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.

SET-VALUED LAW INVARIANT COHERENT AND CONVEX RISK MEASURES

2019 ◽  
Vol 22 (03) ◽  
pp. 1950004 ◽  
Author(s):  
YANHONG CHEN ◽  
YIJUN HU
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Convex Risk Measures ◽  
New Class ◽  
The Relationship ◽  
Distortion Risk Measures

In this paper, we investigate representation results for set-valued law invariant coherent and convex risk measures, which can be considered as a set-valued extension of the multivariate scalar law invariant coherent and convex risk measures studied in the literature. We further introduce a new class of set-valued risk measures, named set-valued distortion risk measures, which can be considered as a set-valued version of multivariate scalar distortion risk measures introduced in the literature. The relationship between set-valued distortion risk measures and set-valued weighted value at risk is also given.

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.

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 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 A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020

2020 ◽  
Vol 13 (9) ◽  
pp. 192
Author(s):  
Beatriz Vaz de Melo Mendes ◽  
André Fluminense Carneiro
Keyword(s):  
Statistical Analysis ◽  
Value At Risk ◽  
Risk Measures ◽  
Value Theory ◽  
Dependence Structure ◽  
Copula Models ◽  
Underlying Distribution ◽  
First Semester ◽  
Long Run ◽  
Daily Data

After more than a decade of existence, crypto-currencies may now be considered an important class of assets presenting some unique appealing characteristics but also sharing some features with real financial assets. This paper provides a comprehensive statistical analysis of the six most important crypto-currencies from the period 2015–2020. Using daily data we (1) showed that the returns present many of the stylized facts often observed for stock assets, (2) modeled the returns underlying distribution using a semi-parametric mixture model based on the extreme value theory, (3) showed that the returns are weakly autocorrelated and confirmed the presence of long memory as well as short memory in the GARCH volatility, (4) used an econometric approach to compute risk measures, such as the value-at-risk, the expected shortfall, and drawups, (5) found that the crypto-coins’ price trajectories do not contain speculative bubbles and that they move together maintaining the long run equilibrium, and (6) using static and dynamic D-vine pair-copula models, assessed the true dependence structure among the crypto-assets, obtaining robust copula based bivariate dynamic measures of association. The analyses indicate that the strength of dependence among the crypto-currencies has increased over the recent years in the cointegrated crypto-market. The conclusions reached will help investors to manage risk while identifying opportunities for alternative diversified and profitable investments. To complete the analysis we provide a brief discussion on the effects of the COVID-19 pandemic on the crypto-market by including the first semester of 2020 data.

RISK MANAGEMENT UNDER A FACTOR STOCHASTIC VOLATILITY MODEL

2011 ◽  
Vol 28 (01) ◽  
pp. 65-80 ◽  
Author(s):  
MARCOS ESCOBAR ◽  
PABLO OLIVARES
Keyword(s):  
Risk Management ◽  
Stochastic Volatility ◽  
Value At Risk ◽  
Factor Model ◽  
Risk Measures ◽  
Stochastic Volatility Model ◽  
Dependence Structure ◽  
Black Scholes Model ◽  
Volatility Model ◽  
Black Scholes

In this paper, we study risk measures and portfolio problems based on a Stochastic Volatility Factor Model (SVFM). We analyze the sensitivity of Value at Risk (VaR) and Expected Shortfall (ES) to the changes in the parameters of the model. We compare the positions of a linear portfolio under assets following a SVFM, a Black–Scholes Model and a model with constant dependence structure. We consider an application to a portfolio of three selected Asian funds.

scholarly journals PERHITUNGAN UKURAN RISIKO UNTUK MODEL KERUGIAN AGREGAT

2020 ◽  
Vol 14 (1) ◽  
pp. 21
Author(s):  
Nadya Pratiwi ◽  
Aprida Siska Lestia ◽  
Nur Salam
Keyword(s):  
Monte Carlo ◽  
Standard Deviation ◽  
Monte Carlo Method ◽  
Value At Risk ◽  
Risk Measure ◽  
Random Variable ◽  
Insurance Companies ◽  
Loss Model ◽  
Aggregate Loss ◽  
Conditional Tail Expectation

In the case of nonlife insurance, insurance companies are very potential to get losses if claims submitted by customers (policyholders) exceeds the reserves of budgeted claims. It is the risk that have to managed properly by insurance companies . One possible disadvantage is the aggregate loss model. The aggregate loss model is a random variable that states the total of all losses incurred in an insurance policy block. This kind of loss can be modeled using a collective risk approach where the number of claims is a discrete random variable and the size of claim is a continuous random variable. The purpose of this study is to determine risk measure of standard deviation premium principle, value at risk (VaR), and conditional tail expectation (CTE) of the aggregate loss model. Standard deviation premium principle risk measure of aggregate loss model is determined analytically by substituted it expected value and varians. Meanwhile, VaR risk measure is determined using numerical method by Monte Carlo method, then the quantile value and it confidence interval for the actual value will estimate. In the CTE calculation, based on the loss data obtained in the Monte Carlo method, the CTE value is estimated by calculating the average loss that exceeds the VaR value. If the data size is large enough, the CTE value estimation will converge to the actual value.Keywords: Aggregate Loss Model, Standard Deviation Premium Principle, Value at Risk (VaR), Conditional Tail Expectation (CTE).

ASYMPTOTIC EXPANSIONS OF GENERALIZED QUANTILES AND EXPECTILES FOR EXTREME RISKS

2015 ◽  
Vol 29 (3) ◽  
pp. 309-327 ◽  
Author(s):  
Tiantian Mao ◽  
Kai Wang Ng ◽  
Taizhong Hu
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Domain Of Attraction ◽  
Random Variable ◽  
Second Order ◽  
Underlying Distribution ◽  
First Order ◽  
Extreme Risk ◽  
Special Cases ◽  
Extreme Risks

Generalized quantiles of a random variable were defined as the minimizers of a general asymmetric loss function, which include quantiles, expectiles and M-quantiles as their special cases. Expectiles have been suggested as potentially better alternatives to both Value-at-Risk and expected shortfall risk measures. In this paper, we first establish the first-order expansions of generalized quantiles for extreme risks as the confidence level α↑ 1, and then investigate the first-order and/or second-order expansions of expectiles of an extreme risk as α↑ 1 according to the underlying distribution belonging to the max-domain of attraction of the Fréchet, Weibull, and Gumbel distributions, respectively. Examples are also presented to examine whether and how much the first-order expansions have been improved by the second-order expansions.

scholarly journals Comparing Different Systemic Risk Measures for European Banking System

2018 ◽  
Vol 12 (1) ◽  
pp. 35 ◽  
Author(s):  
Annalisa Di Clemente
Keyword(s):  
Systemic Risk ◽  
Value At Risk ◽  
Financial Stability ◽  
Risk Measures ◽  
Banking System ◽  
Expected Shortfall ◽  
Conditional Value At Risk ◽  
Dependence Structure ◽  
European Banking ◽  
European Banks

This research examines and compares the performances in terms of systemic risk ranking for three different systemic risk metrics based on daily frequency publicly available data, specifically: Marginal Expected Shortfall (ES), Component Expected Shortfall (CES) and Delta Conditional Value-at-Risk (ΔCoVaR). We compute ΔCoVaR, MES and CES by utilizing EVT principles for modelling marginal distributions and Student’s t copula for describing the dependence structure between every bank and the banking system. Our objective is to attest whether different systemic risk metrics detect the same banks as systemically dangerous institutions with refer to a sample of European banks over the time span 2004-2015. For each bank in the sample we also calculate three traditional market risk measures, like Market VaR, Sharpe’s beta and the correlation between every bank and the banking system (European STOXX 600 Banks Index). Another aim is to explore the existence of a link among systemic risk measures and traditional risk metrics. In addition, the classification results obtained by the different risk metrics are compared with the ranking in terms of systemic riskiness (for European banks) calculated by Financial Stability Board (2015) using end-2014 data and collected in its list of Global Systemically Important Banks (G-SIBs). With refer to the entire sample period, we find a good coherence of ranking results among the three different systemic risk metrics, in particular between CES and ΔCoVaR. Moreover, we find for MES and ΔCoVaR a strong linkage with beta and correlation metrics respectively. Finally, CES metric shows the highest level of concordance with the list of G-SIBs by FSB with refer to European banks.

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.

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