Sharpe Ratios of Securities and Efficient Portfolios
Summary
The question asks whether securities in an efficient portfolio share the portfolio’s Sharpe ratio and how this relates to a covariance-based return measure. The response gives the portfolio standard deviation as the square root of the portfolio weights applied to the return covariance matrix, correcting the variance expression used in the question. It then frames the issue through mean–variance efficiency and the Capital Market Line.
The answer states that individual securities in an efficient portfolio need not have the same Sharpe ratio, while efficient portfolios share the market portfolio’s Sharpe ratio under the stated framework. It also identifies the market portfolio as efficient and invokes the Two-Fund Theorem: efficient portfolios can be formed by combining two efficient portfolios. The post provides conclusions rather than a derivation or worked numerical example, and the claims rely on the usual assumptions behind mean–variance portfolio theory.
Key ideas
- Portfolio risk is computed from the weights and the covariance matrix of asset returns.
- Individual securities in an efficient portfolio need not have equal Sharpe ratios.
- Under the stated mean–variance framework, efficient portfolios share the market portfolio’s Sharpe ratio.
- The Two-Fund Theorem says efficient portfolios can be formed by combining two efficient portfolios.
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# Can someone please verify or disprove this Sharpe Ratio math logic for me
# Can someone please verify or disprove this Sharpe Ratio math logic for me
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?
## Answer by Andrew (score 1)
https://quant.stackexchange.com/a/37473
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)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.