scholarly journals Multivariate Tail Moments for Log-Elliptical Dependence Structures as Measures of Risks

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 559
Author(s):  
Zinoviy Landsman ◽  
Tomer Shushi
Keyword(s):  
Risk Measures ◽  
Short Paper ◽  
Dependence Structure ◽  
Elliptical Distributions ◽  
Tail Conditional Expectation ◽  
Special Cases ◽  
Optimal Portfolio Selection ◽  
Multivariate Risk Measures ◽  
Elliptical Models ◽  
Risk Managers

The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, risk managers confront a system of mutually dependent risks, not only one risk. Thus, it is important to measure risks while capturing their dependence structure. In this short paper, we compute the multivariate risk measures, multivariate tail conditional expectation, and multivariate tail covariance measure for the family of log-elliptical distributions, which captures the dependence structure of the risks while focusing on the tail of their distributions, i.e., on extreme loss events. We then study our result and examine special cases, as well as the optimal portfolio selection using such measures. Finally, we show how the given multivariate tail moments can also be computed for log-skew elliptical models based on similar approaches given for the log-elliptical case.

scholarly journals Tail Variance Premium with Applications for Elliptical Portfolio of Risks

2006 ◽  
Vol 36 (2) ◽  
pp. 433-462 ◽  
Author(s):  
Edward Furman ◽  
Zinoviy Landsman
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Insurance Company ◽  
Risk Capital ◽  
Normal Distributions ◽  
Risk Manager ◽  
Tail Conditional Expectation ◽  
Variance Risk ◽  
Risk Managers ◽  
Variance Risk Measure

In this paper we consider the important circumstances involved when risk managers are concerned with risks that exceed a certain threshold. Such conditions are well-known to insurance professionals, for instance in the context of policies involving deductibles and reinsurance contracts. We propose a new premium called tail variance premium (TVP) which answers the demands of these circumstances. In addition, we suggest a number of risk measures associated with TVP. While the well-known tail conditional expectation risk measure provides a risk manager with information about the average of the tail of the loss distribution, tail variance risk measure (TV) estimates the variability along such a tail. Furthermore, given a multivariate setup, we offer a number of allocation techniques which preserve different desirable properties (sub-additivity and fulladditivity, for instance). We are able to derive explicit expressions for TV and TVP, and risk capital decomposition rules based on them, in the general framework of multivariate elliptical distributions. This class is very popular among actuaries and risk managers because it contains distributions with marginals whose tails are heavier than those of normal distributions. This distinctive feature is desirable when modeling financial datasets. Moreover, according to our results, in some cases there exists an optimal threshold, such that by choosing it, an insurance company minimizes its risk.

scholarly journals Tail Variance Premium with Applications for Elliptical Portfolio of Risks

2006 ◽  
Vol 36 (02) ◽  
pp. 433-462 ◽  
Author(s):  
Edward Furman ◽  
Zinoviy Landsman
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Insurance Company ◽  
Risk Capital ◽  
Normal Distributions ◽  
Risk Manager ◽  
Tail Conditional Expectation ◽  
Variance Risk ◽  
Risk Managers ◽  
Variance Risk Measure

In this paper we consider the important circumstances involved when risk managers are concerned with risks that exceed a certain threshold. Such conditions are well-known to insurance professionals, for instance in the context of policies involving deductibles and reinsurance contracts. We propose a new premium called tail variance premium (TVP) which answers the demands of these circumstances. In addition, we suggest a number of risk measures associated with TVP. While the well-known tail conditional expectation risk measure provides a risk manager with information about the average of the tail of the loss distribution, tail variance risk measure (TV) estimates the variability along such a tail. Furthermore, given a multivariate setup, we offer a number of allocation techniques which preserve different desirable properties (sub-additivity and fulladditivity, for instance). We are able to derive explicit expressions for TV and TVP, and risk capital decomposition rules based on them, in the general framework of multivariate elliptical distributions. This class is very popular among actuaries and risk managers because it contains distributions with marginals whose tails are heavier than those of normal distributions. This distinctive feature is desirable when modeling financial datasets. Moreover, according to our results, in some cases there exists an optimal threshold, such that by choosing it, an insurance company minimizes its risk.

TAIL CONDITIONAL EXPECTATIONS FOR GENERALIZED SKEW-ELLIPTICAL DISTRIBUTIONS

2021 ◽  
pp. 1-14
Author(s):  
Baishuai Zuo ◽  
Chuancun Yin
Keyword(s):  
Conditional Expectation ◽  
Elliptical Distributions ◽  
Tail Conditional Expectation ◽  
Conditional Expectations ◽  
Special Cases ◽  
Skew Normal

This paper deals with the multivariate tail conditional expectation (MTCE) for generalized skew-elliptical distributions. We present tail conditional expectation for univariate generalized skew-elliptical distributions and MTCE for generalized skew-elliptical distributions. There are many special cases for generalized skew-elliptical distributions, such as generalized skew-normal, generalized skew Student-t, generalized skew-logistic and generalized skew-Laplace distributions.

scholarly journals A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 85 ◽  
Author(s):  
Mohamed Lkabous ◽  
Jean-François Renaud
Keyword(s):  
Risk Model ◽  
Risk Measures ◽  
Risk Measure ◽  
Short Paper ◽  
Classical Risk Model ◽  
Parisian Ruin ◽  
The Classical Risk Model

In this short paper, we study a VaR-type risk measure introduced by Guérin and Renaud and which is based on cumulative Parisian ruin. We derive some properties of this risk measure and we compare it to the risk measures of Trufin et al. and Loisel and Trufin.

CAPITAL ALLOCATION WITH MULTIVARIATE RISK MEASURES: AN AXIOMATIC APPROACH

2019 ◽  
Vol 34 (2) ◽  
pp. 297-315
Author(s):  
Linxiao Wei ◽  
Yijun Hu
Keyword(s):  
Standard Deviation ◽  
Portfolio Management ◽  
Risk Measures ◽  
Risk Measure ◽  
Axiomatic Approach ◽  
Capital Allocation ◽  
Axiom System ◽  
Multivariate Risk ◽  
Multivariate Risk Measures ◽  
Multivariate Risk Measure

AbstractCapital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

scholarly journals CMPH: a multivariate phase-type aggregate loss distribution

2017 ◽  
Vol 5 (1) ◽  
pp. 304-315 ◽  
Author(s):  
Jiandong Ren ◽  
Ricardas Zitikis
Keyword(s):  
Negative Binomial ◽  
Risk Measures ◽  
Moment Generating Function ◽  
Insurance Companies ◽  
Joint Distributions ◽  
Special Cases ◽  
Risk Sources ◽  
Computational Aspects ◽  
Aggregate Loss ◽  
Aggregate Loss Distribution

AbstractWe introduce a compound multivariate distribution designed for modeling insurance losses arising from different risk sources in insurance companies. The distribution is based on a discrete-time Markov Chain and generalizes the multivariate compound negative binomial distribution, which is widely used for modeling insurance losses.We derive fundamental properties of the distribution and discuss computational aspects facilitating calculations of risk measures of the aggregate loss, as well as allocations of the aggregate loss to individual types of risk sources. Explicit formulas for the joint moment generating function and the joint moments of different loss types are derived, and recursive formulas for calculating the joint distributions given. Several special cases of particular interest are analyzed. An illustrative numerical example is provided.

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.

scholarly journals Some Results on Measures of Interaction between Paired Risks

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 88 ◽  
Author(s):  
Rui Fang ◽  
Xiaohu Li
Keyword(s):  
Monte Carlo Simulation ◽  
Risk Management ◽  
Monte Carlo ◽  
Risk Measures ◽  
Dependence Structure ◽  
Marginal Distributions ◽  
Numerical Examples ◽  
Risk Contribution ◽  
Actuarial Risk

Co-risk measures and risk contribution measures have been introduced to evaluate the degree of interaction between paired risks in actuarial risk management. This paper attempts to study the ordering behavior of measures on interaction between paired risks. For various co-risk measures and risk contribution measures, we investigate how the marginal distributions and the dependence structure impact on the level of interaction between paired risks. Also, several numerical examples based on Monte Carlo simulation are presented to illustrate the main findings.

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