  Given an operator system $\mathcal{S}$, we define the parameters
$r_k(\mathcal{S})$ (resp. $d_k(\mathcal{S})$) defined as the maximal value of
the completely bounded norm of a unital $k$-positive map from an arbitrary
operator system into $\mathcal{S}$ (resp. from $\mathcal{S}$ into an arbitrary
operator system).
  In the case of the matrix algebras $M_n$, for $1 \leq k \leq n$, we compute
the exact value $r_k(M_n) = \frac{2n-k}{k}$ and show upper and lower bounds on
the parameters $d_k(M_n)$.
  Moreover, when $\mathcal{S}$ is a finite-dimensional operator system,
adapting recent results of Passer and the 4th author, we show that the sequence
$(r_k( \mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ is exact and
that the sequence $(d_k(\mathcal{S}))$ tends to $1$ if and only if
$\mathcal{S}$ has the lifting property.
