efficient portfolio with given risk

Question

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 1

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.