scholarly journals INF-Convolution of Risk Measures on Rearrangement Invariant Spaces

2021 ◽  
Author(s):  
Shengzhong Chen
Keyword(s):  
Risk Measures ◽  
Model Space ◽  
Continuity Condition ◽  
Individual Risk ◽  
Risk Allocation ◽  
Fatou Property ◽  
Economic Agents ◽  
Research Areas ◽  
Optimal Capital ◽  
Invariant Spaces

The problem of optimal capital and risk allocation among economic agents, has played a predominant role in the respective academic and industrial research areas for decades. Typically as risk occurs in face of randomness the risks which are to be measured are identified with real-valued random variables on some probability space (Ω, F, P). Consider a model space X , and n economic agents with initial endowments X1, · · · , Xn ∈ X who assess the riskiness of their positions by means of law-invariant convex risk measures ρi : X → (−∞,∞]. In order to minimize total and individual risk, the agents redistribute the aggregate endowment X = X1 + · · · + Xn among themselves. An optimal capital and risk allocation Y1, · · · , Yn satisfies Y1 + · · · + Yn = X and ρ1(Y1) + · · · + ρ(Yn) = inf nXn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Xn i=1 Xi = X o , (0.1) where n i=1ρi(X) = inf nPn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Pn i=1 Xi = X o is the inf-convolution of ρ1, ..., ρn. In 2008, Filipovi´c and Svindland proved that if X is an L p (P) for some 1 ≤ p ≤ ∞ and ρi satisfy a suitable continuity condition (i.e. Fatou property), then Problem (0.1) always admits a solution. To reflect the fact of randomness of risk, we should consider the model space X chosen for risk evaluations to be as general as possible. The main contribution of this thesis is Theorem 4.10 has been published in [9]. It extends Filipovi´c and Svindland’s result from L p spaces to general rearrangement invariant (r.i.) spaces.

scholarly journals INF-Convolution of Risk Measures on Rearrangement Invariant Spaces

2021 ◽  
Author(s):  
Shengzhong Chen
Keyword(s):  
Risk Measures ◽  
Model Space ◽  
Continuity Condition ◽  
Individual Risk ◽  
Risk Allocation ◽  
Fatou Property ◽  
Economic Agents ◽  
Research Areas ◽  
Optimal Capital ◽  
Invariant Spaces

The problem of optimal capital and risk allocation among economic agents, has played a predominant role in the respective academic and industrial research areas for decades. Typically as risk occurs in face of randomness the risks which are to be measured are identified with real-valued random variables on some probability space (Ω, F, P). Consider a model space X , and n economic agents with initial endowments X1, · · · , Xn ∈ X who assess the riskiness of their positions by means of law-invariant convex risk measures ρi : X → (−∞,∞]. In order to minimize total and individual risk, the agents redistribute the aggregate endowment X = X1 + · · · + Xn among themselves. An optimal capital and risk allocation Y1, · · · , Yn satisfies Y1 + · · · + Yn = X and ρ1(Y1) + · · · + ρ(Yn) = inf nXn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Xn i=1 Xi = X o , (0.1) where n i=1ρi(X) = inf nPn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Pn i=1 Xi = X o is the inf-convolution of ρ1, ..., ρn. In 2008, Filipovi´c and Svindland proved that if X is an L p (P) for some 1 ≤ p ≤ ∞ and ρi satisfy a suitable continuity condition (i.e. Fatou property), then Problem (0.1) always admits a solution. To reflect the fact of randomness of risk, we should consider the model space X chosen for risk evaluations to be as general as possible. The main contribution of this thesis is Theorem 4.10 has been published in [9]. It extends Filipovi´c and Svindland’s result from L p spaces to general rearrangement invariant (r.i.) spaces.

scholarly journals The strong Fatou property of risk measures

2018 ◽  
Vol 6 (1) ◽  
pp. 183-196 ◽  
Author(s):  
Shengzhong Chen ◽  
Niushan Gao ◽  
Foivos Xanthos
Keyword(s):  
Risk Measures ◽  
Lower Semicontinuous ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Fatou Property ◽  
Rearrangement Invariant Spaces ◽  
Dual Representations ◽  
Wide Range ◽  
Invariant Functional ◽  
Invariant Spaces

AbstractIn this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property,whichwas introduced in [19], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on a rearrangement invariant space X with the strong Fatou property is (X, L1) lower semicontinuous and that the converse is true on a wide range of rearrangement invariant spaces. We also study inf-convolutions of law-invariant or surplus-invariant risk measures that preserve the (strong) Fatou property.

scholarly journals Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

2018 ◽  
Vol 22 (2) ◽  
pp. 395-415 ◽  
Author(s):  
Niushan Gao ◽  
Denny Leung ◽  
Cosimo Munari ◽  
Foivos Xanthos
Keyword(s):  
Risk Measures ◽  
Orlicz Spaces ◽  
Fatou Property ◽  
Law Invariant Risk Measures

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 Fair estimation of capital risk allocation

2020 ◽  
Vol 37 (1-2) ◽  
pp. 1-24
Author(s):  
Tomasz R. Bielecki ◽  
Igor Cialenco ◽  
Marcin Pitera ◽  
Thorsten Schmidt
Keyword(s):  
Numerical Study ◽  
Risk Measures ◽  
Estimation Procedure ◽  
Capital Allocation ◽  
Expected Shortfall ◽  
Substantial Part ◽  
Risk Allocation ◽  
Fair Allocation ◽  
Risk Capital ◽  
Risk Capital Allocation

AbstractIn this paper, we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide explicit formulae for fair capital allocations in case when the constituents of the risky portfolio are jointly normally distributed. The main focus of the paper is on the problem of approximating fair portfolio allocations in the case of not fully known law of the portfolio constituents. We define and study the concepts of fair allocation estimators and asymptotically fair allocation estimators. A substantial part of our study is devoted to the problem of estimating fair risk allocations for expected shortfall. We study this problem under normality as well as in a nonparametric setup. We derive several estimators, and prove their fairness and/or asymptotic fairness. Last, but not least, we propose two backtesting methodologies that are oriented at assessing the performance of the allocation estimation procedure. The paper closes with a substantial numerical study of the subject and an application to market data.

scholarly journals Generalized Systematic Risk

2016 ◽  
Vol 8 (2) ◽  
pp. 86-127 ◽  
Author(s):  
Ohad Kadan ◽  
Fang Liu ◽  
Suying Liu
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Broad Class ◽  
Systematic Risk ◽  
Axiomatic Approach ◽  
Security Market ◽  
Risk Allocation ◽  
Multiple Dimensions ◽  
Rare Disasters ◽  
Distribution Moments

We generalize the concept of “systematic risk” to a broad class of risk measures potentially accounting for high distribution moments, downside risk, rare disasters, as well as other risk attributes. We offer two different approaches. First is an equilibrium framework generalizing the Capital Asset Pricing Model, two-fund separation, and the security market line. Second is an axiomatic approach resulting in a systematic risk measure as the unique solution to a risk allocation problem. Both approaches lead to similar results extending the traditional beta to capture multiple dimensions of risk. The results lend themselves naturally to empirical investigation. (JEL D81, G11, G12)

scholarly journals Risk Assessment Method Combining Independent Protection Layers (IPL) of Layer of Protection Analysis (LOPA) and RISKCURVES Software: Case Study of Hydrogen Refueling Stations in Urban Areas

Energies ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 4043
Author(s):  
Byoungjik Park ◽  
Yangkyun Kim ◽  
Kwanwoo Lee ◽  
Shinwon Paik ◽  
Chankyu Kang
Keyword(s):  
Risk Assessment ◽  
Urban Areas ◽  
Risk Mitigation ◽  
Risk Measures ◽  
Assessment Method ◽  
International Standards ◽  
Individual Risk ◽  
Safety Measures ◽  
Risk Assessment Method ◽  
The Impact

The commercialization of eco-friendly hydrogen vehicles has elicited attempts to expand hydrogen refueling stations in urban areas; however, safety measures to reduce the risk of jet fires have not been established. The RISKCURVES software was used to evaluate the individual and societal risks of hydrogen refueling stations in urban areas, and the F–N (Frequency–Number of fatalities) curve was used to compare whether the safety measures satisfied international standards. From the results of the analysis, it was found that there is a risk of explosion in the expansion of hydrogen refueling stations in urban areas, and safety measures should be considered. To lower the risk of hydrogen refueling stations, this study applied the passive and active independent protection layers (IPLs) of LOPA (Layer of Protection Analysis) and confirmed that these measures significantly reduced societal risk as well as individual risk and met international standards. In particular, such measures could effectively reduce the impact of jet fire in dispensers and tube trailers that had a high risk. Measures employing both IPL types were efficient in meeting international standard criteria; however, passive IPLs were found to have a greater risk reduction effect than active IPLs. The combination of RISKCURVES and LOPA is an appropriate risk assessment method that can reduce work time and mitigate risks through protective measures compared to existing risk assessment methods. This method can be applied to risk assessment and risk mitigation not only for hydrogen facilities, but also for hazardous materials with high fire or explosion risk.

Sign in / Sign up

Export Citation Format

Share Document