Interfaces with Other DisciplinesWeak comonotonicity
Introduction
Two functions are said to be comonotonic if the ups and downs of one function follows those of the other function. Hence, though geometric in nature, comonotonicity is also a kind of dependence notion between functions. It is not surprising, therefore, that comonotonicity has given rise to sufficient conditions when solving a variety of problems in economics, banking, and insurance, and in particular those that deal with portfolio diversification, risk aggregation, and premium calculation principles. Our search for necessary and sufficient conditions has revealed that a certain augmentation of the classical (and inherently point-wise) notion of comonotonicity with appropriately constructed measures achieves more advanced goals than those associated with sufficient conditions. As a by-product, the augmented notion of comonotonicity, which we call weak comonotonicity, provides a natural bridge between a host of concepts in the aforementioned areas of application, and also in statistics, including measures of association. In what follows, we methodically develop the notion of weak comonotonicity from first principles, establish its various properties, and demonstrate manifold uses.
Rigorously speaking, two functions g and h are comonotonic whenever the propertyholds for all . This notion of comonotonicity (Schmeidler, 1986) has played a pivotal role in sorting out numerous applications and developing new theories (e.g., Denneberg, 1994, Yaari, 1987). Since then, these advances have been in the mainstream of quantitative finance and economics literature (e.g., Dhaene, Denuit, Goovaerts, Vyncke, 2002a, Dhaene, Denuit, Goovaerts, Vyncke, 2002b, Föllmer, Schied, 2016). In this paper, we shall focus on dependence concepts between uni-dimensional functions (and random variables); for multivariate extensions and further references on comonotonicity, we refer to Puccetti and Scarsini (2010), Carlier, Dana, and Galichon (2012), Ekeland, Galichon, and Henry (2012), and Rüschendorf (2013). Note that if non-negativity in property (1.1) is replaced by non-positivity, the functions g and h are said to be antimonotonic.
Comonotonicity of (Borel) functions g and h is a sufficient condition for non-negativity of the covariance Cov[g(X), h(X)], where X is a random variable such that g(X) and h(X) have finite second moments. This is immediately seen from the equationswhere X′ is an independent copy of X, and FX denotes the cumulative distribution function (cdf) of X. The problem of determining the sign of covariances such as the one above has been of much interest in economics, insurance, banking, reliability engineering, and statistics. Several offshoots have arisen from this type of research, including quadrant dependence (Lehmann, 1966), measures of association (Esary, Proschan, & Walkup, 1967), monotonic (Kimeldorf & Sampson, 1978) and supremum (Gebelein, 1941) correlation coefficients. The following example illustrates the need for such results. Example 1.1 Let X be the severity of a risk, which could, for example, be a profit-and-loss variable. Let g(X) be the cost associated with the risk X, and let be the so-called (knowledge-based) weighted cdf of the original random variable X (e.g., Rao, 1997, and references therein). That is, is defined by the differential equationwhere h is a non-negative function such that . The role of the function h is to modify the probabilities of the original random variable X. For example, in insurance, it is usually designed to lower the left-hand tail of the pdf of X and to lift its right-hand tail, thus making large insurance risks/losses more noticeable and the premiums loaded; we refer to, e.g., Deprez and Gerber (1985) for the Esscher principle of insurance premium calculation, where for some constant t > 0. Under the weighted cdf the average cost is (Furman & Zitikis, 2009)which is not smaller than the average cost under the true cdf FX if and only if the covariance Cov[g(X), h(X)] is non-negative. Several natural questions arise in this context: Under what conditions on the cost function g and the probability weighting function h is the covariance non-negative? Should the functions really be comonotonic, as our earlier arguments would suggest? It is important to note at this point that practical and theoretical considerations may or may not support the latter assumption, due to the complexity of economic agents’ behaviour (e.g., Gillen, Markowitz, 2009, Markowitz, 1952, Pennings, Smidts, 2003).
We have organized the rest of the paper as follows. In Section 2, we define, illustrate, and discuss the notion of weak comonotonicity, first for Borel functions and then for random variables (i.e., generic measurable functions). In Section 3 we elucidate the role of weak comonotonicity in risk aggregation. In particular, we show that a combination of weak comonotonicity and weak antimonotonicity with respect to some sets of measures is sufficient for the maximization of Value-at-Risk (VaR) aggregation, and weak comonotonicity is necessary and sufficient for the Expected Shortfall (ES) aggregation. Both the VaR and the ES aggregation problems have been popular in the recent risk management literature (e.g., Embrechts, Wang, Wang, 2015, McNeil, Frey, Embrechts, 2015, Rüschendorf, 2013). In Section 4, we explore some properties of weak comonotonicity and its relation to other dependence structures and measures of association. As most of this paper deals with weak comonotonicity with respect to product measures, in Section 5 we illuminate the special role of these measures within the general context of joint measures. With the help of the developed theory, in Section 6 we present a detailed solution to a risk-sharing problem by invoking a weak comonotonicity constraint, whose naturalness becomes clear upon noticing that the assumption of arbitrary dependence among admissible allocations might sometimes be too weak, and the assumption of strong comonotonicity might be too strong, and so an intermediate dependence assumption based on weak comonotonicity arises most naturally. Section 7 concludes the paper with a brief overview of main contributions.
Section snippets
Weak comonotonicity
Our efforts to tackle problems like those in the previous section, and in particular those related to risk aggregation (Section 3), have naturally led us to a notion of weak comonotonicity (to be defined in a moment) which naturally bridges the arguments around quantities in (1.1) and (1.2) in the following way: First, note the equationwhere δx and are point masses at the points x and x′, respectively. It now becomes
Risk aggregation and weak comonotonicity
Two of the most popular classes of risk measures used in banking and insurance practice are the Value-at-Risk (VaR) and the Expected Shortfall (ES, also known as TVaR, CTE, CVaR, AVaR). We fix an atomless probability space . For a random variable X, the VaR at level p ∈ (0, 1) is defined asand the ES at level p ∈ (0, 1) is defined as
A classic problem in the field of risk management is risk aggregation with given marginal distributions
Some properties of weak comonotonicity
In this section, we explore some properties of weak comonotonicity, and its relation to notions of dependence structures and measures of association.
Maximality of product measures
Definition 2.2 is based on the product measure π1 × π2, which is a natural choice in view of the examples that have given rise to the notion of weak comonotonicity. There are, however, situations when the need for more generality arises, and for this we introduce an extension of integral (2.4):where, π is a measure on and for any random variable W on Definition 5.1 We say that random variables X and Y are weakly
An application to quantile-based risk sharing
In this section, we illustrate the above developed theory by studying an optimization problem arising in the context of risk sharing, where weak comonotonicity provides a natural constraint on the dependence structure of admissible risk allocations. We follow the framework of Embrechts, Liu, and Wang (2018) and Embrechts, Liu, Mao, and Wang (2019), who studied risk sharing problems with quantile-based risk measures.
Let be the set of all random variables in an atomless probability space. The
Summary and concluding notes
In this paper, we introduced the notion of weak comonotonicity. Via the analysis of several properties and applications, we show the encompassing nature of weak comonotonicity, which contains – as a special case – the classical notion of comonotonicity. The new notion serves a bridge that connects the classical notion of comonotonicity of random variables with a number of well-known notions of (in)dependence and association (e.g., Durante, Sempi, 2015, Joe, 2014). More importantly, we
Acknowledgements
The authors thank Editor Emanuele Borgonovo and four anonymous referees for various helpful comments on an early version of the paper. The authors have been supported by their individual research grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada (RGPIN-2018-03823, RGPAS-2018-522590, RGPIN-2016-427216), as well as by the National Research Organization “Mathematics of Information Technology and Complex Systems” (MITACS) of Canada.
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