= Adjustment coefficient with independent claim expenses
{title2=$M_{X+A}(R)-1=(1+\theta)\mathbb E[X+A]R$}
If each claim includes an independent expense $A$, replace its payment law by the <convolution of independent random variables> $X+A$. Its <moment-generating function> is $M_X(r)M_A(r)$ and its <expected value> is $\mathbb EX+\mathbb EA$. Keeping the <relative safety loading> $\theta$ fixed therefore changes the premium rate as well. The new coefficient solves $M_X(R)M_A(R)-1=(1+\theta)(\mathbb EX+\mathbb EA)R$, within the common finite-transform domain.
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