(1) Ann has vNM utility u1 (x) = x, Bob has utility u2 (x) = √...
(1) Ann has vNM utility u1 (x) = x, Bob has utility u2 (x) = √ x and Carl has utility u3 (x) = x 3 . Who is risk neutral, risk averse and risk loving?
(2) Consider the lottery P again. Find the dollar amount x such that each person is indifferent between the lottery P and $x (x is the certainty equivalent of P)
(3) Calculate the Arrow-Pratt coefficients for everyone. How do they compare? Does this agree with your answers before?
(4) Calculate the Arrow-Pratt coefficient for utility u (x) = −e −ρx where ρ > 0. This type of utility is called Constant Absolute Risk Aversion (CARA). Why do you think it’s called CARA?
(5) Calculate the Arrow-Pratt coefficient for utility u (x) = x 1−ρ 1−ρ where ρ > 0. This type of utility is called Constant Relative Risk Aversion (CRRA). Why do you think it’s called CRRA?
Homework Answers
The attitude of a person towards risk can be judged by the shape of the utility function. The risk preferences of the above persons are mentioned in the image below:
Given a lottery P, let E (P) be the expected value of the lottery P. For...
Given a lottery P, let E (P) be the expected value of the lottery P. For example, if P = ($10, 0.5; $0, 0.5), then E (P) = 0.5 × 10 + 0.5 × 0 = 5 (1) Ann has vNM utility u1 (x) = x, Bob has utility u2 (x) = √ x and Carl has utility u3 (x) = x^3 . Who is risk neutral, risk averse and risk loving? (2) Consider the lottery P again. Find the...
6. A decision maker has a vNM utility function over money of u(x) = x2. This...
6. A decision maker has a vNM utility function over money of u(x) = x2. This decision maker is (a) risk-averse. (b) risk-neutral. (c) risk-loving. (d) none of the above. 7. Consider two lotteries: • Lottery 1: The gamble (0.1, 0.6, 0.3) over the final wealth levels ($1, $2, $3). (The expected value of this lottery equals $2.2) • Lottery 2: Get $2.2 for sure. a) Any risk-averse individual will choose the first lottery. b) Any risk-averse individual will choose...
2. An individual has a vNM utility function over money of u(x) -Vx, where x is final wealth. Assu...
2. An individual has a vNM utility function over money of u(x) -Vx, where x is final wealth. Assume the individual currently has $16. He is offered a lottery with three possible outcomes; he could gain an extra S9, lose $7, or not lose or gain anything. There is a 15% probability that he will win the extra $9, what minimum probability, p, of losing S7 would ensure that the individual chooses to not play the lottery? (a) p >...
4. An individual has a VNM utility function over money of u(x)=x", where x is the...
4. An individual has a VNM utility function over money of u(x)=x", where x is the amount of money won in the lottery. She faces two scenarios: • Scenario 1: With a 50% probability she wins $36. With a 50% probability she wins $16. • Scenario 2: With a 50% probability she wins $0. With a 50% probability she wins $x. For what value of x will the risk premia be identical in these two scenarios? a. O b. 4...
1. a. Two investors, A and B, are evaluating the same investment opportunity, which has an expected value of £100. The u...
1. a. Two investors, A and B, are evaluating the same investment opportunity, which has an expected value of £100. The utility functions of A and B are ln(x) and x2, respectively. Which investor has a certainty equivalent higher than 100? Which investor requires the higher risk premium? b. (i) Describe suitable measures of risk for ‘loss-aversion’ and ‘risk aversion’. (ii) Concisely define the term ‘risk neutral’ with respect to a utility function u (w), where w is the realisation...
2. An individual has a vNM utility function over money of u(x) -Vx, where x is final wealth. Assume the individual currently has $16. He is offered a lottery with three possible outcomes; he could gain an extra S9, lose $7, or not lose or gain anything. There is a 15% probability that he will win the extra $9, what minimum probability, p, of losing S7 would ensure that the individual chooses to not play the lottery? (a) p >...
4. An individual has a VNM utility function over money of u(x)=x", where x is the amount of money won in the lottery. She faces two scenarios: • Scenario 1: With a 50% probability she wins $36. With a 50% probability she wins $16. • Scenario 2: With a 50% probability she wins $0. With a 50% probability she wins $x. For what value of x will the risk premia be identical in these two scenarios? a. O b. 4...
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