## Supermodular ordering and stochastic annuities.(English)Zbl 0942.60008

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., A. Müller [Insur. Math. Econ. 21, No. 3, 219-223 (1997; Zbl 0894.90022)]. By a result due to 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 F. Delbaen [Math. Finance 3, No. 2, 125-134 (1993; Zbl 0884.90023)] closes the paper.

### MSC:

 60E99 Distribution theory 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B28 Finance etc. (MSC2000)

### Keywords:

stop-loss order; supermodular order; stochastic annuities

### Citations:

Zbl 0894.90022; Zbl 0459.62010; Zbl 0884.90023
Full Text:

### References:

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