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
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To determine the investment strategy under different decision criteria, we need to analyze each criterion and select the strategy that maximizes or minimizes the relevant criterion. Let's go through each criterion one by one:
a) Maximin criteria:
Under the maximin criteria, we select the strategy that maximizes the minimum payoff. To find the minimum payoff for each strategy, we look at the lowest value in each column. The strategy with the highest minimum payoff is selected.
In this case, the maximum values for each strategy's minimum payoffs are as follows:
T-Bills: 3.50
Paragon: -22.50
Luster: -20.00
Apex: 5.00
Market: -13.00
Considering these values, we can see that the strategy with the highest minimum payoff is T-Bills (3.50). So, under the maximin criteria, the recommended investment strategy is T-Bills.
b) Maximax criteria:
Under the maximax criteria, we select the strategy that maximizes the maximum payoff. To find the maximum payoff for each strategy, we look at the highest value in each column. The strategy with the highest maximum payoff is selected.
In this case, the maximum values for each strategy's maximum payoffs are as follows:
T-Bills: 3.50
Paragon: 50.00
Luster: 28.00
Apex: 45.00
Market: 38.00
Considering these values, we can see that the strategy with the highest maximum payoff is Paragon (50.00). So, under the maximax criteria, the recommended investment strategy is Paragon.
c) Equally Likely criteria:
Under the equally likely criteria, we assign equal probabilities to each state of nature and calculate the expected payoff for each strategy. The strategy with the highest expected payoff is selected.
To calculate the expected payoff, we multiply each payoff by the corresponding probability and sum the results for each strategy.
Expected payoffs for each strategy are as follows:
T-Bills: 0.05 * 3.50 + 0.35 * 3.50 + 0.30 * 3.50 + 0.20 * 3.50 + 0.10 * 3.50 = 3.50
Paragon: 0.05 * (-22.50) + 0.35 * (-2.00) + 0.30 * 20.00 + 0.20 * 35.00 + 0.10 * 50.00 = 10.85
Luster: 0.05 * 28.00 + 0.35 * 14.70 + 0.30 * 0.00 + 0.20 * (-10.00) + 0.10 * (-20.00) = 5.675
Apex: 0.05 * 10.00 + 0.35 * 5.00 + 0.30 * 7.00 + 0.20 * 45.00 + 0.10 * 30.00 = 8.05
Market: 0.05 * (-13.00) + 0.35 * 5.00 + 0.30 * 22.00 + 0.20 * 38.00 + 0.10 * 47.00 = 15.3
Considering these values, we can see that the strategy with the highest expected payoff is Market (15.3). So, under the equally likely criteria, the recommended investment strategy is Market.
d) Criterion of realism (assume α = 0.6):
In the criterion of realism, we assign a weight (α) to the minimum value and a weight of (1-α) to the maximum value in each column. We then sum the weighted values for each strategy and select the strategy with the highest weighted sum.
To calculate the weighted sum, we multiply the minimum value by α and the maximum value by (1-α) for each column, and sum the results for each strategy.
Weighted sums for each strategy are as follows (assuming α = 0.6):
T-Bills: 0.6 * 3.50 + 0.4 * 3.50 = 3.50
Paragon: 0.6 * (-22.50) + 0.4 * 50.00 = 4.80
Luster: 0.6 * (-20.00) + 0.4 * 28.00 = 0.80
Apex: 0.6 * 5.00 + 0.4 * 45.00 = 20.80
Market: 0.6 * (-13.00) + 0.4 * 38.00 = 13.40
Considering these values, we can see that the strategy with the highest weighted sum is Apex (20.80). So, under the criterion of realism with α = 0.6, the recommended investment strategy is Apex.
e) Minimax regret criteria:
Under the minimax regret criteria, we calculate the regret for each strategy by finding the difference between the maximum payoff for each state of nature and the payoff of the chosen strategy for that same state of nature. We then select the strategy with the lowest maximum regret.
To calculate the maximum regret, we find the maximum value in each row and subtract the payoff of the chosen strategy in that row.
Maximum regrets for each strategy are as follows:
T-Bills: 50.00 - 3.50 = 46.50
Paragon: 50.00 - (-22.50) = 72.50
Luster: 28.00 - (-20.00) = 48.00
Apex: 45.00 - 5.00 = 40.00
Market: 47.00 - (-13.00) = 60.00
Considering these values, we can see that the strategy with the lowest maximum regret is Apex (40.00). So, under the minimax regret criteria, the recommended investment strategy is Apex.
f) Criterion of maximum expected value:
Under the criterion of maximum expected value, we calculate the expected payoff for each strategy using the given probabilities. The strategy with the highest expected payoff is selected.
To calculate the expected payoff, we multiply each payoff by the corresponding probability and sum the results for each strategy (same calculation as in the equally likely criteria).
Expected payoffs for each strategy are as follows (same as calculated in c):
T-Bills: 3.50
Paragon: 10.85
Luster: 5.675
Apex: 8.05
Market: 15.3
Considering these values, we can see that the strategy with the highest expected payoff is Market (15.3). So, under the criterion of maximum expected value, the recommended investment strategy is Market.
Based on the analysis of each decision criterion, we can summarize the recommended investment strategies as follows:
a) Maximin criteria: T-Bills
b) Maximax criteria: Paragon
c) Equally Likely criteria: Market
d) Criterion of realism (assuming α = 0.6): Apex
e) Minimax regret criteria: Apex
f) Criterion of maximum expected value: Market