Choice under uncertainty LECTURE

# Choice under uncertainty LECTURE - 1 Choice under...

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Choice under uncertainty 1. Expected value Expected value is a weighted average of payoffs. [example] You are considering investing in a company that explores for offshore oil. This oil exploration has two possible outcomes, namely, success or failure. It is known that the probability of success is ¼ and probability of failure is ¾. If the exploration turns out to be successful, then the company’s stock price will be \$40, and if the exploration turns out to be failure, then the stock price will be \$20. Then the expected value (price) of the stock will Expected value = (Prob of success) × (Stock price when success) + (Prob of failure) × (Stock price when failure) = ¼ *\$40 + ¾*\$20 = \$25 This idea can be extended to situation where there are many outcomes. Suppose that there are n possible outcomes. Let X 1 , X 2 , X 3 , … X n , be the payoff for each outcome, and Pr 1 Pr 2 Pr 3 … Pr n be the probability associated with each outcome. Then, “Expected Value” will be E(X) = Pr 1 X 1 + Pr 2 X 2 + … + Pr n X n 1

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2. standard deviation and variance Suppose that there is a sales job opportunity that is entirely based on commission - Income earned depend on how much you sell. There are two outcomes: the first possible outcome is “successful” outcome where you earn \$2000, and the second possible outcome is “less successful outcome” where you earn \$1000. Suppose that each outcome occurs with probability ½. Q1: What is the expected income for this job opportunity? Now suppose that there is another sales job opportunity that pays a fixed salary of \$1510. However it is known that there is a slight chance that the firm goes out of business in which case you earn \$510 in severance pay. Suppose that the probability that the company does not go out of business is 0.99 and the probability that firm goes out of business is 0.01. Q2. What is the expected income for this job? From Q1 and Q2, we see that both job opportunities have the same expected income. Although those jobs have the same expected income, it seems that commission based job is more risky. In the following, we will talk about the standard deviation, which is a measure of the risk. a) Variance Using the example of a risky job, variance will be given by the following formula. Suppose a risky job has two outcomes. For the first outcome, you earn \$X 1 , and for the second outcome, you earn 2
2 . Probabilities of getting the first outcome and the second outcome are denoted by Pr 1 and Pr 2 respectively. The variance (of the earnings) is given by the following. Variance = Pr 1 (X 1 - μ ) 2 +Pr 2 (X 2 ) 2 where = the mean of the outcome = Pr 1 X 1 +Pr 2 X 2 b) standard deviation = We use the standard deviation to measure the risk of the job. 3

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Choice under uncertainty LECTURE - 1 Choice under...

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