IMPRECISE PREVISIONS FOR RISK MEASUREMENT

2003 ◽  
Vol 11 (04) ◽  
pp. 393-412 ◽  
Author(s):  
RENATO PELESSONI ◽  
PAOLO VICIG
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Risk Measurement ◽  
Random Numbers ◽  
Coherent Risk Measure ◽  
Coherent Risk ◽  
Consistency Properties ◽  
Special Case

In this paper the theory of coherent imprecise previsions is applied to risk measurement. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. Consistency properties of Value-at-Risk (VaR), currently one of the most used risk measures, are investigated too, showing that it does not necessarily satisfy a weaker notion of consistency called 'avoiding sure loss'. We introduce sufficient conditions for VaR to avoid sure loss and to be coherent. Finally we discuss ways of modifying incoherent risk measures into coherent ones.

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.

COHERENT RISK MEASURES FOR DERIVATIVES UNDER BLACK–SCHOLES ECONOMY

2001 ◽  
Vol 04 (05) ◽  
pp. 819-835 ◽  
Author(s):  
H. YANG ◽  
T. K. SIU
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Fixed Time ◽  
Derivative Securities ◽  
Coherent Risk Measure ◽  
Subjective Probabilities ◽  
Coherent Risk ◽  
Subjective Beliefs ◽  
Black Scholes ◽  
Risk Managers

This paper proposes a risk measure for a portfolio of European-style derivative securities over a fixed time horizon under the Black–Scholes economy. The proposed risk measure is scenario-based along the same line as [3]. The risk measure is constructed by using the risk-neutral probability ([Formula: see text]-measure), the physical probability ([Formula: see text]-measure) and a family of subjective probability measures. The subjective probabilities are introduced by using Girsanov's theorem. In this way, we provide risk managers or regulators with the flexibility of adjusting the risk measure according to their risk preferences and subjective beliefs. The advantages of the proposed measure are that it is easy to implement and that it satisfies the four desirable properties introduced in [3], which make it a coherent risk measure. Finally, we incorporate the presence of transaction costs into our framework.

scholarly journals Spatial Risk Measures and Rate of Spatial Diversification

Risks ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 52 ◽  
Author(s):  
Erwan Koch
Keyword(s):  
Random Fields ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Value Theory ◽  
Spatial Homogeneity ◽  
Spatial Risk ◽  
Spatial Diversification

An accurate assessment of the risk of extreme environmental events is of great importance for populations, authorities and the banking/insurance/reinsurance industry. Koch (2017) introduced a notion of spatial risk measure and a corresponding set of axioms which are well suited to analyze the risk due to events having a spatial extent, precisely such as environmental phenomena. The axiom of asymptotic spatial homogeneity is of particular interest since it allows one to quantify the rate of spatial diversification when the region under consideration becomes large. In this paper, we first investigate the general concepts of spatial risk measures and corresponding axioms further and thoroughly explain the usefulness of this theory for both actuarial science and practice. Second, in the case of a general cost field, we give sufficient conditions such that spatial risk measures associated with expectation, variance, value-at-risk as well as expected shortfall and induced by this cost field satisfy the axioms of asymptotic spatial homogeneity of order 0, −2, −1 and −1, respectively. Last but not least, in the case where the cost field is a function of a max-stable random field, we provide conditions on both the function and the max-stable field ensuring the latter properties. Max-stable random fields are relevant when assessing the risk of extreme events since they appear as a natural extension of multivariate extreme-value theory to the level of random fields. Overall, this paper improves our understanding of spatial risk measures as well as of their properties with respect to the space variable and generalizes many results obtained in Koch (2017).

WEIGHTED COMONOTONIC RISK SHARING UNDER HETEROGENEOUS BELIEFS

2020 ◽  
Vol 50 (2) ◽  
pp. 647-673
Author(s):  
Haiyan Liu
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Sharing ◽  
Optimal Allocation ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Heterogeneous Beliefs ◽  
Optimal Allocations ◽  
Necessary And Sufficient

AbstractWe study a weighted comonotonic risk-sharing problem among multiple agents with distortion risk measures under heterogeneous beliefs. The explicit forms of optimal allocations are obtained, which are Pareto-optimal. A necessary and sufficient condition is given to ensure the uniqueness of the optimal allocation, and sufficient conditions are given to obtain an optimal allocation of the form of excess of loss or full insurance. The optimal allocation may satisfy individual rationality depending on the choice of the weight. When the distortion risk measure is value at risk or tail value at risk, an optimal allocation is generally of the excess-of-loss form. The numerical examples suggest that a risk is more likely to be shared among agents with heterogeneous beliefs, and the introduction of the weight enables us to prioritize some agents as part of a group sharing a risk.

scholarly journals The theory and application of spectral risk measures in Vietnam

2017 ◽  
Vol 24 (04) ◽  
pp. 29-45
Author(s):  
Hai Ho Hong ◽  
Hoa Nguyen Thi
Keyword(s):  
Risk Aversion ◽  
Portfolio Management ◽  
Value At Risk ◽  
Utility Theory ◽  
Risk Measures ◽  
Risk Measure ◽  
Coherent Risk ◽  
Spectral Risk Measures ◽  
Spectral Risk Measure ◽  
Current Risk

This paper aims to provide a new risk measure for portfolio management in Vietnam by incorporating investor’s risk aversion into current risk measures such as value at risk (VaR) and expected shortfall (ES). This measure shares several desirable characteristics with the coherent risk measures, as illustrated in Artzner et al. (1997). In Vietnam, our study makes the first attempt to utilize distortion theory, instead of utility theory, to facilitate the adoption of risk aversion level in the popular risk measures. We find that spectral risk measure is more flexible and effective to different groups of risk-adverse investors, compared to the more monotonic and conventional VaR and ES measures

scholarly journals Coherent Risk Measures and Convex Combinations of the Conditional Value at Risk (C V@R)

2016 ◽  
Vol 31 (1) ◽  
pp. 73-75 ◽  
Author(s):  
Georg Ch. Pflug
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Conditional Value At Risk ◽  
Coherent Risk Measures ◽  
Coherent Risk

The conditional-value-at-risk (C V@R) has been widely used as a risk measure. It is well known, that C V@R is coherent in the sense of Artzner, Delbaen, Eber, Heath (1999). The class of coherent risk measures is convex. It was conjectured, that all coherent risk measures can be represented as convex combinations of C V@R’s. In this note we show that this conjecture is wrong.

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