Formula for the efficient portfolios in mean-variance optimisation?

Consider the setting of mean-variance portfolio optimisation: $$n$$ assets with expected returns $$\overline{r}_1,...,\overline{r}_n$$ and standard deviations $$\sigma_1,...\sigma_n$$.

For a certain fixed $$\sigma$$ I am currently trying to derive a closed formula for highest rate of return (i.e. given $$\sigma$$ I am trying to find the corresponding efficient portfolio on the efficient frontier of the feasible set).

For example, if one wants to compute the minimum variance portfolio we can find the weights by simply computing $$w_i = \frac{\sum_{k=1}^n \sigma_{k,i}^{-1}}{\sum_{j,k=1}^n \sigma_{k,j}^{-1}}$$ where $$\sigma_{k,i}^{-1}$$ is the entry $$(k,i)$$ in the inverse covariance matrix $$\Sigma^{-1}$$.

Now, given $$\sigma$$, how do we find the highest possible $$\overline{r}$$? Using Lagrange multipliers I have derived the following equations, but am unsure whether I am going into the right direction: $$2\cdot\lambda\cdot\sum_{k=1}\sigma_{i,k}w_k = -\mu - \overline{r}_i \quad\quad \forall i=1,...,n$$ $$\sum_{i,j=1}^n w_i w_j \sigma_{i,j}=\sigma^2$$ $$\sum_{i=1}^n w_i = 1$$ How do I find the weights $$w_1,...,w_n$$ for the efficient portfolio given a fixed value of $$\sigma$$?

• Yes, you're on the right track, keep going! :) Remember you can denote $\sum w_i = 1$ as $e^T \cdot w = 1$, where $e$ is a vector of ones. Then it's a simple (but somewhat involved) task to solve the equations by substitutions. In the end you need to solve a qundratic equation for $\lambda$ (or$\mu$). Nov 28, 2013 at 17:41