Econ 423 Michael SalemiDecision Making When Outcomes Are UncertainClass NotesOutline 1. Introduction2. Exercise One: Ranking Artificial SecuritiesStudents will apply the concepts of expected value, variance, and “Marcowitz” risk todescribe the tradeoffs implied in the choice of securities.3. Exercise Two: Risk Aversiona. Students will investigate the meaning of risk aversion and the reasonableness of thehypothesis that the typical agent is risk averse.b. Students will decide on their reservation price for playing the “St. Petersburg” game.4. Expected Utility HypothesisStudents will work with a set of notes that introduces the expected utility hypothesis andprovides an explanation of the St. Petersburg paradox.5. Exercise Three: DiversificationStudents will complete an exercise that introduces them to the concept of diversificationand explains the relationship between diversification and the statistical covariance. IntroductionLet (S t + 1 , S t + 2 , ...S t + M ) be payments that will be received at times t +1, t +2, ..., t +M and let R be thediscount rate. The present value of the stream is:When choosing among streams of payments, it is often reasonable to prefer a stream with a higher presentvalue to one with a lower present value. If the stream of payments is associated with a bond issued by theU.S. Treasury, it is reasonable to assume that there is no uncertainty associated with the size or timing ofthe payments.But the payments made by many assets are uncertain. For example, when we purchase a share ofcommon stock we do not know for sure what dividends we will receive. The purpose of these notes andexercises is to introduce concepts that are used in models describing how agents make decisions whenoutcomes are uncertain.Exercise One: Decision Making When Outcomes Are UncertainIn this exercise, students will decide which of four “securities” they prefer and learn how analystsmeasure the expected pay out and riskiness of securities. For simplicity, we consider securities for whichthere are six possible outcomes. Outcomes are equally likely and chosen by the roll of a fair die. Eachsecurity costs $1.00. Pay outs are given by the following table.Security Table of Pay Outs123456A $0.50 $0.50 $1.00 $1.00 $2.00 $2.00B $0.00 $0.50 $1.00 $1.00 $2.00 $2.50C $0.50 $0.50 $1.00 $1.00 $1.00 $3.00D $0.00 $0.50 $1.00 $1.00 $2.00 $3.00A. Which security is your first choice, second choice, etc.? Ranking: A_________ B________ C_________ D_________Is there a security for which you would not pay a dollar even if it were the only one available?B. Prior to modern finance theory, it was assumed that agents would rank securities on the base of theirexpected payout. Expected payout is the probability weighted average of the payout in each state of theworld. What is the expected payout securities A through D?Expected Payout (:): A_________ B________ C_________ D_________. C. The chief problem with ranking securities on the basis of expected payout alone is that such a rankingcompletely ignores the relative riskiness of the securities. There are two measures of risk which aretypically used in modern finance theory. Variance of the Payout:E (1/6)(yi-:)2Marcowitz Risk Measure: (Probability of a loss)(Average loss given a loss occurs) What are the risk measures associated with each security?Variance: A_________ B________ C_________ D_________Marcowitz: A_________ B________ C_________ D_________D. Discussion: What role does risk play in the ranking you made in part A?Exercise Two: Risk AversionThe purpose of this exercise is to introduce the concept of risk aversion. Risk aversion is important inunderstanding how securities are priced, why markets for insurance exist, and is a key insight into themodern approach to the economics of decision making in the presence of uncertainly.Game 1 (Coin Toss): The following is a set of alternative games. You may choose to play at most onegame. For each game, there will be a single toss of a fair coin. If the outcome is “heads” you win theamount in the win column. If “tails”, you lose the amount in the lose column. Are you willing to play agame? If your answer is “yes”, which game do you prefer?Game Lose Win A $ 1.00 $ 1.00 B $ 2.00 $ 2.00 C $ 5.00 $ 5.10 D $ 10.00 $ 10.40 E $ 20.00 $ 21.00 F $ 50.00 $ 53.00 G $100.00 $110.00 H $500.00 $525.00I am willing to play________________.Which game has the largest expected return? Did you prefer that game? If not, why?Game 2 (St. Petersburg): There will be only one play of the game. Players are required to pay a priceto participate. The price is the only money the player can lose. If a player pays the price, a fair coin istossed until “heads” appears. The player is paid a prize according to the following table.Toss when first head appears 1 2 3 4 5 mPrize .25 .50 1.00 2.00 4.00 (2m-1 ) x (.25)A Reservation Price is the most a buyer would be willing to pay to obtain a good or service. It is the leasta seller would be willing to accept in exchange for a good or service.Question: What is your reservation price for one play of the game?_____________________.Discussion: What is the expected payout of the St. Petersburg game? Is it reasonable to believe thatagents use only expected value calculations when making choices in the face of uncertainty?The Expected Utility HypothesisThe expected utility hypothesis was first put forward by Swiss mathematician Daniel Bernoulli(1700-1782). The hypothesis says that when faced with decisions involving uncertain outcomes, agentschoose the strategy that maximizes their expected utility. The idea was quite novel when Bernoulliintroduced it. Until Bernoulli, the standard rule of thumb was to choose the strategy that provided thehighest expected value. Today, the expected utility hypothesis has become the standard hypothesisused to model decision making in the face of uncertainty.To better understand the hypothesis, we may consider a simple example. Suppose anindividual is faced with the opportunity to invest in a project that has the statistical properties of a faircoin toss. Specifically, the agent will invest $50.00 and has a 50 percent probability that the investmentis a success. If the investment succeeds, the agent receives $100.00; if it does not succeed, she receivesnothing. An expected value maximizer would be indifferent between investing and not investingbecause the expected value of the