Represent Mean-Squared-Prediction error as function of covariance (or Fisher) matrix

Given a simple linear model:

$$y_i = x_i^T \beta + \epsilon_i$$

For simplicity, $$\epsilon_i$$ is Gaussian iid with variance $$\sigma_e^2$$, then the solution for $$\hat{\beta}$$ is given via Ordinary Least Squares:

$$\hat{\beta} = (X^T X)^{-1} X^T y$$

Then it can be shown that the MSE is given as:

$$\mathrm{MSE} = \operatorname{Var}(\hat{\beta} - \beta) = \frac{1}{L} \| \hat{\beta} - \beta \|_2^2 = \frac{1}{L} \sigma_e^2 \operatorname{Tr} (X^T X)^{-1}$$

where $$L$$ is the number of unknowns (length of vector $$\beta$$). That means the MSE for the estimated coefficients can be represented using the trace of the covariance matrix!

Now suppose I pick a new $$X'$$ (which has the same staistics as $$X$$) and want to compare how good my estimator prediction is for a new dataset. I want to represent it using the normalized mean squared prediction error as follows:

$$y' = X' \beta \\ \hat{y}' = X' \hat{\beta} \\ \mathrm{NMSPE} = \frac{\|\hat{y}'-y'\|_2^2}{\|y'\|_2^2}$$

Is there an elegant way to describe the NMSPE in a similar way as the MSE as a function of the covariance matrix (or similar)?