Using Control Variables to Improve the Efficiency of Percentile Estimation in Stochastic Simulation

Control variables may be used in the estimation of percentiles by stochastic stimulation. Hitherto control variables have only been applied to estimating the mean of a response variable; however, in many uses of simulation, information about the tails of the distribution of a response variable are of more interest. The efficacy of the method is demonstrated by application to an example of a simulation study in capital budgeting where exposure to risk is important.     

Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures

In this paper, we extend the concept of tail subadditivity (Belles-Sampera et al., 2014a; Belles-Sampera et al., 2014b) for distortion risk measures and give sufficient and necessary conditions for a distortion risk measure to be tail subadditive. We also introduce the generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure. To further illustrate the applications of the tail subadditivity, we propose multivariate tail distortion (MTD) risk measures and generalize the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016). The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks and explore the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations.     

Extremes for coherent risk measures

Various concepts appeared in the existing literature to evaluate the risk exposure of a financial or insurance firm/subsidiary/line of business due to the occurrence of some extreme scenarios. Many of those concepts, such as Marginal Expected Shortfall or Tail Conditional Expectation, are simply some conditional expectations that evaluate the risk in adverse scenarios and are useful for signaling to a decision-maker the poor performance of its risk portfolio or to identify which sub-portfolio is likely to exhibit a massive downside risk. We investigate the latter risk under the assumption that it is measured via a coherent risk measure, which obviously generalizes the idea of only taking the expectation of the downside risk. Multiple examples are given and our numerical illustrations show how the asymptotic approximations can be used in the capital allocation exercise. We have concluded that the expectation of the downside risk does not fairly take into account the individual risk contribution when allocating the VaR-based regulatory capital, and thus, more conservative risk measurements are recommended. Finally, we have found that more conservative risk measurements do not improve the fairness of the cost of capital allocation when the uncertainty with parameter estimation is present, even at a very high level.     

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.     

Portfolio Selection Problem with Minimax Type Risk Function

The investor's preference in risk estimation of portfolio selection problems is important as it influences investment strategies. In this paper a minimax risk criterion is considered. Specifically, the investor aims to restrict the standard deviation for each of the available stocks. The corresponding portfolio optimization problem is formulated as a linear program. Hence it can be implemented easily. A capital asset pricing model between the market portfolio and each individual return for this model is established using nonsmooth optimization methods. Some numerical examples are given to illustrate our approach for the risk estimation.     

A mean-risk index model for uncertain capital budgeting

This paper discusses the capital budgeting problem of projects using annual cash inflows, cash outflows and initial investment outlays given by experts’ evaluations when no historical data are available. Uncertain variables are used to describe the projects’ parameters. A profit risk index and a capital risk index are proposed, and a mean-risk index model is developed for optimal project selection. In addition, the deterministic forms of the model are given and a solution algorithm is provided. For the sake of illustration, a numerical example is also presented. The results of the example show that both profit risk index and capital risk index are important in investment risk control. However, when the profit risk control requirement is strong, the selected project portfolio may be insensitive to the capital risk constraint; when the profit risk control requirement is moderate, the capital risk constraint plays an important role. The results also show the tendency that when either the tolerable profit risk level or the tolerable capital risk level becomes higher, the obtained expected net present value of the project portfolio becomes larger, which is in agreement with the investment rule that the higher the risk, the higher the return.     

基于风险调整资本收益率下的最优再保险策略

风险调整资本收益率是一个用来描述赚取收益所承担风险的重要指标,是衡量风险调整后的财务绩效的一个有效工具,在银行业中得到广泛采用。近年来,在保险公司的再保险业务中,也越来越多地采用风险调整资本收益率这一指标来衡量收益和风险。本文重新考虑了有再保险控制下的风险调整资本收益率,并得到:在一般再保险自留函数下,我们证明了分层再保险策略是最优再保险形式;进一步,我们获得了最优再保险分层水平及最优风险调整资本收益率;最后,我们也给出了算例及数值分析。     

Longevity risk,cost of capital and hedging for life insurers under Solvency II

The cost of capital is an important factor determining the premiums charged by life insurers issuing life annuities. This capital cost can be reduced by hedging longevity risk with longevity swaps, a form of reinsurance. We assess the costs of longevity risk management using indemnity based longevity swaps compared to costs of holding capital under Solvency II. We show that, using a reasonable market price of longevity risk, the market cost of hedging longevity risk for earlier ages is lower than the cost of capital required under Solvency II. Longevity swaps covering higher ages, around 90 and above, have higher market hedging costs than the saving in the cost of regulatory capital. The Solvency II capital regulations for longevity risk generates an incentive for life insurers to hold longevity tail risk on their own balance sheets, rather than transferring this to the reinsurance or the capital markets. This aspect of the Solvency II capital requirements is not well understood and raises important policy issues for the management of longevity risk.     

信息熵度量风险的探究

本文分析了风险的本质后指出,风险是某一特定行为主体对某一金融投资中损失的不确定性和收益的不确定性的认识。在众多风险度量的方法中,熵函数法有着其独特的度量风险的优势,因此,在本文中重点讨论了熵函数作为风险度量的合理性。同时提出一个新的风险度量模型,剖析其主要的数学特性,阐明该模型可以针对不同行为主体能有效地度量金融风险,并且计算量小,易于操作。     

Capital requirements with defaultable securities

We study capital requirements for bounded financial positions defined as the minimum amount of capital to invest in a chosen eligible asset targeting a pre-specified acceptability test. We allow for general acceptance sets and general eligible assets, including defaultable bonds. Since the payoff of these assets is not necessarily bounded away from zero, the resulting risk measures cannot be transformed into cash-additive risk measures by a change of numéraire. However, extending the range of eligible assets is important because, as exemplified by the recent financial crisis, assuming the existence of default-free bonds may be unrealistic. We focus on finiteness and continuity properties of these general risk measures. As an application, we discuss capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability, the two most important acceptability criteria in practice. Finally, we prove that there is no optimal choice of the eligible asset. Our results and our examples show that a theory of capital requirements allowing for general eligible assets is richer than the standard theory of cash-additive risk measures.     

Some new notions of dependence with applications in optimal allocation problems

Dependence structures of multiple risks play an important role in optimal allocation problems for insurance, quantitative risk management, and finance. However, in many existing studies on these problems, risks or losses are often assumed to be independent or comonotonic or exchangeable. In this paper, we propose several new notions of dependence to model dependent risks and give their characterizations through the probability measures or distributions of the risks or through the expectations of the transformed risks. These characterizations are related to the properties of arrangement increasing functions and the proposed notions of dependence incorporate many typical dependence structures studied in the literature for optimal allocation problems. We also develop the properties of these dependence structures. We illustrate the applications of these notions in the optimal allocation problems of deductibles and policy limits and in capital reserves problems. These applications extend many existing researches to more general dependent risks.     

Continuous‐time stochastic modelling of capital adequacy ratios for banks

Regulation related to capital requirements is an important issue in the banking sector. In this regard, one of the indices used to measure how susceptible a bank is to failure, is the capital adequacy ratio (CAR). We consider two types of such ratios, viz. non‐risk‐based (NRBCARs) and risk‐based (RBCARs) CARs. According to the US Federal Deposit Insurance Corporation (FDIC), we can further categorize NRBCARs into leverage and equity capital ratios and RBCARs into Basel II and Tier 1 ratios. In general, these indices are calculated by dividing a measure of bank capital by an indicator of the level of bank risk. Our primary objective is to construct continuous‐time stochastic models for the dynamics of each of the aforementioned ratios with the main achievement being the modelling of the Basel II capital adequacy ratio (Basel II CAR). This ratio is obtained by dividing the bank's eligible regulatory capital (ERC) by its risk‐weighted assets (RWAs) from credit, market and operational risk. Mainly, our discussions conform to the qualitative and quantitative standards prescribed by the Basel II Capital Accord. Also, we find that our models are consistent with data from FDIC‐insured institutions. Finally, we demonstrate how our main results may be applied in the banking sector. Copyright © 2005 John Wiley & Sons, Ltd.     

Forecasting probabilities of default and loss rates given default in the presence of selection

This paper offers a joint estimation approach for forecasting probabilities of default and loss rates given default in the presence of selection. The approach accommodates fixed and random risk factors. An empirical analysis identifies bond ratings, borrower characteristics and macroeconomic information as important risk factors. A portfolio-level analysis finds evidence that common risk measurement approaches may underestimate bank capital by up to 17% relative to the presented model.     

Optimal reinsurance with regulatory initial capital and default risk

In a reinsurance contract, a reinsurer promises to pay the part of the loss faced by an insurer in exchange for receiving a reinsurance premium from the insurer. However, the reinsurer may fail to pay the promised amount when the promised amount exceeds the reinsurer’s solvency. As a seller of a reinsurance contract, the initial capital or reserve of a reinsurer should meet some regulatory requirements. We assume that the initial capital or reserve of a reinsurer is regulated by the value-at-risk (VaR) of its promised indemnity. When the promised indemnity exceeds the total of the reinsurer’s initial capital and the reinsurance premium, the reinsurer may fail to pay the promised amount or default may occur. In the presence of the regulatory initial capital and the counterparty default risk, we investigate optimal reinsurance designs from an insurer’s point of view and derive optimal reinsurance strategies that maximize the expected utility of an insurer’s terminal wealth or minimize the VaR of an insurer’s total retained risk. It turns out that optimal reinsurance strategies in the presence of the regulatory initial capital and the counterparty default risk are different both from optimal reinsurance strategies in the absence of the counterparty default risk and from optimal reinsurance strategies in the presence of the counterparty default risk but without the regulatory initial capital.     

The optimal operational risk capital requirement by applying the advanced measurement approach

The purpose of this paper is to construct a risk quantification model to achieve the accurate operational risk management and gain the satisfying estimation and control of future possible extreme losses by using capital charges to assess operational risk. The paper takes a case bank as the research object and compares the differences under various circumstances engaging the Basic Indicator Approach, the Standardized Approach, and the Advanced Measurement Approach for the operational risk capital requirement of a bank. The results indicate that it is more appropriate to adopt the Advanced Measurement Approach to estimate the operational risk capital requirement; this way can help a bank enjoy a much lessened capital charge required and subsequently its available capital increases. Hence, it allows a bank to have sufficient funds in operations and reduce the burden of capital costs. Therefore, it will bring the positive benefits to the whole banking industry when enforcing the Advanced Measurement Approach.     

Benefits of working capital sharing in supply chains

Supply chain finance and working capital management are important avenues to reduce supply chain costs. Small suppliers may not have sufficient working capital to finance their operations and efficiently supply their customers. We develop a model that captures the fundamental aspects of financial and operational planning in a two-stage supply chain, with both strong and weak members. A strong member can negotiate for more favorable financing rates, more advantageous payment terms, and shorter lead times than a weaker member. We investigate two working capital allocation scenarios. In the dedicated working capital allocation scenario, the members of the supply chain each have their own working capital. In the joint working capital allocation scenario, the members of the supply chain have a joint pool of working capital. Our results demonstrate significant benefits when the members of the supply chain share the working capital. We also show that extending payment delays to a supplier upstream results in higher overall supply chain costs.     

Capital allocation based on the Tail Covariance Premium Adjusted

The current Solvency II process makes risk capital allocation to different business lines more and more important. This paper considers two business lines with the exponential loss distributions linked by a Farlie–Gumbel–Morgenstern (FGM) copula, modelling the dependence between them. As an allocation principle we use the Tail Covariance Premium Adjusted and obtain expressions for the allocation to the two business lines.     

贸易信用交易下的保险投资策略研究

贸易信用融资被广泛应用于解决中小企业融资困境,而保险正成为解决贸易信用融资风险的重要手段。本文站在核心企业角度,探究贸易信用融资保险的运营策略,运用Stackelberg博弈分析方法分别建立并比较了贸易信用融资、贸易信用融资保险、资金约束无融资、资金充足四类优化模型,探究了博弈均衡下的最优运营决策,并分析了零售商初始资金、生产商风险态度等关键参数的影响。研究表明:融资不仅对供应链有利,还能同时实现生产商及零售商共赢;当生产商风险厌恶程度、保险市场竞争程度较高,零售商初始资金较低时,融资保险能够为生产商及供应链创造价值,否则生产商应当放弃投保。研究结论为工业界合理且高效开展贸易信用融资保险运营提供了策略指导和管理启示。     

基于POT方法的商业银行操作风险极端值估计

对于商业银行而言,操作风险已经成为与市场风险和信用风险同样重要的风险。本文利用极值理论超越样本的估计能力,采用极值理论中对数据要求量较少,可以进行单步预测的超阈值(POT)方法对我国商业银行操作损失极端值分布进行估计,以均值超额函数图和拟合直线的交点确定阈值,估计出给定置信水平之下操作风险损失的分位数,从而使得国内商业银行操作风险监管资本的计算成为可能。     

资本约束对商业银行风险偏好的影响—来自中部某省份若干家商业银行的实证检验

本文以滞后一期的自回归模型为基础,综合运用统计描连、单位根检验、协整检验等技术,针对2002年至2006年中部某省份若干家商业银行的数据,实证检验了资本约束政策的实施对商业银行风险偏好的影响。结论显示:资本约束对商业银行的风险行为选择的影响巨大,但是资本软约束会强烈抵消资本约束带来的正效应;中小型银行比大型银行更具有政策的敏感度,中国的商业银行体系应当进一步进行深入的市场经济改革。本文的创新在于:建立了一个信贷行为偏好指数来刻画商业银行的风险偏好,并把资本约束的实施与资本约束的软化引入计量模型,分别进行了总量检验与分类检验,从而得到了有中国特色的分析结论与政策建议。