ON SOME PROPERTIES OF TWO VECTOR-VALUED VAR AND CTE MULTIVARIATE RISK MEASURES FOR ARCHIMEDEAN COPULAS

2014 ◽  
Vol 44 (3) ◽  
pp. 613-633 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Value At Risk ◽  
Integral Transform ◽  
Risk Measures ◽  
Usual Sense ◽  
Translation Invariance ◽  
Archimedean Copulas ◽  
Coherent Risk ◽  
Multivariate Risk Measures ◽  
Stop Loss ◽  
Conditional Tail Expectation

AbstractWe consider the multivariate Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) risk measures introduced in Cousin and Di Bernardino (Cousin, A. and Di Bernardino, E. (2013) Journal of Multivariate Analysis, 119, 32–46; Cousin, A. and Di Bernardino, E. (2014) Insurance: Mathematics and Economics, 55(C), 272–282). For absolutely continuous Archimedean copulas, we derive integral formulas for the multivariate VaR and CTE Archimedean risk measures. We show that each component of the multivariate VaR and CTE functional vectors is an integral transform of the corresponding univariate VaR measures. For the class of Archimedean copulas, the marginal components of the CTE vector satisfy the following properties: positive homogeneity (PH), translation invariance (TI), monotonicity (MO), safety loading (SL) and VaR inequality (VIA). In case marginal risks satisfy the subadditivity (MSA) property, the marginal CTE components are also sub-additive and hitherto coherent risk measures in the usual sense. Moreover, the increasing risk (IR) or stop-loss order preserving property of the marginal CTE components holds for the class of bivariate Archimedean copulas. A counterexample to the (IR) property for the trivariate Clayton copula is included.

RISK MEASURES DERIVED FROM A REGULATOR’S PERSPECTIVE ON THE REGULATORY CAPITAL REQUIREMENTS FOR INSURERS

2020 ◽  
Vol 50 (3) ◽  
pp. 1065-1092
Author(s):  
Jun Cai ◽  
Tiantian Mao
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Wasserstein Distance ◽  
Capital Requirements ◽  
Translation Invariance ◽  
Coherent Risk Measures ◽  
Regulatory Capital ◽  
Coherent Risk ◽  
Positive Homogeneity ◽  
Stop Loss

AbstractIn this study, we propose new risk measures from a regulator’s perspective on the regulatory capital requirements. The proposed risk measures possess many desired properties, including monotonicity, translation-invariance, positive homogeneity, subadditivity, nonnegative loading, and stop-loss order preserving. The new risk measures not only generalize the existing, well-known risk measures in the literature, including the Dutch, tail value-at-risk (TVaR), and expectile measures, but also provide new approaches to generate feasible and practical coherent risk measures. As examples of the new risk measures, TVaR-type generalized expectiles are investigated in detail. In particular, we present the dual and Kusuoka representations of the TVaR-type generalized expectiles and discuss their robustness with respect to the Wasserstein distance.

scholarly journals Optimal Limited Stop-Loss Reinsurance under VaR, TVaR, and CTE Risk Measures

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Xianhua Zhou ◽  
Huadong Zhang ◽  
Qingquan Fan
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Loss Ratio ◽  
Optimal Reinsurance ◽  
Numerical Examples ◽  
Optimal Values ◽  
Stop Loss ◽  
Conditional Tail Expectation ◽  
Reinsurance Market

This paper aims to provide a practical optimal reinsurance scheme under particular conditions, with the goal of minimizing total insurer risk. Excess of loss reinsurance is an essential part of the reinsurance market, but the concept of stop-loss reinsurance tends to be unpopular. We study the purchase arrangement of optimal reinsurance, under which the liability of reinsurers is limited by the excess of loss ratio, in order to generate a reinsurance scheme that is closer to reality. We explore the optimization of limited stop-loss reinsurance under three risk measures: value at risk (VaR), tail value at risk (TVaR), and conditional tail expectation (CTE). We analyze the topic from the following aspects: (1) finding the optimal franchise point with limited stop-loss coverage, (2) finding the optimal limited stop-loss coverage within a certain franchise point, and (3) finding the optimal franchise point with limited stop-loss coverage. We provide several numerical examples. Our results show the existence of optimal values and locations under the various constraint conditions.

scholarly journals Sum of Bernoulli Mixtures: Beyond Conditional Independence

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Taehan Bae ◽  
Ian Iscoe
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Random Variable ◽  
Dependence Structure ◽  
Sample Distribution ◽  
Mixing Distribution ◽  
Bernoulli Random Variables ◽  
New Class ◽  
Bernoulli Mixtures ◽  
Conditional Tail Expectation

We consider the distribution of the sum of Bernoulli mixtures under a general dependence structure. The level of dependence is measured in terms of a limiting conditional correlation between two of the Bernoulli random variables. The conditioning event is that the mixing random variable is larger than a threshold and the limit is with respect to the threshold tending to one. The large-sample distribution of the empirical frequency and its use in approximating the risk measures, value at risk and conditional tail expectation, are presented for a new class of models which we calldouble mixtures. Several illustrative examples with a Beta mixing distribution, are given. As well, some data from the area of credit risk are fit with the models, and comparisons are made between the new models and also the classical Beta-binomial model.

scholarly journals Quantifying and Correcting the Bias in Estimated Risk Measures

2007 ◽  
Vol 37 (2) ◽  
pp. 365-386 ◽  
Author(s):  
Joseph Hyun Tae Kim ◽  
Mary R. Hardy
Keyword(s):  
At Risk ◽  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Value At Risk ◽  
Risk Measures ◽  
Bootstrap Techniques ◽  
Practical Guideline ◽  
Estimated Risk ◽  
Conditional Tail Expectation

In this paper we explore the bias in the estimation of the Value at Risk and Conditional Tail Expectation risk measures using Monte Carlo simulation. We assess the use of bootstrap techniques to correct the bias for a number of different examples. In the case of the Conditional Tail Expectation, we show that application of the exact bootstrap can improve estimates, and we develop a practical guideline for assessing when to use the exact bootstrap.

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 Teoria de Medidas de Risco: uma revisão abrangente

2014 ◽  
Vol 12 (3) ◽  
pp. 411
Author(s):  
Marcelo Brutti Righi ◽  
Paulo Sergio Ceretta
Keyword(s):  
Risk Management ◽  
Literature Review ◽  
Value At Risk ◽  
Risk Measures ◽  
Fundamental Aspect ◽  
Coherent Risk Measures ◽  
Convex Risk Measures ◽  
Coherent Risk ◽  
Comprehensive Literature Review ◽  
Generalized Deviation

A fundamental aspect of proper risk management is the measurement, especially forecasting of risk measures. Measures such as variance, volatility and Value at Risk had been considered valid because of their practical intuition. However, a solid theoretical framework it is important to ensure better properties for risk measures. Such background is the risk measures theory. This paper presents a comprehensive literature review on risk measures theory, focusing in basic theory and extensions to this fundamental outline. The paper is structured in order to cover the main risk measures classes from literature, which are coherent risk measures, convex risk measures, spectral and distortion risk measures and generalized deviation measures.

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.

Estimating the Variance of Bootstrapped Risk Measures

2009 ◽  
Vol 39 (1) ◽  
pp. 199-223 ◽  
Author(s):  
Joseph H.T. Kim ◽  
Mary R. Hardy
Keyword(s):  
Simulation Study ◽  
Influence Function ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Bootstrap Method ◽  
Analytic Form ◽  
Analytic Approach ◽  
New Method ◽  
Conditional Tail Expectation

AbstractIn Kim and Hardy (2007) the exact bootstrap was used to estimate certain risk measures including Value at Risk and the Conditional Tail Expectation. In this paper we continue this work by deriving the influence function of the exact-bootstrapped quantile risk measure. We can use the influence function to estimate the variance of the exact-bootstrap risk measure. We then extend the result to the L-estimator class, which includes the conditional tail expectation risk measure. The resulting formula provides an alternative way to estimate the variance of the bootstrapped risk measures, or the whole L-estimator class in an analytic form. A simulation study shows that this new method is comparable to the ordinary resampling-based bootstrap method, with the advantages of an analytic approach.

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.

Sign in / Sign up

Export Citation Format

Share Document