Assume Alice’s preferences satisfy the axioms of the expected utility theorem. A risky assetyields $6 utility in a bull market, while it yields $3 utility or $1 utility with equal probability in abear market. Alice’s expected utility from the risky asset is $3. What is the probability of a bullmarket?[Write your answer as a nonnegative number equal or less than 1, with a maximum of twodecimals, like 0.12.]
Question
Assume Alice’s preferences satisfy the axioms of the expected utility theorem. A risky assetyields 3 utility or 3. What is the probability of a bullmarket?[Write your answer as a nonnegative number equal or less than 1, with a maximum of twodecimals, like 0.12.]
Solution
Let's denote the probability of a bull market as p. Then the probability of a bear market is 1-p (since the total probability must sum to 1).
According to the expected utility theorem, Alice's expected utility from the risky asset is the sum of the utilities of each outcome, each weighted by its probability. In this case, the expected utility is $3, and we can set up the following equation:
6 + (1-p)*(1)/2
Solving this equation for p gives us the probability of a bull market.
First, simplify the right side of the equation:
6 + (1-p)*$2
Then, subtract p*2 from both sides to isolate p:
2*(1-p) = p*$6
Simplify to:
$1 + 2p = 6p
Subtract 2p from both sides:
$1 = 4p
Finally, divide by 4 to solve for p:
p = $1/4 = 0.25
So, the probability of a bull market is 0.25, or 25%.
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