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Ordering risk bounds in partially specified factor models
Abstract
For the class of partially specified risk factor models, general ordering results in dependence on the partial specifications are established. In particular, the Schur-order on copula derivatives and the sign-change order are introduced to derive lower orthant ordering conditions on the ✳-product of copulas and supermodular ordering conditions on the upper resp. lower product of bivariate copulas which describe the dependence structure of general specified resp. conditionally comonotonic resp. conditionally countermonotonic random vectors. In the case of internal factor models, it is shown that the standard bivariate dependence orders yield a strongly simplified ordering result for the risk bounds.
The characterization of the supermodular order for multivariate normal distributions is extended to the class of elliptically contoured distributions. As a consequence, for elliptical C-vine models, the upper bound in supermodular order is improved compared to the conditionally comonotonic case if the specifying partial correlations are bounded.
Applications to real market data show the considerable improvement of the standard risk bounds and, in the theory of cost-efficient claims, the dependence of the prices of constrained cost-efficient claims on the specifications.
... for some normalising constant α ∆ > 0 . As a matter of fact, various well-known measures can be represented in the form (1), including the portion of explained variance or Chatterjee's famous coefficient of rank correlation, see Section 2. Comparing conditional distributions is certainly not a new idea, see, e.g., [11,12,31] in the context of differential privacy. However, to the best of our knowledge, the (recently fast growing) literature on dependence measures (also known as measures of predictability, i.e., measures attaining values in [0,1] and being minimal/maximal exclusively for the case of independence/perfect dependence) mainly focuses on comparing conditional and unconditional distributions, so on expressions of the form ...
... where ∆ refers either to a L 2 -distance (see, e.g., [2,4,6,13,15,34,36,9]), the L 1 -distance (see, e.g., [18,22,37]), some optimal transport cost function (see, e.g., [29]), the maximum mean discrepancy (see, e.g., [20]), or the Wasserstein distance (see, e.g., [38]). On the one hand, for ∆ denoting the L 2 -distance between univariate distribution functions the quantities in (1) and (2) can be shown to coincide, see Example 1 in the next section. On the other hand, in general these expressions may differ, so considering functionals of the form Λ ∆ leads to novel measures of association and, in particular, to new dependence measures, which are the main focus of this contribution. ...
... The subsequent two examples illustrate the broadness of the Λ ∆ -approach according to (1) in the context of quantifying (directed) dependence as well as explainability in terms of the sensitivity of conditional distributions. They show that well-known statistical concepts can either be expressed in terms of (1) with adequately chosen ∆. ...
Recently established, directed dependence measures for pairs (X,Y) of random variables build upon the natural idea of comparing the conditional distributions of Y given X=x with the marginal distribution of Y. They assign pairs (X,Y) values in [0,1], the value is 0 if and only if X,Y are independent, and it is 1 exclusively for Y being a function of X. Here we show that comparing randomly drawn conditional distributions with each other instead or, equivalently, analyzing how sensitive the conditional distribution of Y given X=x is on x, opens the door to constructing novel families of dependence measures induced by general convex functions , containing, e.g., Chatterjee's coefficient of correlation as special case. After establishing additional useful properties of we focus on continuous (X,Y), translate to the copula setting, consider the -version and establish an estimator which is strongly consistent in full generality. A real data example and a simulation study illustrate the chosen approach and the performance of the estimator. Complementing the afore-mentioned results, we show how a slight modification of the construction underlying can be used to define new measures of explainability generalizing the fraction of explained variance.
... This motivates to define a version of the Schur order for conditional distributions considering derivatives of bivariate copulas as follows [3,6,56]. ...
... Consequently, these ordering properties explain the monotonicity of Kendall's tau, Spearman's rho and Chatterjee's xi for many copula (sub-)families in Figures 3-6. 3 For example, we know from Table 3 that the Clayton copulas ( ) [ ) ...
Motivated by recently investigated results on dependence measures and robust risk models, this article provides an overview of dependence properties of many well known bivariate copula families, where the focus is on the Schur order for conditional distributions, which has the fundamental property that minimal elements characterize independence and maximal elements characterize perfect directed dependence. We give conditions on copulas that imply the Schur ordering of the associated conditional distribution functions. For extreme-value copulas, we prove the equivalence of the lower orthant order, the Schur order for conditional distributions, and the pointwise order of the associated Pickands dependence functions. Furthermore, we provide several tables and figures that list and illustrate various positive dependence and monotonicity properties of copula families, in particular, from classes of Archimedean, extreme-value, and elliptical copulas. Finally, for Chatterjee’s rank correlation, which is consistent with the Schur order for conditional distributions, we give some new closed-form formulas in terms of the parameter of the underlying copula family.
... For the proof, this author extends the integral representation argument in Müller [22,Theorem 11] in the normal case. We remark that our paper is based on the dissertation of the first author from Apr 09, 2019, see Ansari [3,Theorem 5.2], where Theorem 1 is given in explicit form. ...
A classical result of Slepian (1962) for the normal distribution and extended by Das Guptas et al. (1972) for elliptical distributions gives one-sided (lower orthant) comparison criteria for the distributions with respect to the (generalized) correlations. Müller and Scarsini (2000) established that the ordering conditions even characterize the stronger supermodular ordering in the normal case. In the present paper, we extend this result to elliptical distributions. We also derive a similar comparison result for the directionally convex ordering of elliptical distributions. As application, we obtain several results on risk bounds in elliptical classes of risk models under restrictions on the correlations or on the partial correlations. Furthermore, we obtain extensions and strengthening of recent results on risk bounds for various classes of partially specified risk factor models with elliptical dependence structure of the individual risks and the common risk factor. The moderate dependence assumptions on this type of models allow flexible applications and, in consequence, are relevant for improved risk bounds in comparison to the marginal based standard bounds.
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.
Recent literature has investigated the risk aggregation of a portfolio under the sole assumption that the marginal distributions of the risks are specified, but not their dependence structure. There exists a range of possible values for any risk measure of , and the dependence uncertainty spread, as measured by the difference between the upper and the lower bound on these values, is typically very wide. Obtaining bounds that are more practically useful requires additional information on dependence. Here, we study a partially specified factor model in which each risk has a known joint distribution with the common risk factor Z, but we dispense with the conditional independence assumption that is typically made in fully specified factor models. We derive easy-to-compute bounds on risk measures such as Value-at-Risk () and law-invariant convex risk measures (e.g. Tail Value-at-Risk ()) and demonstrate their asymptotic sharpness. We show that the dependence uncertainty spread is typically reduced substantially and that, contrary to the case in which only marginal information is used, it is not necessarily larger for than for .
Motivated by the problem of sharp risk bounds in partially specified risk factor models and by the method of cost-efficient payoffs with given payoff structure we introduce and describe some stochastic odering problems for conditionally comonotonic resp. antimonotonic random variables. The aim is to describe the influence of the specified dependence of the components of the random vector X with a benchmark Z on the risk bounds in a risk portfolio resp. on the gain of cost efficiency of the optimal payoffs. We obtain in particular explicit results in dependence on distributional parameters for elliptical models in the case of risk bounds and for the multivariate Samuelson model in the case of cost efficient payoffs.
In this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence and uniqueness results for the Esscher parameter and determine cost efficient strategies in particular in the case of price processes driven by multivariate NIG- and VG-processes. From a monotonicity characterization of efficient payoffs we obtain that basket options are generally inefficient in Lévy markets when pricing is based on the Esscher measure.We determine efficient versions of the basket options in real market data and show that the proposed cost efficient strategies are also feasible from a numerical viewpoint. As a result we find that a considerable efficiency loss may arise when using the inefficient payoffs.
Dybvig (1988) introduced the interesting problem of how to construct in the cheapest possible way a terminal wealth with desired distribution. This idea has induced a series of papers concerning generality, consequences and applications. As the optimized claims typically follow the trend in the market, they are not useful for investors who wish to use them to protect an existing portfolio. For this reason, Bernard et al. (2013a) impose additional state-dependent constraints as a way of controlling the payoff structure. The present paper extends this work in various ways. In order to get optimal claims in general models we allow in this paper for extended contracts. We deal with general multivariate price processes and dismiss with several of the regularity assumptions in the previous work (in par-ticular, we omit any continuity assumption). State-dependence is modeled by requiring that terminal wealth has a fixed copula with a benchmark wealth. In this setting, we are able to characterize optimal claims. We apply the theo-retical results to deal with several hedging and expected utility maximization problems of interest.
Copulas are functions that join multivariate distribution functions to their one-dimensional margins. The study of copulas and their role in statistics is a new but vigorously growing field. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions.
With 116 examples, 54 figures, and 167 exercises, this book is suitable as a text or for self-study. The only prerequisite is an upper level undergraduate course in probability and mathematical statistics, although some familiarity with nonparametric statistics would be useful. Knowledge of measure-theoretic probability is not required. The revised second edition includes new sections on extreme value copulas, tail dependence, and quasi-copulas.
Roger B. Nelsen is Professor of Mathematics at Lewis & Clark College in Portland, Oregon. He is also the author of Proofs Without Words: Exercises in Visual Thinking and Proofs Without Words II: More Exercises in Visual Thinking, published by the Mathematical Association of America.
The evaluation of risks and risk bounds for joint portfolios is an important task in connection with the determination of risk capital as induced by regulatory prescriptions in finance and in insurance. It faces two basic problems. One is induced by the model risk arising from the use of specific but possibly incorrect models. On the other hand risk estimates based only on basic information as for example on the marginal (individual) risk distributions may be too wide to be usable in practise. In this paper we survey some recent endeavor to include partial dependence and structural information in order to obtain reliable and usable improved risk bounds.
We find pointwise best-possible bounds on the bivariate distribution function of continuous random variables with given margins and a given value of the population version of a nonparametric measure of association such as Kendall's tau or Spearman's rho.
