Problem 1. A two-factor arbitrage pricing theory. Consider an economy in which the random return ; on each individual asset j follows the equation fj = aj + Bj,M(TM - E[TM]) + Biv(rv - E[rv]) + 8j, where, as we discussed in class, fy is the random return on the market portfolio, Ty is the random return on a "value" portfolio that takes a long position in shares of stock issued by smaller, overlooked companies, or companies with high book-to-market values, &, is an idiosyncratic, firm-specific component, and B,M and Bjv are the "factor loadings" that measure the extent to which the return on asset j is correlated with the return on the market and value portfolios. Assume E[s,] = 0 and E[ME;] = 0 for each individual asset j, and Eejex] = 0 for all asset pairs j and k. Further assume, as Stephen Ross did when developing the arbitrage pricing theory (APT), that there are enough individual assets for investors to form many well-diversified portfolios and that investors act to eliminate all arbitrage opportunities that may arise across all well-diversified portfolios. (a) In the context of APT, define the term well-diversified portfolio. Show that we can write the random return of a well-diversified portfolio as Tp = E[fp] + Bp,M (TM - E[TM]) + Bp,V (TV - E[rv]). What is the key statistical assumption that gives this result? (b) Consider two well-diversified portfolios such that: = E + Bp,M(TM - E[TM]) + Bp,v (TV - E[Tv]), - E + Bp,M(TM - E[FM]) + Bp,V (TV - E[rv]), meaning they have the same factor loadings on the market and value portfolios. Sup- pose that E = E[Fil + A. Show the absence of arbitrage opportunities requires that A = 0