be written: Cov(Ri, Rj) Cov ( i iRM ei, j jRM ej) But since i and j are constants, their covariance with any variable is zero. Further, the firm-specific terms (ei, ej) are assumed uncorrelated with the market and with each other. Therefore, the only source of covariance in the returns between the two stocks derives from their common dependence on the common factor, RM. In other words, the covariance between stocks is due to the fact that the returns on each depend in part on economywide conditions. Thus,   Cov(Ri,Rj) Cov( iRM, j RM) i j 2   (10.4)   These calculations show that if we have   n estimates of the expected excess returns, E(Ri) n estimates of the sensitivity coefficients, i n estimates of the firm-specific variances, 2(ei) 1 estimate for the variance of the (common) macroeconomic factor, 2 ,   then these (3n 1) estimates will enable us to prepare the input list for this single-index security universe. Thus for a 50-security portfolio we will need 151 estimates rather than       4 Practitioners often use a "modified" index model that is similar to equation 10.3 but that uses total rather than excess returns. This practice is most common when daily data are used. In this case the rate of return on bills is on the order of only about .02% per day, so total and excess returns are almost indistinguishable. III. Equilibrium In Capital Markets 10. Single−Index and Multifactor Models The McGraw−Hill Companies, 2001           296 PART III Equilibrium in Capital Markets     1,325; for the entire New York Stock Exchange, about 3,000 securities, we will