loss_mv

loss.loss_mv(sigma_hat, sigma)

The minimum variance loss function of Ledoit and Wolf (2018).

Parameters

sigma_hatnumpy.ndarray

The covariance matrix estimate using the estimator of interest.

sigmanumpy.ndarray

The (true) population covariance matrix.

Returns

outfloat

The minimum variance loss.

Notes

The minimum variance (MV)-loss function is proposed by Engle et al. (2019) as a loss function that is appropriate for covariance matrix estimator evaluation for the problem of minimum variance portfolio allocations under a linear constraint and large-dimensional asymptotic theory.

The loss function is given by:

\[\mathcal{L}_{n}^{\mathrm{MV}}\left(\widehat{\Sigma}_{n}, \Sigma_{n}\right):=\frac{\operatorname{Tr}\left(\widehat{\Sigma}_{n}^{-1} \Sigma_{n} \widehat{\Sigma}_{n}^{-1}\right) / p} {\left[\operatorname{Tr}\left(\widehat{\Sigma}_{n}^{-1}\right) /p\right]^{2}}-\frac{1}{\operatorname{Tr}\left(\Sigma_{n}^{-1}\right)/p}.\]

It can be interpreted as the true variance of the minimum variance portfolio constructed from the estimated covariance matrix.