1. Suppose there is a risk-free asset with return Rf and a risky asset with return . Consider an investor who maximizes expected end-of-period utility of wealth and who has CARA utility and invests Wo. Suppose the investor has labor income Y at the end of the period, so her end-of-period wealth is of Rf + P+ , where of denotes the investment in the risk-free asset and the investment in the risky asset. (a) Suppose 7 and are independent. Show that the optimal o is the same as if there were no labor income. Hine: Use the law of iterated expectations and the fact that if and X Eh any functions g and h. (b) Define b= Cov (, ) /Var (R), a = (E VE) /Rs, and @ = aRy HShow that has a zero mean and is uncorrelated with R. Note: This is an example of an orthogonal projection. (c) Suppose and have a joint normal distribution. Using the result of the previous part, show that the optimal is * b, where * denotes the optimal investment in the risky asset when there is no labor income. 1. Suppose there is a risk-free asset with return Rf and a risky asset with return . Consider an investor who maximizes expected end-of-period utility of wealth and who has CARA utility and invests Wo. Suppose the investor has labor income Y at the end of the period, so her end-of-period wealth is of Rf + P+ , where of denotes the investment in the risk-free asset and the investment in the risky asset. (a) Suppose 7 and are independent. Show that the optimal o is the same as if there were no labor income. Hine: Use the law of iterated expectations and the fact that if and X Eh any functions g and h. (b) Define b= Cov (, ) /Var (R), a = (E VE) /Rs, and @ = aRy HShow that has a zero mean and is uncorrelated with R. Note: This is an example of an orthogonal projection. (c) Suppose and have a joint normal distribution. Using the result of the previous part, show that the optimal is * b, where * denotes the optimal investment in the risky asset when there is no labor income