: 4.45
- Luster: 4.74
- Apex: 16.35
- Market: 17.70
The strategy with the highest expected value is "Market" with an expected value of 17.70. Therefore, under the Criterion of Maximum Expected Value, the investment to make would be "Market".
To determine which investment to make under different decision criteria, we need to calculate the expected payoff for each investment strategy. The expected payoff is the sum of the product of the payoff and the probability for each state of nature.
a) Maximin criteria: Under this criteria, we choose the investment that maximizes the minimum payoff. To find the minimum payoff for each investment strategy, we look at the smallest value in each column (state of nature). The investment with the largest of these minimum payoffs is chosen.
b) Maximax criteria: Under this criteria, we choose the investment that maximizes the maximum payoff. To find the maximum payoff for each investment strategy, we look at the largest value in each column (state of nature). The investment with the largest of these maximum payoffs is chosen.
c) Equally Likely criteria: Under this criteria, we assume that each state of nature is equally likely to occur. We calculate the average payoff for each investment strategy by taking the weighted sum of the payoffs, where the weight is the probability of each state of nature. The investment with the highest average payoff is chosen.
d) Criterion of realism: The criterion of realism considers both the expected value and the variability of the payoffs. To calculate the expected value for each investment strategy, we multiply each payoff by its probability and sum them. Then, we subtract a weight factor, denoted by ε, times the variance of each strategy's payoffs. The investment with the highest value after this adjustment is chosen. In this case, ε is given as 0.6.
e) Minimax regret criteria: For this criteria, we look at the difference between the payoff of each strategy and the maximum possible payoff for each state of nature. We calculate the maximum regret for each investment strategy, which is the largest difference among all the states of nature. The investment with the smallest maximum regret is chosen.
f) Criterion of maximum expected value: Under this criteria, we simply calculate the expected value for each investment strategy, by multiplying each payoff by its probability and summing them. The investment with the highest expected value is chosen.
Now, to determine the specific investment for each criteria, I would need to see the payoffs for each investment strategy, not just the descriptions. Could you provide the payoffs for each strategy?