4. Optimal Portfolio Choice with Time-Varying Expected Return. (50pts) Suppose that the economy has two securities dS, = 4S dt +oS.dz dB; = r Bedt where the expected return on the stock is time-varying and follows de = R(q - Hedt + sd2. Investor maximizes the CRRA utility of the terminal consumption (there is no intermediate consumption) max E 1. Write down the investor's problem as a static problem, and determine the optimal consumption 2. Define by nje = (14 r)/0. Conjecture that investor's wealth at time t can be written as F(74,7x, t) and derive a PDE for F. 3. Assume that F(8,9,4) (17)+/- expl} (447 ) + B (21T 1) + BT 1)n +C(7 ))] Derive ODEs for A, B, C together with boundary conditions. Do not solve for A,B,C. 4. Solve for the fraction of wealth o invested in the risky asset at time t. This expression may depends on A,B,C. 5. Suppose that y = 1. Solve for A, B, C in closed form. Solve for oe in closed form and provide the intuition for the expression 4. Optimal Portfolio Choice with Time-Varying Expected Return. (50pts) Suppose that the economy has two securities dS, = 4S dt +oS.dz dB; = r Bedt where the expected return on the stock is time-varying and follows de = R(q - Hedt + sd2. Investor maximizes the CRRA utility of the terminal consumption (there is no intermediate consumption) max E 1. Write down the investor's problem as a static problem, and determine the optimal consumption 2. Define by nje = (14 r)/0. Conjecture that investor's wealth at time t can be written as F(74,7x, t) and derive a PDE for F. 3. Assume that F(8,9,4) (17)+/- expl} (447 ) + B (21T 1) + BT 1)n +C(7 ))] Derive ODEs for A, B, C together with boundary conditions. Do not solve for A,B,C. 4. Solve for the fraction of wealth o invested in the risky asset at time t. This expression may depends on A,B,C. 5. Suppose that y = 1. Solve for A, B, C in closed form. Solve for oe in closed form and provide the intuition for the expression