A portfolio has an expected rate of return of 0.18 and a standard deviation s = 0.18. The risk-free rate is 9.99 percent. An investor has the following utility function: U = E(r) – (A/2)*s*s. Which value of A makes this investor indifferent between the risky portfolio and the risk-free asset? a. 7b. 5c. 8d. 6
Question
A portfolio has an expected rate of return of 0.18 and a standard deviation s = 0.18. The risk-free rate is 9.99 percent. An investor has the following utility function: U = E(r) – (A/2)ss. Which value of A makes this investor indifferent between the risky portfolio and the risk-free asset? a. 7b. 5c. 8d. 6
Solution
To solve this problem, we need to set the utility of the risky portfolio equal to the utility of the risk-free asset and solve for A.
The utility of the risky portfolio is given by U = E(r) - (A/2)*s^2, where E(r) is the expected return and s is the standard deviation. Substituting the given values, we get U = 0.18 - (A/2)*0.18^2.
The utility of the risk-free asset is simply its return, which is 0.0999.
Setting these two equal to each other, we get 0.18 - (A/2)*0.18^2 = 0.0999.
Solving this equation for A, we get A = 2*(0.18 - 0.0999) / 0.18^2.
Calculating this gives A = 7.41, which is not one of the options. However, it is closest to option a. 7, so that would be the best choice given the options.
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