Convex order approximations in case of cash flows of mixed signs

In Van Weert et al. (2010), results are obtained showing that, when allowing some of the cash flows to be negative, convex order lower bound approximations can still be used to solve general investment problems in a context of provisioning or terminal wealth. In this paper, a correction and further clarification of the reasoning of Van Weert et al. (2010) are given, thereby significantly expanding the scope of problems and cash flow patterns for which the terminal wealth or initial provision can be accurately approximated. Also an interval for the probability level is derived in which the quantiles of the lower bound approximation can be computed. Finally, it is shown how one can move from a context of provisioning of future obligations to a saving and terminal wealth problem by inverting the time axis.