Analytical VaR for international portfolios with common jumps

International portfolios which are composed of domestic assets and foreign assets are popular investment tools for financial institutions in highly integrated global financial markets. However, the focus of past studies had been on either domestic assets or foreign assets, but not both in the same context. They paid no attention to the studies of controlling the market risk of the international portfolios in the risk management literature. In contrast to the existing literature in portfolios, this paper considers not only domestic assets but also foreign assets, and provides an analytical value-at-risk (VaR) with common jump risk and exchange rate risk to manage market risk of international portfolios with exchange rate risk and common jumps over the subprime mortgage crisis. In general, the analytical solution can be used to accurately calculate VaRs by the backtesting criterion in terms of in-sample and out-of-sample fitting for an international portfolio with common jumps.     

基于风险价值约束的动态均值-方差投资组合的研究

研究了基于风险价值约束的动态均值-方差项目投资组合的数学模型,该模型是控制带约束的随机线性二次型（LQ）控制问题.在讨论该随机LQ控制问题的解之后,给出投资组合动态数学模型对应的随机哈密顿-雅克比-贝尔曼方程的解,得出了有效边界和最佳策略,讨论了风险价值约束的影响.最后,针对某油田勘探开发项目的实际情况,应用上述结论求出该实例的解,并讨论了风险价值约束发挥的作用.     

基于历史模拟法的风险价值算法在GPU上的实现和优化

风险价值(Value at Risk,VaR)是风险管理的基本工具,可对现有头寸的下行风险提供量化衡量方法。基于历史模拟法的VaR(Historical VaR)是最流行的计算方法之一,被广泛应用于世界各大金融机构。对金融产品进行实时或准实时的VaR计算,对于及时规避金融风险具有重要意义。由于金融产品日益复杂,产品数量持续增长,现有CPU计算平台上的计算能力已经难以满足VaR的性能需求。为解决这一问题, 在GPU上使用CUDA 对Historical VaR的计算代码进行了实现和优化。通过改进排序算法、基于Multi-stream 隐藏通讯时间、解耦数据依赖并实现细粒度并行等优化方法,CUDA版本的VaR计算性能比优化后的CPU单核性能提升了42.6倍,为快速计算超大数量债券的VaR提供了有效的解决方案。以上优化方法也可以为金融领域内其他算法的GPU化提供思路。     

Extremal financial risk models and portfolio evaluation

It is difficult to find an existing single model which is able to simultaneously model exceedances over thresholds in multivariate financial time series. A new modeling approach, which is a combination of max-stable processes, GARCH processes, and Markov processes, is proposed. Combining Markov processes and max-stable processes defines a new statistical model which has the flexibility of modeling cross-sectional tail dependencies between risk factors and tail dependencies across time. The new model also models asymmetric behaviors of negative and positive returns on financial assets. An important application of the proposed method is to calculate value at risk (VaR) and evaluate portfolio combinations under VaR constraints. Result comparisons between VaRs based on the new approach and VaRs based on some existing methods such as variance-covariance approach and historical simulation approach suggest that some existing methods substantially underestimate the risks during recession and expansion time.     

“报童问题”中风险偏好下的条件风险值及其优化

建立了报童随机利润的分布函数,得到任意风险水平下的VaR解析表达式或应满足的条件。考虑缺货成本,针对风险规避和风险偏爱两种情况,分别建立不同订购量和风险水平下的条件风险值模型,并将模型中对利润变量的积分转换为对随机需求变量的积分,解决了模型中因包含VaR变量而求解困难的问题。分析了给定风险水平下的最优决策,讨论了缺货成本为0的特殊情形。风险中性报童的期望利润与最优决策可由风险规避或风险偏爱的相应公式推导。最后对下一步的研究方向进行了展望。     

The impact of the dependence structure in risk management: a focus on credit-risk

The aim of the paper is to discuss the important role of the dependence structure in risk management. Therefore, we focus on credit-risk and propose an innovative model to value the credit risk of a portfolio. This new approach (HYC for short) is based on a hierarchical hybrid copula and involves a clusterization of the portfolio in several risk's classes. The HYC model is classified as hybrid because the computation of the loss cdf depends on the class's cardinality: for large groups one is justified to apply a limiting approach, while for small ones one applies a procedure preserving the granularity of the group itself. In order to appreciate the impact of the dependence structure in credit-risk evaluation, a VaR analysis based on the HYC loss function is here compared to the CreditMetrics approach in an in-sample exercise and to the empirical VaR in an out-of sample exercise aimed to test the forecasting effectiveness of the model. This comparison allows us to appreciate over/under-valuation of the capital detained from the financial institution. Moreover, the impact of an enlargement of the dependence structure is discussed with respect to the systemic/contagious effects in the context of a portfolio optimisation with constraint on a sub-portfolio's risk.     

Hesitant fuzzy linguistic portfolio model with variable risk appetite and its application in the investment ratio calculation

Qualitative evaluation information is important for financial decision-making and investment when quantitative data are unavailable. Although an alternative ranking is available, specific portfolio and optimal investment ratios cannot be obtained by using the qualitative decision-making methods. To address this issue, this paper proposes a hesitant fuzzy linguistic portfolio model based on the max-score rule and the hesitant fuzzy linguistic element with variable risk appetite (HFLE-RA). The HFLE-RA is able to express qualitative evaluation information by using the hesitant fuzzy linguistic term set and describe the variable investor risk appetites by introducing the asymmetric sigmoid semantics. Thus, different investors can be distinguished by the risk appetite parameters according to the asymmetric sigmoid semantics, and the optimal investment ratios can be obtained by applying the proposed portfolio model. Moreover, the investment opportunities and efficient frontiers of the hesitant fuzzy linguistic portfolio model are investigated. Also, a value-at-risk fitting approach is introduced to calculate the risk appetite parameters. Based on these works, a qualitative investment ratio calculation process is provided in the HFLE-RA environment. Lastly, a real example of calculating the optimal investment ratios for four newly listed stocks in the Growth Enterprises Market board of the Shenzhen Stock Exchange is provided to demonstrate the proposed approaches.     

Approximating the distributions of estimators of financial risk under an asymmetric Laplace law

Explicit expressions are derived for parametric and nonparametric estimators (NPEs) of two measures of financial risk, value-at-risk (VaR) and conditional value-at-risk (CVaR), under random sampling from the asymmetric Laplace (AL) distribution. Asymptotic distributions are established under very general conditions. Finite sample distributions are investigated by means of saddlepoint approximations. The latter are highly computationally intensive, requiring novel approaches to approximate moments and special functions that arise in the evaluation of the moment generating functions. Plots of the resulting density functions shed new light on the quality of the estimators. Calculations for CVaR reveal that the NPE enjoys greater asymptotic efficiency relative to the parametric estimator than is the case for VaR. An application of the methodology in modeling currency exchange rates suggests that the AL distribution is successful in capturing the peakedness, leptokurticity, and skewness, inherent in such data. A demonstrated superiority in the resulting parametric-based inferences delivers an important message to the practitioner.     

Worst-case VaR and robust portfolio optimization with interval random uncertainty set

This paper addresses a new uncertainty set – interval random uncertainty set for worst-case value-at-risk and robust portfolio optimization. The form of interval random uncertainty set makes it suitable for capturing the downside and upside deviations of real-world data. These deviation measures capture distributional asymmetry and lead to better optimization results. We also apply our interval random chance-constrained programming to robust worst-case value-at-risk optimization under interval random uncertainty sets in the elements of mean vector and covariance matrix. Numerical experiments with real market data indicate that our approach results in better portfolio performance.     

A multi-objective robust possibilistic model for technology portfolio optimization considering social impact and different types of financing

With respect to limited financial resources, prioritization of technology fields in order to be supported financially is a matter of paramount significance that governmental organizations, such as “Technology Development Funds (TDFs)”, face with. Innovation and technology development, as the cornerstone of the economic development of countries, requires making decisions in terms of assigning the best-suited form of financial resources mainly by governments. Accordingly, this study addresses a multi-objective portfolio optimization problem in a multi-period setting with the aim of maximizing the created jobs – as a key factor in social welfare – as well as intended profit while minimizing the risk of inappropriate portfolio selection. To formulate the proposed mathematical model, different financing methods, technology readiness levels (TRL), and return on investment (ROI) associated with each technological project are taken into account. Afterward, to deal with the uncertainty arisen from fuzzy parameters, the Multi-Objective Robust Possibilistic Programming approach (MORPP) is applied, the performance of which is examined under several computational tests. Finally, to illustrate the performance of the proposed model and its applicability in practice, the computational results are shown through a real case study in Iran Innovation & Prosperity Fund (IIPF). The results show that selecting small and medium-sized enterprises (SMEs) for being financed, is the best option when increasing job creation is considered in portfolio optimization. Furthermore, the comparison of the MORPP model results with the deterministic model shows that the solutions obtained from the robust possibilistic approach outweighed the deterministic model.     

Cost functions and model combination for VaR-based asset allocationusing neural networks

We introduce an asset-allocation framework based on the active control of the value-at-risk of the portfolio. Within this framework, we compare two paradigms for making the allocation using neural networks. The first one uses the network to make a forecast of asset behavior, in conjunction with a traditional mean-variance allocator for constructing the portfolio. The second paradigm uses the network to directly make the portfolio allocation decisions. We consider a method for performing soft input variable selection, and show its considerable utility. We use model combination (committee) methods to systematize the choice of hyperparameters during training. We show that committees using both paradigms are significantly outperforming the benchmark market performance.     

Modeling default risk via a hidden Markov model of multiple sequences

Default risk in commercial lending is one of the major concerns of the creditors. In this article, we introduce a new hidden Markov model (HMM) with multiple observable sequences (MHMM), assuming that all the observable sequences are driven by a common hidden sequence, and utilize it to analyze default data in a network of sectors. Efficient estimation method is then adopted to estimate the model parameters. To further illustrate the advantages of MHMM, we compare the hidden risk state process obtained by MHMM with that from the traditional HMMs using credit default data. We then consider two applications of our MHMM. The calculation of two important risk measures: Value-at-risk (VaR) and expected shortfall (ES) and the prediction of global risk state. We first compare the performance of MHMM and HMM in the calculation of VaR and ES in a portfolio of default-prone bonds. A logistic regression model is then considered for the prediction of global economic risk using our MHMM with default data. Numerical results indicate our model is effective for both applications.     

Time Series Simulation with Randomized Quasi-Monte Carlo Methods: An Application to Value at Risk and Expected Shortfall

Quasi-Monte Carlo methods are designed to produce efficient estimates of simulated values but the error statistics of these estimates are difficult to compute. Randomized quasi-Monte Carlo methods have been developed to address this shortcoming. In this paper we compare quasi-Monte Carlo and randomized quasi-Monte Carlo techniques for simulating time series. We use randomized quasi-Monte Carlo to compute value-at-risk and expected shortfall measures for a stock portfolio whose returns follow a highly nonlinear Markov switching stochastic volatility model which does not admit analytical solutions for the returns distribution. Quasi-Monte Carlo methods are more accurate but do not allow the computation of reliable confidence intervals about risk measures. We find that randomized quasi-Monte Carlo methods maintain many of the advantages of quasi-Monte Carlo while also providing the ability to produce reliable confidence intervals of the simulated risk measures. However, the advantages in speed of convergence of randomized quasi-Monte Carlo diminish as the forecast horizon increases.     

基于CVaR和Monte Carlo仿真的贷款组合决策模型

提供了一种新的贷款组合决策优化方法,该模型用更能反映贷款组合信用风险特征的CVaR作为风险度量。由于在实际中很难获取各笔贷款的历史数据,为此给出了一种基于Matlab语言的Monte Carlo仿真方法。从而使谊模型可以通过线性规划技术有效的进行求解。最后给出了一个例子。     

不确定环境下基于VaR和CVaR的投资组合优化模型

对不确定环境下的投资组合问题进行研究,使用不确定测度来定义不确定环境下的VaR和CVaR,并用VaR和CVaR度量风险,建立基于VaR和CVaR风险控制的投资组合优化模型,并设计了集成遗传算法、99-方法的混合智能算法来求解此模型,最后通过实例验证了模型和算法的有效性。     

Solving fuzzy p-hub center problem by genetic algorithm incorporating local search

The p-hub center problem has extensive applications in various real-world fields such as transportation and telecommunication systems. This paper presents a new risk aversion p-hub center problem with fuzzy travel times, in which value-at-risk (VaR) criterion is adopted in the formulation of objection function. For trapezoidal and normal fuzzy travel times, we first turn the original VaR p-hub center problem into its equivalent parametric mixed-integer programming problem, then develop a hybrid algorithm by incorporating genetic algorithm and local search (GALS) to solve the parametric mixed-integer programming problem. In our designed GALS, the GA is used to perform global search, while LS strategy is applied to each generated individual (or chromosome) of the population. Finally, we conduct two sets of numerical experiments and discuss the experimental results obtained by general-purpose LINGO solver, standard GA and GALS. The computational results show that the GALS achieves the better performance than LINGO solver and standard GA.     

Intradaily dynamic portfolio selection

A portfolio selection model which allocates a portfolio of currencies by maximizing the expected return subject to Value-at-Risk (VaR) constraint is designed and implemented. Based on an econometric implementation using intradaily data, the optimal portfolio allocation is forecasted at regular time intervals. For the estimation of the conditional variance from which the VaR is computed, univariate and multivariate GARCH models are used. Model evaluation is done using two economic criteria and two statistical tests. The result for each model is given by the best forecasted intradaily investment recommendations in terms of the optimal weights of the currencies in the risky portfolio. The results show that estimating the VaR from multivariate GARCH models improves the results of the forecasted optimal portfolio allocation, compared to using a univariate model.     

Risk management of risk under the Basel Accord: A Bayesian approach to forecasting Value-at-Risk of VIX futures

It is well known that the Basel II Accord requires banks and other Authorized Deposit-taking Institutions (ADIs) to communicate their daily risk forecasts to the appropriate monetary authorities at the beginning of each trading day, using one or more risk models, whether individually or as combinations, to measure Value-at-Risk (VaR). The risk estimates of these models are used to determine capital requirements and associated capital costs of ADIs, depending in part on the number of previous violations, whereby realised losses exceed the estimated VaR. Previous papers proposed a new approach to model selection for predicting VaR, consisting of combining alternative risk models, and comparing conservative and aggressive strategies for choosing between VaR models. This paper, using Bayesian and non-Bayesian combinations of models addresses the question of risk management of risk, namely VaR of VIX futures prices, and extends the approaches given in previous papers to examine how different risk management strategies performed during the 2008–2009 global financial crisis (GFC). The use of time-varying weights using Bayesian methods, allows dynamic combinations of the different models to obtain a more accurate VaR forecasts than the estimates and forecasts that might be produced by a single model of risk. One of these dynamic combinations is endogenously determined by the pass performance in terms of daily capital charges of the individual models. This can improve the strategies to minimize daily capital charges, which is a central objective of ADIs. The empirical results suggest that an aggressive strategy of choosing the Supremum of single model forecasts, as compared with Bayesian and non-Bayesian combinations of models, is preferred to other alternatives, and is robust during the GFC.     

Modelling tail credit risk using transition matrices

Innovative transition matrix techniques are used to compare extreme credit risk for Australian and US companies both prior to and during the global financial crisis (GFC). Transition matrix methodology is traditionally used to measure Value at Risk (VaR), a measure of risk below a specified threshold. We use it to measure Conditional Value at Risk (CVaR) which is the risk beyond VaR. We find significant differences in VaR and CVaR measurements in both the US and the Australian markets. We also find a greater differential between VaR and CVaR for the US as compared to Australia, reflecting the more extreme credit risk that was experienced in the US during the GFC. Traditional transition matrix methodology assumes that all borrowers of the same credit rating transition equally, whereas we incorporate an adjustment based on industry share price fluctuations to allow for unequal transition among industries. Our revised model shows greater change between Pre-GFC and GFC total credit risk than the traditional model, meaning that those industries that were riskiest during the GFC are not the same industries that were riskiest Pre-GFC. Overall, our analysis finds that our innovative modelling techniques are better able to account for the impact of extreme risk circumstances and industry composition than traditional transition matrix techniques.     

Computational Tools for the Analysis of Market Risk

The estimation and management of risk is an important and complex task faced by market regulators and financial institutions. Accurate and reliable quantitative measures of risk are needed to minimize undesirable effects on a given portfolio fromlarge fluctuations in market conditions. To accomplish this, a series of computational tools has beendesigned, implemented, and incorporated into MatRisk, an integratedenvironment for risk assessment developed in MATLAB. Besides standard measures, such as Value at Risk(VaR), the application includes other more sophisticated risk measures that address the inability of VaRproperly to characterize the structure of risk. Conditionalrisk measures can also be estimated for autoregressive models with heteroskedasticity, including some novel mixture models. These tools are illustrated with a comprehensive risk analysis of the Spanish IBEX35 stock index.