Analysis of criteria VaR and CVaR

Criteria VaR (Value-at-Risk) and CVaR (Conditional Value-at-Risk), which are well-known in financial mathematics, are compared. Some connection between them is established. Ways of choice a level of confidence probability for the quantile optimization problem are suggested. The ways are based on some equations of balance between VaR and CVaR. Examples are discussed.     

Minimizing CVaR in global dynamic hedging with transaction costs

This study develops a global derivatives hedging methodology which takes into account the presence of transaction costs. It extends the Hodges and Neuberger [Rev. Futures Markets, 1989, 8, 222–239] framework in two ways. First, to reduce the occurrence of extreme losses, the expected utility is replaced by the conditional Value-at-Risk (CVaR) coherent risk measure as the objective function. Second, the normality assumption for the underlying asset returns is relaxed: general distributions are considered to improve the realism of the model and to be consistent with fat tails observed empirically. Dynamic programming is used to solve the hedging problem. The CVaR minimization objective is shown to be part of a time-consistent framework. Simulations with parameters estimated from the S&P 500 financial time series show the superiority of the proposed hedging method over multiple benchmarks from the literature in terms of tail risk reduction.     

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.     

A comparison of non-Gaussian VaR estimation and portfolio construction techniques

We propose a multivariate model of returns that accounts for four of the stylised facts of financial data: heavy tails, skew, volatility clustering, and asymmetric dependence with the aim of improving the accuracy of risk estimates and increasing out-of-sample utility of investors’ portfolios. We accommodate volatility clustering, the generalised Pareto distribution to capture heavy tails and skew, and the skewed-t copula to provide for asymmetric dependence. The proposed approach produces more accurate VaR estimates than seven competing approaches across eight data sets encompassing five asset classes. We show that this produces portfolios with higher utility, and lower downside risk than alternative approaches including mean–variance. We confirm that investors can substantially increase utility by accounting for departures from normality.     

Optimal portfolio allocation under the probabilistic VaR constraint and incentives for financial innovation

We characterize the investor’s optimal portfolio allocation subject to a budget constraint and a probabilistic VaR constraint in complete markets environments with a finite number of states. The set of feasible portfolios might no longer be connected or convex, while the number of local optima increases exponentially with the number of states, implying computational complexity. The optimal constrained portfolio allocation may therefore not be monotonic in the state–price density. We propose a type of financial innovation, which splits states of nature, that is shown to weakly enhance welfare, restore monotonicity of the optimal portfolio allocation in the state-price density, and reduce computational complexity. We are grateful to Ken Kavajecz and seminar participants at Harvard Business School, London School of Economics, Maastrict University, ZEI Bonn, and Danske Bank Symposium on Asset allocation and Value-at-Risk: Where Theory Meets Practice for comments on an earlier version of this paper. We also benefitted from the suggestions of two anonymous referees. Our papers can be downloaded from www.RiskResearch.org.     

VaR与CVaR在商业银行风险度量中的比较分析及应用

本文系统地阐述了时下银行流行的VaR(ValueatRisk)风险度量技术,并分析了该理论存在的缺陷和使用上的局限性,从而提出以CVaR(ConditionalValueatRisk)模型作为风险度量的替代方法,详细分析了CVaR的原理、特长以及在银行业应用前景,包括风险度量、绩效分析和行为指引等方面的突出作用。最后研究了CVaR在我国商业银行的具体应用。     

Efficient hybrid methods for portfolio credit derivatives

In this paper we discuss the computation of basket credit default swaps and collateralized debt obligation squared transactions. We suggest two hybrid algorithms for these two portfolio credit derivatives. The method combines the analytic approximation to the loss distribution of conditionally independent heterogeneous portfolios with the Monte Carlo simulation. The efficiency and accuracy of the algorithms are illustrated with examples.     

基于GARCH族模型对H股指数期货的VaR与CVaR值的实证研究

本文以港交所H股指数期货的收盘价格数据作为实证载体,基于GARCH族模型中残差的正态分布、T分布和广义误差分布（GED）三种不同情形,分别采用GARCH、EGARCH及PARCH模型,计算H股指数期货收益波动序列的VaR和CVaR值,结果表明基于广义误差分布的PARCH模型（GED-PARCH）无论在计算VaR值还是CVaR值方面都是最优的。     

基于GARCH族模型对H股指数期货的VaR与CVaR值的实证研究

以港交所H股指数期货的收盘价格数据作为实证载体,基于GARCH族模型中残差的正态分布、T分布和广义误差分布（GED）三种不同情形,分别采用GARCH、EGARCH及PARCH模型,计算H股指数期货收益波动序列的VaR和CVaR值,结果表明基于广义误差分布的PARCH模型（GED—PARCH）无论在计算VaR值还是CVaR值方面都是最优的。     

Sensitivity estimates for portfolio credit derivatives using Monte Carlo

Portfolio credit derivatives are contracts that are tied to an underlying portfolio of defaultable reference assets and have payoffs that depend on the default times of these assets. The hedging of credit derivatives involves the calculation of the sensitivity of the contract value with respect to changes in the credit spreads of the underlying assets, or, more generally, with respect to parameters of the default-time distributions. We derive and analyze Monte Carlo estimators of these sensitivities. The payoff of a credit derivative is often discontinuous in the underlying default times, and this complicates the accurate estimation of sensitivities. Discontinuities introduced by changes in one default time can be smoothed by taking conditional expectations given all other default times. We use this to derive estimators and to give conditions under which they are unbiased. We also give conditions under which an alternative likelihood ratio method estimator is unbiased. We illustrate the application and verification of these conditions and estimators in the particular case of the multifactor Gaussian copula model, but the methods are more generally applicable.      

长寿风险对基本医疗保险统筹基金的冲击效应研究——基于VaR与CVaR模型的综合测算

借鉴金融风险管理中VaR和CVaR模型对尾部风险的测量思路,通过构建有限数据Lee-Carter死亡率预测模型,测算了人口的长寿风险及其对基本医疗保险统筹基金的冲击效应,结果表明:2015-2060年基本医疗保险参保人群将面临巨大的长寿风险,极端情况下长寿风险将给统筹基金收支结余超预期下降造成不容忽视的尾部损失;推迟退休年龄、提高生育率、调整个人账户和报销比例、提高职工缴费工资和控制住院费用增长均可以在一定程度上缓解长寿风险的冲击.建议明确政府在基本医疗保险长寿风险管理中的主导作用,构建医疗、养老和长期护理保险的三险联动保障机制.     

Demand for insurance in a portfolio setting

This paper takes an additional step toward analyzing the demand for insurance in the context of a portfolio model. An investor is endowed with a portfolio containing a risky and riskless asset that can be augmented by purchasing insurance. Here, insurance is paid for by reducing the quantity of the risky insurable asset, holding the quantity of the riskless asset fixed. In the standard insurance demand model, insurance is paid for by reducing the amount of the riskless asset. This distinction leads to a different insurance demand function because the opportunity cost of purchasing insurance is now random.     

Consistent loss prediction for a portfolio and its subportfolios

In the present paper, we consider a portfolio of risks consisting of two subportfolios, and we study the problem of whether or not the predictors based on the subportfolios are consistent with those based on the full portfolio. We study this aggregation problem for both the chain-ladder method and the additive method (or incremental loss ratio method). In the case of the chain-ladder method we extend results of Ajne and Klemmt, using the duality of the chain-ladder method applied to incremental losses; we also give a short proof for this duality, which was first observed by Barnett, Zehnwirth & Dubossarky. In the case of the additive method the aggregation problem has not been studied before and its solution is surprisingly simple.     

Applying CVaR for decentralized risk management of financial companies

Over the past decade, financial companies have merged diverse areas including investment banking, insurance, retail banking, and trading operations. Despite this diversity, many global financial firms suffered severe losses during the recent recession. To reduce enterprise risks and increase profits, we apply a decentralized risk management strategy based on a stochastic optimization model. We extend the decentralized approach with the CVaR risk-metric, showing the advantages of CVaR over traditional risk measures such as value-at-risk. An example taken from the earthquake insurance area illustrates the concepts.     

Do ADRs enhance portfolio performance for a domestic portfolio? Evidence from the 1990s

American depository receipts (ADRs) represent an increasingly popular and convenient mechanism for international investing. We analyze ADRs traded throughout the 1990s and find that these securities offer a diversification and portfolio performance benefit when combined with a domestic portfolio (proxied by the S&P 500). While we find that emerging market ADRs are effective instruments for reducing portfolio risk, they do not improve portfolio performance as measured by the Sharpe ratio. Developed market ADRs do improve portfolio performance as measured by the Sharpe ratio. The asset allocation which maximizes the Sharpe ratio is 84 percent domestic stocks, 16 percent developed ADRs, and 0 percent emerging ADRs. Further, due to problems in defining an appropriate market index for ADRs, the Sharpe ratio is viewed to be the preferred performance measure. Other measures such as Jensen’s alpha and the Treynor measure are susceptible to being “gamed” to distort portfolio performance.     

New strategies and a new paradigm for Shariah-compliant portfolio optimization

In this paper we analyze the effects of different strategies to construct Shariah compatible financial portfolios. The difference between conventional and current Shariah portfolio management is the application of sector screens and financial screens by which the asset universe is reduced. Yet, here different schools of scholars define different screening rules leading to significant differences with respect to compliance, but also with respect to performance. After analyzing this discrepancy we propose several new strategies to apply the inconsistent rule systems and a new paradigm for defining Shariah-compliance. Under this new paradigm compliance is attributed to the portfolio and not to the individual assets of the universe. We report results of an empirical study analyzing the potentials of these strategies and of the paradigm. We can show that under the proposed concepts Shariah-compliant portfolios can be realized which have return and risk profiles comparable to the conventional non-constrained portfolios.     

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.     

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.     

Portfolio choices and VaR constraint with a defaultable asset

We consider a Constant Elasticity of Variance (CEV) model for the asset price of a defaultable asset showing the so-called leverage effect (high volatility when the asset price is low). We show that a VaR constraint re-evaluated over time induces an agent more risk averse than a logarithmic utility to take more risk than in the unconstrained setting.