Supermodular ordering and stochastic annuities
The authors propose a method how to deal appropriately from an actuarial point of view with a risk \(V\) whose distribution function is unavailable. Their idea is to replace \(V\) by a `more risky' quantity \(W\) with known distribution where `more risky' means that \(V\) is dominated by \(W\) in stop-loss order, i.e., \(E(V-d)^+ \leq E(W-d)^+\) for all \(d\). The authors give an explicit description of such a dominating risk \(W\) for a primary risk \(V\) of the additive form \(V=\varphi_1(X_1)+\ldots+\varphi_n(X_n)\) where \(\varphi_i:\mathbb{R}\to\mathbb{R^+}\) is decreasing and \(X=(X_1,\ldots,X_n)\) is a random variable with known marginals \(P[X_i \leq x]=F_i(x) (x \in \mathbb{R})\). More precisely, they show that \(W\) may be chosen as \(W:=\varphi_1(Y_1)+\ldots+\varphi_n(Y_n)\) where \(Y=(Y_1,\ldots,Y_n):= (F^{-1}_1(U),\ldots,F^{-1}_n(U))\) with \(U\) uniformly distributed on \([0,1]\). To prove this result, the authors proceed by noting first that risks \(V\) and \(W\) of the above additive form are in stop-loss order if \(X\) is smaller than \(Y\) in the supermodular order; for a definition of this order see, e.g., \textit{A. Müller} [Insur. Math. Econ. 21, No. 3, 219-223 (1997; Zbl 0894.90022)]. By a result due to \textit{A. H. Tchen} [Ann. Probab. 8, 814-827 (1980; Zbl 0459.62010)], the largest random vector \(Y\) with given marginals \(F_i(.)\) is the comonotonic variable \((F^{-1}_1(U), \ldots, F^{-1}_n(U))\). The combination of these results then yields the assertion. The method is illustrated by considering different types of annuities as case studies. They start with \(V_1= \sum_{i=1}^n \alpha_i \exp(-\delta i X_i)\) for nonnegative constants \(\alpha_i\), \(\delta\), and \(X_i \sim N(0,1)\). Passing from this to the continuous time limit \(V_2 = \int_0^t \alpha(s) \exp(-\delta s - \sigma X(s))\) where \(X\) is, e.g., a standard Brownian motion, shows how one may treat continuously composed annuities. Finally, the authors consider the case where \(\alpha(.)\) is discounted using the CIR-model for the short rate. For the infinite horizon case \(t=+\infty\), this allows them to derive upper bounds for the moments of a perpetuity in the CIR-model. The comparison of these bounds with the bounds obtained by \textit{F. Delbaen} [Math. Finance 3, No. 2, 125-134 (1993; Zbl 0884.90023)] closes the paper.
scientific article; zbMATH DE number 1335224