ECO 203: Worksheet 4. Question 1. Question 2. (6 marks)

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1 ECO 203: Worksheet 4 Question 1 (6 marks) Russel and Ahmed decide to play a simple game. Russel has to flip a fair coin: if he gets a head Ahmed will pay him Tk. 10, if he gets a tail he will have to pay Ahmed Tk (a). What is the expected value, E(x), of this game? (2) (b). Would you call this a fair game? Explain. (Refer to Nicholson, Ch-7, pg 203 for definition) (1) Russel decides to make the game a bit more exciting and thus suggests that the payoffs be changed so that when a head comes up the player gets Tk. 10 but if a tail comes up the player loses Tk. 1. (c). What is the expected value of this new game, E(x)? (2) (d). Would you call this new game a fair game? Explain. (1) Question 2 (5 marks) The following game is proposed a coin is flipped until a head appears. If the head first appears on the nth flip, the player is paid $2 n. Let x i denote the prize awarded when the first head appears in the i th trial such that: prize if head appears in the 1 st trial x 1 = 2 1 = $2 prize if head appears in the 2 nd trial x 2 = 2 2 = $4 Prize if head appears in the 3 rd trial x 3 = 2 3 = $8 The probability of getting a head for the first time on the i th trial is 1 i 2 - it is the probability of getting a head after getting tail for (i-1) times. This means the probabilities are: Probability that head appears in the 1 st trial Probability that head appears in the 2 nd trial Probability that head appears in the 3 rd trial π 1 = 1 2 π 2 = 1 2 π 3 = = = = 1 8 (a). What is the expected value, E(x), of this game? (2)

2 (b). Based on the (a), would you be willing to pay the expected value you found in order to play this game? Explain. (1) (c). Your answers in (a) and (b) justifies the use of Expected Utility instead of Expected Value when attempting to estimate how much you will be willing to pay to play the game. Justify this statement. (2) 2 Question 3 (15 marks) Hercules Poirot is extremely scared he will catch a cold if he goes out in the rain. But unfortunately, his car is out of order and he has to walk to the bus stop every day even if it is raining. There is a 35% chance that he will catch a cold since it is monsoon season. When he is healthy he brings home $1000 per week. But when he catches a cold, since he has to spend a substantial amount of money on doctor s visit and medication he brings home $800. His Utility function is U Y = Y 0.25 where Y denotes Income. (a). What is the expected value of his weekly earning? (1) (b). What is the expected utility of his weekly earning? (2) (c). What is the maximum insurance premium Poirot will be willing to pay to avoid the loss brought about by his illness? (4)

3 (d). If the chances of catching a cold increases by 10%, what is the maximum insurance premium he will be willing to pay? (6) 3 (e). What is the actuarially fair insurance premium when : (2) probability of catching a cold is 35% probability of catching a cold is 45% Question 4 (5 marks) Suppose there is a chance that a risk averse individual with a current wealth of $20,000 will fall down the stairs and hurt his back thus suffering a loss of $10,000. (a). Calculate the cost of actuarially fair insurance in this situation. (1) Suppose two types of insurance policies are available:

4 (i) (ii) A fair policy covering the complete loss; and A fair policy covering half of any loss incurred (b). Calculate the cost of the second type of policy. (1) (c). Use a Utility-of-wealth graph to show that the individual will prefer to pay an insurance premium as opposed to accepting the gamble uninsured. (3) 4 Question 5 (30 marks) Find the degree of absolute risk aversion and relative risk aversion for the following Utility Functions :( To find how risk aversion changes with respect to wealth, where necessary, just find the derivative function i.e. dr dw ) U(W) = 1 W U(W) = ln W ( where W > 0)

5 5 U(W) = W γ U W = e 2KW where K is a positive constant

6 6 U W = αw βw 2 where α, β > 0

7 7 Question 6 (9 marks) Sam is a risk averse individual whose initial wealth is Tk Sam is presented with two gambles: in Gamble 1 if he wins he stands to gain Tk 1000, but if he loses he stands to lose Tk In Gamble 2 he stands to gain Tk. 50 if he wins and lose Tk. 50 if he loses. For both gambles there is a 50% chance that he will win. He is offered a third gamble where he gains Tk. 0 with probability of 50% or loses Tk. 0 with probability of 50%. (a) What is the expected value of the three different gambles? (3) (b) Represent the information from (a) in the space provided along with the Certainty Line. (2) (c) Based on what you found in (a), which gamble do you think Sam would choose? Answer in 2 lines. Use a diagram, if needed, to justify your answer. (2)

8 (d) Sam s friend Jim is a risk lover. When faced with the three different gambles, which gamble is Jim most likely to chose. Again explain in two lines using a diagram is needed. (2) 8 Question 7 (5 marks) Using a Utility of Wealth Diagram, show that: (a) a risk averse individual prefers certain wealth of W to a gamble whose expected value is W. Use concept of convex combination to define a risk averse individual. (5) Definition: Explanation: (b) A Risk Neutral individual is indifferent between certain wealth W and a gamble with expected value of W. Use the concept of convex combination to define a risk neutral individual. (5) Definition:

9 Explanation: (c) Can you give an example of when an individual may become risk neutral? (Refer to the first link uploaded on uncertainty and risk it is a power-point presentation) (1) 9