Consider an investor with logarithmic VNM utility and initial wealth Wo. The investor chooses a portfolio...

Transcribed Image Text:

Consider an investor with logarithmic VNM utility and initial wealth Wo. The investor chooses a portfolio at date 0 to maximize the expected utility of wealth at date 1. There are two assets: a riskless asset with (gross) rate of return Rf and a risky asset with rate of return Å that takes values R₁ and R₂ (0 < R₁ < R₂) with probability and (1-7) respectively. Denote E(R) by R, and the amount invested in the risky asset by a € R. Note that there are no restrictions on short sales. 1. (5 points) Determine a condition on rates of return such that the investor's portfolio problem has a solution. Interpret this condition. Assume that it holds for the remainder of this question. 2. (5 points) Solve explicitly for the optimal amount a* invested in the risky asset. You should be able to answer the remaining questions, at least partially, even if you are not able to obtain an explicit expression for a*. 3. (5 points) Show that (R-Rf)a* ≥ 0. да 4. (5 points) Show that w > 0, provided a* is positive. Relate this finding to a property of the coefficient of absolute risk aversion that characterizes the logarithmic VNM utility function. (State the relevant result; do not prove it). 5. (5 points) Suppose R₁ falls and R₂ increases, keeping R constant. What happens to a*? Interpret. 6. (5 points) Calculate the certainty equivalent of the portfolio in which the agent invests all his initial wealth in the risky asset. Consider an investor with logarithmic VNM utility and initial wealth Wo. The investor chooses a portfolio at date 0 to maximize the expected utility of wealth at date 1. There are two assets: a riskless asset with (gross) rate of return Rf and a risky asset with rate of return Å that takes values R₁ and R₂ (0 < R₁ < R₂) with probability and (1-7) respectively. Denote E(R) by R, and the amount invested in the risky asset by a € R. Note that there are no restrictions on short sales. 1. (5 points) Determine a condition on rates of return such that the investor's portfolio problem has a solution. Interpret this condition. Assume that it holds for the remainder of this question. 2. (5 points) Solve explicitly for the optimal amount a* invested in the risky asset. You should be able to answer the remaining questions, at least partially, even if you are not able to obtain an explicit expression for a*. 3. (5 points) Show that (R-Rf)a* ≥ 0. да 4. (5 points) Show that w > 0, provided a* is positive. Relate this finding to a property of the coefficient of absolute risk aversion that characterizes the logarithmic VNM utility function. (State the relevant result; do not prove it). 5. (5 points) Suppose R₁ falls and R₂ increases, keeping R constant. What happens to a*? Interpret. 6. (5 points) Calculate the certainty equivalent of the portfolio in which the agent invests all his initial wealth in the risky asset.

Answer rating: 100% (QA)

Explanation 1 To determine a condition for a solution to the investors portfolio problem we need to ... View the full answer