The series expansion introduced by Frey and Schmidt (1996) [Taylor Series expansion for multivariate characteristics of
classical risk processes. Insurance: Mathematics and Economics 18, 1–12.] constitutes an original approach in approximating
multivariate cThe series expansion introduced by Frey and Schmidt (1996) [Taylor Series expansion for multivariate characteristics of
classical risk processes. Insurance: Mathematics and Economics 18, 1–12.] constitutes an original approach in approximating
multivariate characteristics of classical ruin processes, specially ruin probabilities within finit time with certain surplus prior
to ruin and severity of ruin. This approach can be considered alternative to inversion of Laplace transforms for particular claim
size distributions [Gerber, H., Goovaerts, M., Kaas, R., 1987. On the probability and severity of ruin. ASTIN Bulletin 17(2),
151–163; Dufresne, F., Gerber, H., 1988a. The probability and severity of ruin for combinations of exponential claim amount
distributions and their translations. Insurance: Mathematics and Economics 7, 75–80; Dufresne, F., Gerber, H., 1988b. The
surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics 7,
193–199.] or discretization of the claim size and time [Dickson, C., 1989. Recursive calculation of the probability and severity
of ruin. Insurance: Mathematics and Economics 8, 145–148; Dickson, C., Waters, H., 1992. The probability and severity of
ruin in finit and infinit time. ASTIN Bulletin 22(2), 177–190; Dickson, C., 1993. On the distribution of the claim causing
ruin. Insurance: Mathematics and Economics 12, 143–154.] applying the so-called Panjer’s recursive algorithm [Panjer, H.H.,
1981. Recursive calculation of a family of compound distributions. ASTIN Bulletin 12, 22–26.].
We will prove that the recursive relation involved in the calculations of the the nth derivative with respect to – average
number of claims in the time unit – of the multivariate finit time ruin probability (developed in the original paper by Frey
and Schmidt (1996) can be simplified The cited simplificatio leads to a substantial reduction in the number of multiple
integrals used in the calculations and makes the series expansion approach more appealing for practical implementation[+][-]