# Managerial Economics

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1 Managerial Economics Unit 9: Risk Analysis Rudolf Winter-Ebmer Johannes Kepler University Linz Winter Term 2012 Managerial Economics: Unit 9 - Risk Analysis 1 / 1

2 Objectives Explain how managers should make strategic decisions when faced with incomplete or imperfect information Study how economists make predictions about individual s or firm s choices under uncertainty Study the standard assumptions about attitudes towards risk Managerial Economics: Unit 9 - Risk Analysis 2 / 1

3 Management tools Expected value Decision trees Techniques to reduce uncertainty Expected utility Managerial Economics: Unit 9 - Risk Analysis 3 / 1

4 Uncertainty Consumer and firms are usually uncertain about the payoffs from their choices. Some examples... Example 1: A farmer chooses to cultivate either apples or pears When she takes the decision, she is uncertain about the profits that she will obtain. She does not know which is the best choice. This will depend on rain conditions, plagues, world prices... Managerial Economics: Unit 9 - Risk Analysis 4 / 1

5 Uncertainty Example 2: playing with a fair dice We will winac2 if 1, 2, or 3 We neither win nor lose if 4, or 5 We will loseac6 if 6 Example 3: John s monthly consumption: AC3000 if he does not get ill AC500 if he gets ill (so he cannot work) Managerial Economics: Unit 9 - Risk Analysis 5 / 1

6 Lottery Economists call a lottery a situation which involves uncertain payoffs: Cultivating apples is a lottery Cultivating pears is another lottery Playing with a fair dice is another one Monthly consumption another Each lottery will result in a prize Managerial Economics: Unit 9 - Risk Analysis 6 / 1

7 Risk and probability Risk: Hazard or chance of loss Probability: likelihood or chance that something will happen The probability of a repetitive event happening is the relative frequency with which it will occur probability of obtaining a head on the fair-flip of a coin is 0.5 Managerial Economics: Unit 9 - Risk Analysis 7 / 1

8 Probability Frequency definition of probability: An event s limit of frequency in a large number of trials Probability of event A = P(A) = r/r R = Large number of trials r = Number of times event A occurs Rules of probability Probabilities may not be less than zero nor greater than one. Given a list of mutually exclusive, collectively exhaustive list of the events that can occur in a given situation, the sum of the probabilities of the events must be equal to one. Managerial Economics: Unit 9 - Risk Analysis 8 / 1

9 Probability Subjective definition of probability: The degree of a manager s confidence or belief that the event will occur Probability distribution: A table that lists all possible outcomes and assigns the probability of occurrence to each outcome Managerial Economics: Unit 9 - Risk Analysis 9 / 1

10 Probability If a lottery offers n distinct prizes and the probabilities of winning the prizes are p i (i = 1,...,n) then n p i = p 1 +p p n = 1 i=1 Managerial Economics: Unit 9 - Risk Analysis 10 / 1

11 Expected value of a lottery The expected value of a lottery is the average of the prizes obtained if we play the same lottery many times If we played 600 times the lottery in Example 2 We obtained a times, a times... We would win AC2 300 times, win AC0 200 times, and lose AC6 100 times Average prize = ( )/600 Average prize = (1/2) 2+(1/3) 0 (1/6) 6 = 0 Notice, we have the probabilities of the prizes multiplied by the value of the prizes Managerial Economics: Unit 9 - Risk Analysis 11 / 1

12 Expected Value. Formal definition For a lottery (X) with prizes x 1,x 2,...,x n and the probabilities of winning p 1,p 2,...p n, the expected value of the lottery is E(X) = p1 x 1 +p 2 x p n x n n E(X) = p i x i i=1 The expected value is a weighted sum of the prizes the weights are the respective probabilities Managerial Economics: Unit 9 - Risk Analysis 12 / 1

13 Comparisons of expected profit Example: Jones Corporation is considering a decision involving pricing and advertising. The expected value if they raise price is Profit Probability (Probability)(Profit) \$ 800, \$ 400, , ,000 Expected Profit = \$ 100,000 The payoff from not increasing price is \$ 200,000, so that is the optimal strategy. Managerial Economics: Unit 9 - Risk Analysis 13 / 1

14 Road map to decisions Decision tree: A diagram that helps managers visualize their strategic future Figure 15.1: Decision Tree, Jones Corporation Managerial Economics: Unit 9 - Risk Analysis 14 / 1

15 Constructing a decision tree Managerial Economics: Unit 9 - Risk Analysis 15 / 1

16 Remarks Decision fork: a juncture representing a choice where the decision maker is in control of the outcome Chance fork: a juncture where chance controls the outcome Managerial Economics: Unit 9 - Risk Analysis 16 / 1

17 Expected value of perfect information Expected Value of Perfect Information (EVPI) The increase in expected profit from completely accurate information concerning future outcomes. Jones Example (Figure 15.1) Given perfect information, the company will increase price if the campaign will be successful and will not increase price if the campaign will not be successful. Expected profit = \$500,000 so EVPI = \$500,000 \$200,000 = \$300,000 Why is this useful? Managerial Economics: Unit 9 - Risk Analysis 17 / 1

18 Simple decision rule Use expected value of a project How do people really decide? Managerial Economics: Unit 9 - Risk Analysis 18 / 1

19 Is the expected value a good criterion to decide between lotteries? Does this criterion provide reasonable predictions? Let s examine a case... Lottery A: GetAC3125 for sure (i.e. expected value =AC3125) Lottery B: getac4000 with probability 0.75, and getac500 with probability 0.25 (i.e. expected value also AC3125) Probably most people will choose Lottery A because they dislike risk (risk averse). However, according to the expected value criterion, both lotteries are equivalent. The expected value does not seem a good criterion for people that dislike risk. If someone is indifferent between A and B it is because risk is not important for him/her (risk neutral). Managerial Economics: Unit 9 - Risk Analysis 19 / 1

20 Measuring attitudes toward risk: the utility approach Another example A small business is offered the following choice: 1 A certain profit of \$2,000,000 2 A gamble with a change of \$4,100,000 profit or a \$60,000 loss. The expected value of the gamble is \$2,020,000. If the business is risk averse, it is likely to take the certain profit. Utility function: Function used to identify the optimal strategy for managers conditional on their attitude toward risk Managerial Economics: Unit 9 - Risk Analysis 20 / 1

21 Expected Utility: The standard criterion to choose among lotteries Individuals do not care directly about the monetary values of the prizes they care about the utility that the money provides U(x) denotes the utility function for money We will always assume that individuals prefer more money than less money, so: U (x i ) > 0 Managerial Economics: Unit 9 - Risk Analysis 21 / 1

22 Expected Utility: The standard criterion to choose among lotteries The expected utility is computed in a similar way to the expected value However, one does not average prizes (money) but the utility derived from the prizes n EU = p i U(x i ) = p 1 U(x 1 )+p 2 U(x 2 )+...+p n U(x n ) i=1 The sum of the utility of each outcome times the probability of the outcome s occurrence The individual will choose the lottery with the highest expected utility Managerial Economics: Unit 9 - Risk Analysis 22 / 1

23 Can we construct a utility function? Example Utility function is not unique: you can add a constant term you can multiply by a constant factor the general shape is important Managerial Economics: Unit 9 - Risk Analysis 23 / 1

24 How do you get these points? Start with any values: e.g. U( 90) = 0, U(500) = 50 Then ask the decision maker questions about indifference cases Find value for 100 Do you prefer the certainty of a \$100 gain to a gamble of \$500 with probability P and \$-90 with probability (1 P)? Try several values of P until the respondent is indifferent Suppose outcome is P = 0.4 Then it follows U(100) = 0.4U(500) + 0.6U( 90) U(100) = 0.4(50)+0.6(0) = 20 Managerial Economics: Unit 9 - Risk Analysis 24 / 1

25 Attitudes towards risk Risk-averse: expected utility of lottery is lower than utility of expected profit - the individual fears a loss more than she values a potential gain Risk-neutral: the person looks only at expected value (profit), but does not care if the project is high- or low-risk. Risk-seeking: expected utility is higher than utility of expected profit - the individual prefers a gamble with a less certain outcome to one with a certain outcome Managerial Economics: Unit 9 - Risk Analysis 25 / 1

26 Attitudes toward risk Managerial Economics: Unit 9 - Risk Analysis 26 / 1

27 Attitudes towards risk What attitude towards risk do most people have? (maybe you want to differentiate between long-term investment and, say, Lotto) What attitude towards risk should a manager of a big (publicly traded) company have? What s the effect of a managers risk attitude? Managerial Economics: Unit 9 - Risk Analysis 27 / 1

28 Example A risk averse person gets Y 1 or Y 2 with probability of 0.5 Expected Utility < Utility of expected value Managerial Economics: Unit 9 - Risk Analysis 28 / 1

29 Measure of Risk: Standard deviation and Coefficient of Variation as a measure of risk we often use the standard deviation N σ = ( P i [π i E(π)] 2 ) 0.5 i=1 to consider changes in the scale of projects, use the coefficient of variation V = σ/e(π) Figure 15.4: Probability Distribution of the Profit from an Investment in a New Plant Managerial Economics: Unit 9 - Risk Analysis 29 / 1

30 How can we measure risk? Probability Distributions of the Profit from an Investment in a New Plant Managerial Economics: Unit 9 - Risk Analysis 30 / 1

31 Adjusting for risk Certainty equivalent approach: When a manager is indifferent between a certain payoff and a gamble, the certainty equivalent (rather than the expected profit) can identify whether the manager is a risk averter, lover, or risk neutral. Managerial Economics: Unit 9 - Risk Analysis 31 / 1

32 Definition of certainty equivalent The certainty equivalent of a lottery m, ce(m), leaves the individual indifferent between playing the lottery m or receiving ce(m) for certain. U(ce(m)) = E[U(m)] Managerial Economics: Unit 9 - Risk Analysis 32 / 1

33 Adjusting for risk Certainty equivalent approach If the certainty equivalent is less than the expected value, then the decision maker is risk averse. If the certainty equivalent is equal to the expected value, then the decision maker is risk neutral. If the certainty equivalent is greater than the expected value, then the decision maker is risk loving. Managerial Economics: Unit 9 - Risk Analysis 33 / 1

34 Adjusting for risk The present value of future profits, which managers seek to maximize, can be adjusted for risk by using the certainty equivalent profit in place of the expected profit. Managerial Economics: Unit 9 - Risk Analysis 34 / 1

35 Adjusting for risk Indifference curves Figure 15.5: Manager s Indifference Curve between Expected Profit and Risk With expected value on the horizontal axis, the horizontal intercept of an indifference curve is the certainty equivalent of the risky payoffs represented by the curve. If a decision maker is risk neutral, indifference curves will be vertical. Managerial Economics: Unit 9 - Risk Analysis 35 / 1

36 Manager s Indifference Curve Managerial Economics: Unit 9 - Risk Analysis 36 / 1

37 Definition of risk premium Risk premium = E[m] ce(m) The risk premium is the amount of money that a risk-averse person would sacrifice in order to eliminate the risk associated with a particular lottery. In finance, the risk premium is the expected rate of return above the risk-free interest rate. Managerial Economics: Unit 9 - Risk Analysis 37 / 1

38 Lottery m. Prizes m 1 and m 2 Managerial Economics: Unit 9 - Risk Analysis 38 / 1

39 Risk Premium Managerial Economics: Unit 9 - Risk Analysis 39 / 1

40 Examples of commonly used Utility functions for risk averse individuals U(x) = ln(x) U(x) = x U(x) = x a where 0 < a < 1 U(x) = exp( a x) where a > 0 Managerial Economics: Unit 9 - Risk Analysis 40 / 1

41 Measuring Risk Aversion The most commonly used risk aversion measure was developed by Pratt r(x) = U (X) U (X) For risk averse individuals, U (X) < 0 r(x) will be positive for risk averse individuals Managerial Economics: Unit 9 - Risk Analysis 41 / 1

42 Risk Aversion If utility is logarithmic in consumption U(X) = ln(x) where X > 0 Pratt s risk aversion measure is r(x) = U (X) U (X) = 1 X Risk aversion decreases as wealth increases Managerial Economics: Unit 9 - Risk Analysis 42 / 1

43 Risk Aversion If utility is exponential U(X) = e ax = exp( ax) where a is a positive constant Pratt s risk aversion measure is r(x) = U (X) U (X) = a2 e ax ae ax = a Risk aversion is constant as wealth increases Managerial Economics: Unit 9 - Risk Analysis 43 / 1

44 Example Lotteries A and B Lottery A: GetAC3125 for sure (i.e. expected value =AC3125) Lottery B: getac4000 with probability 0.75, and getac500 with probability 0.25 (i.e. expected value also AC3125) Suppose also that the utility function is U(X) = sqrt(x) where X > 0 U(A) = certainty equivalent: E(U(B)) = 0.75*U(4000) *U(500) = ( ) 2 = = U(ce(B))) risk premium: = Managerial Economics: Unit 9 - Risk Analysis 44 / 1

45 Willingness to Pay for Insurance Consider a person with a current wealth ofac100,000 who faces a 25% chance of losing his car worth AC20,000 Suppose also that the utility function is U(X) = ln(x) where X > 0 the person s expected utility will be E(U) = 0.75U(100,000) U(80,000) E(U) = 0.75 ln(100,000) ln(80,000) E(U) = Managerial Economics: Unit 9 - Risk Analysis 45 / 1

46 Willingness to Pay for Insurance What is the maximum insurance premium the individual is willing to pay? E(U) = U(100,000 - y) = ln(100,000 - y) = ,000 - y = exp( ) y= 5,426 The maximum premium he is willing to pay isac5,426. Managerial Economics: Unit 9 - Risk Analysis 46 / 1

47 Example Roy Lamb has an option on a particular piece of land, and must decide whether to drill on the land before the expiration of the option or give up his rights. If he drills, he believes that the cost will be \$200,000. If he finds oil, he expects to receive \$1 million; if he does not find oil, he expects to receive nothing. a) Can you tell wether he should drill on the basis of the available information? Why or why not? Managerial Economics: Unit 9 - Risk Analysis 47 / 1

48 Example cont d No, there are no probabilities given. Mr. Lamb believes that the probability of finding oil if he drills on this piece of land is 1 4, and the probability of not finding oil if he drills here is 3 4. b) Can you tell wether he should drill on the basis of the available information. Why or why not? c) Suppose Mr. Lamb can be demonstrated to be a risk lover. Should he drill? Why? d) Suppose Mr. Lamb is risk neutral. Should he drill or not. Why? Managerial Economics: Unit 9 - Risk Analysis 48 / 1

49 Example cont d b) 1/4(800) 3/4(200) = 50 > 0, so a person who is risk neutral would drill. However, if very risk averse, the person would not want to drill. c) Yes, since the project has both a positive expected value and contains risk, Mr. Lamb will be doubly pleased. d) Yes, Mr. Lamb cares only about expected value, which is positive for this project. Managerial Economics: Unit 9 - Risk Analysis 49 / 1

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