Regulation and risk management in banks depend on underlying risk measures. In general this is the only purpose that is seen for risk measures. In this paper we suggest that the reporting of risk measures can be used to determine the loss distribution function for a financial entity. We demonstrate that a lack of sufficient information can lead to ambiguous risk situations. We give examples, showing the need for the reporting of multiple risk measures in order to determine a bank's loss distribution. We conclude by suggesting a regulatory requirement of multiple risk measures being reported by banks, giving specific recommendations.
We axiomatically introduce risk-consistent conditional systemic risk measures defined on multidimensional risks. This class consists of those conditional systemic risk measures which can be decomposed into a state-wise conditional aggregation and a univariate conditional risk measure. Our studies extend known results for unconditional risk measures on finite state spaces. We argue in favor of a conditional framework on general probability spaces for assessing systemic risk. Mathematically, the problem reduces to selecting a realization of a random field with suitable properties. Moreover, our …
Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include set-valued ver…
To find a trade-off between profitability and prudence, financial practitioners need to choose appropriate risk measures. Two key points are: Firstly, investors' risk attitudes under uncertainty conditions should be an important reference for risk measures. Secondly, risk attitudes are not absolute. For different market performance, investors have different risk attitudes. We proposed a new risk measure named SlideVaR which sufficiently reflects the different subjective attitudes of investors and the impact of market changes on investors' attitudes. We proposed the concept of risk-tail region …
In the present contribution we characterize law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (2006), we show that these risk measures can be identified with a class of generalized shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (2014) and Bellini and Bignozzi (2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterization of robustness for convex risk measures in terms…
Risk measures connect probability theory or statistics to optimization, particularly to convex optimization. They are nowadays standard in applications of finance and in insurance involving risk aversion. This paper investigates a wide class of risk measures on Orlicz spaces. The characterizing function describes the decision maker's risk assessment towards increasing losses. We link the risk measures to a crucial formula developed by Rockafellar for the Average Value-at-Risk based on convex duality, which is fundamental in corresponding optimization problems. We characterize the dual and prov…
The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk measure. Examples of multivariate risk measures satisfying the supermartingale property are given. Crucial for obtaining the results are dual representations of scalarizations of set-valued dynamic risk measures, which are of independent interest in the fast growing literature on …
The inf-convolution of risk measures is directly related to risk sharing and general equilibrium, and it has attracted considerable attention in mathematical finance and insurance problems. However, the theory is restricted to finite sets of risk measures. This study extends the inf-convolution of risk measures in its convex-combination form to a countable (not necessarily finite) set of alternatives. The intuitive meaning of this approach is to represent a generalization of the current finite convex weights to the countable case. Subsequently, we extensively generalize known properties and re…
An Integrated Approach to Importance Sampling and Machine Learning for Efficient Monte Carlo Estimation of Distortion Risk Measures in Black Box Models
This survey gives an introduction to monetary measures of risk as monotone and cash additive functions on spaces of univariate random variables. Primal and dual representation results as well as several examples are discussed. Principal ways to construct risk measures are given and extensions to more general situations indicated.
The intuition of risk is based on two main concepts: loss and variability. In this paper, we present a composition of risk and deviation measures, which contemplate these two concepts. Based on the proposed Limitedness axiom, we prove that this resulting composition, based on properties of the two components, is a coherent risk measure. Similar results for the cases of convex and co-monotone risk measures are exposed. We also provide examples of known and new risk measures constructed under this framework in order to highlight the importance of our approach, especially the role of the Limitedn…
This paper introduces and fully characterizes the novel class of quasi-logconvex measures of risk, to stand on equal footing with the rich class of quasi-convex measures of risk. Quasi-logconvex risk measures naturally generalize logconvex return risk measures, just like quasi-convex risk measures generalize convex monetary risk measures. We establish their dual representation and analyze their taxonomy in a few (sub)classification results. Furthermore, we characterize quasi-logconvex risk measures in terms of properties of families of acceptance sets and provide their law-invariant representa…
The paper mentioned in the title introduces the entropic value at risk. I give some extra comments and using the general theory make a relation with some commonotone risk measures.
Distortion risk measures are extensively used in finance and insurance applications because of their appealing properties. We present three methods to construct new class of distortion functions and measures. The approach involves the composting methods, the mixing methods and the approach that based on the theory of copula. Subadditivity is an important property when aggregating risks in order to preserve the benefits of diversification. However, Value at risk (VaR), as the most well-known example of distortion risk measure is not always globally subadditive, except of elliptically distribute…
We propose a novel class of convex risk measures, based on the concept of the Fréchet mean, designed in order to handle uncertainty which arises from multiple information sources regarding the risk factors of interest. The proposed risk measures robustly characterize the exposure of the firm, by filtering out appropriately the partial information available in individual sources into an aggregate model for the risk factors of interest. Importantly, the proposed risks can be expressed in closed analytic forms allowing for interesting qualitative interpretations as well as comparative statics and…
We give an axiomatic framework for conditional generalized deviation measures. Under financially reasonable assumptions, we give the correspondence between conditional coherent risk measures and generalized deviation measures. Moreover, we establish the notion of continuous-time risk contribution for conditional coherent risk measures and generalized deviation measures. With the help of the correspondence between these two different types of risk measures, we give a microscopic interpretation of their risk contributions. Particularly, we show that the risk contributions of time-consistent risk…
In this paper, we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution ($Δ$CoD) measures as measures of systemic risk and analyze their properties and representations. The classes include the well-known conditional Value-at-Risk, conditional Expected Shortfall, and risk contribution measures in terms of the VaR and ES as special cases. Sufficient conditions are presented for two random vectors to be ordered by the proposed CoD-risk measures and distortion risk contribution measures. These conditions are expressed using the conventional stoch…
Trier University of Applied Sciences, Business School（特里尔应用科学大学商学院）；Technical University of Munich, School of Computation, Information and Technology（慕尼黑工业大学计算、信息与技术学院）
The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast with scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilize this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multiportfolio time consistency between set-valued risk measures for pro…
In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1…
A risk measure that is consistent with the second-order stochastic dominance and additive for sums of independent random variables can be represented as a weighted entropic risk measure (WERM). The expected utility maximization problem with risk controlled by WERM and a related risk minimization problem are investigated in this paper. The latter is same to a problem of maximizing a weighted average of constant-absolute-risk-aversion (CARA) certainty equivalents. The solutions of all the optimization problems are explicitly characterized and an iterative method of the solutions is provided.
We show how risk measures originally defined in a model free framework in terms of acceptance sets and reference assets imply a meaningful underlying probability structure. Hereafter we construct a maximal domain of definition of the risk measure respecting the underlying ambiguity profile. We particularly emphasise liquidity effects and discuss the correspondence between properties of the risk measure and the structure of this domain as well as subdifferentiability properties. Keywords: Model free risk assessment, extension of risk measures, continuity properties of risk measures, subgradient…
In this paper monetary risk measures that are positively superhomogeneous, called star-shaped risk measures, are characterized and their properties studied. The measures in this class, which arise when the controversial subadditivity property of coherent risk measures is dispensed with and positive homogeneity is weakened, include all practically used risk measures, in particular, both convex risk measures and Value-at-Risk. From a financial viewpoint, our relaxation of convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive …
This paper introduces a new type of risk measures, namely regime switching entropic risk measures, and study their applicability through simulations. The state of the economy is incorporated into the entropic risk formulation by using a Markov chain. Closed formulae of the risk measure are obtained for futures on crude oil derivatives. The applicability of these new types of risk measures is based on the study of the risk aversion parameter and the convenience yield. The numerical results show a term structure and a mean-reverting behavior of the convenience yield.
We present a general framework for measuring the liquidity risk. The theoretical framework defines a class of risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defined as the risk that a given security or a portfolio of securities cannot be easily sold or bought by the financial institutions without causing significant changes in prices. The new risk measures present some differences with respect to the standard risk measures. In particular, they are increasing monotonic and convex cash sub-…
In this paper, we study general monetary risk measures (without any convexity or weak convexity). A monetary (respectively, positively homogeneous) risk measure can be characterized as the lower envelope of a family of convex (respectively, coherent) risk measures. The proof does not depend on but easily leads to the classical representation theorems for convex and coherent risk measures. When the law-invariance and the SSD (second-order stochastic dominance)-consistency are involved, it is not the convexity (respectively, coherence) but the comonotonic convexity (respectively, comonotonic coh…
