2
$\begingroup$

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.

$\endgroup$
0

1 Answer 1

3
$\begingroup$

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.