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.

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.

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 Time-consistency of risk measures with GARCH volatilities and their estimation

2016 ◽  
Vol 32 (2) ◽  
Author(s):  
Claudia Klüppelberg ◽  
Jianing Zhang
Keyword(s):  
At Risk ◽  
Stock Prices ◽  
Value At Risk ◽  
Risk Measures ◽  
Analytical Formula ◽  
Time Consistency ◽  
Value Theory ◽  
Data Set ◽  
Average Value ◽  
Average Value At Risk

AbstractIn this paper we study time-consistent risk measures for returns that are given by a GARCH(1,1) model. We present a construction of risk measures based on their static counterparts that overcomes the lack of time-consistency. We then study in detail our construction for the risk measures Value-at-Risk (VaR) and Average Value-at-Risk (AVaR). While in the VaR case we can derive an analytical formula for its time-consistent counterpart, in the AVaR case we derive lower and upper bounds to its time-consistent version. Furthermore, we incorporate techniques from extreme value theory (EVT) to allow for a more tail-geared statistical analysis of the corresponding risk measures. We conclude with an application of our results to a data set of stock prices.

scholarly journals Return Based Risk Measures for Non-Normally Distributed Returns: An Alternative Modelling Approach

2021 ◽  
Vol 14 (11) ◽  
pp. 540
Author(s):  
Eyden Samunderu ◽  
Yvonne T. Murahwa
Keyword(s):  
At Risk ◽  
Normal Distribution ◽  
Hedge Funds ◽  
Value At Risk ◽  
Financial Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Extensive Literature ◽  
Data Set ◽  
Normally Distributed

Developments in the world of finance have led the authors to assess the adequacy of using the normal distribution assumptions alone in measuring risk. Cushioning against risk has always created a plethora of complexities and challenges; hence, this paper attempts to analyse statistical properties of various risk measures in a not normal distribution and provide a financial blueprint on how to manage risk. It is assumed that using old assumptions of normality alone in a distribution is not as accurate, which has led to the use of models that do not give accurate risk measures. Our empirical design of study firstly examined an overview of the use of returns in measuring risk and an assessment of the current financial environment. As an alternative to conventional measures, our paper employs a mosaic of risk techniques in order to ascertain the fact that there is no one universal risk measure. The next step involved looking at the current risk proxy measures adopted, such as the Gaussian-based, value at risk (VaR) measure. Furthermore, the authors analysed multiple alternative approaches that do not take into account the normality assumption, such as other variations of VaR, as well as econometric models that can be used in risk measurement and forecasting. Value at risk (VaR) is a widely used measure of financial risk, which provides a way of quantifying and managing the risk of a portfolio. Arguably, VaR represents the most important tool for evaluating market risk as one of the several threats to the global financial system. Upon carrying out an extensive literature review, a data set was applied which was composed of three main asset classes: bonds, equities and hedge funds. The first part was to determine to what extent returns are not normally distributed. After testing the hypothesis, it was found that the majority of returns are not normally distributed but instead exhibit skewness and kurtosis greater or less than three. The study then applied various VaR methods to measure risk in order to determine the most efficient ones. Different timelines were used to carry out stressed value at risks, and it was seen that during periods of crisis, the volatility of asset returns was higher. The other steps that followed examined the relationship of the variables, correlation tests and time series analysis conducted and led to the forecasting of the returns. It was noted that these methods could not be used in isolation. We adopted the use of a mosaic of all the methods from the VaR measures, which included studying the behaviour and relation of assets with each other. Furthermore, we also examined the environment as a whole, then applied forecasting models to accurately value returns; this gave a much more accurate and relevant risk measure as compared to the initial assumption of normality.

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