Bounds for functions of multivariate risks

Abstract. Extending the approach of Jouini, Meddeb, and Touzi [Finance Stoch., 8 (2004) 1. Introduction. The concept of set-valued coherent measures of risk has been introduced recently by Jouini, Meddeb, and Touzi [16]. The basic question is how to evaluate the (financial) risk of a multivariate random outcome in terms of more than one reference instrument, for example if the regulator accepts deposits in more than one currency. This is of particular importance if transaction costs have to be paid for each transaction between assets including the reference instruments.This question became unexpectedly topical as the European Central Bank decreed [7] that temporarily, until the end of 2009, "the list of assets eligible as collateral in Eurosystem credit operations will be expanded" by "marketable debt instruments denominated in other currencies than the euro, namely the US dollar, the British pound and the Japanese yen, and issued in the euro area. These instruments will be subject to a uniform haircut add-on of 8%." See also [8].The following exposition is based on the model used in [16]; in particular, we assume the presence of proportional transaction costs modeled via a closed convex cone which goes back to [17]. A special feature of this model is that the risk of a d-dimensional random variable is evaluated in terms of m reference instruments with 1 ≤ m ≤ d. Usually m d will be true, but the case m = d is not excluded (and also, of course, neither m = 1, d > 1, nor m = d = 1).We shall make a few generalizations compared to [16]. First, we do not assume a "substitutability condition"; i.e., we do not assume that everything could be transferred into one distinguished currency. Second, we consider a general subspace M of R d as the collection

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“…In [5], two extensions of the scalar value at risk are defined. The "multivariate lower-orthant Value-at-Risk" V aR α is given by…”

Section: Is Translative Andmentioning

confidence: 99%

Duality for Set-Valued Measures of Risk

Hamel¹,

Heyde²

2010

SIAM J. Finan. Math.

124

142

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“…de Haan and Huang [11] model a risk-problem of flood in the bivariate setting using an estimator of level curves ∂L(c) of the bivariate distribution function. Furthermore, as noticed by Embrechts and Puccetti [19], ∂L(c) can be viewed as a natural multivariate version of the univariate quantile. The interested reader is also referred to Tibiletti [29], Belzunce et al [3], Nappo and Spizzichino [24].…”

Section: Introductionmentioning

confidence: 98%

Estimating covariate functions associated to multivariate risks: a level set approach

Bernardino

Laloë

Servien

2014

Metrika

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The aim of this paper is to study the behavior of a covariate function in a multivariate risks scenario. The first part of this paper deals with the problem of estimating the c-upper level sets L(c) = {F (x) ≥ c}, with c ∈ (0, 1), of an unknown distribution function F on R d + . A plug-in approach is followed. We state consistency results with respect to the volume of the symmetric difference. In the second part, we obtain the Lp-consistency, with a convergence rate, for the regression function estimate on these level sets L(c).We also consider a new multivariate risk measure: the Covariate-Conditional-Tail-Expectation. We provide a consistent estimator for this measure with a convergence rate. We propose a consistent estimate when the regression cannot be estimated on the whole data set. Then, we investigate the effects of scaling data on our consistency results. All these results are proven in a non-compact setting. A complete simulation study is detailed. Finally, a real environmental application of our risk measure is provided.

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“…To consider the risk of both the cedent and the reinsurer, we propose to use the bivariate lower orthant VaR introduced by [26], which is:…”

Section: Preliminariesmentioning

confidence: 99%

“…A natural starting point for measuring the (joint) risk of the cedent and the reinsurer is a bivariate risk measure, such as the bivariate VaR ( [26]) of the pair C f and R f . However, since the ceded loss f (X) and the retained loss I f (X) are comonotonic (see [27,28] for a very detailed discussion of the concept of comonotonicity with applications), the set of values of the bivariate VaRs of C f and R f is determined by values of the univariate VaR of C f and R f .…”

Section: Introductionmentioning

confidence: 99%

Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account

Jiang

Ren

Zitikis

2017

Risks

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Optimal forms of reinsurance policies have been studied for a long time in the actuarial literature. Most existing results are from the insurer's point of view, aiming at maximizing the expected utility or minimizing the risk of the insurer. However, as pointed out by Borch (1969), it is understandable that a reinsurance arrangement that might be very attractive to one party (e.g., the insurer) can be quite unacceptable to the other party (e.g., the reinsurer). In this paper, we follow this point of view and study forms of Pareto-optimal reinsurance policies whereby one party's risk, measured by its value-at-risk (VaR), cannot be reduced without increasing the VaR of the counter-party in the reinsurance transaction. We show that the Pareto-optimal policies can be determined by minimizing linear combinations of the VaRs of the two parties in the reinsurance transaction. Consequently, we succeed in deriving user-friendly, closed-form, optimal reinsurance policies and their parameter values.

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Bounds for functions of multivariate risks

Cited by 90 publications

References 14 publications

Duality for Set-Valued Measures of Risk

Duality for Set-Valued Measures of Risk

Estimating covariate functions associated to multivariate risks: a level set approach

Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account

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