05/05/2011. Degree of Risk. Degree of Risk. BUSA 4800/4810 May 5, Uncertainty

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1 BUSA 4800/4810 May 5, 2011 Uncertainty We must believe in luck. For how else can we explain the success of those we don t like? Jean Cocteau Degree of Risk We incorporate risk and uncertainty into our models of decision making because they can cause consumers and firms to modify decisions about consumption and investment choices. Risk is the when the likelihood of each possible outcome is known or can be estimated, and no single possible outcome is certain to occur. Estimates of how risky each outcome is allows us to estimate the most likely outcome. Degree of Risk A probability is a number between 0 and 1 that indicates the likelihood that a particular outcome will occur. We can estimate probability with frequency, the number of times that one particular outcome occurred (n) out of the total number of times an event occurred (N). If we don t have a history of the event that allows us to calculate frequency, we can use our best estimate or subjective probability. 1

2 Degree of Risk A probability distribution relates the probability of occurrence to each possible outcome. Degree of Risk Expected value is the value of each possible outcome (V i ) times the probability of that outcome ( ), summed over all n possible outcomes: θ i How is expected value used to measure risk? Variance measures the spread of the probability distribution or how much dispersion there is between the actual value and the expected value. σ Standard deviation ( ) is the square root of the variance and is a more commonly reported measure of risk. Decision Making Under Uncertainty Example: Greg schedules an outdoor event If it doesn t rain, he ll make $15 in profit If it does rain, he ll make -$5 in profit (loss) There is a 50% chance of rain. Greg s expected value (outdoor event): Variance = 100 Standard deviation = $10 2

3 Decision Making Under Uncertainty Example, continued: Greg schedules an indoor event If it doesn t rain, he ll make $10 in profit If it does rain, he ll make $0 in profit There is still a 50% chance of rain. Greg s expected value (indoor event) is the same! Variance (indoor event) is much smaller: 25 Standard deviation = $5 Much less risky to schedule event indoors! Decision Making Under Uncertainty Although indoor and outdoor events have the same expected value, the outdoor event involves more risk. He ll schedule the event outdoors only if he likes to gamble. People can be classified according to attitudes toward risk. A fair bet is a wager with an expected value of zero. Example: You receive $1 if a flipped coin comes up heads and you pay $1 if a flipped coin comes up tails. Someone who is unwilling to make a fair bet is risk averse. Someone who is indifferent about a fair bet is risk neutral. Someone who is risk preferring will make a fair bet. One more element: Risk Aversion Consider the following gamble: Prospect a prob = θ G(a, b:θ, 1- θ) prospect b prob = 1- θ Question: Will we prefer the expected value of the gamble with certainty, or will we prefer the gamble itself? Example: consider the gamble with 10% chance of winning $100 90% chance of winning $0 E(gamble) = $10 would you prefer the $10 for sure or would you prefer the gamble? if prefer the gamble, you are risk seeking if indifferent to the options, you are risk neutral if prefer the expected value over the gamble, risk averse 3

4 Utility Function Utility is an Index that measures preferences. Example (let wealth be W = $100) U(W) = W 2 = 10,000 Convex U(W) = 2W = 200 inear U(W) = W 1/2 = 10 Concave Utility Graphically U(W) U(W) U(W) U(b) U(b) U(b) U(a) U(a) U(a) a b W a b W a b W U(W) = W 2 U(W) = 2W U(W) = W 1/2 Risk Averse Utility U(W) = W 1/2 Skippy has $17 and can win or lose $8 with probability of 0.5 W = 25 (win) or W = 9 (lose) EV = 0.5(25) + 0.5(9) = 17 U(17) = 4.2 (not taking bet) U(25) = 5 and U(9) = 3 EU = 0.5(5) + 0.5(3) = 4 (less than not betting) Note: winning adds 0.8 utility but losing takes away 1.2 utility. 4

5 U(W) The Utility Function 3.0 et U(W) = W 1/2 (square root) 2 MU = Change in Utility MU positive But diminishing W The Certainty Equivalent The Expected wealth is 17 The E[U(W)] = 4 How much would this individual accept with certainty and be indifferent with the gamble? et W 0 = certainty equivalent U(W 0 ) = 4 W 0 = 16 This individual would take 16 with certainty rather than the gamble with expected payoff of 17. The difference (17 16 ) = 1, can be viewed as a risk premium an amount that would be paid to avoid risk. Attitudes Toward Risk Example: Risk-aversion and wealth Irma has initial wealth of $40 Option 1: keep the $40 and do nothing U($40) = 120 Option 2: buy a vase that she thinks is a genuine Ming vase with probability of 50% If she is correct, wealth = $70 U($70) = 140 If she is wrong, wealth = $10 U($10) = 70 Expected value of wealth remains $40 = (½ $10) + (½ $70) Expected value of utility is 105 = (½ 70) + (½ 140) Although both options have the same expected value of wealth, the option with risk has lower expected utility. 5

6 Attitudes Toward Risk Irma is risk-averse and would pay a risk premium to avoid risk. Attitudes Toward Risk Risk-neutral and risk-preferring utilities. The Risk Premium Risk Premium = an individual's expected wealth, given he plays the gamble - level of wealth the individual would accept with certainty if the gamble were removed (ie the certainty equivalent) In general, if U[E(W)] > E[U(W)] then person is risk averse (RP > 0) if U[E(W)] = E[U(W)] then person is risk neutral (RP = 0) if U[E(W)] < E[U(W)] then person is risk seeking (RP < 0) risk aversion occurs when the utility function is strictly concave risk neutrality occurs when the utility function is linear risk loving occurs when the utility function is convex 6

7 Avoiding Risk There are four primary ways for individuals to avoid risk: 1. Just say no Abstaining from risky activities is the simplest way to avoid risk. 2. Obtain information Armed with information, people may avoid making a risky choice or take actions to reduce probability of a disaster. 3. Diversify Don t put all your eggs in one basket. 4. Insure Insurance is like paying a risk premium to avoid risk. Avoiding Risk Via Diversification Diversification can eliminate risk if two events are perfectly negatively correlated. If one event occurs, then the other won t occur. Diversification does not reduce risk if two events are perfectly positively correlated. If one even occurs, then the other will occur, too. Example: investors reduce risk by buying shares in a mutual fund, which is comprised of shares of many companies. Avoiding Risk Via Insurance A risk-averse individual will fully insure by buying enough insurance to eliminate risk if the insurance company offers a fair bet, or fair insurance. In this scenario, the expected value of the insurance is zero; the policyholder s expected value with and without the insurance is the same. Insurance companies never offer fair insurance. Reason: insurance premiums and other receipts from policyholders equal the total dollar value of all insurance claims (if all insurance is fair); hence, firms would earn losses equal to their fixed costs. Thus, insurance firms do not offer fair insurance, so most people do not fully insure. 7

8 The Airbus Case: Adding uncertainty Each Airbus team will find different estimates for key variables. Results and conclusions were sensitive to the values given these variables. Teams are to look explicitly at how changes to variables would impact their conclusion Airbus Case Demand forecasts Demand forecasts played an important role in the decision to aunch or Not aunch Extending the Analysis: Consider two demands: High and ow et P be the probability of high demand et 1-P be the probability of low demand Use reasonable assumptions for all other variables Airbus Case High Demand Payoff Matrix Demand = 1645 Airbus dominant Strategy: aunch Boeing dominant strategy: aunch Nash Equilibrium:, AIRBUS 8, 12 10, 10 0, 24 0, 9 8

9 Airbus Case ow Demand Payoff Matrix Demand = 800 Airbus: No dominant Strategy Boeing: No dominant strategy: Two Nash Equilibria:, and, AIRBUS -1, 4 1, 6 0, 9 0, 6 Payoff Matrix with Uncertainty -1, 4 1, 6 AIRBUS 0, 9 0, 6 High Demand: P ow Demand: 1-P AIRBUS 8P + (1-P)(-1) 10P + (1-P) 12P + 4(1-P) 10P + 6(1-P) P + 9(1-P) 9P + 6(1-P) Simplified Payoff Matrix Airbus Payoffs: If 9P-1 > 0 then P > 1/9 9P + 1 >0 always Boeing Payoffs: If 8P + 4 > 4P +6 then P > ½ 15P + 9 > 3P + 6 always 9

10 Airbus: Best Response Functions: Boeing: If Boeing = then If 9P-1 > 0 or P > 1/9 Then Airbus = If P < 1/9 Airbus = If Boeing =, Then Airbus = If Airbus =, then If 8P + 4 > 9P +1 or P > ½ Then Boeing = If P < ½ Boeing = If Airbus =, Then Boeing = Equilibrium Conditions If P > ½ then, Airbus and Boeing both have as dominant Strategy If ½> P > 1/9, then, Only Airbus has a dominant strategy If P < 1/9, then 2 Nash equilibria, and, Sequential Game AIRBUS P 1-P P 1-P P 1-P P 1-P High ow High ow High ow High ow 10