  We derive the mapping between two of the most pervasive utility functions,
the mean square error ($MSE$) and the concordance correlation coefficient (CCC,
$\rho_c$). Despite its drawbacks, $MSE$ is one of the most popular performance
metrics (and a loss function); along with lately $\rho_c$ in many of the
sequence prediction challenges. Despite the ever-growing simultaneous usage,
e.g., inter-rater agreement, assay validation, a mapping between the two
metrics is missing, till date. While minimisation of $L_p$ norm of the errors
or of its positive powers (e.g., $MSE$) is aimed at $\rho_c$ maximisation, we
reason the often-witnessed ineffectiveness of this popular loss function with
graphical illustrations. The discovered formula uncovers not only the
counterintuitive revelation that `$MSE_1<MSE_2$' does not imply
`$\rho_{c_1}>\rho_{c_2}$', but also provides the precise range for the $\rho_c$
metric for a given $MSE$. We discover the conditions for $\rho_c$ optimisation
for a given $MSE$; and as a logical next step, for a given set of errors. We
generalise and discover the conditions for any given $L_p$ norm, for an even p.
We present newly discovered, albeit apparent, mathematical paradoxes. The study
inspires and anticipates a growing use of $\rho_c$-inspired loss functions
e.g., $\left|\frac{MSE}{\sigma_{XY}}\right|$, replacing the traditional
$L_p$-norm loss functions in multivariate regressions.
