Criteria for the Stochastic Ordering of Random Sums,with Actuarial Applications

It is shown that vectors ( S M 1 , … , S Mn ) and ( S' M'1 , …, S' M'n ) of random sums of positive random variables are stochastically ordered by upper orthant dependence, lower orthant dependence, concordance or by the supermodular ordering whenever their corresponding random numbers of terms ( M 1 , … , M n ) and ( M' 1 , … , M' n ) are themselves ordered in this fashion. Actuarial applications of these results are given to different dependence structures for the collective risk model with several classes of business.     

Tail mutual exclusivity and Tail-VaR lower bounds

In this paper, we extend the concept of mutual exclusivity proposed by [Dhaene, J. & Denuit, M. (1999). The safest dependence structure among risks. Insurance: Mathematics and Economics 25, 11–21] to its tail counterpart and baptize this new dependency structure as tail mutual exclusivity. Probability levels are first specified for each component of the random vector. Under this dependency structure, at most one exceedance over the corresponding Value-at-Risks (VaRs) is possible, the other components being zero in such a case. No condition is imposed when all components stay below the VaRs. Several properties of this new negative dependence concept are derived. We show that this dependence structure gives rise to the smallest value of Tail-VaR (TVaR) of a sum of risks within a given Fréchet space, provided that the probability level of the TVaR is close enough to one.     

Ruin probabilities and optimal capital allocation for heterogeneous life annuity portfolios

A large part of the actuarial literature is devoted to the derivation of ruin probabilities in various non-life insurance risk models. On the contrary, very few papers deal with ruin probabilities for life insurance portfolios. The difficulties arise from the dependence and non-stationarity of the annual payments made by the insurance company. This paper shows that the ruin probability in case of life annuity portfolios can be computed from algorithms derived by De Pril (1989) and Dhaene & Vandebroek (1995). Approximations for ruin probabilities are discussed. The present article complements the works of Frostig et al. (2003) who considered whole life, endowment, and temporary assurances, and of Denuit & Frostig (2008) who considered homogeneous life annuities portfolios. Here, heterogeneous portfolios (with respect to age and/or face amounts) are studied. Particular attention is paid to the capital allocation problem. The total amount of reserve is shared among the risk classes in order to minimize the ruin probability. It is then fair to charge a higher margin to the risk classes requiring more capital.     

Correlated risks,bivariate utility and optimal choices

In this paper, we consider a decision-maker facing a financial risk flanked by a non-financial background risk such as health or environmental risk. A decision has to be made about the amount of an investment (in the financial dimension) resulting in a future benefit either in the same dimension (savings) or in the other dimension (environmental quality or health improvement). In this framework, we study the impact of the correlation between the two risks on optimal choices. In the saving problem, we find conditions ensuring that positive correlation between the two risks implies that the optimal amount of savings increases. These conditions involve specific requirements on the direct and cross derivatives of the two-argument utility function. Similarly, we find a different and specific set of conditions ensuring that the same conclusion on optimal investment for health (environmental) improvement is reached. The two sets of conditions determined support the conclusion that the signs of the derivatives of the two-argument utility function should alternate.     

When Ross meets Bell: The linex utility function

At first glance, there would appear to be no relationship between Bell’s (1988) concept of one-switch utility functions and that of a stronger measure of risk aversion due to Ross (1981). We show however that specific assumptions about the behavior of the stronger measure of risk aversion also give rise to the linex utility function which belongs to the class of one-switch utility functions. In particular, this utility class is the only one that satisfies a stronger version of Kimball’s (1993) standard risk aversion over all levels of wealth. We apply our results to consider n$n$th-degree deteriorations in background risk and their effect on risk taking behavior.     

Bivariate almost stochastic dominance

Univariate almost stochastic dominance has been widely studied and applied since its introduction by Leshno and Levy (Manag Sci 48:1074–1085, 2002). This paper extends this construction to the bivariate case by means of suitable two-attribute utility functions. After having confined correlation aversion and correlation loving to some acceptable levels, bivariate almost stochastic dominance rules are introduced for the preferences exhibiting confined correlation aversion and confined correlation loving. The impact of a change in risk in terms of bivariate almost stochastic dominance on optimal saving is analyzed as an application, as well as the effect of envy and altruism on income distributions. Finally, alternative definitions of bivariate almost stochastic dominance are discussed, as well as testing procedures for such dominance rules in financial problems.     

Tradeoffs for Downside Risk-Averse Decision-Makers and the Self-Protection Decision

In addition to risk aversion, decision-makers tend to be also downside risk averse. Besides the usual size for risk trade-off, this allows several other trade-offs to be considered. The decision to increase the level of self-protection generates five trade-offs each involving an unfavourable downside risk increase and an accompanying beneficial change. Five stochastic orders that correspond to these trade-offs are defined, characterised and used to prove comparative static theorems that provide information concerning the self-protection decision. The five stochastic orders are general in nature and can be applied in any decision model where downside risk aversion is assumed.     

Extreme Value Analysis of Mortality at the Oldest Ages: A Case Study Based on Individual Ages at Death

In this article, the force of mortality at the oldest ages is studied using the statistical tools from extreme value theory. A unique data basis recording all individual ages at death above 95 for extinct cohorts born in Belgium between 1886 and 1904 is used to illustrate the relevance of the proposed approach. No leveling off in the force of mortality at the oldest ages is found, and the analysis supports the existence of an upper limit to human lifetime for these cohorts. Therefore, assuming that the force of mortality becomes ultimately constant, that is, that the remaining lifetime tends to the Negative Exponential distribution as the attained age grows is a conservative strategy for managing life annuities.     

Properties of a Risk Measure Derived from Ruin Theory

This paper studies a risk measure inherited from ruin theory and investigates some of its properties. Specifically, we consider a value-at-risk (VaR)-type risk measure defined as the smallest initial capital needed to ensure that the ultimate ruin probability is less than a given level. This VaR-type risk measure turns out to be equivalent to the VaR of the maximal deficit of the ruin process in infinite time. A related Tail-VaR-type risk measure is also discussed.