Conditional lognormal layer recovery ($C(\mu ,\sigma ;M,A)=\mathbb{E}[L_{M,A}(X)\mid X>M]$)

For a lognormal distribution, let $d_j(t)=(\log t-\mu -j{\sigma }^2)/\sigma$ for $j=0,1$. The truncated lognormal moment formula gives the recovery numeratorDivide by $1-\Phi (d_0(M))$ to condition on $X>M$. This handles both the partially penetrating claims and the capped payments beyond the upper attachment.

For positive inflation factor $h$, $L_{M,A}(hX)=h{L}_{M/h,A/h}(X)$. The conditioning event also changes to $X>M/h$. If claims have a lognormal distribution, the same result follows by replacing their log-mean $\mu$ by $\mu +\log h$ and keeping the contractual attachment and limit fixed. Merely multiplying the original conditional mean by $h$ would hold those effective thresholds incorrectly fixed.