1 / 54

Chapter 12 Risk Attitude and Utility Theory

Chapter 12 Risk Attitude and Utility Theory . Utility Theory  Maximize Expected Utility Motivation Rescale all values to between 0 and 1 based on risk attitude Certainty Equivalent Manage Risk – Utility Theory explains Insurance Partnership Risk Aversion assessment

armina
Download Presentation

Chapter 12 Risk Attitude and Utility Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 12 Risk Attitude and Utility Theory • Utility Theory  Maximize Expected Utility • Motivation • Rescale all values to between 0 and 1 based on risk attitude • Certainty Equivalent • Manage Risk – Utility Theory explains • Insurance • Partnership • Risk Aversion assessment • Standard function with parameters • Direct assessment • Problems with assessment • Critique Utility Theory: Paradox and Prospect Theory

  2. Utility Theory - Overview • Incorporates decision-maker’s risk attitude  Value of alternatives cannot be captured by E(X) because risk attitude • Certainty Equivalent (CE): The payoff amount we would accept in lieu of under-going the uncertain situation, taking a gamble. • Risk Premium: A situation’s risk premium (RP) is the difference between expected payoff (EP) and certainty equivalent (CE) • Risk Aversion: CE < E(X)  Concave function • Profit – prefer sure profit that is less than E(X) of a profit gamble • Cost – prefer sure cost that is higher than E(Y) of a loss gamble • Risk Prone: CE > E(X) Try for more money than average • With high guaranteed minimum, may be will to try for home run even if offered a value somewhat higher than the expected value

  3. TV show “Deal or No Deal” – Activate WEB GAMEGame Description and Risk Attitude • Game starts with 26 suitcases, each with a dollar value ranging from $0.01 to $1M • Contestant picks one case as his to be opened at the end, and selects 6 cases to be opened immediately. As they are opened, their stored dollar value is deleted from the list. • After opening these six cases, the player is offered a sure amount or the option to keep playing. • At each the step the number of boxes to open at this stage decreases until only one at a time is opened each time. • For example, I played the game until there were 3 unopened boxes left ($25, $50,000, and $1,000,000). One of these boxes I had selected as my box. • The offer was for $220,000. This is much less than the expected value • E(X) = (1/3)(25)+ (1/3)((50,000) + (1/3)(1,000,000) = (1/3) (1,000,025) = $350,008 • If you prefer to take less than the expected value  risk averse

  4. Simple gamble (lottery) – Two OutcomesTwo boxes are left: $10,000 and $20,000. • You have a 50–50 chance of winning $10,000 or $20,000 depending upon what is in your selected box.  E(X) = $15,000 • Would you accept an offer of $12,000 or continue Yes __ or No __ • Would you accept an offer of $14,500 or continue Yes __ or No __ • What is the minimum you would accept to stop the game? ___  this is Certainty Equivalent (CE) of the gamble • If the CE is less than the expected value, you are a risk averse. • Risk Premium is the difference between your CE and the expected value • Assume you would accept $12,500 • Risk Premium = E(X) – CE = $15,000 - $12,500 = $2,500 • If the CE is more than the expected value, you are a risk prone. • Gamble is a set of uncertain outcomes with probabilities • Not necessarily just 2 outcomes • All outcomes may not have the same probability.

  5. Expected Value is Inadequate Generally Risk Averse • Extremely Rare Catastrophic Events • Law of averages no help for individual • Motivates insurance industry • Companies may self-insure up to extreme costs • Can tolerate moderate risks that are not devastating • Deductible on insurance • Large investments relative to size of company • Cannot afford the loss • Has need for certain profit more than high upside potential  sell development rights

  6. Determine function U(X) • U(X) function may be determined • Select from standard functional forms – exponential or logarithmic and estimate parameters from questions • Brute force graphing of the results of series of Certainty Equivalent responses to gambles • Assume exponential U(X) = 1-e(-x/15000) • U(20,000) = 1-e(-20000/15000) = .7364

  7. (0.5) 0.4866 0.7364 (0.5) (0.5) $10,000 $20,000 (0.5) Calculate Expected Utility  E(U(X)) and Not E(X)Assume exponential utility function for this example E(X) = 0.5($10000) + 0.5($20000) = $12,500 • Assume U(X) = 1-e(-x/15000) • U(10,000) = 1-e(-10000/15000) = 0.4866 • U(20,000) = 1-e(-20000/15000) = 0.7364 E(U(X)) = 0.5(0.487) + 0.5(0.736) = 0.6115 invert exponential function  CE(0.6115) = $14,182 $14,250 would be preferred to gamble $14,000 would not be preferred

  8. Invert function U(X)  U-1(X) • 0.6115 = 1-e(-x/15000) • e(-x/15000) = 1- 0.6115 = 0.3885 • ln of both sides  (-X/15000) = ln(0.3885) = -0.9455 • X = 15000 (0.9455) = 14,182 • CE of 50-50 gamble of 10,000 and 20,000 is $14,182 • Risk Premium = $15,000 - $14,182 = $818

  9. Risk Aversion • CONCAVE for Negative (Cost) • Prefer to PAY More than expected value to be sure Cost does not get too large. • 50-50 COST Gamble $90,000 or $10,000 => prefer sure $60,000cost • Basis for Insurance Industry • CONCAVE for Maximization (Profit) • Prefer to EARN less than expected value to be sure of earning at least that profit. • 50-50 PROFIT Gamble $90,000 or $10,000 ===> would prefer sure $40,000profit to the gamble. • Basis for Selling off bank Debts such as mortgages

  10. Risk Aversion – Concave Curve Profit Cost 0.5 for $10,000 or 0.5 for $90,000 E(X) = $50,000 Accept $40,000 = CE  U(40000)=.5 Risk Premium $10,000 0.5 for -$10,000 or 0.5 for -$90,000 E(X) = -$50,000 Accept -$60,000 = CE  U(60000)=.5 Risk Premium $10,000

  11. Activity – Risk Aversion • Pay to avoid loss ___________________________________________ ___________________________________________ • Accept less profit but be sure (or more certain) of profit ___________________________________________ ___________________________________________ • Other Examples ___________________________________________ ___________________________________________ ___________________________________________ ___________________________________________

  12. Utility Function Assessment Brute Force • Series of 3 lotteries • Obtain Certainty Equivalent of utility values of .25, .5, & .75 • Extrapolate over the range of 0 to 1. • Create your own utility function for $0M to $10M • set U(min) = 0 & U(max) = 1 • set U(0) = 0 & U(10) = 1

  13. 40.0% 0 30% Take 9.9 1.9 FALSE Take Rate Low -8 5.86 60.0% 0 50% Take 16.5 8.5 How Much Automation Investment 6.32 40.0% 0.4 30% Take 13.8 0.8 TRUE Take Rate High -13 6.32 60.0% 0.6 50% Take 23 10 Decision tree Boss Controls automation investmentNeed utility function for four values (0.8, 1.9, 8.5, 10)

  14. (0.5) XL ≈ CE25 (0.5) CE50 (0.5) XH ≈ CE75 (0.5) CE50 (0.5) XL ≈ CE50 (0.5) XH Certainty equivalents

  15. (0.5) (0.5) (0.5) 0 10 0 ≈ CE25 = 1.6 ≈ CE75 = 6.2 ≈ CE50 = 3.6 (0.5) (0.5) (0.5) 3.6 3.6 10 BC automation investment certainty equivalent for 0.50, 0.25, and 0.75 utility

  16. Utility assessment for BC automation investment example Table ‎12‑4: Utility scores for BC automation investment

  17. 40% 0 0.13 0.13 FALSE Take Rate High 0.652 60% 0 1.0 1.0 Decision 0.657 40% 0.4 0.29 0.29 TRUE Take Rate Low 0.657 60% 0.6 0.90 0.9 Decision tree for BC automation investment using utility function 30% Take Rate 50% Take Rate Automation Investment 30% Take Rate 50% Take Rate CE: $5.142 (0.652) vs. $5.196 million (0.657)

  18. Utility Function: Simplify Interview • Goal: simplify interview process  One question for each parameter • Exponential U(x) = 1 – exp(-x/R)  Constant Risk Aversion • Implies that the magnitude of cash on hand does not affect attitude towards Risk • R = Risk Tolerance • R = Sum of Money about which you are indifferent between a 50-50 chance of gaining the whole sum, R and losing the half sum, R/2

  19. Constant Risk AversionRisk Premium is Constant R=35 U(X) = 1 – exp(-x/35)

  20. Determine R = Risk Tolerance 50- 50 gamble to determine R Personal

  21. Determine R = Risk Tolerance 50- 50 gamble to determine R Corporate

  22. ENCO Project Selection An energy company will select one of two projects. If the company chooses Project A, it will undertake the development of a new power plant in one developing country. The company estimates that the investment cost will be $50 million and total revenue for five-year after the operation cost will be $80, $90, or $110 million. There is a 20% chance that the local government will take over the plant once it is finished due to some legal issues and just repay the original investment cost ($50 million). Project B also requires a $50 million investment. Total revenue for five-year after the operation cost will be $66, $80, or $90 million. In this second country, there is no chance that the government will take over this project when it is completed.

  23. 0.2 20.0% Yes 0 50 TRUE Government Take Project? Project A 34.4 -50 0.24 30.0% Low 30 80 Revenue 80.0% No 43 0 0.32 40.0% Medium 40 90 0.24 30.0% High 60 110 Project Decision Project 34.4 0 30.0% Low 16 66 FALSE Revenue Project B 28.8 -50 0 40.0% Medium 30 80 0 30.0% High 40 90 Maximize expected valueDecision tree for ENCO project selection

  24. Figure 12.11: Cumulative risk profile for ENCO project selection Redraw diagram without student version

  25. A utility score was calculated for each value. R= $30 million EU(Project A) = 0*20% + (0.632*30%+0.736*40%+0.865*30%)*80% = 0.595 EU(Project B) = 0.413*30%+0.632*40%+0.736*30% = 0.598

  26. 20.0% 20.0% 0 0 Yes Yes 0.000 0.000 50 50 FALSE FALSE Government Take Project? Project A Project A - - 50 50 30.0% 30.0% 0 0 0.595 Low Low 80 80 0.632 0.632 80.0% 80.0% Revenue Revenue No No 0 0 0.744 0.744 40.0% 40.0% 0 0 Medium Medium 0.736 0.736 90 90 30.0% 30.0% 0 0 High High 110 110 Project Decision Project Decision Project Project 30.0% 30.0% 0.3 0.3 Low Low 0.413 0.413 66 66 TRUE TRUE Revenue Revenue Project B Project B 0.598 0.598 - - 50 50 40.0% 40.0% 0.4 0.4 Medium Medium 80 80 0.632 0.632 30.0% 30.0% 0.3 0.3 High High 90 90 0.736 0.736 Decision tree for ENCO: using expected utility 0.865 0.598

  27. Impact of risk tolerance on certainty equivalent

  28. Risk Sharing Strategies • Buy insurance • Partnerships • Example: Transmission plant by GM and Ford • Joint ventures with foreign companies • Diversification • Diversification with independent investments • Example: Dual sourcing, investment in different stocks • Diversification with dependent investments (Hedging)

  29. Risk Sharing through Buy Insurance: Project Selection • Buy insurance against a government takeover of project A. • $4 million premium the company will receive a $10 million payment if the government takes the project. • Add new decision branch to tree as part of Project A. (top of next tree)

  30. 20.0% 0 6 60 FALSE Government Take Project? 32.400 -4 30.0% 0 26 80 80.0% Revenue 39.000 0 40.0% 0 90 36 0 30.0% 56 TRUE 110 Insurance Decicion 34.400 -50 20.0% 0.2 0 50 TRUE Government Take Project? 34.400 0 30.0% 0.24 30 80 Revenue 80.0% 43.000 0 40.0% 0.32 40 90 0.24 30.0% 60 110 Project Decision 34.400 30.0% 0 16 66 FALSE Revenue 28.800 -50 40.0% 0 80 30 30.0% 0 40 90 ENCO: insurance reporting expected value Yes • Buy Insurance Low No Medium High • Project A Yes • Do Not Buy Low No Medium High • Project Insurance Low • Project B Medium High

  31. 20.0% 0.2 TRUE 60 6 Government Take Project? -4 27.630 30.0% 0.24 80 26 80.0% Revenue 0 36.830 40% 0.32 90 36 Insurance Decision 27.630 30.0% 0.24 TRUE 110 56 -50 20.0% 0 FALSE 50 0 Government Take Project? 27.107 0 30.0% 0 30 80 80.0% Revenue 0 40.830 40.0% 0 Project Decision 90 40 27.630 30.0% 0 60 110 30.0% 0 16 66 FALSE Revenue 27.322 -50 40.0% 0 30 80 30.0% 0 40 90 ENCO : insurance option reporting CE Yes • Buy Insurance Low No Medium High • Project A Yes • Do Not Buy Low No Medium High • Project Insurance Low • Project B Medium High

  32. Decision Tree for Insurance companyTree Not in Textbook

  33. Share Risk through Partnership • Concept: Commit to only a percentage of the cost (liability) and accrue the equivalent percentage of revenue • Common amongst insurance companies to reinsure and share risk of catastrophic event – Lloyd’s of London • Decision Tree Method – reduce costs and revenues proportionate to share and calculate new utility scores.

  34. Share Risk through Partnership:Project Selection • Outside investor shares 50% of cost and benefit of each project. • By sharing the investment, the company can now be involved in more projects. Let’s evaluate a 50% share. As a result the company can invest in BOTH A and B.

  35. Government takes away project A? TRUE Project A + 34.400 -50 Project Decision 34.400 FALSE Revenue Project B + 28.800 -50 0 30.0% 8 33 20.0% Revenue from B 14.400 25 0 40.0% 15 40 0 30.0% 20 45 FALSE Government takes away project A? 31.600 -50 0 30.0% 23 33 Revenue from B 30.0% 29.400 40 0 40.0% 30 40 0 30.0% 35 45 Revenue from A 80.0% 35.900 0 Revenue from B 40.0% + 34.400 45 Revenue from B 30.0% + 44.400 55 ENCO with 50% partnership reporting expected profit Project - Diversification Low Yes Medium High 50% of Both Projects Low Low Medium High No Medium High

  36. FALSE Government Take Project? Project A + 27.107 -50 Project Decision Project - Diversification 29.450 FALSE Revenue Project B + 27.322 -50 0.06 30.0% Low 8 33 Revenue from B 20.0% Yes 14.032 25 0.08 40.0% Medium 15 40 0.06 30.0% High 20 45 TRUE Government Take Project A? 50% of Both Projects 29.450 -50 0.072 30.0% Low 23 33 Revenue from B 30.0% Low 29.032 40 0.096 40.0% Medium 30 40 0.072 30.0% High 35 45 Revenue from A 80.0% No 34.966 0 Revenue from B 40.0% Medium + 34.032 45 Revenue from B 30.0% High + 44.032 55 ENCO decision tree with partnership option reporting certainty equivalent

  37. Optimize PERCENT Share of Project • Can optimize based on risk attitude • Percent share of option A • Percent share of option B • Different companies with different risk attitudes will have different optimum preferences

  38. Phillips Petroleum and Onshore US Oil ExplorationProspect Ranking: R equal to $25 million

  39. Following two slides not in the textbook • Impacts of the target setting on the risk attitude • Consistent with prospect theory

  40. $47 .5 $46 OR $40 .5 Risk Attitude and TargetsComponent Cost Target $46 – Goal: Minimize Cost You are near completion on a design and are almost certain that you will be able to reach the variable cost target of $46. A suggestion comes along that by changing materials the component should be easier to manufacture with the cost dropping to $40 per part. However, time is short and there is a concern that without proper testing of Design for Manufacture the change could, in fact, increase the cost to $47 per component and miss the target. Would you make the change assuming the two outcomes were equally likely?

  41. $47 .5 $42 OR $40 .5 Risk Attitude and TargetsComponent Cost Target $40 You are near completion on a design and are almost certain that you will NOT be able to reach the variable cost target of $40 and that the part will cost $42. A suggestion comes along that by changing materials the part should be easier to manufacture with the cost dropping to $40 per part. However, time is short and there is a concern that without proper testing of Design for Manufacture the change could, in fact, increase the cost to $47 per part and miss the target. Would you make the change assuming the two outcomes were equally likely?

  42. Problems With Carrying Out Risk Aversion Assessment in Real-World • Managers are uncomfortable with this abstract concept • Managers are inconsistent -- Prefer ranges for CE and not specific values • People are inconsistent • risk prone for small values ( i.e. will buy lottery tickets) risk averse for large values (i.e. buy insurance) • May be risk prone if values are negative but risk averse if all values are positive and vice versa • Attitudes towards risk are influenced by artificial targets • Target secure – take no gamble to improve • Target at risk – take extreme gambles to reach target

  43. Risk Assessment In Practice (Done Less than 10% of the time) • Howard Rule of Thumb for R in major decisions: • R = 6.4% of sales or 1.25 Net Income – Your company value? • Example: R = $1 Billion dollar investment or purchase decision for a company with over $15 Billion in sales. • First carry out analysis without Risk Aversion • Display Comparative Risk Profiles and discuss • Insert exponential function & determine whether or not the optimal solution is Sensitive to Risk Tolerance Value within a realistic range for your company. • If optimal decision can change within a reasonable range of R then assess actual utility function.

  44. Assumptions of Utility Theory • Expectation: U(x1,p1;…;xn,pn) = p1u(x1)+…+pnu(xn) • Asset Integration: The domain of the utility function is final states (which include one’s asset position) rather than gains or losses. • Risk Aversion: u is concave (u’’< 0)

  45. Critique of Utility Theory • Expectation Assumption - Allais paradox Problem 1: Choose between A: $1 Million with certainty B: $1 Million with probability 0.89 $5 Million with probability 0.10 0 with probability 0.01 Problem 2: Choose between X: $1 Million with probability 0.11 Y: $5 Million with probability 0.10 0 with probability 0.89 0 with probability 0.90

  46. Critique of Utility Theory- Actual preferences • Expectation Assumption - Allais paradox Problem 1: 82% choose A A: $1 Million with certainty B: $1 Million with probability 0.89 $5 Million with probability 0.10 0 with probability 0.01 Problem 2: 83% Choose Y X: $1 Million with probability 0.11 Y: $5 Million with probability 0.10 0 with probability 0.89 0 with probability 0.90 • Conclusion: People overweight “certain” outcomes more than merely probable outcomes.

  47. Utility Function Stability and Accuracy • Affected by mood • Quality of life QOL • Assessment of QOL living with serious disease.

  48. Prospect Theory (Kahneman and Tversky)Example not in textbook • Value Function • Concave for gains and Convex for losses; • Steeper for losses than gains (Loss Aversion)

  49. Prospect Theory (Kahneman and Tversky) Example not in textbook • Weighting Function (The impact of events on desirability of prospects, not merely the likelihood of events) • Overweight certainty and small probabilities; • Underweight moderate and high probabilities;

  50. Prospect Theory (Kahneman and Tversky) Example not in textbook • Fourfold risk attitude:

More Related