450 The journal of business Rk= BuRm+(1-B)R2 Taking expected values of both sides of equation (20), and rewriting ha E(Rx)=E(R:)+BIE(Rm)-E(RD] (21) Equation(21) says that the expected return on an efficient portfolio k is a linear function of its Bk. From(1), we see that the corresponding elationship when there is a riskless asset and riskless borrowing and lending are allowed is )=R+BAE(Rm)一R (22) Thus the relation between the expected return on an efficient portfolio k and its Bk is the same whether or not there is a riskless asset. If there is, then the intercept of the relationship is Ry. If there is not, then the intercept is E(r) We can now show that equation(21)applies to individual securi ties as well as to efficient portfolios. Subtracting equation (10)from itself after permuting the indexes, we get cov(R,R)一cov(R,R)=SAE(R)一E(点)](23) Since the market is an efficient portfolio, we can put m for k, and since i and j can be taken to be portfolios as well as assets, we can put z for Then equation (23)becomes cov(R, Rm)=Sm[E(R)-E(R) (24) Equation (24) may be rewritten as E(R, =E(R)+[var(Rm)/S, 2 var(Rnm)/Sm=E(点n)一E(R) So equation(25) becomes E(RD=E(R,)+B,[E(Rm)-E(R) Thus the expected return on every asset, even when there is no riskless asset and riskless borrowing is not allowed, is a linear function of its B Comparing equation (27)with equation (1), we see that the introduc tion of a riskless asset simply replaces E(R,) with R, ow we can derive another property of portfolio z. Equation (27) holds for any asset and thus for any portfolio Setting B:=0, we see that every portfolio with B equal to zero must have the same expected eturn as portfolio z. Since the return on portfolio z is independent of the return on portfolio m, and since weighted combinations of portfolios m and z must be efficient, portfolio z must be the minimum-variance zero-B portfolio his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions450 The Journal of Business Rk PkRm + (1 -/k)Rz. (20) Taking expected values of both sides of equation (20), and rewriting slightly, we have E(Rk) =E(Rz) +I/8k[E(Rm) -E(Rz)]. (21) Equation (21) says that the expected return on an efficient portfolio k is a linear function of its /3k. From (1), we see that the corresponding relationship when there is a riskless asset and riskless borrowing and lending are allowed is E(Rk) Rf + 1k3[E(Rm) - Rf]. (22) Thus the relation between the expected return on an efficient portfolio k and its /3k is the same whether or not there is a riskless asset. If there is, then the intercept of the relationship is Rf. If there is not, then the intercept is E(R~z). We can now show that equation (21) applies to individual securi￾ties as well as to efficient portfolios. Subtracting equation (10) from itself after permuting the indexes, we get cov(Ri, Rk) - cov(Rj, R) = Sk[E(Ri) - E(Rj)]. (23) Since the market is an efficient portfolio, we can put m for k, and since i and j can be taken to be portfolios as well as assets, we can put z for j. Then equation (23) becomes cov(Ai, Am) Sm[E(Ri) - E(Rz)]. (24) Equation (24) may be rewritten as E(Ri) = E(Rz) + [var(Rm)/Sm]/3i. (25) Putting m for i in equation (25), we find var(Rm)/SIn = E(Rm) - E(Rz). (26) So equation (25) becomes E(ftj =E(Rz) + /3i[E(Rm) - E(Rz)]. (27) Thus the expected return on every asset, even when there is no riskless asset and riskless borrowing is not allowed, is a linear function of its /3. Comparing equation (27) with equation (1), we see that the introduc￾tion of a riskless asset simply replaces E(RZ) with Rf. Now we can derive another property of portfolio z. Equation (27) holds for any? asset and thus for any portfolio. Setting /i3 0, we see that every portfolio with /8 equal to zero must have the same expected return as portfolio z. Since the return on portfolio z is independent of the return on portfolio m, and since weighted combinations of portfolios m and z must be efficient, portfolio z must be the minimum-variance zero-/3 portfolio. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions