Question

Carol's risk preference is represented by the following expected utility formula: U(π,c₁ ;1−π,c₂ )=π√c₁ +(1−π) √c₂

i) Suppose Carol is indifferent between the following two options: the first option A returns $64 with probability 1/2 and $36 with probability 1/2 , and the second option B returns $X for sure. Determine X. Is X smaller than the expected return of A ? Explain why.
ii) Consider the following three lotteries: L₁ =(0.8,$100;0.2,$25),L₂ =(0.6,$100;0.4,$36), and L₃ =(0.5,$225;0.5,$1). What is the ranking of these lotteries for Carol? Calculate the risk premiums of these lotteries for Carol.

Answer 1

In option A, X should be $50, which is equal to the expected return of option A. The ranking of the lotteries for Carol is L₂ > L₃ > L₁. The risk premiums of the lotteries are $41, $41.6, and $42.

Step-by-step explanation:

i) To determine the value of X, we need to find the expected return of option A. The expected return can be calculated by multiplying the possible outcomes by their respective probabilities and summing them:
Expected return of option A = (0.5 * 64) + (0.5 * 36) = 32 + 18 = $50.
Since Carol is indifferent between options A and B, the expected return of option B must be $50 as well. Therefore, X = $50.

ii) To rank the lotteries, we can calculate the expected utility for each lottery using Carol's expected utility formula.
Expected utility of L₁ = (0.8 * √100) + (0.2 * √25) = 0.8 * 10 + 0.2 * 5 = 8 + 1 = 9.
Expected utility of L₂ = (0.6 * √100) + (0.4 * √36) = 0.6 * 10 + 0.4 * 6 = 6 + 2.4 = 8.4.
Expected utility of L₃ = (0.5 * √225) + (0.5 * √1) = 0.5 * 15 + 0.5 * 1 = 7.5 + 0.5 = 8.
Therefore, the ranking of the lotteries for Carol is L₂ > L₃ > L₁.
The risk premium of a lottery is the difference between its expected value and the expected utility.
Risk premium of L₁ = 50 - 9 = $41.
Risk premium of L₂ = 50 - 8.4 = $41.6.
Risk premium of L₃ = 50 - 8 = $42.