Markowitz Mean-Variance Portfolios and the Efficient Frontier

Summary

The document asks how to choose portfolio weights that maximize expected return for a specified level of variance. The response identifies the problem as Markowitz mean-variance optimization and describes the efficient frontier using expected returns and the return covariance matrix. It presents formulas for two named portfolios and says that combinations of these portfolios span the frontier when there is no risk-free asset. With a risk-free asset, it describes the efficient set as a straight line from the risk-free rate, with its slope determined by the Sharpe ratio.

This is a compact conceptual answer rather than a full derivation or implementation guide. The portfolio labels and formulas in the response appear inconsistent with conventional naming, so they should be checked against the intended constraints before use. It also does not discuss estimation error, short-selling restrictions, transaction costs, or other practical limits that can materially change optimized weights.

Key ideas

• Mean-variance optimization uses expected returns and a covariance matrix to construct portfolios.
• The efficient frontier represents portfolios with the best expected return for each level of risk.
• The response describes spanning the frontier by combining two portfolios when no risk-free asset is available.
• A risk-free asset leads to a linear risk-return opportunity set whose slope is the Sharpe ratio.
• The formulas and portfolio labels should be verified, and practical constraints are not addressed.

Tags

# efficient portfolio with given risk


# efficient portfolio with given risk












Is there a formula to derive an efficient portfolio to maximise the return, x'mu, for a given risk, x'S x (where x are the portfolio coefficients, mu is the mean return for each asset and S is the var-cov matrix of the assets) ?

## Answer by phdstudent (score 2)

https://quant.stackexchange.com/a/21860

Yes, this is the simple markowitz optimization.

Denote $\Sigma$ as the variance-covariance matrix of returns and $\bar{R}$ as a vector of expected returns. The tangency portfolio weights are given by $\omega_{tan}=\frac{\textbf{{1}'}\Sigma^{-1}}{\textbf{{1}'}\Sigma^{-1}\textbf{1}}$.

The minimum-variance portfolio is $\omega_{mv}=\frac{\Sigma^{-1}\bar{R}}{\textbf{{1}'}\Sigma^{-1}\bar{R}}$.

Using the two mutual fund theorem one can span the entire efficient frontier without a risk-free asset by linearly combining this two portfolios. Letting $\alpha \in [-\infty , +\infty]$, the efficient weights of any frontier portolio can be obtained as: $\alpha \times (\omega_{tan}) + (1-\alpha) \times (\omega_{MV})$ .

Using the risk-free asset, the mean-variance efficient frontier becomes a straight line starting from the risk-free rate and with slope equal to the sharpe ratio.

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.