Global Minimum-Variance Weights for Uncorrelated Assets

Summary

For assets with equal expected returns and no correlations, the response derives the global minimum-variance portfolio weights. It begins with the general solution: multiply the vector of ones by the inverse covariance matrix, then normalize so the weights sum to one.

Because the covariance matrix is diagonal in this case, its inverse has reciprocal variances on the diagonal. Thus each asset’s weight is its inverse variance divided by the sum of all assets’ inverse variances. The result assigns larger weights to lower-variance assets. The document states the weights but does not explicitly derive the portfolio’s resulting minimum variance, despite the question’s request for a formula using that quantity. The solution also assumes uncorrelated assets and a fully invested portfolio with weights summing to one.

Key ideas

• The global minimum-variance portfolio weights are proportional to the inverse covariance matrix applied to a vector of ones.
• With uncorrelated assets, the covariance matrix is diagonal and its inverse contains reciprocal variances.
• Each asset receives a weight equal to its inverse variance divided by the sum of all inverse variances.
• Lower-variance assets therefore receive larger weights under the stated assumptions.
• The answer gives weights but does not explicitly provide the resulting minimum portfolio variance.

Tags

# How to calculate a hypothetical minimum-variance point?


# How to calculate a hypothetical minimum-variance point?












If we have $N$ assets which are uncorrelated, but have the same mean return of $\mu$ but the variances are different where $\sigma_i^2$ is the variance of each asset $i = 1, 2,...,N$ how can you write a formula for the minimum-variance point? Write the result in terms of $\sigma_p^2=\sum_{i=1}^N{1/\sigma_i^2}$.

I tried solving the minimization problem by minimizing the portfolio variance subject to the weights summing to one, however when taking the inverse of the matrix to get the weights I cannot seem to write an elegant solution. Any help would be appreciated.

## Answer by Alex C (score 3, accepted)

https://quant.stackexchange.com/a/21531

The general formula for the global minimum variance portfolio is $w=\frac{C^{-1} 1}{1^T C^{-1} 1}$ where C is the covariance matrix and 1 is a vector of 1's. In this case the covariance matrix is diagonal with $\sigma_i^2$ in the ith diagonal element. Its inverse is also diagonal and has $\frac{1}{\sigma_i^2}$ in the ith diagonal element.

Evaluating the expression above we get that the weight for asset $i$ is

$$\frac{1/\sigma_i^2}{\sum_k 1/\sigma_k^2}$$

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