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Two acts are comonotonic if they co-vary in the same direction. The main purpose of this paper is to derive a new characterization of Cumulative Prospect Theory (CPT) through simple properties involving comonotonicity. The main novelty is a concept dubbed gain–loss hedging: mixing positive and negative acts creates hedging possibilities even when acts are comonotonic. This allows us to clarify in which sense CPT differs from Choquet expected utility. Our analysis is performed under the assumption that acts are real-valued functions. This entails a simple (piece-wise) constant marginal utility representation of CPT, which allows us to clearly separate the perception of uncertainty from the evaluation of outcomes. • Cumulative Prospect Theory (CPT) is a prominent behavioral model in decision-making. • We mathematically characterize the CPT formula. • We propose a novel preference axiomatization of CPT. • We highlight the central role played by comonotonicity. • We introduce a new property termed gain–loss hedging. [ABSTRACT FROM AUTHOR]

Wüthrich (Eur Actuar J, https://doi.org/10.1007/s13385-022-00339-9, 2023) established that the Gini index is a consistent scoring rule in the class of autocalibrated predictors. This note further explores performances criteria in this class. Elementary Pearson's correlation turns out to be consistent when restricted to autocalibrated predictors. Also, any performance measure that is minimized for predictors that are comonotonic with the true regression model is consistent under autocalibration. This provides a new proof of the consistency for Gini index. In addition, it is established that the concentration curve of the true model is the lowest possible concentration curve under autocalibration and that the same property holds true for Lorenz curve. [ABSTRACT FROM AUTHOR]

The alternating direction method of multipliers (ADMM) is a powerful splitting algorithm for linearly constrained convex optimization problems. In view of its popularity and applicability, a growing attention is drawn toward the ADMM in nonconvex settings. Recent studies of minimization problems for nonconvex functions include various combinations of assumptions on the objective function including, in particular, a Lipschitz gradient assumption. We consider the case where the objective is the sum of a strongly convex function and a weakly convex function. To this end, we present and study an adaptive version of the ADMM which incorporates generalized notions of convexity and penalty parameters adapted to the convexity constants of the functions. We prove convergence of the scheme under natural assumptions. To this end, we employ the recent adaptive Douglas–Rachford algorithm by revisiting the well-known duality relation between the classical ADMM and the Douglas–Rachford splitting algorithm, generalizing this connection to our setting. We illustrate our approach by relating and comparing to alternatives, and by numerical experiments on a signal denoising problem. [ABSTRACT FROM AUTHOR]

This paper offers a systematic treatment of risk‐sharing rules for insurance losses, based on a list of relevant properties. A number of candidate risk‐sharing rules are considered, including the conditional mean risk‐sharing rule proposed in Denuit and Dhaene and the newly introduced quantile risk‐sharing rule. Their compliance with the proposed properties is established. Then, methods for building new risk‐sharing rules are discussed. The results derived in this paper are helpful in the development of peer‐to‐peer insurance (or crowdsurance), as well as to manage contingent risk funds where a given budget is distributed among claimants. [ABSTRACT FROM AUTHOR]

The properties of fuzzy relations have been extensively studied, and the preservation of their properties plays a fundamental role in the various applications. However, either sufficient or necessity conditions for the preservation requires the aggregated functions of fuzzy relations to dominate or to be dominated by the corresponding operations, which constructs a significant limitation on applicable functions. This work concentrates on the preservation of transitivities and Ferrers property for the aggregation of comonotone or countermonotone fuzzy relations. Firstly, definitions of comonotonicity and countermonotonicity for binary functions are initially proposed. On the foundation of that, the relations of commuting and bisymmetry between min/max and commonly used increasing/decreasing functions are found. Afterwards, with the condition that underlying fuzzy relations are pair-wisely comonotone or countermonotone, theorems on the aggregation functions which can preserve the transitivities and the Ferrers property are proposed. Moreover, an interesting conclusion that the equivalent condition for the min-Ferrers property of fuzzy relations is clarified. [ABSTRACT FROM AUTHOR]

The properties of fuzzy relations have been extensively studied, and the preservation of their properties plays a fundamental role in the various applications. However, either sufficient or necessity conditions for the preservation requires the aggregated functions of fuzzy relations to dominate or to be dominated by the corresponding operations, which constructs a significant limitation on applicable functions. This work concentrates on the preservation of transitivities and Ferrers property for the aggregation of comonotone or countermonotone fuzzy relations. Firstly, definitions of comonotonicity and countermonotonicity for binary functions are initially proposed. On the foundation of that, the relations of commuting and bisymmetry between min/max and commonly used increasing/decreasing functions are found. Afterwards, with the condition that underlying fuzzy relations are pair-wisely comonotone or countermonotone, theorems on the aggregation functions which can preserve the transitivities and the Ferrers property are proposed. Moreover, an interesting conclusion that the equivalent condition for the min-Ferrers property of fuzzy relations is clarified.

Discussions on low volatility effects often highlight the advantage of low volatility stocks outperforming high volatility stocks. Using comonotonicity tests, our study provides evidence of the downside of this effect: stock returns do not increase monotonically with low volatility, but volatility increases monotonically with specific risks. We find that, counterintuitively, the low volatility effect is mostly driven by high volatility stocks with high specific risks. Our empirical analysis addresses a cross section of stock returns across 23 developed countries and employs comonotonicity tests to show that expected stock returns do not increase monotonically with lower volatility. In addition, by decomposing volatility into its individual risk components, we show that volatility increases monotonically with its specific risk component. Finally, we also confirm that returns obtained from the low volatility effect in stocks are principally driven by the specific risk component rather than the systematic risk component. [ABSTRACT FROM AUTHOR]

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.

We review recent research results on stochastic arrangement increasing risks in financialand actuarial risk management, including allocation of deductibles and coverage limits concerned withmultiple dependent risks in an insurance policy, the independence model and the threshold models fora portfolio of defaults risks with dependence, and the optimal capital allocation for a financial institutewith multiple line of business.

This paper studies a peer-to-peer (P2P) insurance scheme where participants share the first layer of their respective losses while the higher layer is transferred to a (re-)insurer. The conditional mean risk sharing rule proposed by Denuit and Dhaene (Insur Math Econ 51:265–270, 2012) appears to be a very convenient way to distribute retained losses among participants, as shown by Denuit (ASTIN Bull 49:591–617, 2019). The amount of contributions paid by participants is determined by splitting it into the price of the stop-loss protection limiting the community's total payout and an appropriate provision for the coverage of the lower layer which is mutualized inside the P2P community. As an application, the paper considers the case of a P2P insurance scheme when losses are modeled by independent compound Poisson sums with integer-valued severities (resulting from discretization). Some extensions are also discussed. [ABSTRACT FROM AUTHOR]

This article contains various results on a class of nonmonotone, 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. [ABSTRACT FROM AUTHOR]

For the class of (partially specified) internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models. This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor. The proof of our main comparison result is not standard. It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers. An application to real market data shows considerable improvement over the standard method. [ABSTRACT FROM AUTHOR]

We study optimal risk sharing among n agents endowed with distortion risk measures. Our model includes market frictions that can either represent linear transaction costs or risk premia charged by a clearing house for the agents. Risk sharing under third-party constraints is also considered. We obtain an explicit formula for Pareto optimal allocations. In particular, we find that a stop-loss or deductible risk sharing is optimal in the case of two agents and several common distortion functions. This extends recent result of Jouini et al. (Adv Math Econ 9:49–72, 2006) to the problem with unbounded risks and market frictions.

Herd behavior is an important economic phenomenon, especially in the context of the recent financial crises. Prior studies propose several measures to quantify herd behavior. In this paper, we show that these measures reflect different perspectives on this behavior, and hence, their interpretation requires great care. Taking a critical attitude toward existing herd behavior measures, we study their properties and pitfalls in detail.

The so-called risk diversification principle is analyzed, showing that its convenience depends on individual characteristics of the risks involved and the dependence relationship among them. || Se analiza el principio de diversificación de riesgos y se demuestra que no siempre resulta mejor que no diversificar, pues esto depende de características individuales de los riesgos involucrados, así como de la relación de dependencia entre los mismos.

We study the principal component analysis based approach introduced by Reisinger and Wittum (2007) and the comonotonic approach considered by Hanbali and Linders (2019) for the approximation of American basket option values via multidimensional partial differential complementarity problems (PDCPs). Both approximation approaches require the solution of just a limited number of low-dimensional PDCPs. It is demonstrated by ample numerical experiments that they define approximations that lie close to each other. Next, an efficient discretisation of the pertinent PDCPs is presented that leads to a favourable convergence behaviour.

In this article, we introduce three new classes of multivariate risk statistics, which can be considered as data-based versions of multivariate risk measures. These new classes are multivariate convex risk statistics, multivariate comonotonic convex risk statistics and multivariate empirical-law-invariant convex risk statistics, respectively. Representation results are provided. The arguments of proofs are mainly developed by ourselves. It turns out that all the relevant existing results in the literature are special cases of those obtained in this article. [ABSTRACT FROM AUTHOR]

We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent in basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the European-type comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte-Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method. [ABSTRACT FROM AUTHOR]

Many copula families, including the classes of Archimedean, elliptical and Liouville copulas, may be written as the survival copula of a random vector R × (Y 1 , Y 2) , where R is a strictly positive random variable independent of the random vector (Y 1 , Y 2). A unified framework is presented for studying the dependence structure underlying this stochastic representation, which is called the background risk model. Formulas for the copula, Kendall's tau and tail dependence coefficients are obtained and special cases are detailed. The usefulness of the construction for model building is illustrated with an extension of Archimedean copulas with completely monotone generators, based on the Farlie–Gumbel–Morgenstern copula. In particular, explicit expressions for the distribution and the Tail-Value-at-Risk of the aggregated risk R Y 1 + R Y 2 are available in a generalization of the widely used multivariate Pareto-II model. [ABSTRACT FROM AUTHOR]

In this work, we propose a definition of comonotonicity for elements of B (H) s a , i.e. bounded self-adjoint operators defined over a complex Hilbert space H. We show that this notion of comonotonicity coincides with a form of commutativity. Intuitively, comonotonicity is to commutativity as monotonicity is to bounded variation. We also define a notion of Choquet expectation for elements of B (H) s a that generalizes quantum expectations. We characterize Choquet expectations as the real-valued functionals over B (H) s a which are comonotonic additive, c -monotone, and normalized. [ABSTRACT FROM AUTHOR]

Following previous theoretical development based on the assumption of a rank correlated spectrum, the Rank Correlated Full Spectrum k -distribution (RC-FSK) method is proposed. The method proves advantageous in modeling radiation transfer in high temperature gases in non-uniform media in two important ways. First, and perhaps most importantly, the method requires no specification of a reference gas thermodynamic state. Second, the spectral construction of the RC-FSK model is simpler than original correlated FSK models, requiring only two cumulative k -distributions. Further, although not exhaustive, example problems presented here suggest that the method may also yield improved accuracy relative to prior methods, and may exhibit less sensitivity to the blackbody source temperature used in the model predictions. This paper outlines the theoretical development of the RC-FSK method, comparing the spectral construction with prior correlated spectrum FSK method formulations. Further the RC-FSK model's relationship to the Rank Correlated Spectral Line Weighted-sum-of-gray-gases (RC-SLW) model is defined. The work presents predictions using the Rank Correlated FSK method and previous FSK methods in three different example problems. Line-by-line benchmark predictions are used to assess the accuracy. [ABSTRACT FROM AUTHOR]

We revisit the general problem of minimizing a separable convex function with both a budget constraint and a set of box constraints. This optimization problem arises naturally in many resource allocation problems in engineering, economics, finance and insurance. Existing literature tackles this problem by using the traditional Kuhn–Tucker theory, which leads to either iterative schemes or yields explicit solutions only under some special classes of convex functions owe to the presence of box constraints. This paper presents a novel approach of solving this constrained minimization problem by using the theory of comonotonicity. The key step is to apply an integral representation result to express each convex function as the stop-loss transform of some suitable random variable. By using this approach, we can derive and characterize not only the explicit solution, but also obtain its geometric meaning and some other qualitative properties. [ABSTRACT FROM AUTHOR]

We characterize the set W of possible joint laws of terminal values of a nonnegative submartingale X of class (D), starting at 0, and the predictable increasing process (compensator) from its Doob-Meyer decomposition. The set of possible values remains the same under certain additional constraints on X, for example, under the condition that X is an increasing process or a squared martingale. Special attention is paid to extremal (in a certain sense) elements of the set W and to the corresponding processes. We relate also our results with Rogers's results on the characterization of possible joint values of a martingale and its maximum. [ABSTRACT FROM AUTHOR]

Measuring market fear is an important way of understanding fundamental economic phenomena related to financial crises. There have been several approaches to measure market fear or panic level in a financial market. Recently, herd behavior has gained its popularity as important economic phenomena explaining the fear in the financial market. In this paper, we investigate herd behavior in global stock markets with a focus on intercontinental comparison. While various risk measures are available for the detection of herd behavior in the market, we use the standardized herd behavior index in Dhaene et al. (Insurance: Mathematics and Economics, 50, 357-370, 2012b) and Lee and Ahn (Dependence Modeling, 5, 316-329, 2017) for the comparison of herd behaviors in global stock markets. A global stock market data from Morgan Stanley Capital International is used to study herd behavior especially during periods of financial crises. [ABSTRACT FROM AUTHOR]

In this note we establish some appropriate conditions for stochastic equality of two random variables/ vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result are also considered. [ABSTRACT FROM AUTHOR]

We study the worst portfolios for a class of law invariant dynamic monetary utility functions with domain in a class of stochastic processes. The concept of comonotonicity is introduced for these processes in order to prove the existence of worst portfolios. Using robust representations of monetary utility function processes in discrete time, a relation between the worst portfolios at different periods of time is presented. Finally, we study conditions to achieve the maximum in the representation theorems for concave monetary utility functions that are continuous for bounded decreasing sequences. [ABSTRACT FROM AUTHOR]

Two variables are called comonotone when there is an increasing relation between them, in the sense that when one of them increases (decreases), so does the other one. This notion has been widely investigated in probability theory, and is related to copulas. This contribution studies how the notion of comonotonicity can be extended to an imprecise setting on discrete spaces, where probabilities are only known to belong to a convex set. We define comonotonicity for such sets and investigate its characterizations in terms of lower probabilities, as well as its connection with copulas. As this theoretical characterization can be tricky to apply to general lower probabilities, we also investigate specific models of practical importance. In particular, we provide some sufficient conditions for a comonotone belief function with fixed marginals to exist, and characterize comonotone bivariate p-boxes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Copulas are one of the most powerful tools in modeling dependence structure of multivariate variables. In Tran et al. (Integrated uncertainty in knowledge modelling and decision making. Springer, Berlin, pp 126-137, 2015), we have constructed a new measure of dependence, $$\lambda (C),$$ based on Sobolev norm for copula C which can be used to characterize comonotonicity, countermonotonicity and independence of random vectors. This paper aims to use the measure $$ \lambda (C) $$ to study how dependence structure of a distorted copula after being transformed by a distortion function is changed. Firstly, we propose two methods to estimate the measure $$\lambda (C)$$ , one for known copula C using conditional copula-based Monte Carlo simulation and the latter for unknown copula dealing with empirical data. Thereafter, PH-transform $$g_{ PH }$$ of extreme value copulas and Wang's transform $$ g_\gamma $$ of normal and product copula are studied, and we observe their dependence behaviors changing through variability of the measure $$ \lambda (C) $$ . Our results show that dependence structure of distorted copulas is subject to comonotonicity as increasing the parametric $$ \gamma $$ . [ABSTRACT FROM AUTHOR]

Fréchet–Hoeffding 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 Fréchet–Hoeffding upper bound is the maximum of any multivariate copulas, there is no minimum copula available for dimensions d ≥ 3 . 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. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for a maxitive function to be the upper probability of a bivariatep-box, in terms of its associated possibility distribution and its focal sets. This allows us to derive conditions in terms of the lower and upper distribution functions of the bivariatep-box. In particular, we prove that only bivariatep-boxes with a non-informative lower or upper distribution function may induce a maxitive function. In addition, we also investigate the extension of Sklar’s theorem to this context. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]