scholarly journals Time consistency for scalar multivariate risk measures

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zachary Feinstein ◽  
Birgit Rudloff
Keyword(s):  
Transaction Costs ◽  
Systemic Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Time Consistency ◽  
Entire Family ◽  
Dual Representations ◽  
Multivariate Risk ◽  
Usual Scalar ◽  
Multivariate Risk Measures

Abstract 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 (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).

scholarly journals Scalar Multivariate Risk Measures with a Single Eligible Asset

2021 ◽  
Author(s):  
Zachary Feinstein ◽  
Birgit Rudloff
Keyword(s):  
Transaction Costs ◽  
Risk Measures ◽  
Risk Measure ◽  
Time Consistency ◽  
Capital Requirements ◽  
Market Model ◽  
Dual Representation ◽  
Pricing Process ◽  
Usual Scalar ◽  
Equivalent Martingale

In this paper, we present results on scalar risk measures in markets with transaction costs. Such risk measures are defined as the minimal capital requirements in the cash asset. First, some results are provided on the dual representation of such risk measures, with particular emphasis given on the space of dual variables as (equivalent) martingale measures and prices consistent with the market model. Then, these dual representations are used to obtain the main results of this paper on time consistency for scalar risk measures in markets with frictions. It is well known from the superhedging risk measure in markets with transaction costs that the usual scalar concept of time consistency is too strong and not satisfied. We will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process. We will prove the equivalence of this weaker notion of time consistency and a certain type of backward recursion with respect to the underlying risk measure with a fixed consistent pricing process. Several examples are given, with special emphasis on the superhedging risk measure.

CAPITAL ALLOCATION WITH MULTIVARIATE RISK MEASURES: AN AXIOMATIC APPROACH

2019 ◽  
Vol 34 (2) ◽  
pp. 297-315
Author(s):  
Linxiao Wei ◽  
Yijun Hu
Keyword(s):  
Standard Deviation ◽  
Portfolio Management ◽  
Risk Measures ◽  
Risk Measure ◽  
Axiomatic Approach ◽  
Capital Allocation ◽  
Axiom System ◽  
Multivariate Risk ◽  
Multivariate Risk Measures ◽  
Multivariate Risk Measure

AbstractCapital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

scholarly journals Multivariate risk measures in the non-convex setting

2019 ◽  
Vol 36 (1-4) ◽  
pp. 25-35
Author(s):  
Andreas Haier ◽  
Ilya Molchanov
Keyword(s):  
Finite Number ◽  
Transaction Costs ◽  
Risk Measures ◽  
Proportional Transaction Costs ◽  
Closed Set ◽  
Multivariate Risk ◽  
The Family ◽  
Random Closed Set ◽  
Multivariate Risk Measures ◽  
Convex Setting

Abstract The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g., in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions based on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex portfolios are presented.

A Theory for Measures of Tail Risk

2021 ◽  
Author(s):  
Fangda Liu ◽  
Ruodu Wang
Keyword(s):  
Value At Risk ◽  
Financial Regulation ◽  
Risk Measures ◽  
Risk Measure ◽  
Continuity Condition ◽  
Tail Risk ◽  
Dual Representations ◽  
Joint Properties ◽  
Risk Aggregation ◽  
Popular Classes

The notion of “tail risk” has been a crucial consideration in modern risk management and financial regulation, as very well documented in the recent regulatory documents. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures that quantify the tail risk, that is, the behaviour of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, value at risk (VaR) and expected shortfall, are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk measures, risk aggregation, elicitability, and dual representations. In particular, there is no elicitable tail convex risk measure other than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.

scholarly journals VECTOR-VALUED COHERENT RISK MEASURE PROCESSES

2014 ◽  
Vol 17 (02) ◽  
pp. 1450011 ◽  
Author(s):  
IMEN BEN TAHAR ◽  
EMMANUEL LÉPINETTE
Keyword(s):  
Continuous Time ◽  
Risk Measures ◽  
Risk Measure ◽  
Time Consistency ◽  
Axiomatic Characterization ◽  
Coherent Risk Measure ◽  
Coherent Risk ◽  
Dynamic Version ◽  
Vector Valued

Introduced by Artzner et al. (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini et al. (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results, we study different notions of time consistency and we give examples of vector-valued risk measure processes.

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