4. Consider a two-period economy (/ = 0, 1). The operating profit of firm j can be expressed asis the exogenous profitability shock; At is a technology shook at time t; is private capital of fimm j. a > 0 is assumed. The depreciation rate of capital k for firm / is δj. Thus, the investment of firm, firm j has a liquidation value of. The investment in private capital involves adjustment cost . The stochastic discount factor between time 0 and 1 is m, where log(m) = logβ + γ(x0 -x1), where 0 <β< 1, and γ> 0. Firm j then chooses ij.0 to maximize its value, which is the sum of the discounted cash flows from the two periods: E[.] is an expectation operator. From standard asset pricing equation, -Cov0(r1,m). Specifically, the cash flows from t = 0 are equal to the operating profit minus the investment cost and the adjustment cost. The cash flows from ! = I are equal to the operating profit plus the liquidation value.
(3) (10 points) Please derive expected excess return:, which is a function of the covariance between the operating profit at t = 1 of firm y and the aggregate productivity shock.