Consider the Arrow's portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of an investor is: u(w) = In w, where w represents wealth, and in natural log. Denote as a the amount invested in the risky asset. Let the initial wealth be $10,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%,5%; 0.55, 0.45). 1.1 Write the equation of the expected utility of final wealth 1.2 Take the first derivative of the expected utility with respect to a. 1.3 Now evaluate the first derivative, that you found in 1.2 above, at a = 0. What can you conclude about the optimal value of a? Why is that so? Consider the Arrow's portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of an investor is: u(w) = (w)1/2, where w represents wealth. Let the initial wealth be $10,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%, 10%; 0.55, 0.45). 2.1 Consider two portfolios. Portfolio 1 (P1) consists in investing $3,000 in the risky asset, and in portfolio 2 (P2) the investment in the risky asset is $4,000. Find the probability distribution of each portfolio, calculate the expected utility of each, and then tell what the preferred portfolio to the investor is. 2.2 Find the portfolio that maximizes the expected utility of the investor 2.3 What would be your prediction about the optimal (in the sense that maximizes the expected utility) amount invested in the risky asset if the wealth of the investor goes up? Justify your prediction. 2.4 What would be your prediction about the optimal (in the sense that maximizes the expected utility) amount invested in the risky asset if the distribution of the random return is now 1 = (1%, 10%; 0.547, 0.453)? Justify your prediction Consider the Arrow's portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of an investor is: u(w) = In w, where w represents wealth, and in natural log. Denote as a the amount invested in the risky asset. Let the initial wealth be $10,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%,5%; 0.55, 0.45). 1.1 Write the equation of the expected utility of final wealth 1.2 Take the first derivative of the expected utility with respect to a. 1.3 Now evaluate the first derivative, that you found in 1.2 above, at a = 0. What can you conclude about the optimal value of a? Why is that so? Consider the Arrow's portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of an investor is: u(w) = (w)1/2, where w represents wealth. Let the initial wealth be $10,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%, 10%; 0.55, 0.45). 2.1 Consider two portfolios. Portfolio 1 (P1) consists in investing $3,000 in the risky asset, and in portfolio 2 (P2) the investment in the risky asset is $4,000. Find the probability distribution of each portfolio, calculate the expected utility of each, and then tell what the preferred portfolio to the investor is. 2.2 Find the portfolio that maximizes the expected utility of the investor 2.3 What would be your prediction about the optimal (in the sense that maximizes the expected utility) amount invested in the risky asset if the wealth of the investor goes up? Justify your prediction. 2.4 What would be your prediction about the optimal (in the sense that maximizes the expected utility) amount invested in the risky asset if the distribution of the random return is now 1 = (1%, 10%; 0.547, 0.453)? Justify your prediction