= Conditional lognormal layer recovery
{title2=$C(\mu,\sigma;M,A)=\mathbb E[L_{M,A}(X)\mid X>M]$}
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 numerator
$$
e^{\mu+\sigma^2/2}[\Phi(d_1(M+A))-\Phi(d_1(M))]-M[\Phi(d_0(M+A))-\Phi(d_0(M))]+A[1-\Phi(d_0(M+A))].
$$
Divide 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.
Back to article page
