2. (80 points total) You manage money for a company that provides you with capital and takes profits in the following way: If your return is positive in a given period, your capital is increased by a times the return. For example, if you make 10% in one period and a = }, you would then be managing 1.05 times the amount you had previously managed. . If your return is positive in a given period, your capital is decreased by B times the negative of the return. For example, if you lose 10% in one period and B = 2, you would then be managing 0.8 times the amount you had previously managed. Let w be portfolio weights, that is, the vector of percentages of your captial invested in each asset and Rt the IID vector of returns in each asset in each period. You can assume that Rt is sufficiently well-behaved in terms of the existence of any moments required by the following problems. Let C be the amount of capital managed after period t and Let Co = 1. (a) (20 points) What is the value of: lim - In(n) nn in terms of a, b, R, and w? You may also use expected values of arbitrary functions of these variables including the positive part function given byxt = max(x,0) and the negative part function given by x = max(-x,0). (b) (28 points) You can invest some portion of your capital in a risky stock. In each period, there are two possible scenarios: scenario 1, which occurs with probability P, and scenario 2, which occurs with probability 1 - pi. If scenario i occurs, then Rt = ri, that is, the return on the risky stock is ri. Let w be the fraction of capital invested in the risky stock. What is the value of w which optimizes the long-term growth in capital in terms of a, B, P1, r and r2 assuming that r2