SET-VALUED DYNAMIC RISK MEASURES FOR BOUNDED DISCRETE-TIME PROCESSES

2020 ◽  
Vol 23 (03) ◽  
pp. 2050017
Author(s):  
YANHONG CHEN ◽  
YIJUN HU
Keyword(s):  
Discrete Time ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Dynamic Risk Measures ◽  
Time Value Of Money ◽  
Average Value ◽  
Dynamic Risk ◽  
Entropic Risk Measure ◽  
Average Value At Risk

In this paper, we study how to evaluate the risk of a financial portfolio, whose components may be dependent and come from different markets or involve more than one kind of currencies, while we also take into consideration the uncertainty about the time value of money. Namely, we introduce a new class of risk measures, named set-valued dynamic risk measures for bounded discrete-time processes that are adapted to a given filtration. The time horizon can be finite or infinite. We investigate the representation results for them by making full use of Legendre–Fenchel conjugation theory for set-valued functions. Finally, some examples such as the set-valued dynamic average value at risk and the entropic risk measure for bounded discrete-time processes are also given.

scholarly journals Distribution-Invariant Risk Measures, Entropy, and Large Deviations

2007 ◽  
Vol 44 (1) ◽  
pp. 16-40 ◽  
Author(s):  
Stefan Weber
Keyword(s):  
Large Deviations ◽  
Value At Risk ◽  
Empirical Distribution ◽  
Risk Measures ◽  
Risk Measure ◽  
Rate Function ◽  
Average Value ◽  
Simulated Distribution ◽  
Average Value At Risk ◽  
Risk Measurements

The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.

scholarly journals Distribution-Invariant Risk Measures, Entropy, and Large Deviations

2007 ◽  
Vol 44 (01) ◽  
pp. 16-40
Author(s):  
Stefan Weber
Keyword(s):  
Large Deviations ◽  
Value At Risk ◽  
Empirical Distribution ◽  
Risk Measures ◽  
Risk Measure ◽  
Rate Function ◽  
Average Value ◽  
Simulated Distribution ◽  
Average Value At Risk ◽  
Risk Measurements

The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.

scholarly journals SET-VALUED SHORTFALL AND DIVERGENCE RISK MEASURES

2017 ◽  
Vol 20 (05) ◽  
pp. 1750026 ◽  
Author(s):  
ÇAĞIN ARARAT ◽  
ANDREAS H. HAMEL ◽  
BIRGIT RUDLOFF
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Preference Relation ◽  
Market Risk ◽  
Dual Representation ◽  
Average Value ◽  
Multivariate Risk Measures ◽  
Average Value At Risk ◽  
Incomplete Preference Relation

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 versions of the entropic risk measure and the average value at risk. As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved.

scholarly journals Set-Valued Haezendonck-Goovaerts Risk Measure and Its Properties

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Yu Feng ◽  
Yichuan Dong ◽  
Jia-Bao Liu
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Equivalent Representation ◽  
Average Value ◽  
Average Value At Risk

We propose a new set-valued risk measure, which is called set-valued Haezendonck-Goovaerts risk measure. First, we construct the set-valued Haezendonck-Goovaerts risk measure and then provide an equivalent representation. The properties of the set-valued Haezendonck-Goovaerts risk measure are investigated, which show that the set-valued Haezendonck-Goovaerts risk measure is coherent. Finally, an example of set-valued Haezendonck-Goovaerts risk measure is given, which exhibits the fact that the set-valued average value at risk is a particular case of the set-valued Haezendonck-Goovaerts risk measures.

REPRESENTATION OF BSDE-BASED DYNAMIC RISK MEASURES AND DYNAMIC CAPITAL ALLOCATIONS

2014 ◽  
Vol 17 (05) ◽  
pp. 1450032 ◽  
Author(s):  
EDUARD KROMER ◽  
LUDGER OVERBECK
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Static Case ◽  
Coherent Risk Measures ◽  
Dynamic Risk Measures ◽  
Convex Risk Measures ◽  
Coherent Risk ◽  
Dynamic Risk ◽  
Entropic Risk Measure ◽  
Representation Result

In this paper, we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations (BSDEs). We derive this representation from a classical differentiability result for BSDEs and the full allocation property of the Aumann–Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our results are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gâteaux-differentiable.

Optimal expected utility risk measures

2018 ◽  
Vol 35 (1-2) ◽  
pp. 73-87 ◽  
Author(s):  
Sebastian Geissel ◽  
Jörn Sass ◽  
Frank Thomas Seifried
Keyword(s):  
At Risk ◽  
Expected Utility ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Point Of View ◽  
Financial Loss ◽  
Certainty Equivalents ◽  
Average Value ◽  
Average Value At Risk

AbstractThis paper introduces optimal expected utility (OEU) risk measures, investigates their main properties and puts them in perspective to alternative risk measures and notions of certainty equivalents. By taking the investor’s point of view, OEU maximizes the sum of capital available today and the certainty equivalent of capital in the future. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility functionuhas constant relative risk aversion. We present several different risk measures that can be derived with special choices ofuand illustrate that OEU is more sensitive than value at risk and average value at risk with respect to changes of the probability of a financial loss.

scholarly journals CONSTRUCTION AND APPLICATION OF THE CLASSIFICATION SCHEME OF DYNAMIC RISK MEASURES ESTIMATING

2016 ◽  
Vol 5 ◽  
pp. 67-80 ◽  
Author(s):  
Nataly Zrazhevska
Keyword(s):  
Value At Risk ◽  
Classification Scheme ◽  
Risk Measures ◽  
Estimation Method ◽  
Daily Basis ◽  
Stock Index ◽  
Dynamic Risk Measures ◽  
Dynamic Risk ◽  
Risk Managers

The most popular methods for dynamic risk measures – Value-at-Risk (VaR) and Conditional VaR (CVaR) estimating were analyzed, description and comparative analysis of the methods were fulfilled, recommendations on the use were given. Results of the research were presented in the form of a classification scheme of dynamic risk measures estimating that facilitates the choice of an estimation method. The GARCH-based models of dynamic risk measures VaR and CVaR evaluation for artificially generated series and two time series of log return on a daily basis of the most well-known Asian stock indexes Nikkey225 Stock Index and CSI30 were constructed to illustrate the effectiveness of the proposed scheme. A qualitative analysis of the proposed models was conducted. To analyze the quality of the dynamic VaR estimations the Cupets test and the Cristoffersen test were used. For CVaR estimations the V-test was used as quality test. The tests results confirm the high quality of obtained estimations. The proposed classification scheme of dynamic risk measures VaR and CVaR estimating may be useful for risk managers of different financial institutions.

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