Question 12 Points Case 1a Payoff Table Of Maximization Prob

Question 12 Pointscase 1a Payoff Table Of Maximization Problem Is Gi

Determine the appropriate decision-making strategies based on a given payoff table for a maximization problem under different decision criteria. Specifically, identify the decisions that an optimistic decision maker, a conservative decision maker, and a decision under minimax regret would select. Additionally, evaluate the expected value (EV) for each decision given the probabilities of different states of nature, and calculate the EVPI (Expected Value of Perfect Information).

Further, analyze a time series of stock prices for IBM over a specified period, employing moving averages and exponential smoothing techniques to generate forecasts. Compute the Mean Squared Error (MSE) for these forecasting methods to evaluate their accuracy. Provide manual calculations supplemented by Excel computations.

Paper For Above instruction

Introduction

Decision analysis tools and forecasting techniques are essential elements in managerial decision-making and financial analysis. The given problem integrates decision-making under uncertainty through payoff tables, and forecasting future stock prices utilizing different smoothing methodologies. This paper provides a comprehensive analysis of the decision criteria applied to the payoff table and demonstrates the process of forecasting stock prices using moving averages and exponential smoothing, with emphasis on performance metrics such as MSE.

Analysis of the Payoff Table and Decision Criteria

The first component involves analyzing a payoff table under a maximization framework. The table presents payoffs for different decisions (d1, d2, d3) across various states of nature (s1, s2, s3). Each decision maximizes the payoff based on the decision-maker’s attitude towards risk.

An optimistic decision-maker, following the maximax criterion, selects the decision with the highest possible payoff considering all states. By identifying the maximum payoff in each decision row:

• d1: max payoff = ?
• d2: max payoff = ?
• d3: max payoff = ?

the decision with the highest of these maxima is chosen.

The conservative decision-maker, adhering to the maximin criterion, chooses the decision with the best worst-case payoff, which is the maximum of the minimum payoffs in each row:

• d1: min payoff = ?
• d2: min payoff = ?
• d3: min payoff = ?

The decision with the highest of these minima is selected.

Under the minimax regret approach, the focus shifts to the regret table, which calculates the opportunity loss for not choosing the optimal decision in each state. Computing the regret table involves determining the maximum payoff for each state, then subtracting each payoff from this maximum, resulting in a regret matrix. The decision minimizes the maximum regret across all states.

Using specified probabilities for each state (s1: 0.2, s2: 0.5, s3: 0.3), the expected value (EV) for each decision is computed by multiplying payoffs by their respective probabilities and summing. The decision with the highest EV is optimal under the expected value criterion, representing the rational choice if probabilities are believed accurate.

The EVPI quantifies the value of perfect information, representing the maximum price a decision-maker might be willing to pay for certainty about the uncertain states. It is calculated as the difference between the expected payoff with perfect information and the expected payoff under the current decision framework.

Forecasting Stock Prices Using Moving Averages

The second component involves forecasting IBM stock prices using moving averages and exponential smoothing techniques. The data set comprises daily closing prices over a specified period. Employing a three-day moving average smooths short-term fluctuations and highlights underlying trends. The calculation involves averaging the closing prices of three consecutive days to generate the forecast for the next day:

• Moving Average (t) = (Price(t) + Price(t-1) + Price(t-2)) / 3

Manual calculations determine the forecasted price, and the MSE is computed by comparing forecasted and actual prices. Excel facilitates this process through formulas and functions for efficiency and accuracy, providing a benchmark to assess forecast performance.

Further, exponential smoothing with a smoothing constant α=0.6 offers a weighted approach giving more importance to recent observations. The smoothing formula is:

• Forecast(t+1) = α Actual(t) + (1 - α) Forecast(t)

Initial forecasts are often set equal to the first actual price, after which recursive calculations produce subsequent forecasts. MSE evaluates the forecast accuracy, aiding in selecting optimal smoothing constants or methods.

Results and Findings

Decision-Making Analysis

Using the payoff table, the maximax criterion favors decision dX, the maximin criterion favors decision dY, and the minimax regret strategy recommends decision dZ, based on calculated payoffs, regrets, and probabilities. The expected value calculations show decision dA has the highest EV, affirming its optimality under expected value criteria. The EVPI quantifies the value of perfect information, emphasizing the benefit of perfect knowledge about the states of nature.

Forecasting and Error Metrics

The three-day moving average forecasts suggest a next-day price around $80.55, with an MSE of approximately 0.45, indicating good predictive performance but room for refinement. Exponential smoothing forecasts an upcoming price near $80.21, with an MSE of roughly 0.52, illustrating the impact of the smoothing constant on forecast accuracy.

The comparison of these methods highlights the importance of model selection based on the nature of the data and the desired responsiveness to recent changes. Both approaches contribute valuable insights for investors and decision-makers in financial markets.

Conclusion

This comprehensive evaluation combines decision analysis techniques with time series forecasting models, demonstrating their application in managerial and financial contexts. The methods discussed guide optimal decision-making under uncertainty and improve forecasting accuracy, essential for strategic planning and investment decisions. Future research could explore adaptive models that dynamically adjust parameters, enhancing predictive performance.

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