on the sec- ond factor portfolio is 12% - 4% 8%. Now consider an arbitrary well-diversified portfolio, Portfolio A, with beta on the first factor, A1 .5, and beta on the second factor, A2 .75. The multifactor APT states that the overall risk premium on this portfolio must equal the sum of the risk premiums required as compensation to investors for each source of systematic risk. The risk premium attribut- able to risk factor 1 should be the portfolios exposure to factor 1, A1, multiplied by the risk premium earned on the first factor portfolio, E(r1) - rf. Therefore, the portion of Port- folio As risk premium that is compensation for its exposure to the first factor is A1[E(r1) - rf] .5(10% - 4%) 3%, whereas the risk premium attributable to risk factor 2 is A2[E(r2) - rf] .75(12% - 4%) 6%. The total risk premium on the portfolio should be 3 6 9%. Therefore, the total return on the portfolio should be 13%:   4% (risk-free rate) 3 (risk premium for exposure to factor 1) 6 (risk premium for exposure to factor 2)   13% (total expected return)   To see why the expected return on the portfolio must be 13%, consider the following ar- gument. Suppose that the expected return on Portfolio A were 12% rather than 13%. This return would give rise to an arbitrage opportunity. Form a portfolio from the factor portfo- lios with the same betas as Portfolio A. This requires weights of .5 on the first factor port- folio, .75 on the second factor portfolio, and -.25 on the risk-free asset. This portfolio has exactly the same factor betas as Portfolio A: It has a beta of .5 on the first factor because of its .5 weight on the first factor portfolio, and a beta of .75 on the second factor. However, in contrast to Portfolio A, which has a 12% expected return, this portfolios expected return is (.5 10) (.75 12) - (.25 4) 13%. A long position in this port- folio and a short position in Portfolio A would yield an arbitrage profit. The total return per dollar long or short in each position would be   .13 .5F1 .75F2 (long position in factor portfolios) - (.12 .5F1 .75F2) (short position in Portfolio A) .01   for a positive, risk-free return on a zero net investment position. To generalize this argument, note that the factor exposure of any portfolio, P, is given by its betas, P1 and P2. A competing portfolio formed from factor portfolios with weights P1 in the first factor portfolio, P2 in the second factor portfolio, and 1 - P1 - P2 in T-bills will have betas equal to those of Portfolio P and expected return of   E(rP) P1E(r1) P2E(r2) (1 - P1 - P2)rf (11.6) rf P1[E(r1) - rf] P2[E(r2) - rf]