An optimization approach to adaptive multi-dimensional capital management

Firms should keep capital to offer sufficient protection against the risks they are facing. In the insurance context methods have been developed to determine the minimum capital level required, but less so in the context of firms with multiple business lines including allocation. The individual capital reserve of each line can be represented by means of classical models, such as the conventional Cramér–Lundberg model, but the challenge lies in soundly modelling the correlations between the business lines. We propose a simple yet versatile approach that allows for dependence by introducing a common environmental factor. We present a novel Bayesian approach to calibrate the latent environmental state distribution based on observations concerning the claim processes. The calibration approach is adjusted for an environmental factor that changes over time. The convergence of the calibration procedure towards the true environmental state is deduced. We then point out how to determine the optimal initial capital of the different business lines under specific constraints on the ruin probability of subsets of business lines. Upon combining the above findings, we have developed an easy-to-implement approach to capital risk management in a multi-dimensional insurance risk model.     

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.     

Top-down approaches for integrated risk management: How accurate are they?

Banks and other financial institutions try to compute the necessary amount of total capital that they need for absorbing stochastically dependent losses from different risk types (e.g., credit risk and market risk). Two sophisticated procedures of this so-called integrated risk management are the top-down and the bottom-up approaches. When banks apply a more sophisticated risk integration approach at all, it is usually the top-down approach where copula functions are employed for linking the marginal distributions of profit and losses resulting from different risk types. However, it is not clear at all how accurate this approach is. Assuming that the bottom-up approach corresponds to the real-word data-generating process and using a comprehensive simulation study, it is shown that the top-down approach can underestimate the necessary amount of total capital for lower credit qualities. Furthermore, the direction and strength of the stochastic dependence between the risk types, the copula function employed, and the loss definitions all have an impact on the performance of the top-down approach. In addition, a goodness-of-fit test shows that, based on time series of loss data with realistic length, it is rather difficult to decide which copula function is the right one.     

Multivariate risk models under heavy‐tailed risks

In this paper, we consider four common types of ruin probabilities for a discrete‐time multivariate risk model, where the insurer is assumed to be exposed to a vector of net losses resulting from a number of business lines over each period. By assuming a large initial capital for the risk model and regularly varying distributions for the net losses, we establish some interesting asymptotic estimates for ruin probabilities in terms of the upper tail dependence function of the net loss vector. Our results insightfully characterize how the dependence structure among the individual net losses affect the ruin probabilities in an asymptotic sense, and more importantly, from our main results, explicit asymptotic estimates for those ruin probabilities can be obtained via specifying a copula for the net loss vectors. Copyright © 2013 John Wiley & Sons, Ltd.     

Tail behaviour of credit loss distributions for general latent factor models

Using a limiting approach to portfolio credit risk, we obtain analytic expressions for the tail behavior of credit losses. To capture the co‐movements in defaults over time, we assume that defaults are triggered by a general, possibly non‐linear, factor model involving both systematic and idiosyncratic risk factors. The model encompasses default mechanisms in popular models of portfolio credit risk, such as CreditMetrics and CreditRisk⁺. We show how the tail characteristics of portfolio credit losses depend directly upon the factor model's functional form and the tail properties of the model's risk factors. In many cases the credit loss distribution has a polynomial (rather than exponential) tail. This feature is robust to changes in tail characteristics of the underlying risk factors. Finally, we show that the interaction between portfolio quality and credit loss tail behavior is strikingly different between the CreditMetrics and CreditRisk⁺ approach to modeling portfolio credit risk.     

基于贝叶斯分析的中国商业银行内部欺诈分析

内部欺诈事件类型是中国商业银行最严重的操作风险类型。但由于操作风险本质特征和中国商业银行内部欺诈损失数据收集年度较短,数据匮乏,小样本数据容易导致参数结果不稳定。为了在小样本数据下进行更准确的度量,本文采用贝叶斯马尔科夫蒙特卡洛模拟方法,在损失分布法框架下,假设损失频率服从泊松-伽马分布,而损失强度服从广义帕累托-混合伽马分布,分析后验分布的形式,获得中国商业银行不同业务线的内部欺诈损失频率和损失强度的后验分布估计,并进行蒙特卡罗模拟获得不同业务线内部欺诈的风险联合分布。结果表明,拟合结果很好,与传统极值分析法相比,基于利用贝叶斯的分析获得的后验分布可以作为未来的先验分布,有利于在较小样本下获得较真实的参数估计,本方法有助于银行降低监管资本要求。     

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.     

供应链融资畅想

如果建立一个基于电子发票系统的综合IT系统平台,连接银行信贷管理平台、第三方电子商务平台和企业ERP平台,那么,就可以将供应链中的物流、资金流、信息流打通,不仅能解决无数中小企业的融资难题,还可以将我国企业的财务与信用管理拉上新的台阶。     

Stress testing credit card portfolios: an application in South Africa

Motivated by a real problem, this study aims to develop models to conduct stress testing on credit card portfolios. Two modelling approaches were extended to include the impact of lenders’ actions within the model. The first approach was a regression model of the aggregate losses based on economic variables with autocorrelations of the errors. The second approach was a set of vintage-level models that highlighted the months-on-book effect on credit losses. A case study using the models was described using South African credit card data. In this case, the models were used to stress test the credit card portfolio under several economic scenarios.     

The development of a simple and intuitive rating system under Solvency II

Regulatory authorities pay considerable attention to setting minimum capital levels for different kinds of financial institutions. Solvency II, the European Commission’s planned reform of the regulation of insurance companies is well underway. One of its consequences will be a shift in focus to internally based models in determining the regulatory capital needed to cover unexpected losses. This evolution emphasises the importance of credit risk assessment through internal ratings. In light of this new prudential regulation, this paper suggests a Basel II compliant approach to predicting credit ratings for non-rated corporations and evaluates its performance compared to external ratings. The paper provides an interesting modelling of non-financial European companies rated by S&P. In developing the model, broad applicability is set as an important boundary condition. Even though the model developed is fairly simple and maintains a high level of granularity, it gives high rates of accuracy and is very interpretable.     

Assessing the cost of capital for longevity risk

The cost of capital is a key element of the embedded value methodology for the valuation of a life business. Further, under some solvency approaches (in particular, the Swiss Solvency Test and the developing Solvency 2 project) assessing the cost of capital constitutes a step in determining the required capital allocation.Whilst the cost of capital is usually meant as a reward for the risks encumbering a given life portfolio, in actuarial practice the relevant parameter has been traditionally chosen, at least to some extent, inconsistently with such risks. The adoption of market-consistent valuations has then been advocated to reach a common standard.A market-consistent value usually acknowledges a reward to shareholders’ capital as long as the market does, namely if the risk is systematic or undiversifiable. When dealing with a life annuity portfolio (or a pension plan), an important example of systematic risk is provided by the longevity risk, i.e. the risk of systematic deviations from the forecasted mortality trend. Hence, a market-consistent approach should provide appropriate valuation tools.In this paper we refer to a portfolio of immediate life annuities and we focus on longevity risk. Our purpose is to design a framework for a valuation of the portfolio which is market-consistent, and therefore based on a risk-neutral argument, while involving some of the basic items of a traditional valuation, viz best estimate future flows and allocated capital. This way, we try to reconcile the traditional with a market-consistent (or risk-neutral) approach. This allows us, in particular, to translate the results obtained under the risk-neutral approach in terms of a properly redefined embedded value.     

Precautionary measures for credit risk management in jump models

Sustaining efficiency and stability by properly controlling the equity to asset ratio is one of the most important and difficult challenges in bank management. Due to unexpected and abrupt decline of asset values, a bank must closely monitor its net worth as well as market conditions, and one of its important concerns is when to raise more capital so as not to violate capital adequacy requirements. In this paper, we model the trade-off between avoiding costs of delay and premature capital raising, and solve the corresponding optimal stopping problem. In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by Lévy processes, and solve explicitly for the double exponential jump-diffusion process and for a general spectrally negative Lévy process.     

基于供应链金融的供应商链式融资策略研究

供应链中由于核心企业延期付款而给上游供应商带来资金压力,供应商、核心企业以及银行组成供应商链式融资系统以缓解供应商资金压力。针对供应商初始资金以及银行给定信用期的不同情况,分别建立了供应商不贷款、贷款且在信用期内收回所有货款以及贷款但在信用期内未收回所有货款三种情形下的数学模型。通过分析得到了核心企业的最优订货策略和供应商的最优生产策略。研究表明供应商链式融资在一定程度上可以达到供应商、核心企业、银行三方共赢的效果:缓解供应商、核心企业资金压力并增加其利润,降低银行贷款风险扩大其贷款业务。     

On the distribution of the (un)bounded sum of random variables

We propose a general treatment of random variables aggregation accounting for the dependence among variables and bounded or unbounded support of their sum. The approach is based on the extension to the concept of convolution to dependent variables, involving copula functions. We show that some classes of copula functions (such as Marshall-Olkin and elliptical) cannot be used to represent the dependence structure of two variables whose sum is bounded, while Archimedean copulas can be applied only if the generator becomes linear beyond some point. As for the application, we study the problem of capital allocation between risks when the sum of losses is bounded.     

出口海陆仓融资决策多目标优化模型与算法

授信额度分配决策是出口海陆仓融资业务中的重要问题,为权衡该项业务中的风险和收益,综合考虑出口国家政治风险、风险价值约束以及信用风险等因素,构建以收益最高和风险最低为目标的优化模型。针对所构建模型的特点,提出一种将模拟退火与NSGA-II算法相结合的混合算法对模型进行求解。最后,通过具体实例,分别以现实中常见的1~5个月质押期为例,验证了模型和算法的适用性及有效性。     

Optimal capital allocation in a hierarchical corporate structure

We consider capital allocation in a hierarchical corporate structure where stakeholders at two organizational levels (e.g., board members vs line managers) may have conflicting objectives, preferences, and beliefs about risk. Capital allocation is considered as the solution to an optimization problem whereby a quadratic deviation measure between individual losses (at both levels) and allocated capital amounts is minimized. Thus, this paper generalizes the framework of Dhaene et al. (2012), by allowing potentially diverging risk preferences in a hierarchical structure. An explicit unique solution to this optimization problem is given. In several examples, it is shown how the optimal capital allocation achieves a compromise between conflicting views of risk within the organization.     

Efficient frontier cutoff policies in credit portfolios

Historically, account acquisition in scored retail credit and loan portfolios has focused on risk management in the sense of minimizing default losses. We believe that acquisition policies should focus on a broader set of business measures that explicitly recognize tradeoffs between conflicting objectives of losses, volume and profit. Typical business challenges are: ‘How do I maximize portfolio profit while keeping acceptance rate (volume, size) at acceptable levels?’ ‘How do I maximize profit without incurring default losses above a given level?’ ‘How do I minimize the risk of large loss exposures for a given market share?’ In this paper we are not concerned with which combination of objectives are appropriate, but rather focus on the cutoff policies that allow us to capture a number of different portfolio objectives. When there are conflicting objectives we show that optimal policies yield meaningful tradeoffs and efficient frontiers and that optimal shadow prices allow us to develop risk-adjusted tradeoffs between profit and market share. Some of the graphical solutions that we obtain are simple to derive and easy to understand without explicit mathematical formulations but even simple constraints may require formal use of non-linear programming techniques. We concentrate on models and insights that yield decision strategies and cutoff policies rather than the techniques for developing good predictors.     

Bank-sourced credit transition matrices: Estimation and characteristics

This study proposes and analyses a novel alternative to credit transition matrices (CTMs) developed by credit rating agencies - bank-sourced CTMs. It provides a unique insight into estimation of bank-sourced CTMs by assessing the extent to which the CTMs depend on the characteristics of the underlying credit risk datasets and the aggregation method and outlines that the choice of aggregation approach has a substantial effect on credit risk model results. Further, we show that bank-sourced CTMs are more dynamic than those of credit rating agencies, with higher off-diagonal transition rates and higher propensity to upgrade. Finally, we create a set of industry-specific CTMs, otherwise unobtainable due to the data sparsity faced by credit rating agencies, and highlight the implications of their differences, signalling the existence of industry-specific business cycles. The study uses a unique and large dataset of internal credit risk estimates from 24 global banks covering monthly observations on more than 26,000 large corporates and employs large-scale Monte Carlo simulations. This approach can be replicated by regulators (e.g., data collected by the European Central Bank in the AnaCredit project) and used by organisations aiming to improve their credit risk models.     

确定性需求下的最优保理融资与生产策略

企业为下游买方提供赊销,由于大量的资金被应收账款占用,企业可能因资金不足而无法生产足够的产品。企业可以通过保理(出售应收账款)进行融资,减小需求损失。在离散时间多周期的确定需求下,使用决策变量描述各周期的系统状态及其状态转移方程,将此问题建模为线性规划。通过分析此问题的结构特点,再提出了一种新颖且等价的建模方法,可以有效减少决策变量和约束条件的数量。在连续时间模型和混合模型中,这种建模方法同样适用,将优化问题写为连续线性规划,极大地降低了优化问题的复杂度。此连续线性规划问题可通过适当的区间划分进行离散化,用分片常量函数代替优化模型中的一般函数(无限维)决策变量,通过求解有限维线性规划得到原问题的可行近似解。最后,通过数值例子分析了贴现率对企业利润的影响。     

Pricing industry loss warranties in a Lévy–Frailty framework

We propose a novel risk-neutral pricing approach for industry loss warranties. In doing so, we explicitly take into account the statistical dependence of the losses on individual policies in the underlying insurance portfolio, caused by the occurrence of a natural catastrophe. Inspired by recent advances in the structured credit literature, we model joint claim events in a Lévy–Frailty framework with a stochastic time change. Event time is driven by rare and large jumps of a compound Poisson subordinator and thus elapses more quickly when a natural catastrophe has struck, leading to a clustering of losses. We estimate the model on historical ILW quotes and obtain encouraging fit statistics.