应用复合极值理论计算VaR1

本文介绍了一种复合极值理论,并将其应用到VaR的计算上。实际中大的损失发生的频率也是风险的一种度量,在应用复合极值理论方法计算VaR时,我们第一次将在一定时期内金融资产的损失率超过一定阈值的次数的分布和收益率的分布结合了起来,对欧元/人民币、日元/人民币两种汇率进行了VaR的计算,经过实证分析,得到了一些有意义的结果。     

Characterizing optimal allocations in quantile-based risk sharing

Unlike classic risk sharing problems based on expected utilities or convex risk measures, quantile-based risk sharing problems exhibit two special features. First, quantile-based risk measures (such as the Value-at-Risk) are often not convex, and second, they ignore some part of the distribution of the risk. These features create technical challenges in establishing a full characterization of optimal allocations, a question left unanswered in the literature. In this paper, we address the issues on the existence and the characterization of (Pareto-)optimal allocations in risk sharing problems for the Range-Value-at-Risk family. It turns out that negative dependence, mutual exclusivity in particular, plays an important role in the optimal allocations, in contrast to positive dependence appearing in classic risk sharing problems. As a by-product of our main finding, we obtain some results on the optimization of the Value-at-Risk (VaR) and the Expected Shortfall, as well as a new result on the inf-convolution of VaR and a general distortion risk measure.     

Orderings and Probability Functionals Consistent with Preferences

This paper unifies the classical theory of stochastic dominance and investor preferences with the recent literature on risk measures applied to the choice problem faced by investors. First, we summarize the main stochastic dominance rules used in the finance literature. Then we discuss the connection with the theory of integral stochastic orders and we introduce orderings consistent with investors' preferences. Thus, we classify them, distinguishing several categories of orderings associated with different classes of investors. Finally, we show how we can use risk measures and orderings consistent with some preferences to determine the investors' optimal choices.     

Value at risk calculation through ARCH factor methodology: Proposal and comparative analysis

In this paper we put forward a new method to estimate value at risk (VaR), autoregressive conditional heteroskedastic (ARCH) factor, which combines multivariate analysis with ARCH models. Firstly, from a set of correlated portfolio risk factors, we derive a smaller uncorrelated risk factors set, by applying multivariate analysis. Secondly, we use ARCH schemes to model uncorrelated factors historical behaviour. Thirdly, we use the estimated models to predict future values for factors standard deviation. From them, VaR calculation is immediate. In this way, ARCH factor methodology overcomes the multivariate ARCH models drawbacks, which, in practice, make these unworkable for VaR calculation purposes. We apply the proposed methodology over a set of foreign exchange risk exposed portfolios, obtaining better results than those reached when J.P. Morgan’s Riskmetrics is used.     

Stochastic dominance and mean–variance measures of profit and loss for business planning and investment

In this paper, we first extend the stochastic dominance (SD) theory by introducing the first three orders of both ascending SD (ASD) and descending SD (DSD) to decisions in business planning and investment to risk-averse and risk-loving decision makers so that they can compare both return and loss. We provide investors with more tools for empirical analysis, with which they can identify the first-order ASD and DSD prospects and discern arbitrage opportunities that could increase his/her utility as well as wealth and set up a zero dollar portfolio to make huge profit. Our tools also enable investors and business planners to identify the third order ASD and DSD prospects and make better choices.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASUREgS

In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

Bayesian parameter inference for models of the Black and Scholes type

In this paper, we describe a general method for constructing the posterior distribution of the mean and volatility of the return of an asset satisfying dS=SdX for some simple models of X. Our framework takes as inputs the prior distributions of the parameters of the stochastic process followed by the underlying, as well as the likelihood function implied by the observed price history for the underlying. As an application of our framework, we compute the value at risk (VaR) and conditional VaR (CVaR) measures for the changes in the price of an option implied by the posterior distribution of the volatility of the underlying. The implied VaR and CVaR are more conservative than their classical counterpart, since it takes into account the estimation risk that arises due to parameter uncertainty. Copyright © 2008 John Wiley & Sons, Ltd.     

期货套期保值C VaR风险的敏感度分析

把条件风险价值应用于期货组合套期保值的风险管理,分析条件风险价值对期货部位的敏感性.在一般的概率分布下,分空头套期保值和多头套期保值两种情况,导出期货组合套期保值的条件风险价值关于套期比的一阶和二阶变化率,并研究其经济意义.投资者可以根据条件风险价值的敏感度增减期货头寸,把握好用于套期保值的期货量,帮助投资者管理套期保值风险.     

GARCH模型在股票市场风险计量中的应用

本文以上证综指的日收益率为研究对象,运用GARCH模型簇分析上海股市日收益率波动的条件异方差性,计算每天的V aR值.实证研究表明,GARCH模型的V aR计算方法对我国股市风险的管理有较好的效果.     

t-分布与不同风险度量下有效投资组合及其边缘的相互关系研究

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASURES

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivari-ate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

A New Robust Risk Measure: Quantile Shortfall

Among recent measures for risk management, value at risk (VaR) has been criticized because it is not coherent and expected shortfall (ES) has been criticized because it is not robust to outliers. Recently,[Math. Oper. Res., 38, 393-417 (2013)] proposed a risk measure called median shortfall (MS) which is distributional robust and easy to implement. In this paper, we propose a more generalized risk measure called quantile shortfall (QS) which includes MS as a special case. QS measures the conditional quantile loss of the tail risk and inherits the merits of MS. We construct an estimator of the QS and establish the asymptotic normality behavior of the estimator. Our simulation shows that the newly proposed measures compare favorably in robustness with other widely used measures such as ES and VaR.     

Optimal reinsurance under VaR and TVaR risk measures in the presence of reinsurer’s risk limit

In most studies on optimal reinsurance, little attention has been paid to controlling the reinsurer’s risk. However, real-world insurance markets always place a limit on coverage, otherwise the insurer will be subjected to under a heavy financial burden when the insured suffers a large unexpected covered loss. In this paper, we revisit the optimal reinsurance problem under the optimality criteria of VaR and TVaR risk measures when the constraints for the reinsurer’s risk exposure are presented. Two types of constraints are considered that have been proposed by Cummins and Mahul (2004) and Zhou et al. (2010), respectively. It is shown that two-layer reinsurance is always the optimal reinsurance policy under both VaR and TVaR risk measures and under both types of constraints. This implies that the two-layer reinsurance policy is more robust. Furthermore, the optimal quantity of ceded risk depends on the confidence level, the safety loading and the tolerance level, as well as on the relation between them.     

基于Copula的VsR度量与事后检验

估计VaR的传统方法有三种:协方差矩阵法、历史模拟法和蒙特仁洛模拟法。通常,文献中认为刚蒙特卡洛模拟法度量VaR有很多方面的优点。但是,本文通过实证检验发现,使用传统蒙特卡洛模拟法估计的VaR偏小,事后检验效果很不理想。本文引入Copula函数来改进传统的蒙特卡洛模拟法。Copula函数能将单个边际分布和多元联合分布联系起来,能处理非正态的边际分布,并且它度量的相关性不再局限于线性相关性。实证检验表明,基于Copula的蒙特卡罗模拟法可以更加准确地度量资产组合的VaR。     

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In the situation of \rho-mixing dependent sequences, this paper studied the mean square error and the optimal bandwidth of distribution kernel estimator nu_{p,h} of VaR. And the optimal bandwidth minimized the mean square error. The density function of Laplace distribution is used in the calculation of bandwidth and we adopt the method of interpolation to compute specific value of bandwidth in this paper. According to the numerical simulations, the distribution kernel estimator is more accurate by comparing the performance of VaR distribution kernel estimation with a common order statistic. Finally, Shangzheng A-share index and Shenzheng B-share index are chosen for an empirical research, which concludes that the risk of the latter is significantly higher than that of the former.     

The practice of Delta–Gamma VaR: Implementing the quadratic portfolio model

This paper intends to critically evaluate state-of-the-art methodologies for calculating the value-at-risk (VaR) of non-linear portfolios from the point of view of computational accuracy and efficiency. We focus on the quadratic portfolio model, also known as “Delta–Gamma”, and, as a working assumption, we model risk factor returns as multi-normal random variables. We present the main approaches to Delta–Gamma VaR weighing their merits and accuracy from an implementation-oriented standpoint. One of our main conclusions is that the Delta–Gamma-Normal VaR may be less accurate than even Delta VaR. On the other hand, we show that methods that essentially take into account the non-linearity (hence gammas and third or higher moments) of the portfolio values may present significant advantages over full Monte Carlo revaluations. The role of non-diagonal terms in the Gamma matrix as well as the sensitivity to correlation is considered both for accuracy and computational effort. We also qualitatively examine the robustness of Delta–Gamma methodologies by considering a highly non-quadratic portfolio value function.     

Entropic risk measures and their comparative statics in portfolio selection: Coherence vs. convexity

We conduct a decision-theoretic analysis of optimal portfolio choices and, in particular, their comparative statics under two types of entropic risk measures, the coherent entropic risk measure (CERM) and the convex entropic risk measure (ERM). Starting with the portfolio selection between a risky and a risk free asset (framework of Arrow (1965) and Pratt (1964)), we find a restrictive all-or-nothing investment decision under the CERM, while the ERM yields diversification. We then address a portfolio problem with two risky assets, and provide comparative statics with respect to the investor’s risk aversion (framework of Ross (1981)). Here, both the CERM and the ERM exhibit closely interrelated inconsistencies with respect to the interpretation of their risk parameters as a measure of risk aversion: for any two investors with different risk parameters, it may happen that the investor with the higher risk parameter invests more in the riskier one of the two assets. Finally, we analyze the portfolio problem “risky vs. risk free” in the presence of an independent background risk, and analyze the effect of changes in this background risk (framework of Gollier and Pratt (1996)). Again, we find questionable predictions: under the CERM, the optimal risky investment is always increasing instead of decreasing when a background risk is introduced, while under the ERM it remains unaffected.     

GlueVaR risk measures in capital allocation applications

GlueVaR risk measures defined by Belles-Sampera et al. (2014) generalize the traditional quantile-based approach to risk measurement, while a subfamily of these risk measures has been shown to satisfy the tail-subadditivity property. In this paper we show how GlueVaR risk measures can be implemented to solve problems of proportional capital allocation. In addition, the classical capital allocation framework suggested by Dhaene et al. (2012) is generalized to allow the application of the Value-at-Risk (VaR) measure in combination with a stand-alone proportional allocation criterion (i.e., to accommodate the Haircut allocation principle). Two new proportional capital allocation principles based on GlueVaR risk measures are defined. An example based on insurance claims data is presented, in which allocation solutions with tail-subadditive risk measures are discussed.     

Estimating Value-at-Risk and Expected Shortfall using the intraday low and range data

Value-at-Risk (VaR) is a popular measure of market risk. To convey information regarding potential exceedances beyond the VaR, Expected Shortfall (ES) has become the risk measure for trading book bank regulation. However, the estimation of VaR and ES is challenging, as it requires the estimation of the tail behaviour of daily returns. In this paper, we take advantage of recent research that develops joint scoring functions for VaR and ES. Using these functions, we present a novel approach to estimating the two risk measures based on intraday data. We focus on the intraday range, which is the difference between the highest and lowest intraday log prices. In contrast to intraday observations, the intraday low and high are widely available for many financial assets. To alleviate the challenge of modelling extreme risk measures, we propose the use of the intraday low series. We draw on a theoretical result for Brownian motion to show that a quantile of the daily returns can be estimated as the product of a constant term and a less extreme quantile of the intraday low returns, which we define as the difference between the lowest log price of the day and the log closing price of the previous day. In view of this, we use estimates of the VaR and ES of the intraday low returns to estimate the VaR and ES of the daily returns. We provide empirical support for the new proposals using data for five stock indices and five individual stocks.     

基于几种风险测度的多阶段组合优化研究

基于均值-方差(MV)、VaR(Value at Risk)、CVaR(Conditional VaR)、HMCR(p=1,2,3)(Higher Moment Coherent Risk)几种风险测度进行多阶段组合优化研究。首先从一致性公理和随机占优一致性角度分析几种风险测度的风险识别能力,认为HMCR(p=2,3)的风险识别能力最高,然后给出静态和动态下的风险规避型的规划函数及多阶段CVaR和HMCR模型,最后依据单阶段和多阶段优化模型,对上证50指数成份股进行实证分析。对比单阶段和多阶段下几种风险测度优化组合的累计收益率及几种风险测度之间的关系,结合上证50指数收益率发现,多阶段优化组合要整体优于单阶段优化组合,且HMCR(p=2,3)要优于指数收益率和其它几种风险测度。从投资者投资决策方面来分析,HMCR(p=2,3)型积极投资策略比较适用于股市平稳期、顶峰期和下降期,被动投资策略比较适用于股市上升期。