a question about CAPM, beta of risky asset, return of the market portfolio

There are two equally likely states of nature that can occur at date t + 1, either there is in an expansion or there is a recession. An investor has preferences over consumption at dates If and 33+ 1 that can be represented by the utility function 1_ CT c1TT U(Ctict+l) : 1t_7 + EP ( Hl) 17 where \"y 7 1 and 19 E (0,1). The investor chooses her consumption plan to maximise U(ct, (334.1) taking prices of the financial assets as given. Let p? and pf denote the price of a bond and a share of the firm in period t, respectively. A riskfree bond issued at date if is a promise to pay 1 unit at date t + 1. The payoff of a share at date t + 1 is p+1 + dt-ll- The growth rate of consumption takes values 93 in an expansion and 9,; in a recession, and the investor does not face any binding borrowing constraint. (a) Use the consumption based model to derive the basic pricing equation for any payoff. Justify your answer. (10 marks) (b) (C) (6') Find the riskfree rate as a function of 9H, 9L, 6 and \"y. (10 marks) How does the riskfree rate adjust so that the optimal consumption plan does not change when 5 changes? What is the economic intuition? Justify your answer. (10 marks) Suppose the dividend growth rate between dates t and t+ consumption growth rate between dates t and t + 1. Find the share price dividend ratio at date t, ZZ, as a function of 9;; 9L, [3 and \"y and the share price dividend ratio at date t+ 1. (10 marks) Let 9;; : ,u + a and 9L : ,u a. Show that, for or small enough, the share's Sharpe ratio is bounded from above by 7%. Suppose the Sharpe ratio is 0.5, u : 1 and cr : 0.01 and use the bound on the Sharpe ratio to describe the equity premium puzzle. (Hint: Use that for a small enough WNW\" 9 where SD stands for the CtIl Ct standard deviation). (10 marks)