Model uncertainty and VaR aggregation

Despite well-known shortcomings as a risk measure, Value-at-Risk (VaR) is still the industry and regulatory standard for the calculation of risk capital in banking and insurance. This paper is concerned with the numerical estimation of the VaR for a portfolio position as a function of different dependence scenarios on the factors of the portfolio. Besides summarizing the most relevant analytical bounds, including a discussion of their sharpness, we introduce a numerical algorithm which allows for the computation of reliable (sharp) bounds for the VaR of high-dimensional portfolios with dimensions d possibly in the several hundreds. We show that additional positive dependence information will typically not improve the upper bound substantially. In contrast higher order marginal information on the model, when available, may lead to strongly improved bounds. Several examples of practical relevance show how explicit VaR bounds can be obtained. These bounds can be interpreted as a measure of model uncertainty induced by possible dependence scenarios.     

VaR模型比较技术及其评价——理论、实证回顾及其应用初探

随着VaR模型的不断创新,各类VaR模型的比较技术也由采用单一的回顾测试转变为设计一套较完整的比较评价体系来进行比较研究。近十年来,通过将指标评价工具、假设检验工具和比较评价工具三类VaR模型的比较工具进行不同的组合,学术界涌现出了大量相关的理论和实证文献。本文尝试对其进行较全面的回顾,并结合中国实际进行初步探索。     

风险度量中的VaR模型

金融风险一旦发生,将可能引起金融市场资产价值贬值,对整个金融市场和宏观经济将造成严重危害。因此,对金融市场风险的分析就变得十分重要。     

Varying the VaR for unconditional and conditional environments

Accurate forecasting of risk is the key to successful risk management techniques. Using the largest stock index futures from 12 European bourses, this paper presents VaR measures based on their unconditional and conditional distributions for single and multi-period settings. These measures underpinned by extreme value theory are statistically robust explicitly allowing for fat-tailed densities. Conditional tail estimates accounting for volatility clustering are obtained by adjusting the unconditional extreme value procedure with GARCH filtered returns. The conditional modelling results in iid returns allowing for the use of a simple and efficient multi-period extreme value scaling law. The paper examines the properties of these distinct conditional and unconditional trading models. The paper finds that the biases inherent in unconditional single and multi-period estimates assuming normality extend to the conditional setting.     

Ambiguity intolerance and financial reporting alternatives

Authoritative accounting pronouncements almost always require more, rather than less, financial reporting disclosures and accountants rarely, if ever, act to overturn the required disclosures. Since the personality literature suggests that desire for more information is linked to an individual's intolerance for ambiguity, this study tested selected hypotheses about the relationship between accountants' intolerance for ambiguity and their desire for financial reporting alternatives. The findings of the study did not disclose a significant relationship between accountants' intolerance for ambiguity and their desire for financial reporting alternatives. However, the results did disclose a relationship between education level and two dependent variables: desire for disclosure and consistency of desired disclosures with generally accepted accounting principles. Therefore, additional research should pay more attention to the effects of demographic information, which may in fact be more informative than many previously tested variables such as personality characteristics.     

Evaluation of volatility predictions in a VaR framework

The evaluation of volatility forecasts is not straightforward and some issues can arise. A standard approach relies on statistical loss functions. Another approach bases the evaluation of the volatility predictions on utility functions or Value at Risk (VaR) measures. This work aims to combine the two approaches, using the VaR measures within the loss functions. By means of this method, the VaR measures obtained from a set of competing models are plugged into two loss functions, the magnitude loss function and a proposed new one. This latter loss function more heavily penalizes the models with a number of VaR violations greater than the expected one. The loss function values are evaluated against a benchmark obtained from the inclusion of a consistent estimate of the VaR measures in the loss function. In order to investigate the performance of the proposed method and the new loss function, a Monte Carlo experiment and an empirical analysis of a stock listed on the New York Stock Exchange are provided. The proposed strategy helps with the selection of a superior model, in terms of forecast accuracy, when the cited approaches do not clearly and uniquely identify it. Moreover, the new asymmetric loss function allows a greater discrimination with regard to models, helping to find the best volatility model.     

Pitfalls in backtesting Historical Simulation VaR models

Historical Simulation (HS) and its variant, the Filtered Historical Simulation (FHS), are the most popular Value-at-Risk forecast methods at commercial banks. These forecast methods are traditionally evaluated by means of the unconditional backtest. This paper formally shows that the unconditional backtest is always inconsistent for backtesting HS and FHS models, with a power function that can be even smaller than the nominal level in large samples. Our findings have fundamental implications in the determination of market risk capital requirements, and also explain Monte Carlo and empirical findings in previous studies. We also propose a data-driven weighted backtest with good power properties to evaluate HS and FHS forecasts. A Monte Carlo study and an empirical application with three US stocks confirm our theoretical findings. The empirical application shows that multiplication factors computed under the current regulatory framework are downward biased, as they inherit the inconsistency of the unconditional backtest.     

VaR：非对称成分GARCH模型的应用

采用ACGARCH模型在正态分布和广义误差分布下对上证综合指数的VaR值进行估计,然后把它与应用GARCH模型的估计结果进行比较分析,并进行了Kupiec失败率检验。     

On the performance of the minimum VaR portfolio

Alexander and Baptista [2002. Economic implications of using a mean-value-at-risk (VaR) model for portfolio selection: A comparison with mean–variance analysis. Journal of Economic Dynamics and Control 26: 1159–93] develop the concept of mean-VaR efficiency for portfolios and demonstrate its very close connection with mean–variance efficiency. In particular, they identify the minimum VaR portfolio as a special type of mean–variance efficient portfolio. Our empirical analysis finds that, for commonly used VaR breach probabilities, minimum VaR portfolios yield ex post returns that conform well with the specified VaR breach probabilities and with return/risk expectations. These results provide a considerable extension of evidence supporting the empirical validity and tractability of the mean-VaR efficiency concept.     

Optimal Insurance Under the Insurer's VaR Constraint

In this paper, we impose the insurer's Value at Risk (VaR) constraint on Arrow's optimal insurance model. The insured aims to maximize his expected utility of terminal wealth, under the constraint that the insurer wishes to control the VaR of his terminal wealth to be maintained below a prespecified level. It is shown that when the insurer's VaR constraint is binding, the solution to the problem is not linear, but piecewise linear deductible, and the insured's optimal expected utility will increase as the insurer becomes more risk-tolerant. Basak and Shapiro (2001) showed that VaR risk managers often choose larger risk exposures to risky assets. We draw a similar conclusion in this paper. It is shown that when the insured has an exponential utility function, optimal insurance based on VaR constraint causes the insurer to suffer larger losses than optimal insurance without insurer's risk constraint.     

估计VaR的一种新方法及实证分析

在VaR计算的众多方法中,历史模拟法以其概念直观、计算简单、容易实施,被越来越广泛地应用到市场风险的管理中.然而采用历史模拟法计算VaR时,会出现VaR高估现象,另外,历史模拟法在计算VaR时没有很好的反映金融数据的聚集现象和厚尾现象.有经济学家在1999年曾提出过一种改进方法.本文提出了一种新的估计方法,实证分析显示新的估计方法能更好地揭示资产的风险水平.     

国际银行业VaR信贷风险技术发展与应用

以VaR为基础的信贷风险模型使用金融工程方法，定量测定管理信贷风险，使得贷款等信贷资产可定价出售或证券化后转移，同时注意全部信贷资产的组合风险，充分考虑不同信贷资产具有的联动或对冲关系。信贷风险模型中，信用评级模型及信用等级转移矩阵是信贷风险险度量的基础，违约和盯市等模型是信贷风险度量的主要方法，信用衍生工具等模型则直接为信贷风险管理经营服务。国际银行业非常重视，已开始将以VaR为基础的信贷风险模型作为先进信贷风险评估管理技术应用于银行业实务。     

VaR limits for pension funds: an evaluation

In a series of papers during the last ten years an interest rate theory with models which are driven by Lévy or more general processes has been developed. In this paper we derive explicit formulas for the correlations of interest rates as well as zero coupon bonds with different maturities. The models considered in this general setting are the forward rate (HJM), the forward process and the LIBOR model as well as the multicurrency extension of the latter. Specific subclasses of the class of generalized hyperbolic Lévy motions are studied as driving processes. Based on a data set of parametrized yield curves derived from German government bond prices we estimate correlations. In a second step the empirical correlations are used to calibrate the Lévy forward rate model. The superior performance of the Lévy driven models becomes obvious from the graphs.     

Skewness and leptokurtosis in GARCH-typed VaR estimation of petroleum and metal asset returns

This paper utilizes the most flexible skewed generalized t (SGT) distribution for describing petroleum and metal volatilities that are characterized by leptokurtosis and skewness in order to provide better approximations of the reality. The empirical results indicate that the forecasted Value-at-Risk (VaR) obtained using the SGT distribution provides the most accurate out-of-sample forecasts for both the petroleum and metal markets. With regard to the unconditional and conditional coverage tests, the SGT distribution produces the most appropriate VaR estimates in terms of the total number of rejections; this is followed by the nonparametric distribution, generalized error distribution (GED), and finally the normal distribution. Similarly, in the dynamic quantile test, the VaR estimates generated by the SGT and nonparametric distributions perform better than that generated by other distributions. Finally, in the superior predictive test, the SGT distribution has significantly lower capital requirements than the nonparametric distribution for most commodities.     

基于超高频数据的日内风险价值度量研究

随着金融市场的快速发展,传统的以日为单位的风险价值（VaR）已无法满足金融风险管理的需求,计算持有期小于1天的日内风险价值（Intraday VaR）显得愈加重要。文中对日内风险价值测度的方法进行了梳理、细化和改进,对测度中细节给予更加充分的考虑,最后结合我国股市特点对日内风险价值测度进行了实证研究。     

信用风险度量——VaR值的计算方法研究

信用风险是银行业所面临的最古老也是最主要的一类风险.在金融市场迅猛发展同时也复杂多变的今天,迫切需要度量和管理信用风险的先进方法.CreditMetnics模型是以VaR理论为依据,以信用转移分析为核心的高级信用风险度量模型.本文着重阐述了计算信用风险VaR值的方差一协方差法和蒙特卡洛模拟法,比较了两种方法的计算结果,并基于仿真试验说明了后者可以更好地解决信用风险的"厚尾现象".     

Large Sample Behavior of the CTE and VaR Estimators under Importance Sampling

Abstract

The α-level value at risk (Var) and the α-level conditional tail expectation (CTE) of a continuous random variable X are defined as its α-level quantile (denoted by q_α ) and its conditional expectation given the event {X > q_α }, respectively. Var is a popular risk measure in the banking sector, for both external and internal reporting purposes, while the CTE has recently become the risk measure of choice for insurance regulation in North America. Estimation of the CTE for company assets and liabilities is becoming an important actuarial exercise, and the size and complexity of these liabilities make inference procedures with good small sample performance very desirable. A common situation is one in which the CTE of the portfolio loss is estimated using simulated values, and in such situations use of variance reduction techniques such as importance sampling have proved to be fruitful. Construction of confidence intervals for the CTE relies on the availability of the asymptotic distribution of the normalized CTE estimator, and although such a result has been available to actuaries, it has so far been supported only by heuristics. The main goal of this paper is to provide an honest theorem establishing the convergence of the normalized CTE estimator under importance sampling to a normal distribution. In the process, we also provide a similar result for the Var estimator under importance sampling, which improves upon an earlier result. Also, through examples we motivate the practical need for such theoretical results and include simulation studies to lend insight into the sample sizes at which these asymptotic results become meaningful.     

外汇风险度量研究——基于GARCH类模型及VaR方法

本文应用GARCH模型和VaR方法对美元、欧元、日元和港币对人民币的四种外汇汇率进行了实证分析,得出如下主要结论:四种汇率波动率序列均为非正态平稳序列,存在显著的ARCH效应,GARCH类模型可以有效地刻画其非线性动态波动特性.四种汇率均具有自我稳定功能,且美元和港币汇率波动的持续性显著地高于欧元和日元.在日元汇率中存在显著的非对称效应,而在美元和港币汇率中存在显著的风险补偿效应,欧元和日元的汇率风险大约为美元和港币的6-7倍;欧元汇率最理想的估计模型为GARCH(1,1)和IGARCH(1,1),而美元和港币汇率的首选模型为GARCH-M(1,1)-t和GARCH-M(1,1)-g,日元汇率的首选模型为PARCH(1,1)-t和EGARCH(1,1)-g.     

A robust VaR model under different time periods and weighting schemes

This paper analyses several volatility models by examining their ability to forecast Value-at-Risk (VaR) for two different time periods and two capitalization weighting schemes. Specifically, VaR is calculated for large and small capitalization stocks, based on Dow Jones (DJ) Euro Stoxx indices and is modeled for long and short trading positions by using non parametric, semi parametric and parametric methods. In order to choose one model among the various forecasting methods, a two-stage backtesting procedure is implemented. In the first stage the unconditional coverage test is used to examine the statistical accuracy of the models. In the second stage a loss function is applied to investigate whether the differences between the models, that calculated accurately the VaR, are statistically significant. Under this framework, the combination of a parametric model with the historical simulation produced robust results across the sample periods, market capitalization schemes, trading positions and confidence levels and therefore there is a risk measure that is reliable.