Stochastic ordering of Gini indexes for multivariate elliptical risks

In this paper, we show that the conjecture, made by Samanthi et al. (2016), on the ordering of Gini indexes of multivariate normal risks with respect to the strength of dependence, is not true. By using the positive semi-definite ordering of covariance matrices, we can obtain the usual stochastic order of the Gini indexes for multivariate normal risks. This can be generalized to multivariate elliptical risks. We also investigate the monotonicity of the Gini indexes in the usual stochastic order when the covariance (dispersion, resp.) matrices of multivariate normal (elliptical, resp) risks increase componentwise. In addition, we derive a large deviation result for the Gini indexes of multivariate normal risks.     

Optimization incentive and relative riskiness in experimental stag-hunt games

We compare the experimental results of three stag-hunt games. In contrast to Battalio et al. (Econometrica 69:749–764, 2001), our design keeps the riskiness ratio of the two strategies at a constant level as the optimization premium is increased. We define the riskiness ratio as the relative payoff range of the two strategies. We find that decreasing the riskiness ratio while keeping the optimization premium constant decreases sharply the frequency of the payoff-dominant equilibrium strategy. On the other hand an increase of the optimization premium with a constant riskiness ratio has no effect on the choice frequencies. Finally, we confirm the dynamic properties found by Battalio et al. that increasing the optimization premium favours best-response and sensitivity to the history of play.     

A note on likelihood ratio ordering of order statistics from two samples

The purpose of this paper is to establish the stochastic comparisons of order statistics from two samples in the sense of likelihood ratio order. We strengthen and complement some results in Zhao and Balakrishnan (2012) and Ding et al. (2013).     

Optimal portfolio selection for general provisioning and terminal wealth problems

In Dhaene et al. (2005), multiperiod portfolio selection problems are discussed, using an analytical approach to find optimal constant mix investment strategies in a provisioning or a savings context. In this paper we extend some of these results, investigating some specific, real-life situations. The problems that we consider in the first section of this paper are general in the sense that they allow for liabilities that can be both positive or negative, as opposed to Dhaene et al. (2005), where all liabilities have to be of the same sign. Secondly, we generalize portfolio selection problems to the case where a minimal return requirement is imposed. We derive an intuitive formula that can be used in provisioning and terminal wealth problems as a constraint on the admissible investment portfolios, in order to guarantee a minimal annualized return. We apply our results to optimal portfolio selection.     

On the optimal product mix in life insurance companies using conditional value at risk

This paper proposes a Conditional Value-at-Risk Minimization (CVaRM) approach to optimize an insurer’s product mix. By incorporating the natural hedging strategy of Cox and Lin (2007) and the two-factor stochastic mortality model of Cairns et al. (2006b), we calculate an optimize product mix for insurance companies to hedge against the systematic mortality risk under parameter uncertainty. To reflect the importance of required profit, we further integrate the premium loading of systematic risk. We compare the hedging results to those using the duration match method of Wang et al. (forthcoming), and show that the proposed CVaRM approach has a narrower quantile of loss distribution after hedging—thereby effectively reducing systematic mortality risk for life insurance companies.     

Stochastic orders in time transformed exponential models with applications

This paper studies expectations of a supermodular function of bivariate random risks following TTE models. Comparison of such expectations are conducted based on some stochastic orders of the involved univariate survival functions in the models, and also the upper orthant-convex order between two bivariate random risks in TTE models is built. This corrects Theorem 2.3 of Mulero et al. (2010) and invalidates some results there. Some applications in actuarial science are presented as well.     

The role of the dependence between mortality and interest rates when pricing Guaranteed Annuity Options

In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider some multifactor models for the mortality and interest rates based on more general affine models which remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.     

风险相依的保险投资组合研究

在个人和团体风险模型中,用s=Em=1x.表示保险投资组合的累积索赔,其中,X1,I>/1表示第i个保单产生的损失,N表示保险公司在一定时期内(例如,一年)总的索赔项数,每一个保单可能包含若干个不同的项目,假定一个保单的损失是这些不同项目索赔的总和,而一个保单的不同项目的索赔往往是相关的.Marceau et al.(1999)建立了一个相依风险模型,其中考虑了保险投资组合中个体风险之间的相依性.最近,Denuit(2001)证明了两个不同参数投资组合的索赔向量之间拉普拉斯变换序成立.本文将证明,事实上更强的随机序是成立的,并将该模型推广到团体风险模型的情形.     

Optimal insurance with belief heterogeneity and incentive compatibility

People may evaluate risk differently in the insurance market. Motivated by this, we examine an optimal insurance problem allowing the insured and the insurer to have heterogeneous beliefs about loss distribution. To reduce ex post moral hazard, we follow Huberman et al. (1983) to assume that alternative insurance contracts satisfy the principle of indemnity and the incentive-compatible constraint. Under the assumption that the insurance premium is calculated by the expected value principle, we establish a necessary and sufficient condition for an optimal insurance solution and provide a practical scheme to improve any suboptimal insurance strategy under an arbitrary form of belief heterogeneity. By virtue of this condition, we explore qualitative properties of optimal solutions, and derive optimal insurance contracts explicitly for some interesting forms of belief heterogeneity. As a byproduct of this investigation, we find that Theorem 3.6 of Young (1999) is not completely true.     

Erratum and addendum to “Two-point b.v.p. for multivalued equations with weakly regular r.h.s.”

In this paper, we define a topological index for compact multivalued maps in convex metrizable subsets of a locally convex topological vector space in order to correct the proofs of Theorems 4.1 and 4.2 in Benedetti et al. (2011) [1].     

Characterizations of counter-monotonicity and upper comonotonicity by (tail) convex order

In this paper, we characterize counter-monotonic and upper comonotonic random vectors by the optimality of the sum of their components in the senses of the convex order and tail convex order respectively. In the first part, we extend the characterization of comonotonicity by  Cheung (2010) and show that the sum of two random variables is minimal with respect to the convex order if and only if they are counter-monotonic. Three simple and illuminating proofs are provided. In the second part, we investigate upper comonotonicity by means of the tail convex order. By establishing some useful properties of this relatively new stochastic order, we prove that an upper comonotonic random vector must give rise to the maximal tail convex sum, thereby completing the gap in  Nam et al. (2011)’s characterization. The relationship between the tail convex order and risk measures along with conditions under which the additivity of risk measures is sufficient for upper comonotonicity is also explored.     

Optimal insurance under rank-dependent expected utility

We re-visit the problem of optimal insurance design under Rank-Dependent Expected Utility (RDEU) examined by Bernard et al. (2015), Xu (2018), and Xu et al. (2018). Unlike the latter, we do not impose the no-sabotage condition on admissible indemnities, that is, that indemnity and retention functions be nondecreasing functions of the loss. Rather, in a departure from the aforementioned work, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, hence automatically ruling out any ex post moral hazard that could otherwise arise from possible misreporting of the loss by the insured. We fully characterize the optimal indemnity schedule and discuss how our results relate to those of Bernard et al. (2015) and Xu et al. (2018). We then extend the setting by allowing for a distortion premium principle, with a distortion function that differs from that of the insured, and we provide a characterization of the optimal retention in that case.     

A neural network approach to efficient valuation of large portfolios of variable annuities

Managing and hedging the risks associated with Variable Annuity (VA) products require intraday valuation of key risk metrics for these products. The complex structure of VA products and computational complexity of their accurate evaluation have compelled insurance companies to adopt Monte Carlo (MC) simulations to value their large portfolios of VA products. Because the MC simulations are computationally demanding, especially for intraday valuations, insurance companies need more efficient valuation techniques. Recently, a framework based on traditional spatial interpolation techniques has been proposed that can significantly decrease the computational complexity of MC simulation (Gan and Lin, 2015). However, traditional interpolation techniques require the definition of a distance function that can significantly impact their accuracy. Moreover, none of the traditional spatial interpolation techniques provide all of the key properties of accuracy, efficiency, and granularity (Hejazi et al., 2015). In this paper, we present a neural network approach for the spatial interpolation framework that affords an efficient way to find an effective distance function. The proposed approach is accurate, efficient, and provides an accurate granular view of the input portfolio. Our numerical experiments illustrate the superiority of the performance of the proposed neural network approach compared to the traditional spatial interpolation schemes.     

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traub’s conjecture [7] for four function evaluations per iteration. Notice that Bi et al.’s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

Analysis of survivorship life insurance portfolios with stochastic rates of return

A general portfolio of survivorship life insurance contracts is studied in a stochastic rate of return environment with a dependent mortality model. Two methods are used to derive the first two moments of the prospective loss random variable. The first one is based on the individual loss random variables while the second one studies annual stochastic cash flows. The distribution function of the present value of future losses at a given valuation time is derived. For illustrative purposes, an AR(1) process is used to model the stochastic rates of return, and the future lifetimes of a couple are assumed to follow a copula model. The effects of the mortality dependence, the portfolio size and the policy type, as well as the impact of investment strategies on the riskiness of portfolios of survivorship life insurance policies are analyzed by means of moments and probability distributions.     

Equiseparability on terminal Wiener index

The aim of this work is to explore the properties of the terminal Wiener index, which was recently proposed by Gutman et al. (2004) [3], and to show the fact that there exist pairs of trees and chemical trees which cannot be distinguished by using it. We give some general methods for constructing equiseparable pairs and compare the methods with the case for the Wiener index. More specifically, we show that the terminal Wiener index is degenerate to some extent.     

A characterization of the multivariate excess wealth ordering

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernández-Ponce et al. (1996), is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernández-Ponce et al. (1998). The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are described.     

A new discrete dynamical system of signed integer partitions

In Brylawski (1973) Brylawski described the covering property for the domination order on non-negative integer partitions by means of two rules. Recently, in Bisi et al. (in press), Cattaneo et al. (2014), Cattaneo et al. (2015) the two classical Brylawski covering rules have been generalized in order to obtain a new lattice structure in the more general signed integer partition context. Moreover, in Cattaneo et al. (2014), Cattaneo et al. (2015), the covering rules of the above signed partition lattice have been interpreted as evolution rules of a discrete dynamical model of a two-dimensional p–n semiconductor junction in which each positive number represents a distribution of holes (positive charges) located in a suitable strip at the left semiconductor of the junction and each negative number a distribution of electrons (negative charges) in a corresponding strip at the right semiconductor of the junction. In this paper we introduce and study a new sub-model of the above dynamical model, which is constructed by using a single vertical evolution rule. This evolution rule describes the natural annihilation of a hole–electron pair at the boundary region of the two semiconductors. We prove several mathematical properties of such new discrete dynamical model and we provide a discussion of its physical properties.     

Comparisons on aggregate risks from two sets of heterogeneous portfolios

In this paper, we stochastically compare the aggregate risks from two heterogeneous portfolios. It is shown that under suitable conditions the more heterogeneities among aggregate risks would result in larger aggregate risks in the sense of the stochastic order. The stochastic properties of aggregate risks when the claims follow proportional hazard rates models or scale models are studied. We also provide sufficient conditions for comparing the aggregate risks arising from two sets of heterogeneous portfolios with claims having gamma distributions. In particular, the aggregate risks of portfolios from dependent samples with comonotonic dependence structures or arrangement increasing density functions are discussed. The new results established strengthen and generalize several results known in the literature including Ma (2000), Khaledi and Ahmadi (2008), Xu and Hu (2011), Xu and Balakrishnan (2011), Pan et al. (2013) and Barmalzan et al. (2015).     

On the invariant properties of notions of positive dependence and copulas under increasing transformations

Notions of positive dependence and copulas play important roles in modeling dependent risks. The invariant properties of notions of positive dependence and copulas under increasing transformations are often used in the studies of economics, finance, insurance and many other fields. In this paper, we examine the notions of the conditionally increasing (CI), the conditionally increasing in sequence (CIS), the positive dependence through the stochastic ordering (PDS), and the positive dependence through the upper orthant ordering (PDUO). We first use counterexamples to show that the statements in Theorem 3.10.19 of Müller and Stoyan (2002) about the invariant properties of CIS and CI under increasing transformations are not true. We then prove that the invariant properties of CIS and CI hold under strictly increasing transformations. Furthermore, we give rigorous proofs for the invariant properties of PDS and PDUO under increasing transformations. These invariant properties enable us to show that a continuous random vector is PDS (PDUO) if and only of its copula is PDS (PDUO). In addition, using the properties of generalized left-continuous and right-continuous inverse functions, we give a rigorous proof for the invariant property of copulas under increasing transformations on the components of any random vector. This result generalizes Proposition 4.7.4 of Denuit et al. (2005) and Proposition 5.6. of McNeil et al. (2005).