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I am referring to the book Sharpe et al. (1998), Investments, 6th Edition. I am trying to wrap my head around some lines from the book, pertaining to Security Market Line. It reads:

Earlier it was established that the expected return of a portfolio is a weighted average of the expected returns of component securities, where the proportions invested are the weights. Therefore every portfolio plots on the SML because every security plots on SML. To put it more broadly, not only every security but also every portfolio must plot on the upward sloping straight line in a diagram with expected return on vertical axis and beta on horizontal axis. So efficient portfolios plot on both CML and SML, although inefficient portfolios plot on the SML but below CML.

I understand the first two sentences. A portfolio is a convex combination of the individual securities and hence I can imagine that a line that plots individual securities on a standard deviation-expected return plane will also contain the portfolio made out of the securities. What I can infer from the above paragraph is that efficiency and inefficiency has nothing to do with plotting of SML since all portfolios will lie on it. However, CML plots only efficient sets. Is this correct? I seem to have some issues with understanding the relationship. I would be great help if someone could help me with it. Thanks a lot!

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In equilibrium, all securities and portfolios (i.e. convex combinations of securities) lie on the SML, which plots expected return as a function of beta. Note that outside of equilibrium, if a security was undervalued, it would lie above the SML and vice versa.

The efficient frontier consists of all efficient portfolios, i.e. all portfolios that yield the maximum expected return given their standard deviation of return. Basically, for every point along the sigma-axis, it is the topmost portfolio - or equivalently, for every point along the expected return-axis, it is the leftmost portfolio.

The CML is the combination of all portfolios for which the sharpe ratio is maximized (i.e. the risk-adjusted excess return is the largest). This will always be a combination of the risk free security and the market (tangent) portfolio. Hence, the CML will intersect the second axis at the risk free rate and go through the market (tangent) portfolio. It is important to note that all portfolios on the CML offer a superior risk-reward profile to any portfolio on the efficient frontier. This is evident when drawn out, since the CML is above or to the left of the efficient frontier at all points (except for the tangent portfolio).

Hence, while all portfolios on the CML are efficient, the CML does not contain all efficient portfolios.

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  • $\begingroup$ Thank you for the answer. The line you stated, "while all portfolios on CML are efficient, all efficient portfolios are not on CML"- what I understand is that some efficient portfolios lie on the curved efficient set (bullet shaped) but below the straight line joining the market portfolio and the risk free asset return which corresponds to the CML. Therefore all efficient sets do not lie on the CML. Is this correct? $\endgroup$ – S.Rana Mar 8 at 16:53
  • $\begingroup$ @ShinjiniRana that's correct. the 'optimal' portfolio is then that which achieves both objectives, namely the combination of risk-free and risky assets that achieves the maximum Sharpe ratio, or the tangent of the CML and SML. $\endgroup$ – Chris Mar 8 at 23:16
  • $\begingroup$ Exactly, Matthew. Thanks for clearing up my nomenclature. $\endgroup$ – AdB Mar 9 at 21:55
  • $\begingroup$ @ShinjiniRana Sharpe's passage appears to refer to the mean-variance frontier over all securities (including the risk free security) while AdB calls a portfolio mean-variance efficient iff it lies on the mean-variance frontier formed by risky securities (i.e. excluding the risk free security). The mean-variance frontier among risky securities forms a sideways parabola. The efficient side of the mean-variance frontier over all securities is a ray originiating at $(0, r_f)$ and passing through the mean and variance of the tangency portfolio. $\endgroup$ – Matthew Gunn Mar 9 at 22:21
  • $\begingroup$ Given a set of securities $\mathcal{R}$, a point $(\sigma^2, \mu)$ is said to lie on the mean-variance frontier if $\sigma^2$ is the minimum return variance of all portfolios formed over $\mathcal{R}$ which achieve expected return $\mu$. An important distinction is whether the risk free security belongs $\mathcal{R}$. $\endgroup$ – Matthew Gunn Mar 9 at 22:21

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