All investors are mean variance optimizers but a

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All investors are mean-variance optimizers (but a speci´c functional form is not necessary). All investors perceive the same means, variances, and covariances for returns. John Y. Campbell (Ec1723) Lecture 7 September 23, 2014 30 / 44
Statement of the CAPM To state the CAPM, we need to de´ne the market portfolio . This is the value-weighted index that contains all risky assets in proportion to their market value. The return on the market portfolio is given by: R m = N X i = 1 V i V R i , (7) where V i is the market value (i.e., price times number of shares) of the risky asset i , and V = P N i = 1 V i . Recall also the optimal risky portfolio, p . Its return can be denoted by: R p = N X i = 1 w i R i , where X i w i = 1. John Y. Campbell (Ec1723) Lecture 7 September 23, 2014 31 / 44
Statement of the CAPM Theorem The assumptions of the CAPM (stated two slides ago) imply: 1 Mutual Fund Theorem : All investors hold a combination of the risk-free asset, f , and the same risky portfolio, p. In particular, all investors hold the risky assets in the same proportion. Thus, the total investment in risky asset i relative to risky asset j is given by w i w j , where w i and w j denote their shares in portfolio, p. 2 The main result of the CAPM : In equilibrium, the market portfolio is the optimal risky portfolio, i.e., m = p. Put di/erently, the market portfolio, m, is mean-variance e¢ cient. John Y. Campbell (Ec1723) Lecture 7 September 23, 2014 32 / 44
Proof of the ´rst part of theorem The assumptions imply that: All investors look at the same mean-standard deviation diagram. All investors hold a mean-variance e¢ cient portfolio. Since all e¢ cient portfolios combine the risk-free asset with the optimal risky portfolio, p , all investors hold risky assets in the same proportions to one another (mutual fund theorem). Thus, the total investment in risky assets is also proportional to their shares in p . This proves the ´rst part. John Y. Campbell (Ec1723) Lecture 7 September 23, 2014 33 / 44
The aggregate investment in risky assets is proportional to their shares in p John Y. Campbell (Ec1723) Lecture 7 September 23, 2014 34 / 44
Proof of the second part (the main result) But in equilibrium demand for assets must equal supply: Total investment in asset i = Market value of asset i ( V i ). This implies: Total investment in asset i Total investment in asset j = V i V j . From the ´rst part, we know the left hand side is equal to w i w j . Thus, for any pair of risky assets, we have w i w j = V i V j . Since portfolio weights sum up to 1, this also implies: R p = N X i = 1 w i R i = N X i = 1 V i V R i = R m . Thus, m = p . Put di/erently, the market portfolio is mean-variance e¢ cient , proving the second part. John Y. Campbell (Ec1723) Lecture 7 September 23, 2014 35 / 44
Pictorial illustration for the CAPM The Capital Market Line (CML) is just the CAL corresponding to the market portfolio. John Y. Campbell (Ec1723) Lecture 7 September 23, 2014 36 / 44
Intuition for the CAPM The ´rst part just says that the demand for risky assets nicely aggregates (as long as we assume people have the same beliefs). The second part is more subtle: It says that in equilibrium the