An Excel TUTORIAL for Introductory Statistics Applications

When monetary value does not necessarily lead to the most preferred decision, expressing the value (or worth) of a consequence in terms of its utility will permit the use of expected utility to identify the most desirable decision alternative, Utility is a measure of the total worth or relative desirability of a particular outcome. It reflects the decision maker’s attitude toward a collection of factors such as profit, loss, and risk

Example of a situation where utility can help in selecting the best decision alternative:

• Swofford Inc. currently has two investment opportunities that require approximately the same cash outlay
• The cash requirements necessary prohibit Swofford from making more than one investment at this time
• Consequently, three possible decision alternatives may be considered

Utility and Decision Analysis

A decision maker who would choose a guaranteed payoff over a lottery with a superior expected payoff is a risk avoider

The following steps state in general terms the procedure used to solve the Swofford investment problem:

• Step 1: Develop a payoff table using monetary values
• Step 2: Identify the best and worst payoff values in the table and assign each a utility, with u(best payoff)> u(worst payoff)
• Step 3: For every other monetary value min the original payoff table, do the following to determine its utility:
  1. Define the lottery such that there is a probability pof the best payoff and a probability (1 - p) of the worst payoff
  2. Determine the value of p such that the decision maker is indifferent between a guaranteed payoff of mand the lottery defined in step 3(a)
  3. Calculate the utility of mas follows: \[ U(M) = p\,U(\mbox{best payoff}) + \left( 1-p \right) U(\mbox{Worst payoff}) . \]
• Steo 4: Convert each monetary value in the payoff table to a utility
• Step 5: Apply the expected utility approach to the utility table developed in Step 4 and select the decision alternative with the highest expected utility We can compute the expected utility (EU) of the utilities in a similar fashion as we computed expected value

Subsection: Utility Functions

Different decision makers may approach risk in terms of their assessment of utility, A risk taker is a decision maker who would choose a lottery over a guaranteed payoff when the expected value of the lottery is inferior to the guaranteed payoff

Analyze the decision problem faced by Swofford from the point of view of a decision maker who would be classified as a risk taker Compare the conservative point of view of Swofford’s president (a risk avoider) with the behavior of a decision maker who is a risk taker

Example: Figure 15.11: Utility Function for Money for Risk-Avoider, Risk-Taker, and Risk-Neutral Decision Makers

• Utility function for a risk avoider shows a diminishing marginal return for money
• Utility function for a risk taker shows an increasing marginal return
• These values can be plotted on a graph (Figure 15.11) as the utility function for money
• Top curve is utility function for risk avoider
• Bottom curve is utility function for risk taker
• Utility function for a decision maker neutral to risk shows a constant return (middle line)

Subsection: Exponential Utility Function

Used as an alternative to assume that the decision maker’s utility is defined when decision maker provides enough indifference values to create a utility function All the exponential utility functions indicate that the decision maker is risk averse

Figure 15.12: Exponential Utility Functions with Different Risk Tolerance (R) Values

The R parameter in equation (15.7) represents the decision maker’s risk tolerance; it controls the shape of the exponential utility function Larger R values create flatter exponential functions, indicating that the decision maker is less risk averse (closer to risk neutral) Smaller R values indicate that the decision maker has less risk tolerance (is more risk averse)