Copulas and the distribution of cash flows with mixed signs

In a paper of 2000, Kaas, Dhaene and Goovaerts investigate the present value of a rather general cash flow. Making use of comonotonic risks, they derive upper and lower bounds for the distribution of the present value. These bounds are very close to the real distribution in case all payments have the same sign: however, if there are both positive and negative payments, the upper bounds perform rather badly. In the present contribution we show what happens when solving this problem by means of copulas. The idea consists of splitting up the total present value in the difference of two present values with positive payments. Making use of a copula as an approximation for the joint distribution of the two sums, an approximation for the distribution of the original present value can be derived.