I want to start by stating a problem that I wanted to figure out initially so that this all ties in somehow. I initially wanted to figure out if individual securities in an efficient portfolio all had the same Sharpe Ratio as the portfolio itself which I soon easily figured out was not the case as $$Sharpe = \frac{E[R_{P}] - r_{f}}{SD(R_{P})}= \frac{x_{1}E[R_{1}]+x_{2}E[R_{2}]+...+x_{n}E[R_{n}]-r_{f}}{\sqrt{x_{1}Cov(R_{1},R_{M})+x_{2}Cov(R_{2},R_{M}),+...+x_{n}Cov(R_{n},R_{M})}}$$ which indicates that the Sharpe Ratio of the portfolio is a combination of the expected returns and covariance of all individual securities. Next I stumbled upon a statement which said that in an efficient portfolio the following holds true $$\frac{E[R_1] - r_f}{Cov(R_{1},R_{M})} = \frac{E[R_2] - r_f}{Cov(R_{2},R_{M})} = ... = \frac{E[R_M] - r_f}{Var(R_{M})} $$ This makes sense to me and after playing with the equation I got $$\frac{E[R_1] - r_f}{SD(R_1)Corr(R_1,R_M)}=\frac{E[R_2] - r_f}{SD(R_2)Corr(R_2,R_M)}=...=\frac{E[R_{eff}] - r_f}{SD(R_{eff})}$$ Where we assume the market portfolio is efficient.

So my question is: In the last equation is that the ratio which all securities in an efficient portfolio must have equal and is it true that the Sharpe ratio of individual securities in an efficient portfolio need not be the same? Or am I just confusing an efficient portfolio with the market portfolio which is simply a type of efficient portfolio?

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    $\begingroup$ The equation you have written down for the volatility of your portfolio looks wrong to me. The volatility of a portfolio in CAPM is given by $\sigma_{R_{P}} = \sqrt{\sum_{i}^{n}\sum_{j}^{n}w_{i}w_{j}cov\left ( i,j \right )}$ .If you only know the covariances of the assets to the market you can only set limits on the value of the portfolio's volatility. you have to know the intra-asset covariance matrix for the portfolio to calculate its volatility. $\endgroup$
    – RobAbMo
    Dec 22 '17 at 10:38
  • $\begingroup$ There is nothing wrong with computing a Sharpe Ratio for individual securities (it is not illegal or impossible or in bad taste) but it is not often done and the Sharpe Ratio is a concept that is generally applied to Portfolios (i.e. it looks at what an investor holds (a portfolio of N securities), after internal portfolio diversification has taken place. For stocks,, which have risk that could be diversified away, it is misapplied). $\endgroup$
    – noob2
    Dec 22 '17 at 18:00
  • $\begingroup$ The desirability of a portfolio is measured by Sharpe Ratio, the desirability of an individual security is measured by Excess Return divided by Undiversifiable RIsk, which in the CAPM is $\frac{E[R_i]-r_f}{Cov(R_i,R_M)}$ $\endgroup$
    – noob2
    Dec 22 '17 at 18:26
  • $\begingroup$ @RobAbMo isn't $R_m = \sum_jx_jR_j$ so by using properties of covariance we just arrive at my equation? In reverse $\sum_ix_iCov(R_i,R_m) =\sum_ix_iCov(R_i,\sum_jx_jR_j) = \sum_i\sum_jx_ix_jCov(R_i,R_j) $ $\endgroup$
    – Dmitriy
    Dec 22 '17 at 20:10
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    $\begingroup$ I think that you are mistaken $R_m$ with $R_p$ in your equation. $\endgroup$
    – Andrew
    Dec 23 '17 at 16:21

First of all RobAbMo is correct. If $X \in \mathbb{R}^{n * n}$ is the VCV-Matrix of the returns, and $w=(w_1,...,w_n) \in \mathbb{R}^n$ the vector of the portfolio-weights, then the Standarddeviation is given by $SD(R_P)=\sqrt{w^\top X w}$.

All your questions are regarding the Capital Market Line (look it up).

  • The Sharpe Ratio of individual securities in an efficient portfolio is not the same (construct a counterexample for simple markets, use Two-Fund-Theorem)

  • The market portfolio is an efficient portfolio

  • Every efficient portfolio has the same Sharpe Ratio as the market portfolio

  • Every efficient portfolio can be constructed by combining two efficient portfolios (Two-Fund-Theorem)

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