Problem 4 (CAPM from Linear Pricing) We will derive a formula that looks like CAPM from linear pricing. Assume there exists a pricing kernel 4 so that fo = S fredP = E[ fTQ]. (a) Let R; denote the total return of asset i from time 0 to T, and let Rf = (1 + rf) denote the total return on the risk free asset. What is E[(] in terms of the risk free rate Rf? (b) Derive a formula for E[R;] - R in terms of a covariance with the pricing kernel. (c) From this, derive bounds for the Sharpe ratio of any asset in the market. (d) Let R* be an asset that achieves the Sharpe ratio upper-bound, then how must it be related to the pricing kernel? (e) Using R*, eliminate 4 from the formula for E[R;] - R and write it in a form that looks like CAPM. (f) If this were actually CAPM, how would the market portfolio relate to Q? (Note: This problem shows that the structure of CAPM comes from linear pricing, and nothing more. In fact, any market without arbitrage has a CAPM relationship, but we just can't necessarily say that R* is the market portfolio, as in CAPM.)