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This paper considers a peer-to-peer (P2P) insurance scheme where the higher layer is transferred to a (re-)insurer and retained losses are distributed among participants according to the conditional mean risk sharing rule proposed by Denuit and Dhaene (2012) . The global retention level of the pool of participants grows proportionally with their number. We study the asymptotic behavior of the individual retention levels, as well as individual cash-backs and stop-loss premiums, as the number of participants increases. The probability that the total loss hits the upper layer protected by the stop-loss treaty is also considered. The results depend on the proportional rate of increase of the global retention level with the number of participants, as well as on the existence of the Esscher transform of the losses brought to the pool.

In this paper, we are concerned with the valuation of Catastrophic Mortality Bonds and, in particular, we examine the case of the Swiss Re Mortality Bond 2003 as a primary example of this class of assets. This bond was the first Catastrophic Mortality Bond to be launched in the market and encapsulates the behaviour of a well-defined mortality index to generate payoffs for bondholders. Pricing these type of bonds is a challenging task and no closed form solution exists in the literature. In our approach, we express the payoff of such a bond in terms of the payoff of an Asian put option and present a new approach to derive model-independent bounds exploiting comonotonic theory as illustrated in Albrecher (2008), Dhaene (2002) and Simon (2000) for the pricing of Asian options. We carry out Monte Carlo simulations to estimate the bond price and illustrate the quality of the bounds.

This paper studies properties of functions having monotone tails. We extend Theorem 1 of Dhaene et al. (2002a) and show how the tail quantiles of a random variable transformed with a monotone tail function can be expressed as the transformed tail quantiles of the original random variable. The main result is intuitive, in that Dhaene et al. (2002a)’s properties still hold, but only for certain quantile values. However, the proof presents some complications that arise especially when the function involved has discontinuities.We consider different situations where monotone tail functions occur and can be useful, such as the evaluation of the payoff of option trading strategies and the present value of insurance contracts providing both death and survival benefits. The paper also applies monotone tail functions to study quadrant perfect dependence, and shows how this dependence structure integrates within the framework of monotone tail functions. Moreover, we apply the theory to the problem of risk reduction and investigate conditions on a hedger ensuring efficient reductions of required economic capital.

The Basel Accord requires commercial banks to meet a capital requirement for operational risk. Therefore, a reliable operational risk measurement is of great significance for financial institutions. In this paper, based on the framework of LDA, we propose a more appropriate model (Double Correlation model) in correlation structure construction, which considers both frequency correlation and mean severity correlation simultaneously. On this basis, annual loss scenarios for each risk cell are restored and VaR is also estimated by Monte Carlo simulation. Empirical study adopts the data of Chinese commercial banks disclosed to the public and contains the comparison of double correlation model and the other four correlation models. The result indicates that the VaR estimated by the double correlation model is significantly lower than that upon the assumption of comonotonicity, proving the existence of risk diversification. Meanwhile, the VaR estimated by double correlation model is significantly lower than that of the other two correlation models when confidence level is lower than 99% but the relationships between them will reverse if the confidence level rises to more than 99%.

This paper derives several novel properties of conditional quantiles viewed as nonlinear operators. The results are organized in parallel to the usual properties of the expectation operator. We first define a τ-conditional quantile random set, relative to any sigma-algebra, as a set of solutions of an optimization problem. Then, well-known properties of unconditional quantiles, as translation invariance, comonotonicity, and equivariance to monotone transformations, are generalized to the conditional case. Moreover, a simple proof for Jensen’s inequality for conditional quantiles is provided. We also investigate continuity of conditional quantiles as operators with respect to different topologies and obtain a novel Fatou’s lemma for quantiles. Conditions for continuity in Lp and weak continuity are also provided. We also investigate differentiability properties of quantiles. We demonstrate the validity of the Leibniz’s rule for conditional quantiles for the cases of monotone, as well as separable functions. Finally, although the law of iterated quantiles does not hold in general, we characterize the maximum set of random variables for which this law holds, and investigate its consequences for the infinite composition of conditional quantiles.

We devise a method to circumvent the complexity that arises from the option multi-dimensionality. That is, we transform the model to make it as simple as the one-dimensional case. Furthermore, the assumption of comonotonicity and other assumptions regarding the structure of the underlying asset become needless.

Suppose that a decision maker faces a random outcome which is the sum of several risky components. If she is indifferent to the dependence structure of the risky components, then we say that she (or her preference) is dependence neutral. Obviously, if the decision maker is risk neutral, i.e., her preference is numerically represented by the expectation, then she is dependence neutral. We show the converse direction is also true: dependence neutrality and risk neutrality are indeed equivalent. Moreover, the decision maker may be averse to strong positive dependence representing extreme comovement (comovement aversion), and she may prefer less to more positive dependence (dependence monotonicity). Under a continuity assumption on the preference, we show that comovement aversion, dependence monotonicity, and strong risk aversion are all equivalent. Our results bridge the gap between dependence and risk attitudes, connecting two prominent concepts in statistics and decision theory.

Scoring rules are traditional techniques to measure the association between a reported belief and an observed outcome. The condition that a scoring rule is proper means that an agent maximizes his expected score when he reports a belief that equals his true belief. The implicit assumption that the agent is risk neutral is, however, often unrealistic, at least when the underlying agent is a human. Modern decision theories based on rank-dependent utilities, such as cumulative prospect theory, have been shown to be more effective at describing how human beings make decisions under risk and uncertainty. Traditional proper scoring rules are, however, incompatible with cumulative prospect theory because they fail to satisfy a property called comonotonicity. In this paper, we provide novel insights on why comonotonicity is crucial to make proper scoring rules indeed proper when eliciting beliefs from cumulative prospect theory agents. After suggesting strategies to create comonotonic proper scoring rules, we propose calibration procedures to obtain an agent’s true belief by removing the influence of the agent’s value function and weighting functions from his reported belief, when beliefs are elicited by means of comonotonic proper scoring rules.

We present a mixed-type multivariate distributional model generalizing the model of k-independent identically distributed random variables, and give necessary and sufficient conditions for a distribution of a random vector to be the joint distribution of the numbers of successes of q categories in n such dependent trials. The model includes, as particular cases, mixtures of multivariate extremal distributions and models corresponding to ordinal sums of independence and comonotonicity copulas. As an application, we study the extremal properties of order statistics based on random vectors with known multivariate marginals.

A variant of the bivariate Poisson common shock model is proposed which, contrary to the original, spans all possible degrees of dependence. Its basic distributional properties are described, moment-based estimation is studied, and its use is illustrated on real data.

We introduce a pricing framework for a guaranteed annuity option (GAO) where both the interest and mortality risks are correlated. We assume that the short rate and the force of mortality follow the Cox–Ingersoll–Ross (CIR) and Lee–Carter models, respectively. Employing the change of measure technique, we decompose the pure endowment into the product of the bond price and survival probability, thereby facilitating the evaluation of the annuity expression. With the aid of the dynamics of interest and mortality processes under the forward measure, we construct an algorithm based on comonotonicity theory to estimate the quantiles of survival probability and annuity rate. The comonotonic upper and lower bounds in the convex order are used to approximate the annuity and GAO prices and henceforth avoiding the simulation-within-simulation problem. Numerical illustrations show that our algorithm gives an efficient and practical method to estimate GAO values.

The aim of the present work is to provide a universal theoretical formulation of global methods for radiative heat transfer in non-uniform gaseous media. Starting from the definition of an arbitrary probability measure on the wavenumber axis, it is shown that no gas reference state is required to develop rigorously a full spectrum model, both in uniform and non-uniform media. This general formulation, which constitutes a novel mathematical modeling of gas radiation, is then applied for: (1) the theoretical justification of new developments introduced recently in the so-called Rank Correlated SLW method in non-uniform media, (2) emphasizing the differences and similarities between the SLW and FSK methods, from the point of view of the way these two techniques treat path non-uniformities. The theoretical results provided in the present work can also be used to enlighten the concept of “spectral correlation”, widely encountered in gas radiation modeling.

FrchetHoeffding upper and lower bounds play an important role in various bivariate optimization problems because they are the maximum and minimum of bivariate copulas in concordance order, respectively. However, while the FrchetHoeffding upper bound is the maximum of any multivariate copulas, there is no minimum copula available for dimensions d3. Therefore, multivariate minimization problems with respect to a copula are not straightforward as the corresponding maximization problems. When the minimum copula is absent, minimal copulas are useful for multivariate minimization problems. We illustrate the motivation of generalizing the joint mixability to d-countermonotonicity defined in Lee and Ahn (2014) through variance minimization problems and show that d-countermonotonic copulas are minimal copulas.

The insurer usually solicits the insured through granting a certain amount of deductible to multiple risks according to his/her own will. Due to the nonlinear nature of the concerned optimization problem, in the literature on the optimal allocations of deductibles researchers usually assume independence or comonotonicity among concerned risks and ignore the impact due to frequency. In this study we build two sufficient conditions for the decreasing optimal allocation of deductibles, relaxing the stochastic arrangement increasing or right tail weakly stochastic arrangement increasing discount factors in Cai and Wei (2014, Theorems 6.3 and 6.6) to the conditionally upper orthant arrangement increasing or weak conditionally upper orthant arrangement increasing frequencies.

The computation of various risk metrics is essential to the quantitative risk management of variable annuity guaranteed benefits. The current market practice of Monte Carlo simulation often requires intensive computations, which can be very costly for insurance companies to implement and take so much time that they cannot obtain information and take actions in a timely manner. In an attempt to find low-cost and efficient alternatives, we explore the techniques of comonotonic bounds to produce closed-form approximation of risk measures for variable annuity guaranteed benefits. The techniques are further developed in this paper to address in a systematic way risk measures for death benefits with the consideration of dynamic policyholder behavior, which involves very complex path-dependent structures. In several numerical examples, the method of comonotonic approximation is shown to run several thousand times faster than simulations with only minor compromise of accuracy.

The class of distortion riskmetrics is defined through signed Choquet integrals, and it includes many classic risk measures, deviation measures, and other functionals in the literature of finance and actuarial science. We obtain characterization, finiteness, convexity, and continuity results on general model spaces, extending various results in the existing literature on distortion risk measures and signed Choquet integrals. This paper offers a comprehensive toolkit of theoretical results on distortion riskmetrics which are ready for use in applications.

Several inequalities for the pan-integral are investigated. It is shown that the Chebyshev inequality holds for an arbitrary subadditive measure if and only if the integrands f, g are comonotone. Thus, we provide a new characterization for nonnegative comonotone functions. It is also shown that the Holder and Minkowski inequalities for the pan-integral hold if the monotone measure µ is subadditive. Since the pan-integral coincides with the concave integral when µ is subadditive, our results can also be applied to the concave integral.

Gini index is a well-known tool in economics that is often used for measuring income inequality. In insurance, the index and its modifications have been used to compare the riskiness of portfolios, to order reinsurance contracts, and to summarize insurance scores (relativities). In this paper, we establish several stochastic orders between the Gini indexes of multivariate elliptical risks with the same marginals but different dependence structures. This work is motivated by the applied studies of Brazauskas et al. (2007) and Samanthi et al. (2015), who employed the Gini index to compare the riskiness of insurance portfolios. Based on extensive Monte Carlo simulations, these authors have found that the power function of the associated hypothesis test increases as portfolios become more positively correlated. The comparison of the Gini indexes (of empirically estimated risk measures) presented in this paper provides a theoretical explanation to this statistical phenomenon. Moreover, it enriches the studies of the problem of central concentration of elliptical distributions and generalizes the pd-1 order proposed by Shaked and Tong (1985).

Comonotonicity has been successfully applied to derive various approximations in the single-factor mortality projection model proposed by Lee and Carter (1992), after Denuit and Dhaene (2007). However, this approach appears to lead to inaccurate approximations in the multi-factor mortality projection models developed by Cairns et?al. (2006). Therefore, we propose in this paper an alternative approach which consists in keeping the correlation structure within the time factors unchanged for every calendar year. Proceeding in this way, we provide accurate approximations useful for insurance practitioners avoiding them the simulations within simulations problem whatever the size of their portfolios.

Diversification of assets by an investor offers reduced exposure to risk compared to investing in a single asset. A multi-asset option gives an investor this advantage as its payout depends on the overall performance of several underlying assets. This study uses an information-based model to derive an approximate price for European call multi-asset options. The single asset price is derived using the risk-neutral pricing approach, and the multi-asset case uses the notion of comonotonicity. A numerical illustration is looked at to validate the theoretical results and to show the accuracy of the information-based model. The results show that prices from the information-based model provide a close fit to the empirical prices using a suitable information flow rate parameter. Hence, by making use of the information available in the market, an investor can price multi-asset European call options.

The use of copulas is proposed as a way of modelling dependencies between different agents' probability judgements when carrying out probability pooling. This is combined with an established Bayesian model in which pooling is viewed as a form of updating on the basis of probability values provided by different individuals. Adopting the Frank family of copulas we investigate the effect of different assumed levels of comonotonic dependence between individuals, in the context of a collective learning problem in which a population of agents must reach consensus on which of two mutually exclusive and exhaustive hypotheses is true. In this scenario agents receive evidence from two sources; directly from the environment and also from other agents in the form of probability judgements. They then apply Bayesian updating to the former and probability pooling to the latter. We carry out multi-agent simulation experiments and show that optimal population level performance is obtained under the assumption of some degree of comonotonicity between agents, and consequently show that the standard assumption of agent independence is suboptimal. This is found to be particularly true of scenarios where there is a large amount of noise and very low amounts of direct evidence. Finally, we investigate dynamic environments in which the true state of the world changes and show that identifying the optimal level of agent dependency has an even greater effect on performance than for static environments in which the true state remains constant.

Stochastic orders are mathematical methods allowing the comparison of random quantities. Probably the most usual one is stochastic dominance, which is based on the comparison of univariate cumulative distribution functions. Although it has been commonly applied, it does not consider the dependence between the random variables. This paper introduces a new stochastic order that slightly modifies stochastic dominance preserving its philosophy but taking into account the dependence between the random variables. This new stochastic order is based on the comparison of the cumulative distribution functions of the differences of the random variables, and it is closely related to regret theory. Since it uses the joint distribution, the copula gathering the dependence plays a crucial role. We present a theoretical study of this new stochastic order, delving into its connection with regret theory, investigating the role of the copula that links the random variables and establishing some connections with stochastic dominance.

For an arbitrage-free market with a single underlying asset, we investigate conditions under which the consecutive price levels are comonotonic. Furthermore, for an arbitrage-free market with n assets we investigate the consequences of assuming comonotonicity of the vector containing the price levels of each asset at a single future date T . Although being of a theoretical nature, the results of this paper give insight in the reachability of the comonotonic upper bounds for Asian options and for basket options that can be found in Simon et al. (2000), Dhaene et al. (2002) [1], Hobson et al. (2005) and Chen et al. (2008).

This article contains various results on a class of non-monotone law-invariant risk functionals, called the signed Choquet integrals. A functional characterization via comonotonic additivity is established, along with some theoretical properties including six equivalent conditions for a signed Choquet integral to be convex. We proceed to address two practical issues currently popular in risk management, namely, robustness (continuity) issues and risk aggregation with dependence uncertainty, for signed Choquet integrals. Our results generalize in several directions those in the literature of risk functionals. From the results obtained in this paper, we see that many profound and elegant mathematical results in the theory of risk measures hold for the general class of signed Choquet integrals; thus they do not rely on the assumption of monotonicity.

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. (2015). Unlike the latter, we do not impose the no sabotage condition on admissible indemnities, that is, the comonotonicity of indemnity functions and retention functions with 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. Hence, monotonicity properties of indemnification schedules become of second-order concern. We fully characterize the optimal indemnity schedule and discuss how our results relate to those of Bernard et al. (2015) and Xu et al. (2015). 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.

It has been a challenge to characterize the set of all possible sums of random variables with given marginal distributions, referred to as an aggregation set in this paper. We study the aggregation set via its connection to the corresponding lower-convex set, which is the set of all sums of random variables that are smaller than the respective marginal distributions in convex order. Theoretical properties of the two sets are discussed, assuming that all marginal distributions have finite mean. In particular, an aggregation set is always a subset of its corresponding lower-convex set, and the two sets are identical in the asymptotic sense after scaling. We also show that a lower-convex set is identical to the set of comonotonic sums with the same marginal constraint. The main theoretical results contribute to the field of multivariate distributions with fixed margins.

This paper extends a useful property of the increasing convex order to the multivariate orthant convex order. Specifically, it is shown that vectors of sums of comonotonic random variables dominate in the orthant convex order vectors of sums of random variables that are smaller in the increasing convex sense, whatever their dependence structure. This result is then used to derive orthant convex order bounds on random vectors of sums of random variables. Extensions to vectors of compound sums are also discussed.

In portfolios of life annuity contracts, the payments made by an annuity provider (an insurance company or a pension fund) are driven by the random number of survivors. This paper aims to provide accurate approximations for the present value of the payments made by the annuity provider. These approximations account not only for systematic longevity risk but also for the diversifiable fluctuations around the unknown life table. They provide the practitioner with a useful tool avoiding the problem of simulations within simulations in, for instance, Solvency 2 calculations, valid whatever the size of the portfolio.

Intuition based on the usual interpretation of the covariance of two random variables suggests that the inequality cov [ f ( X ) , g ( X ) ] ≥ 0 should hold for any random variable X and any two increasing functions f and g . The inequality holds indeed, but a proof is hard to find in the literature. In this paper we provide an elementary proof of a more general inequality for moments and we present several applications in actuarial mathematics.

Dependence structures of multiple risks play an important role in optimal allocation problems for insurance, quantitative risk management, and finance. However, in many existing studies on these problems, risks or losses are often assumed to be independent or comonotonic or exchangeable. In this paper, we propose several new notions of dependence to model dependent risks and give their characterizations through the probability measures or distributions of the risks or through the expectations of the transformed risks. These characterizations are related to the properties of arrangement increasing functions and the proposed notions of dependence incorporate many typical dependence structures studied in the literature for optimal allocation problems. We also develop the properties of these dependence structures. We illustrate the applications of these notions in the optimal allocation problems of deductibles and policy limits and in capital reserves problems. These applications extend many existing researches to more general dependent risks.

In this article, we show that some important implications concerning comonotonic couples and corresponding convex order relations for their sums cannot be translated to counter-monotonicity in general. In a financial context, it amounts to saying that merging counter-monotonic positions does not necessarily reduce the overall level of risk. We propose a simple necessary and sufficient condition for such a merge to be effective. Natural interpretations and various characterizations of this condition are given. As applications, we develop cancelation laws for convex order and identify desirable structural properties of insurance indemnities that make an insurance contract universally marketable, in the sense that it is appealing to both the policyholder and the insurer.

This short note revisits the classical Theorem of Borch on the characterization of Pareto optimal risk exchange treaties under the expected utility paradigm. Our objective is to approach the optimal risk exchange problem by a new method, which is based on a Breeden–Litzenberger type integral representation formula for increasing convex functions and the theory of comonotonicity. Our method allows us to derive Borch’s characterization without using Kuhn–Tucker theory, and also without the need of assuming that all utility functions are continuously differentiable everywhere. We demonstrate that our approach can be used effectively to solve the Pareto optimal risk-sharing problem with a positivity constraint being imposed on the admissible allocations when the aggregate risk is positive.

We consider the valuation of a guaranteed annuity option (GAO) under a generalised modelling setup where both interest and mortality risks are stochastic and correlated. Changes of probability measures are employed to obtain more implementable valuation formulae for mortality-linked contracts. Comonotonicity theory is applied to derive upper and lower bounds for the annuity rate in the convex order sense. These bounds provide accurate approximations for the value of GAOs. Numerical illustrations are included to show the accuracy and practical applicability of our comonotonic approximations for the GAO values.

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.

In this paper we derive expressions for the Tail Variance and the Tail Variance Premium of risks in a multivariate log-elliptical setting. The theoretical results are illustrated by considering lognormal and log-Laplace distributions. We also derive approximate expressions for a Tail Variance-based allocation rule in a multivariate lognormal setting. A numerical example illustrates the accurateness of the proposed approximations.

The overall comonotonicity has become popular in actuarial science and finance over the last decade. As a further step, the notion of upper comonotonicity has recently been proposed. Using the technique of distributional representation we provide a unified method to extend the notion of comonotonicity further to lower comonotonicity, tail comonotonicity, and interval comonotonicity respectively. Numerical illustrations are provided to make a comparison among these different types of dependence structures. The numerical results can be explained to some extent by the sum of uniform ( 0 , 1 ) random variables, for which we obtain explicit formulae for the probability density functions of the sum of two random variables in partial comonotonicity cases. For higher dimension cases, it becomes complicated to find the corresponding explicit formulae.

Probability statements about future evolutions of …nancial and actuarial risks are expressed in terms of the ‘real-world’ probability measure P, whereas in an arbitrage-free environment, the prices of these traded risks can be expressed in terms of an equivalent martingale measure Q. The assumption of independence between …nancial and actuarial risks in the real world may be quite reasonable in many situations. Making such an independence assumption in the pricing world however, may be convenient but hard to understand from an intuitive point of view. In this pedagogical paper, we investigate the conditions under which it is possible (or not) to transfer the independence assumption from P to Q. In particular, we show that an independence relation that is observed in the P-world can often not be maintained in the Q-world.

We consider the problem of determining the minimal requirement one must establish in order to meet a series of future random payments. It is shown in a very general setting that this problem can be recast as a chance constrained model and how the technique of Sample Average Approximation can be employed to find solutions. We also use comonotonic theory to analyze analytical approximations in a restricted Gaussian setting. Our numerical illustrations demonstrate that the Sample Average Approximation is a viable and efficient way to solve the stated problem generally and outperforms the analytical approximations. In passing we present a result that is related to Stein’s famous lemma ( Stein, 1981 ) and is of interest in itself.

We investigate properties of a version of tail comonotonicity that can be applied to absolutely continuous distributions, and give several methods for constructions of multivariate distributions with tail comonotonicity or strongest tail dependence. Archimedean copulas as mixtures of powers, and scale mixtures of a non-negative random vector with the mixing distribution having slowly varying tails, lead to a tail comonotonic dependence structure. For random variables that are in the maximum domain of attraction of either Frechet or Gumbel, we prove the asymptotic additivity property of Value at Risk and Conditional Tail Expectation.

Using a standard reduction argument based on conditional expectations, this paper argues that risk sharing is always beneficial (with respect to convex order or second degree stochastic dominance) provided the risk-averse agents share the total losses appropriately (whatever the distribution of the losses, their correlation structure and individual degrees of risk aversion). Specifically, all agents hand their individual losses over to a pool and each of them is liable for the conditional expectation of his own loss given the total loss of the pool. We call this risk sharing mechanism the conditional mean risk sharing. If all the conditional expectations involved are non-decreasing functions of the total loss then the conditional mean risk sharing is shown to be Pareto-optimal. Explicit expressions for the individual contributions to the pool are derived in some special cases of interest: independent and identically distributed losses, comonotonic losses, and mutually exclusive losses. In particular, conditions under which this payment rule leads to a comonotonic risk sharing are examined.

We introduce a new and easy-to-calculate measure for the expected degree of herd behavior or co-movement between stock prices. This forward looking measure is model-independent and based on observed option data. It is baptized the Herd Behavior Index (HIX). The degree of co-movement in a stock market can be determined by comparing the observed market situation with the extreme (theoretical) situation under which the whole system is driven by a single factor. The HIX is then defined as the ratio of an option-based estimate of the risk-neutral variance of the market index and an option-based estimate of the corresponding variance in case of the extreme single factor market situation. The HIX can be determined for any market index provided an appropriate series of vanilla options is traded on this index as well as on its components. As an illustration, we determine historical values of the 30-days HIX for the Dow Jones Industrial Average, covering the period January 2003 to October 2009.

We supply a Chebyshev type inequality for Choquet integral and link this inequality with comonotonicity.

International audience; In finite markets with short-selling, conditions on agents' utilities insuring the existence of efficient allocations and equilibria are by now well understood. In infinite markets, a standard assumption is to assume that the individually rational utility set is compact. Its drawback is that one does not know whether this assumption holds except for very few examples as strictly risk averse expected utility maximizers with same priors. The contribution of the paper is to show that existence holds for the class of strictly concave second order stochastic dominance preserving utilities. In our setting, it coincides with the class of strictly concave law-invariant utilities. A key tool of the analysis is the domination result of Lansberger and Meilijson that states that attention may be restricted to comonotone allocations of aggregate risk. Efficient allocations are characterized as the solutions of utility weighted problems with weights expressed in terms of the asymptotic slopes of the restrictions of agents' utilities to constants. The class of utilities which is used is shown to be stable under aggregation.

In this paper, we show a characterization of upper comonotonicity via tail convex order. For any given marginal distributions, a maximal random vector with respect to tail convex order is proved to be upper comonotonic under suitable conditions. As an application, we consider the computation of the Haezendonck risk measure of the sum of upper comonotonic random variables with exponential marginal distributions.

In this paper we discuss multiperiod portfolio selection problems related to a specific provisioning problem. Our results are an extension of Dhaene et al. (2005) [14], where optimal constant mix investment strategies are obtained in a provisioning and savings context, using an analytical approach based on the concept of comonotonicity. We derive convex bounds that can be used to estimate the provision to be set up at a specified time in future, to ensure that, after having paid all liabilities up to that moment, all liabilities from that moment on can be fulfilled, with a high probability.

echet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shu e of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums.

A life annuity contract is an insurance instrument which pays pre-scheduled living benefits conditional on the survival of the annuitant. In order to manage the risk borne by annuity providers, one needs to take into account all sources of uncertainty that affect the value of future obligations under the contract. In this paper, we define the concept of annuity rate as the conditional expected present value random variable of future payments of the annuity, given the future dynamics of its risk factors. The annuity rate deals with the non-diversifiable systematic risk contained in the life annuity contract, and it involves mortality risk as well as investment risk. While it is plausible to assume that there is no correlation between the two risks, each affects the annuity rate through a combination of dependent random variables. In order to understand the probabilistic profile of the annuity rate, we apply comonotonicity theory to approximate its quantile function. We also derive accurate upper and lower bounds for prediction intervals for annuity rates. We use the Lee–Carter model for mortality risk and the Vasicek model for the term structure of interest rates with an annually renewable fixed-income investment policy. Different investment strategies can be handled using this framework.

International audience; This paper deals with risk-sharing problems between many agents, each of whom having a strictly concave law invariant utility. In the special case where every agent's utility is given by a concave integral functional of the quantile of her individual endowment, we fully characterize the optimal risk-sharing rules. When there are many agents, these rules cannot be computed analytically. We therefore give a simple convergent algorithm and illustrate it on several examples.

When the dependence structure among several risks is unknown, it is common in the actuarial literature to study the worst dependence structure that gives rise to the riskiest aggregate loss. A central result is that the aggregate loss is the riskiest with respect to convex order when the underlying risks are comonotonic. Many proofs were given before. The objective of this article is to present a new proof using the notions of decreasing rearrangement and the majorization theorem, and give clear explanation of the relation between convex order, the theory of majorization and comonotonicity.