Convex order approximations in the case of cash flows of mixed signs

Abstract

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.

A stochastic order for the analysis of investments affected by the time value of money

“…There is a broad diversity of economic concepts which reflect that idea, like the present value, the present value of an annuity, the present value of a perpetuity, the net present value, the future value, the future value of an annuity, etc. Problems in relation to this concept concern the net value of cash flows at different moments in time (see, for instance, De Schepper et al (2002) or Dhaene et al (2012)).…”

Convex Ordering for Insurance Preferences

“…For instance, it can be applied to compare the aggregate risk of a portfolio, in which the comonotonicity structure among the risks attains the upper bound of the convex order. For comprehensive studies and other applications in convex ordering, see Denuit et al (2005), Denuit and Dhaene (2012), Dhaene et al (2002Dhaene et al ( , 2006Dhaene et al ( , 2012, Kaas et al (1994Kaas et al ( , 2008, Müller and Stoyan (2002), Rüschendorf (2013), Shaked and Shanthikumar (2007), and the references therein.…”