Basic Business Statistics 10 th Edition Chapter 17

Learning Objectives In this chapter, you learn: § § To use payoff tables and decision trees to evaluate alternative courses of action To use several criteria to select an alternative course of action To use Bayes’ theorem to revise probabilities in light of sample information About the concept of utility Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 2

Steps in Decision Making § List Alternative Courses of Action § § List Uncertain Events § § Possible events or outcomes Determine ‘Payoffs’ § § Choices or actions Associate a Payoff with Each Event/Outcome combination Adopt Decision Criteria § Evaluate Criteria for Selecting the Best Course of Action Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 3

List Possible Actions or Events Two Methods of Listing Payoff Table Basic Business Statistics,

A Payoff Table A payoff table shows alternatives, states of nature, and payoffs Investment Choice (Action) Profit in $1, 000’s (Events) Strong Stable Weak Economy Large factory Average factory Small factory Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 200 90 40 50 120 30 -120 -30 20 5

Sample Decision Tree Large factory Average factory Small factory Strong Economy 200 Stable Economy 50 Weak Economy -120 Strong Economy 90 Stable Economy 120 Weak Economy -30 Strong Economy 40 Stable Economy 30 Weak Economy 20 Payoffs Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 6

Opportunity Loss Opportunity loss is the difference between an actual payoff for an action and the optimal payoff, given a particular event Investment Choice (Action) Large factory Average factory Small factory Payoff Table Profit in $1, 000’s (Events) Strong Economy Stable Economy Weak Economy 200 90 40 50 120 30 -120 -30 20 The action “Average factory” has payoff 90 for “Strong Economy”. Given “Strong Economy”, the choice of “Large factory” would have given a payoff of 200, or 110 higher. Opportunity loss = 110 for this cell. Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 7

Opportunity Loss (continued) Investment Choice (Action) Large factory Average factory Small factory Payoff Table Profit in $1, 000’s (States of Nature) Strong Economy Stable Economy Weak Economy 200 90 40 50 120 30 -120 -30 20 Investment Choice (Action) Large factory Average factory Small factory Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. Opportunity Loss Table Opportunity Loss in $1, 000’s (Events) Strong Economy Stable Economy Weak Economy 0 110 160 70 0 90 140 50 0 8

Decision Criteria § Expected Monetary Value (EMV) § § Expected Opportunity Loss (EOL) § § The expected profit for taking action Aj The expected opportunity loss for taking action Aj Expected Value of Perfect Information (EVPI) § The expected opportunity loss from the best decision Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 9

Expected Monetary Value Solution Goal: Maximize expected value § The expected monetary value is the weighted average payoff, given specified probabilities for each event Where EMV(j) = expected monetary value of action j xij = payoff for action j when event i occurs Pi = probability of event i Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 10

Expected Monetary Value Solution (continued) § The expected value is the weighted average payoff, given specified probabilities for each event Profit in $1, 000’s (Events) Investment Choice (Action) Large factory Average factory Small factory Strong Economy (. 3) Stable Economy (. 5) Weak Economy (. 2) 200 90 40 50 120 30 -120 -30 20 Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. Suppose these probabilities have been assessed for these three events 11

Expected Monetary Value Solution (continued) Goal: Maximize expected value Payoff Table: Profit in $1, 000’s (Events) Investment Choice (Action) Large factory Average factory Small factory Strong Economy (. 3) Stable Economy (. 5) Weak Economy (. 2) 200 90 40 50 120 30 -120 -30 20 Expected Values (EMV) 61 81 31 Maximize expected value by choosing Average factory Example: EMV (Average factory) = 90(. 3) + 120(. 5) + (-30)(. 2) = 81 Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 12

Decision Tree Analysis § A Decision tree shows a decision problem, beginning with the initial decision and ending will all possible outcomes and payoffs. Use a square to denote decision nodes Use a circle to denote uncertain events Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 13

Add Probabilities and Payoffs (continued) Large factory Strong Economy (. 3) 200 Stable Economy (. 5) 50 Weak Economy Average factory Small factory -120 Strong Economy (. 3) 90 Stable Economy (. 5) 120 Weak Economy Decision (. 2) -30 Strong Economy (. 3) 40 Stable Economy (. 5) 30 Weak Economy (. 2) 20 Uncertain Events Probabilities Payoffs Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 14

Fold Back the Tree EMV=200(. 3)+50(. 5)+(-120)(. 2)=61 Large factory Strong Economy (. 3) 200 Stable Economy (. 5) 50 Weak Economy EMV=90(. 3)+120(. 5)+(-30)(. 2)=81 Average factory Small factory 90 Stable Economy (. 5) 120 (. 2) -30 Strong Economy (. 3) 40 Stable Economy (. 5) 30 Weak Economy Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. -120 Strong Economy (. 3) Weak Economy EMV=40(. 3)+30(. 5)+20(. 2)=31 (. 2) 20 15

Make the Decision EV=61 Large factory Strong Economy (. 3) 200 Stable Economy (. 5) 50 Weak Economy EV=81 Average factory Strong Economy (. 3) Stable Economy (. 5) Weak Economy EV=31 Small factory (. 2) -120 90 120 40 Stable Economy (. 5) 30 (. 2) Maximum EMV=81 -30 Strong Economy (. 3) Weak Economy Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. (. 2) 20 16

Expected Opportunity Loss Solution Goal: Minimize expected opportunity loss § The expected opportunity loss is the weighted average loss, given specified probabilities for each event Where EOL(j) = expected monetary value of action j Lij = opp. loss for action j when event i occurs Pi = probability of event i Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 17

Expected Opportunity Loss Solution Goal: Minimize expected opportunity loss Opportunity Loss Table Opportunity Loss in $1, 000’s (Events) Investment Choice (Action) Large factory Average factory Small factory Strong Economy (. 3) Stable Economy (. 5) Weak Economy (. 2) 0 110 160 70 0 90 140 50 0 Expected Op. Loss (EOL) 63 43 93 Minimize expected op. loss by choosing Average factory Example: EOL (Large factory) = 0(. 3) + 70(. 5) + (140)(. 2) = 63 Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 18

Expected Opportunity Loss vs. Expected Monetary Value § § The Expected Monetary Value (EMV) and the Expected Opportunity Loss (EOL) criteria are equivalent. Note that in this example the expected monetary value solution and the expected opportunity loss solution both led to the choice of the average size factory. Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 19

Value of Information § Expected Value of Perfect Information, EVPI Expected Value of Perfect Information EVPI = Expected profit under certainty – expected monetary value of the best alternative (EVPI is equal to the expected opportunity loss from the best decision) Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 20

Expected Profit Under Certainty § Expected profit under certainty = expected value of the best decision, given perfect information Profit in $1, 000’s (Events) Investment Choice (Action) Strong Economy (. 3) Stable Economy (. 5) Weak Economy (. 2) 200 90 40 50 120 30 -120 -30 20 Value of best decision 200 for each event: 120 20 Large factory Average factory Small factory Example: Best decision given “Strong Economy” is “Large factory” Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 21

Expected Profit Under Certainty (continued) Profit in $1, 000’s (Events) Investment Choice (Action) § Now weight these outcomes with their probabilities to find the expected value: Large factory Average factory Small factory Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. Strong Economy (. 3) Stable Economy (. 5) Weak Economy (. 2) 200 90 40 50 120 30 -120 -30 20 200 120 20 200(. 3)+120(. 5)+20(. 2) = 124 Expected profit under certainty 22

Value of Information Solution Expected Value of Perfect Information (EVPI) EVPI = Expected profit under certainty – Expected monetary value of the best decision Recall: Expected profit under certainty = 124 EMV is maximized by choosing “Average factory”, where EMV = 81 so: EVPI = 124 – 81 = 43 (EVPI is the maximum you would be willing to spend to obtain perfect information) Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 23

Accounting for Variability Consider the choice of Stock A vs. Stock B Percent Return (Events) Stock Choice (Action) Strong Economy (. 7) Weak Economy (. 3) Stock A 30 -10 18. 0 Stock B 14 8 12. 2 Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. Expected Return: Stock A has a higher EMV, but what about risk? 24

Accounting for Variability (continued) Calculate the variance and standard deviation for Stock A and Stock B: Percent Return (Events) Stock Choice (Action) Strong Economy (. 7) Weak Economy (. 3) Stock A 30 -10 18. 0 336. 0 18. 33 Stock B 14 8 12. 2 7. 56 2. 75 Expected Standard Return: Variance: Deviation: Example: Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 25

Accounting for Variability (continued) Calculate the coefficient of variation for each stock: Stock A

Return-to-Risk Ratio (RTRR): Expresses the relationship between the return (expected payoff) and the risk

Return-to-Risk Ratio You might want to consider Stock B if you don’t like risk. Although Stock A has a higher Expected Return, Stock B has a much larger return to risk ratio and a much smaller CV Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 28

Decision Making with Sample Information Prior Probability § Permits revising old probabilities based on new information New Information Revised Probability Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 29

Revised Probabilities Example Additional Information: Economic forecast is strong economy § When the economy was strong, the forecaster was correct 90% of the time. § When the economy was weak, the forecaster was correct 70% of the time. F 1 = strong forecast F 2 = weak forecast E 1 = strong economy = 0. 70 Prior probabilities from stock choice example E 2 = weak economy = 0. 30 P(F 1 | E 1) = 0. 90 Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. P(F 1 | E 2) = 0. 30 30

Revised Probabilities Example § (continued) Revised Probabilities (Bayes’ Theorem) Basic Business Statistics, 10 e

EMV with Revised Probabilities Pi Event Stock A xij. Pi Stock B xij. Pi . 875 strong 30 26. 25 14 12. 25 . 125 weak -10 -1. 25 8 1. 00 Σ = 25. 0 Revised probabilities Σ = 11. 25 EMV Stock B = 11. 25 EMV Stock A = 25. 0 Maximum EMV Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 32

EOL Table with Revised Probabilities Pi Event Stock A xij. Pi Stock B xij. Pi . 875 strong 0 0 16 14. 00 . 125 weak 18 2. 25 0 0 Σ = 2. 25 Revised probabilities Σ = 14. 00 EOL Stock B = 14. 00 EOL Stock A = 2. 25 Minimum EOL Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 33

Accounting for Variability with Revised Probabilities Calculate the variance and standard deviation for Stock A and Stock B: Percent Return (Events) Stock Choice (Action) Strong Economy (. 875) Weak Economy (. 125) Stock A 30 -10 25. 0 175. 0 13. 229 Stock B 14 8 13. 25 3. 94 1. 984 Expected Standard Return: Variance: Deviation: Example: Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 34

Accounting for Variability with Revised Probabilities (continued) The coefficient of variation for each stock using the results from the revised probabilities: Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 35

Return-to-Risk Ratio with Revised Probabilities With the revised probabilities, both stocks have higher expected returns, lower CV’s, and larger return to risk ratios Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 36

Utility § § Utility is the pleasure or satisfaction obtained from an action. The utility of an outcome may not be the same for each individual. Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 37

Utility § § Example: each incremental $1 of profit does not have the same value to every individual: A risk averse person, once reaching a goal, assigns less utility to each incremental $1. A risk seeker assigns more utility to each incremental $1. A risk neutral person assigns the same utility to each extra $1. Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 38

$ Risk Averter Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. Utility Three

Maximizing Expected Utility § Making decisions in terms of utility, not $ § § § Translate $ outcomes into utility outcomes Calculate expected utilities for each action Choose the action to maximize expected utility Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 40

Chapter Summary § Described the payoff table and decision trees § § Provided criteria for decision making § § § Opportunity loss Expected monetary value Expected opportunity loss Return to risk ratio Introduced expected profit under certainty and the value of perfect information Discussed decision making with sample information Addressed the concept of utility Basic Business Statistics, 10 e © 2006 Prentice-Hall, Inc. 41