Deriving Efficient Portfolios at a Fixed Volatility

Summary

The document asks how to find the highest expected return among fully invested portfolios with a specified standard deviation. It sets out the standard mean-variance ingredients: asset expected returns, a covariance matrix, a target portfolio variance, and the constraint that weights sum to one. It also gives a formula for the minimum-variance portfolio and presents first-order conditions obtained from Lagrange multipliers for the fixed-risk problem.

The text is a derivation question rather than a completed solution. It supplies no numerical inputs, computed weights, or test of the equations. To obtain an efficient portfolio, the constraints must be handled together with the covariance matrix and return vector; the resulting frontier depends on those inputs and generally requires appropriate nonsingularity assumptions. The note is useful as a setup for constrained portfolio optimization, but it does not explain the closed-form solution or address practical constraints such as bounds, transaction costs, or estimation error.

Key ideas

• The objective is to maximize expected portfolio return subject to fixed variance and weights summing to one.
• The covariance matrix captures portfolio risk through the quadratic form in the asset weights.
• Lagrange multipliers yield first-order conditions alongside the two portfolio constraints.
• The document poses the derivation but provides no solution, data, or numerical validation.
• Practical portfolio constraints and estimation uncertainty are outside the stated setup.

Tags

# Formula for the efficient portfolios in mean-variance optimisation?


# 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 $\sigma$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.