A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is U = $\sqrt { I }$ , what is the risk premium associated with this choice?
A) $1.59
B) $2.52
C) $0
D) $3.95

Question

Quizplus · Accepted Answer

First, we need to find the expected utility of each option:

Expected utility of the lottery:
U(0.3*$30 + 0.7*$80) = U($62) = √62 ≈ 7.87

Expected utility of the guaranteed payoff:
U($65) = √65 ≈ 8.06

So, the expected utility of the guaranteed payoff is higher than that of the lottery.

The risk premium is the amount of money that a decision maker is willing to pay to avoid the risk. It can be calculated by finding the difference between the expected payoffs of the two options and the minimum amount of money that would make the decision maker indifferent between the two options. In this case, the minimum amount of money is the risk premium.

Expected payoff of the lottery: 0.3*$30 + 0.7*$80 = $62

Risk premium = Expected payoff of the guaranteed option - Expected payoff of the lottery
= $65 - $62
= $3

However, we need to convert this monetary value into a utility value in order to compare it to the expected utilities. Using the utility function, we can find that the utility of $3 is:

U($3) = √3 ≈ 1.73

Since the expected utility of the guaranteed option is higher than that of the lottery, the decision maker would choose the guaranteed option. Therefore, the best choice is the guaranteed option with a payoff of $65.

A Decision-Maker Is Faced with a Choice Between a Lottery I\sqrt { I }I​

A Decision-Maker Is Faced with a Choice Between a Lottery $\sqrt { I }$