4. (The asset-market economy) Suppose there are two assets available to investors: a risk-free asset with...

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4. (The asset-market economy) Suppose there are two assets available to investors: a risk-free asset with a price of p1 = 1 and that pays Ry with certainty at date 1, and a risky asset with a price of p2 = 1 and that pays R at date 1, where R is normally distributed with mean R and variance o2. An expected utility maximizer with von Neumann-Morgenstern utility function u(c) =-e- and date 0 wealth of w holds a portfolio of these two assets. For simplicity, we will allow consumption at date 1 to take negative values. (a) Calculate the investor's expected utility if she holds a proportion of her date 0 wealth in the risky asset. Hint: If y is normally distributed with mean p and variance 2 then Ele-ae-au+fa²d² (b) What level of a maximizes her expected utility? (c) How does her portfolio change as her Arrow-Pratt measure of absolute risk aversion increases? (d) Now suppose that the investor also has normally-distributed labor income L at date 1, with mean L, variance 02, and covariance with the risky asset's payoff equal to pare. That is, her date 1 consumption equals the sum of her portfolio payoff and her labor income. Derive the expected utility maximizing portfolio z and discuss how it is affected by labor income. 4. (The asset-market economy) Suppose there are two assets available to investors: a risk-free asset with a price of p1 = 1 and that pays Ry with certainty at date 1, and a risky asset with a price of p2 = 1 and that pays R at date 1, where R is normally distributed with mean R and variance o2. An expected utility maximizer with von Neumann-Morgenstern utility function u(c) =-e- and date 0 wealth of w holds a portfolio of these two assets. For simplicity, we will allow consumption at date 1 to take negative values. (a) Calculate the investor's expected utility if she holds a proportion of her date 0 wealth in the risky asset. Hint: If y is normally distributed with mean p and variance 2 then Ele-ae-au+fa²d² (b) What level of a maximizes her expected utility? (c) How does her portfolio change as her Arrow-Pratt measure of absolute risk aversion increases? (d) Now suppose that the investor also has normally-distributed labor income L at date 1, with mean L, variance 02, and covariance with the risky asset's payoff equal to pare. That is, her date 1 consumption equals the sum of her portfolio payoff and her labor income. Derive the expected utility maximizing portfolio z and discuss how it is affected by labor income.

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