449 Capital Market equilibrium Equation(12), then, shows that the efficient portfolio held by in- vestor k consists of a weighted combination of the basic portfolios p and q. Note, however, that the two basic portfolios are not unique. Supl that we transform the basic portfolios p and q into two different folios u and v, using weights wup, Wug, Wop, and wog. Then we have xui=wuprpi+ wuorgi; (15) Normally, we will be able to solve equations(15) for xps and xge. Let us write the resulting coefficients wpw, Wpu, Wou, and wov. Then we will rud+ (16 xai=wgului+ woori Substituting equations(16) into equation(12), we see that we can write the efficient portfolio k as a linear combination of the new basic port- folios u and v as follows Xut=wkulut+ Whorf. (17) In equation(17), the weights whu and wrw sum to one. Thus the basic portfolios u and v can be any pair of different port folios that can be formed as weighted combinations of the original pair of basic portfolios p and Every efficient portfolio can be expressed as a weighted combination of portfolios u and v, but they need not be efficient themselves Portfolios p and q must have different B's, if it is to be possible generate every efficient portfolio as a weighted combination of these two portfolios. But if they have different B's, then it will be possible to generate new basic portfolios u and v with arbitrary Bs, by choosing ap- propriate weights. In particular, let us choose weights such that n=1;B=0. Multiplying equation (12)by the fraction xmk of total wealth held by investor k, and summing over all investors (k= 1, 2, L),we obtain the weights xmi of each asset in the market portfolio tmt= xmkWip xp+ ImkWkar (19) Since the market portfolio is a weighted combination of portfolios p and q, and since Bm is one, portfolio u must be the market portfolio. Thus we can rename the portfolios u and v specified by(18)portfolios m and z, for the market portfolio and the zero-s basic portfolio. When we write the return on an efficient portfolio k as a weighted combination of the returns on portfolios m and z, the coefficient of the return on portfolio m must be Be. Thus we can write his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions449 Capital Market Equilibrium Equation (12), then, shows that the efficient portfolio held by in￾vestor k consists of a weighted combination of the basic portfolios p and q. Note, however, that the two basic portfolios are not unique. Suppose that we transform the basic portfolios p and q into two different port￾folios u and v, using weights wu,, Wuq, wa, and Wvq. Then we have XUi- WUVXi + WuqXqi; (15) Xvi WV- X i + WvqXqi. Normally, we will be able to solve equations (15) for xiA and Xqj. Let us write the resulting coefficients wvu, wpv, Wqu, and Wqv. Then we will have x-i WVUX1i + wpVXvi; (16) Xqi WquXui + WqvXvi. Substituting equations ( 16) into equation ( 12), we see that we can write the efficient portfolio k as a linear combination of the new basic port￾folios u and v as follows: Xki =WkuXui + WkvXvi. (17) In equation (17), the weights Wku and Wkv sum to one. Thus the basic portfolios u and v can be any pair of different port￾folios that can be formed as weighted combinations of the original pair of basic portfolios p and q. Every efficient portfolio can be expressed as a weighted combination of portfolios u and v, but they need not be efficient themselves. Portfolios p and q must have different /3's, if it is to be possible to generate every efficient portfolio as a weighted combination of these two portfolios. But if they have different /3's, then it will be possible to generate new basic portfolios u and v with arbitrary /3's, by choosing ap￾propriate weights. In particular, let us choose weights such that Flu= 1; ,l]V =0. (18) Multiplying equation (12) by the fraction Xmk of total wealth held by investor k, and summing over all investors (k - 1, 2, . . . , L), we obtain the weights xmi of each asset in the market portfolio: L L Xri (ZE XmkWkp) Xi + (Z XmnkWkq )Xqi. (19) k=1 k=1 Since the market portfolio is a weighted combination of portfolios p and q, and since Pm is one, portfolio u must be the market portfolio. Thus we can rename the portfolios u and v specified by (18) portfolios m and z, for the market portfolio and the zero-,8 basic portfolio. When we write the return on an efficient portfolio k as a weighted combination of the returns on portfolios m and z, the coefficient of the return on portfolio m must be /k. Thus we can write 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