Chapter 15 - Risk and information. This chapter presents the following content: Introduction (Amazon.com), describing risky outcome – basic tools, evaluating risky outcomes, avoiding and bearing risk.
Chapter 15 Copyright (c)2014 John Risk and Information Chapter Fifteen Overview Introduction: Amazon.com Describing Risky Outcome – Basic Tools • • Evaluating Risky Outcomes • Lotteries and Probabilities Expected Values Variance Risk Preferences and the Utility Function Avoiding and Bearing Risk • • • The Demand for Insurance and the Risk Premium Asymmetric Information and Insurance The Value of Information and Decision Trees Chapter Fifteen Copyright (c)2014 John • Tools for Describing Risky Outcomes Definition: A lottery is any event with an uncertain outcome Definition: A probability of an outcome (of a lottery) is the likelihood that this outcome occurs Example: The probability often is estimated by the historical frequency of the outcome Chapter Fifteen Copyright (c)2014 John Examples: Investment, Roulette, Football Game Probability Distribution Definition: The probability distribution of the lottery depicts all possible payoffs in the lottery and their associated probabilities • • The probability of any particular outcome is between and The sum of the probabilities of all possible outcomes equals Definition: Probabilities that reflect subjective beliefs about risky events are called subjective probabilities Chapter Fifteen Copyright (c)2014 John Property: Probability Distribution Probability 67% chance of losing Payoff $25 Chapter Fifteen Copyright (c)2014 John 90 80 70 60 50 40 30 20 10 Probability Distribution Probability 67% chance of losing 33% chance of winning $25 Payoff $100 Chapter Fifteen Copyright (c)2014 John 90 80 70 60 50 40 30 20 10 Expected Value EV = Pr(A)xA + Pr(B)xB + Pr(C)xC Where: Pr(.) is the probability of (.) A,B, and C are the payoffs if outcome A, B or C occurs Chapter Fifteen Copyright (c)2014 John Definition: The expected value of a lottery is a measure of the average payoff that the lottery will generate Expected Value Copyright (c)2014 John In our example lottery, which pays $25 with probability 67 and $100 with probability 0.33, the expected value is: EV = 67 x $25 + 33 x 100 = $50 Notice that the expected value need not be one of the outcomes of the lottery Chapter Fifteen Variance & Standard Deviation Var = (A - EV)2(Pr(A)) + (B - EV)2(Pr(B)) + (C EV)2(Pr(C)) Definition: The standard deviation of a lottery is the square root of the variance It is an alternative measure of risk Chapter Fifteen Copyright (c)2014 John Definition: The variance of a lottery is the sum of the probabilityweighted squared deviations between the possible outcomes of the lottery and the expected value of the lottery It is a measure of the lottery's riskiness Variance & Standard Deviation For the example lottery Copyright (c)2014 John The squared deviation of winning is: • ($100 - $50)2 = 502 = 2500 The squared deviation of losing is: • ($25 - $50)2 = 252 = 625 The variance is: • (2500 x 33)+ (625 x 67) = 1250 Chapter Fifteen 10 The Demand for Insurance If you are risk averse, you prefer to insure this way over no insurance Why? Definition: A fairly priced insurance policy is one in which the insurance premium (price) equals the expected value of the promised payout i.e.: 500 = 05(10,000) + 95(0) Chapter Fifteen 31 Copyright (c)2014 John Full coverage ( no risk so prefer all else equal) The Supply of Insurance Copyright (c)2014 John Insurance company expects to break even and assumes all risk – why would an insurance company ever offer this policy? Definition: Asymmetric Information is a situation in which one party knows more about its own actions or characteristics than another party Chapter Fifteen 32 Adverse Selection & Moral Hazard Copyright (c)2014 John Definition: Adverse Selection is opportunism characterized by an informed person's benefiting from trading or otherwise contracting with a less informed person who does not know about an unobserved characteristic of the informed person Definition: Moral Hazard is opportunism characterized by an informed person's taking advantage of a less informed person through an unobserved action Chapter Fifteen 33 Adverse Selection & Market Failure Lottery: • • $50,000 if no blindness (p = 95) $40,000 if blindness (1-p = 05) EV = $49,500 Copyright (c)2014 John • (fair) insurance: • Coverage = $10,000 • Price = $500 • $500 = 05(10,000) + 95(0) Chapter Fifteen 34 Adverse Selection & Market Failure Now, p' = 10 so that: EV of payout = 1(10,000) + 9(0) = $1000 while price of policy is only $500 The insurance company no longer breaks even Chapter Fifteen 35 Copyright (c)2014 John Suppose that each individual's probability of blindness differs [0,1] Who will buy this policy? Adverse Selection & Market Failure Now, p'' = 20 so that EV of payout = 2(10,000) + 8(0) = $2000 So the insurance company still does not break even and thus the Market Fails Chapter Fifteen 36 Copyright (c)2014 John Suppose we raise the price of policy to $1000 Decision Trees Decision Nodes Chance Nodes Probabilities Payoffs Key Elements We analyze decision problems by working backward along the decision tree to decide what the optimal decision would Be Chapter Fifteen 37 Copyright (c)2014 John Definition: A decision tree is a diagram that describes the options available to a decision maker, as well as the risky events that can occur at each point in time Copyright (c)2014 John Decision Trees Chapter Fifteen 38 Decision Trees Map out the decision and event sequence Identify the alternatives available for each decision Identify the possible outcomes for each risky event Assign probabilities to the events Identify payoffs to all the decision/event combinations Find the optimal sequence of decisions Chapter Fifteen 39 Copyright (c)2014 John Steps in constructing and analyzing the tree: Definition: The value of perfect information is the increase in the decision maker's expected payoff when the decision maker can at no cost obtain information that reveals the outcome of the risky event Chapter Fifteen 40 Copyright (c)2014 John Perfect Information Perfect Information Example: • Expected payoff to conducting test: $35M Expected payoff to not conducting test: $30M The value of information: $5M The value of information reflects the value of being able to tailor your decisions to the conditions that will actually prevail in the future It should represent the agent's willingness to pay for a "crystal ball" Chapter Fifteen 41 Copyright (c)2014 John • Auctions - Types English Auction – An auction in which participants cry out their bids and each participant can increase his or her bid until the auction ends with the highest bidder winning the object being sold Second-Price Sealed-Bid Auction – An auction in which each bidder submits one bid, not knowing the other bids The highest bidder wins the object but pays a price equal to the second-highest bid Dutch Descending Auction – An auction in which the seller of the object announces a price which is then lowered until a buyer announces a desire to buy the item at that price Chapter Fifteen 42 Copyright (c)2014 John First-Price Sealed-Bid Auction – An auction in which each bidder submits one bid, not knowing the other bids The highest bidder wins the object and pays a price equal to his or her bid Auctions Private Values – A situation in which each bidder in an auction has his or her own personalized valuation of the object Common Values – A situation in which an item being sold in an auction has the same intrinsic value to all buyers, but no buyer knows exactly what that value is Winner’s Curse – A phenomenon whereby the winning bidder in a common-values auction might bid an amount that exceeds the item’s intrinsic value Chapter Fifteen 43 Copyright (c)2014 John Revenue Equivalence Theorem – When participants in an auction have private values, any auction format will, on average, generate the same revenue for the seller Summary We can think of risky decisions as lotteries Individuals differ in their attitudes towards risk: those who prefer a sure thing are risk averse Those who are indifferent about risk are risk neutral Those who prefer risk are risk loving Insurance can help to avoid risk The optimal amount to insure depends on risk attitudes Chapter Fifteen 44 Copyright (c)2014 John We can think of individuals maximizing expected utility when faced with risk Summary The provision of insurance by individuals does not require risk lovers We can calculate the value of obtaining information in order to reduce risk by analyzing the expected payoff to eliminating risk from a decision tree and comparing this to the expected payoff of maintaining risk The main types of auctions are private Chapter Fifteen 45 values auctions and common values Copyright (c)2014 John Adverse Selection and Moral Hazard can cause inefficiency in insurance markets ... premium Chapter Fifteen 25 Copyright (c)2014 John pU(I1) + (1-p)U(I2) = U(pI1 + (1-p)I2 - RP) Computing Risk Premium Example: Computing a Risk Premium • • U = I(1/2); p = I1 = $104,000 I2 = $4,000 Chapter. .. John 5(104,000)1/2 + 5(4,000)1/2 = (.5(104,000) + 5(4,000) - RP)1/2 $192.87 = ($54,000 - RP)1/2 $37,198 = $54,000 - RP RP = $16,802 Chapter Fifteen 27 Computing Risk Premium B Let I1 = $108,000... expected value need not be one of the outcomes of the lottery Chapter Fifteen Variance & Standard Deviation Var = (A - EV)2(Pr(A)) + (B - EV)2(Pr(B)) + (C EV)2(Pr(C)) Definition: The standard deviation