Bayesian Value-at-Risk with product partition models

In this paper we propose a novel Bayesian methodology for Value-at-Risk computation based on parametric Product Partition Models. Value-at-Risk is a standard tool for measuring and controlling the market risk of an asset or portfolio, and is also required for regulatory purposes. Its popularity is partly due to the fact that it is an easily understood measure of risk. The use of Product Partition Models allows us to remain in a Normal setting even in the presence of outlying points, and to obtain a closed-form expression for Value-at-Risk computation. We present and compare two different scenarios: a product partition structure on the vector of means and a product partition structure on the vector of variances. We apply our methodology to an Italian stock market data set from Mib30. The numerical results clearly show that Product Partition Models can be successfully exploited in order to quantify market risk exposure. The obtained Value-at-Risk estimates are in full agreement with Maximum Likelihood approaches, but our methodology provides richer information about the clustering structure of the data and the presence of outlying points.     

(Non-)robustness of maximum likelihood estimators for operational risk severity distributions

The quality of operational risk data sets suffers from missing or contaminated data points. This may lead to implausible characteristics of the estimates. Outliers, especially, can make a modeler's task difficult and can result in arbitrarily large capital charges. Robust statistics provides ways to deal with these problems as well as measures for the reliability of estimators. We show that using maximum likelihood estimation can be misleading and unreliable assuming typical operational risk severity distributions. The robustness of the estimators for the Generalized Pareto distribution, and the Weibull and Lognormal distributions is measured considering both global and local reliability, which are represented by the breakdown point and the influence function of the estimate.     

Haar wavelets-based approach for quantifying credit portfolio losses

This paper proposes a new methodology to compute Value at Risk (VaR) for quantifying losses in credit portfolios. We approximate the cumulative distribution of the loss function by a finite combination of Haar wavelet basis functions and calculate the coefficients of the approximation by inverting its Laplace transform. The Wavelet Approximation (WA) method is particularly suitable for non-smooth distributions, often arising in small or concentrated portfolios, when the hypothesis of the Basel II formulas are violated. To test the methodology we consider the Vasicek one-factor portfolio credit loss model as our model framework. WA is an accurate, robust and fast method, allowing the estimation of the VaR much more quickly than with a Monte Carlo (MC) method at the same level of accuracy and reliability.     

On a subjective approach to risk measurement

This study is based on the analogy between hedging a risky asset and keeping reserves to meet an unknown demand. The optimal hedging level, which depends on individual preferences, is regarded as a measure of risk. We determine the set of optimal levels and investigate the properties of the associated risk measures. This approach provides a new insight into Value at Risk (VaR). We consider it as a solution of a certain optimal inventory problem with linear cost and loss functions. We show that these functions determine the confidence level of VaR. In this way we obtain a simple model that helps us to choose a proper confidence level α and explains why supervisory institutions (such as the Basle Committee) choose a higher α than financial institutions themselves.     

中国股票市场风险度量的模型选择与比较研究

分别采用等权移动平均方法、指教加权移动平均方法、GARCH(1,1)方法、GARCH(1,1)-t方法和Pareto型极值分布方法计算上海和深圳股票日收益率的VaR.向后检验表明,Pareto型极值分布方法比其他方法更能准确地反映我国股市的风险.     

论银行市场风险的资本计提——兼评内部模型法的适用性

市场风险及其监管资本要求的计量历来为业界和监管当局所关注。近期,次贷危机爆发导致的市场动荡使得全球银行业和监管当局开始重新审视其市场风险管理和监管资本要求。文章结合国际银行业和监管机构计量市场风险及其监管资本要求的当前做法,针对我国银行业的实际情况,重点探索了内部模型法在我国银行业的适用性,尤其是从方法论、特殊风险计量、验证等角度探讨了内部模型法的主要工具——风险价值体系在我国银行业计量市场风险及其监管资本要求的适用性,并从方法论和应用层面提出了相应的政策建议。     

A smooth non-parametric estimation framework for safety-first portfolio optimization

In this paper, we adopt a smooth non-parametric estimation to explore the safety-first portfolio optimization problem. We obtain a non-parametric estimation calculation formula for loss (truncated) probability using the kernel estimator of the portfolio returns’ cumulative distribution function, and embed it into two types of safety-first portfolio selection models. We numerically and empirically test our non-parametric method to demonstrate its accuracy and efficiency. Cross-validation results show that our non-parametric kernel estimation method outperforms the empirical distribution method. As an empirical application, we simulate optimal portfolios and display return-risk characteristics using China National Social Security Fund strategic stocks and Shanghai Stock Exchange 50 Index components.     

The disclosure of risk in financial statements

The accounting information currently issued by firms is not wholly adequate when used for decision making purposes, and within that process, for forecasting, for which additional information on risks is required. Therefore a reform of the current framework becomes necessary. Within this reform an adequate scheme and typology for the risks facing firms must be established and a set of specific risk quantification models must be designed. This paper focuses on both issues, showing all the risks that can affect firms and proposing a quantification model for each one.     

收益率曲线动态与银行经济价值的利率风险

本文在收益率曲线动态的主成分分析基础上,运用MonteCarlo模拟的主成分VaR方法,以我国五家商业银行为样本研究银行账户经济价值利率风险的计量方法,并与巴塞尔委员会标准久期法的结果进行比较。同时,对VaR模型的有效性进行了样本外的返回检验。研究发现,五家银行的经济价值面临的是利率上升的风险;非正态主成分VaR模型估计的经济价值利率风险,都要大于正态主成分VaR模型的结果,这反映了利率波动的厚尾特征,正态假设有可能低估风险。     

Portfolio optimization for student t and skewed t returns

We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential Lévy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors—one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of Fouque et al. [Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, 2011] to models of the exponential Lévy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility, as demonstrated in a calibration to S&;P500 options data.     

On Tail Value-at-Risk for sums of non-independent random variables with a generalized Pareto distribution

Recently in actuarial literature several authors have derived lower and upper bounds in the sense of convex order for sums of random variables with given marginal distributions and unknown dependency structure. In this paper, we derive convex bounds for sums of non-independent and identically distributed random variables when marginal distributions are mixture models. In particular, we examine some well-known risk measures and we find approximations for Tail Value-at-Risk of the sums considered when marginal distributions are generalized Pareto distributions. By numerical examples we illustrate the goodness of the presented approximations.      

Estimation of multiple period expected shortfall and median shortfall for risk management

With the regulatory requirements for risk management, Value at Risk (VaR) has become an essential tool in determining capital reserves to protect the risk induced by adverse market movements. The fact that VaR is not coherent has motivated the industry to explore alternative risk measures such as expected shortfall. The first objective of this paper is to propose statistical methods for estimating multiple-period expected shortfall under GARCH models. In addition to the expected shortfall, we investigate a new tool called median shortfall to measure risk. The second objective of this paper is to develop backtesting methods for assessing the performance of expected shortfall and median shortfall estimators from statistical and financial perspectives. By applying our expected shortfall estimators and other existing approaches to seven international markets, we demonstrate the superiority of our methods with respect to statistical and practical evaluations. Our expected shortfall estimators likely provide an unbiased reference for setting the minimum capital required for safeguarding against expected loss.     

Portfolio diversification and value at risk under thick-tailedness†

This paper focuses on the study of portfolio diversification and value at risk analysis under heavy-tailedness. We use a notion of diversification based on majorization theory that will be explained in the text. The paper shows that the stylized fact that portfolio diversification is preferable is reversed for extremely heavy-tailed risks or returns. However, the stylized facts on diversification are robust to heavy-tailedness of risks or returns as long as their distributions are moderately heavy-tailed. Extensions of the results to the case of dependence, including convolutions of α-symmetric distributions and models with common shocks are provided.     

Optimal Insurance Design Under a Value-at-Risk Framework

This study designs an optimal insurance policy form endogenously, assuming the objective of the insured is to maximize expected final wealth under the Value-at-Risk (VaR) constraint. The optimal insurance policy can be replicated using three options, including a long call option with a small strike price, a short call option with a large strike price, and a short cash-or-nothing call option. Additionally, this study also calculates the optimal insurance levels for these models when we restrict the indemnity to be one of three common forms: a deductible policy, an upper-limit policy, or a policy with proportional coinsurance. JEL Classification No: G22     

The impact of the choice of VaR models on the level of regulatory capital according to Basel II

The Basel II framework allows the calculation of the capital requirements for market risk with Value-at-Risk models. Since no special model is prescribed in the framework, banks may use simple models with questionable assumptions concerning their underlying distributions. Our numerical analysis reveals that simple VaR models that perform noticeably worse than comparable simple models with more realistic assumptions may lead to a lower level of regulatory capital for banks. For this reason, banks have a major incentive to implement bad models. This is obviously contrary to the interests of regulatory authorities.     

Nonparametric Inference of Value-at-Risk for Dependent Financial Returns

The article considers nonparametric estimation of value-at-risk(VaR) and associated standard error estimation for dependentfinancial returns. Theoretical properties of the kernel VaRestimator are investigated in the context of dependence. Thepresence of dependence affects the variance of the VaR estimatesand has to be taken into consideration in order to obtain adequateassessment of their variation. An estimation procedure of thestandard errors is proposed based on kernel estimation of thespectral density of a derived series. The performance of theVaR estimators and the proposed standard error estimation procedureare evaluated by theoretical investigation, simulation of commonlyused models for financial returns, and empirical studies onreal financial return series.     

Systemic risk and asymmetric responses in the financial industry

To date, an operational measure of systemic risk capturing nonlinear tail-comovements between system-wide and individual bank returns has not yet been developed. This paper proposes an extension of the CoVaR methodology in Adrian and Brunnermeier (2011) to capture the asymmetric response of the banking system to positive and negative shocks to the market-valued balance sheets of individual banks. Building on a comprehensive sample of U.S. banks in the period 1990–2010, the evidence in this paper shows that ignoring asymmetries that feature tail-interdependences may lead to a severe underestimation of systemic risk. On average, the relative impact on the system of a fall in individual market value is sevenfold that of an increase. Moreover, the downward bias in systemic-risk measuring from ignoring this asymmetric pattern increases with bank size. In particular, the conditional tail-comovement between the banking system and a bank that is losing market value belonging to the top size-sorted decile is nearly 5.5 times larger than the unconditional tail-comovement versus 3.3 times for banks in the bottom decile. The asymmetric model also produces much better fitting, with the restriction that gives rise to the standard symmetric model being rejected for most firms in the sample, particularly, in the segment of large-scale banks. This result is important from a regulatory and supervisory perspective, since the asymmetric generalization enhances the capacity to monitor systemic interdependences.     

基于VaR的开放式股票型基金市场风险的测量与评价

通过采用半参数法计算投资组合VaR,得到相应VaR的近似置信区间,并结合成分VaR、边际VaR对投资组合vaR进行分解,结果发现,VaR作为风险管理工具同样可以有效应用于开放式股票型基金市场风险的测量与评价.     

Learning about Risk: Some Lessons from Insurance

This paper argues that in the fundamental subject of financialrisk analysis, some valuable lessons may be drawn from insurance.The probability of ruin, defined as a first passage time, carriesa dynamic element whose absence in Value at Risk is one liability,among others. Extreme value theory, which has been successfullyapplied to insurance shortly after it was introduced in probability,may offer a coherent framework for analyzing the extreme movessuch as the ones observed in recent foreign exchange and financialcrises. Lastly, we show that the genuine hazards generated byglobal capital markets and illustrated by the events of summer1998, generate a market incompleteness that existing modelsof defaultable bonds do not fully address. In contrast, thelong experience of risk premium analysis in the insurance andreinsurance industry, as well as the existence of historicaldata on natural disasters, render the valuation of catastrophebonds less perilous than that of defaultable bonds.     

Financial networks and systemic risk vulnerabilities: A tale of Indian banks

This study identifies the nature and direction of unprecedented upheavals in the Indian banking sector which is linked to credit market asymmetry. A tail-driven network approach with a mixed sample of banks and firms exhibits the characteristics of the twin-balance-sheet syndrome. We construct the networks with a degree of interconnectedness at different quantiles and identify major systemic risk emitters and receivers. Furthermore, we find a spillover of the riskiness of deep-in-debt firms to banks. Smaller banking institutions evince a greater connection to banks and firms than larger ones. Our results are valuable for policymakers formulating financial stabilization policies and investors considering Indian markets for various opportunities.