How to calculate a hypothetical minimum-variance point?

Question

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 1

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

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