On two approaches to coherent risk contribution

We compare two approaches to risk contribution estimation for coherent risk measures:

• directional risk contribution, defined as
  $$ \rho^d(X;Y) = \lim_{\varepsilon \searrow 0} \frac{\rho(Y+\varepsilon X) - \rho(Y)}{\varepsilon}, $$
  where $\rho(X)$ is a coherent risk measure
• linear risk contribution defined through a set of axioms, one of which is linearity in $X$.

The closed formula for both risk contributions is given for $\textrm{MINV@R}_N$, defined by
$$ \textrm{MINV@R}_N(X) = -E\min\{ X_1, \dots, X_N \}, $$
where $X_1, \dots, X_N$ are independent copies of $X$.
Also we study the asymptotic behavior of two empirical estimates of the coherent risk measure $\textrm{MINV@R}_N$.