How does innovation''''s tail risk determine marginal tail risk of a stationary financial time series?

We discuss the relationship between the marginal tail risk probability and theinnovation's tail risk probability for some stationary financial time series models. We firstgive the main results on the tail behavior of a class of infinite weighted sums of randomvariables with heavy-tailed probabilities. And then, the main results are applied to threeimportant types of time series models; infinite order moving averages, the simple bilineartime series and the solutions of stochastic difference equations. The explicit formulasare given to describe how the marginal tail probabilities come from the innovation's tailprobabilities for these time series. Our results can be applied to the tail estimation of timeseries and are useful for risk analysis in finance.     

Upper tail probabilities of integrated Brownian motions

We obtain new upper tail probabilities of m-times integrated Brownian motions under the uniform norm and the Lp norm. For the uniform norm, Talagrand’s approach is used, while for the Lp norm, Zolotare’s approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities(large ball probabilities) for general Gaussian random variables in Banach spaces. As applications,explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of m-times integrated Brownian motions are presented as well.     

带常利率的Erlang(2)风险模型的罚金折现期望函数

The purpose of this paper is to consider the expected value of a discounted penalty due at ruin in the Erlang（2） risk process under constant interest force. An integro-differential equation satisfied by the expected value and a second-order differential equation for the Laplace transform of the expected value are derived. In addition, the paper will present the recursive algorithm for the joint distribution of the surplus immediately before ruin and the deficit at ruin. Finally, by the differential equation, the defective renewal equation and the explicit expression for the expected value are given in the interest-free case.     

ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved.Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given.Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced.As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems,two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.     

On the Expected Present Value of Total Dividends in a Risk Model with Potentially Delayed Claims

In this paper, we consider a risk model in which two types of individual claims, main claims and by-claims, are defined. Every by-claim is induced by the main claim randomly and may be delayed for one time period with a certain probability. The dividend policy that certain amount of dividends will be paid as long as the surplus is greater than a constant dividend barrier is also introduced into this delayed claims risk model. By means of the probability generating functions, formulae for the expected present value of total dividend payments prior to ruin are obtained for discrete-type individual claims. Explicit expressions for the corresponding results are derived for K n claim amount distributions. Numerical illustrations are also given.     

Portfolio Optimization Model with Transaction Costs

Abstract The purpose of the article is to formulate,under the l_∞ risk measure,a model of portfolio selectionwith transaction costs and then investigate the optimal strategy within the proposed.The characterization of aoptimal strategy and the efficient algorithm for finding the optimal strategy are given.     

Tail behavior of supremum of a random walk when Cramér’s condition fails

We investigate tail behavior of the supremum of a random walk in the case that Cramer＇s condition fails, namely, the intermediate case and the heavy-tailed ease. When the integrated distribution of the increment of the random walk belongs to the intersection of exponential distribution class and O-subexponential distribution class, under some other suitable conditions, we obtain some asymptotic estimates for the tail probability of the supremum and prove that the distribution of the supremum also belongs to the same distribution class. The obtained results generalize some corresponding results of N. Veraverbeke. Finally, these results are applied to renewal risk model, and asymptotic estimates for the ruin probability are presented.     

Uniform tail asymptotics for the aggregate claims with stochastic discount in the renewal risk models

Considering an insurer who is allowed to make risk-free and risky investments, as in Tang et al.(2010), the price process of the investment portfolio is described as a geometric L′evy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of extended regular variation, we obtain an asymptotically equivalent formula which holds uniformly for all time horizons, and furthermore, the same asymptotic formula holds for the finite-time ruin probabilities. The results extend the works of Tang et al.(2010).     

OPTIMAL REINSURANCE UNDER EXPECTED VALUE PRINCIPLE

The paper concerns the problem how to purchase the reinsurance in order to make the insurer and the reinsurance company＇s total risk to be least under the expected value principle. When the insurer and reinsurance company take arbitrary risk measures, sufficient con- ditions for optimality of reinsurance contract are given within the restricted class of admissible contracts. Further, the explicit forms of optimal reinsurance contract under several special risk measures are given, and the method to decide parameters as well.     

On two actuarial quantities for the compound Poisson risk model with tax and a threshold dividend strategy

In this paper, we consider a compound Poisson risk model with taxes paid according to a loss-carry-forward system and dividends paid under a threshold strategy. First, the closed-form expression of the probability function for the total number of taxation periods over the lifetime of the surplus process is derived. Second, analytical expression of the expected accumulated discounted dividends paid between two consecutive taxation periods is provided. In addition, explicit expressions are also given for the exponential individual claims.     

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.     

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.     

变动基数投资组合中的系统误差与估计误差权衡

投资组合选择中的系统误差与估计误差是决定样本期外绩效的重要因素,其权衡受到资产基数N的影响。本文在变动基数的设定下,将Bootstrapping和样本期外滚动的方法应用到均权重、最小方差组合及其误差修正策略的绩效和尾部风险检验过程中,并在不同的市场状态下进行分组讨论。研究发现：(1)最小方差组合与均权重策略的样本期外夏普比率差异与N存在倒U型的关系。(2)最小方差组合的尾部风险随N的扩大而迅速降低,总体来看最小方差组合的尾部风险低于均权重策略。(3)最小方差组合的换手率与N存在正相关关系,盲目增加投资组合选择中的资产基数会带来无谓损失。研究结果表明,投资者应理性选择资产基数,充分利用最小方差组合带来的分散化收益。     

TVaR-based capital allocation with copulas

Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.     

基于鞍点逼近的投资组合极端风险贡献度测度

在当今金融市场资产价格高波动的背景下,度量投资组合中各资产对总体风险的风险贡献度对探析投资组合风险波动不定的深层次原因有重要意义。关于风险贡献度的测算,目前运用较广泛的是历史数据法,其主要适用于存在大量数据样本且持续期较短的情况。特别地,极端情况下的风险贡献度估计主要由处于分布尾部的少量观测值决定,因此历史数据法估计的准确性此时较难保证,为此,本文对鞍点逼近模型优化并考察上述情形。通过对中国股市进行实证分析发现,与传统历史数据法相比,鞍点逼近模型呈现下列优点:投资组合分布函数简洁、风险贡献度计算效率和准确性较高,压力测试表明该方法具有较好的稳健性。因此该方法有望对投资组合的风险预警与防范起到决策支持作用。     

Haezendonck–Goovaerts risk measure with a heavy tailed loss

Recently Haezendonck–Goovaerts (H–G) risk measure has received much attention in (re)insurance and portfolio management. Some nonparametric inferences have been proposed in the literature. When the loss variable does not have enough moments, which depends on the involved Young function, the nonparametric estimator in Ahn and Shyamalkumar (2014) has a nonnormal limit, which challenges interval estimation. Motivated by the fact that many loss variables in insurance and finance could have a heavier tail such as an infinite variance, this paper proposes a new estimator which estimates the tail by extreme value theory and the middle part nonparametrically. It turns out that the proposed new estimator always has a normal limit regardless of the tail heaviness of the loss variable. Hence an interval with asymptotically correct confidence level can be obtained easily either by the normal approximation method via estimating the asymptotic variance or by a bootstrap method. A simulation study and real data analysis confirm the effectiveness of the proposed new inference procedure for estimating the H–G risk measure.     

基于Copula函数的我国股指期货风险及套期保值实证研究

运用SJC-Copula-GJR模型,计算了持有沪深300股指期货多头和空头两种组合的VaR值和最优投资比例,模型的特点是能够准确地描述尾部相关关系,且其对尾部相关性的描述是非对称的,所得结论为投资者进行风险管理提供了可靠的依据.同时,通过构造加权的非线性相关系数来计算沪深300股指期货最优套期保值比率,解决了分布非正态、期货与现货非线性的问题,准确地度量了股指期货收益率序列的动态相依关系,实证研究表明基于Copula函数的套期保值有效性明显地优于传统模型.     

Conditional tail risk measures for the skewed generalised hyperbolic family

This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.     

V@R representation theorems in ambiguous frameworks

The value at risk (V@R) is a very important risk measure with significant applications in finance (risk management, pricing, hedging, portfolio theory, etc), insurance (premium principles, optimal reinsurance, etc), production, marketing (newsvendor problem), etc. It also plays a critical role in regulation about risk (Basel, Solvency, etc), it is very appreciated by practitioners due to its intuitive interpretation, and it is the unique popular risk measure remaining finite for heavy tailed risks with unbounded expectation. Besides, ambiguous frameworks are becoming more and more usual in applications of risk analysis. Lack of data or committed errors may provoke discrepancies between real probabilities and estimated ones. This paper combines both V@R and ambiguous settings, and a new representation theorem for V@R is given. Consequently, inspired by previous studies dealing with coherent risk measures and their representation, we will give new methods to compute and optimize V@R under ambiguity. This seems to be a relevant finding because the analytical properties of V@R are very weak if one compares with a coherent risk measure. Indeed, V@R is neither continuous nor convex, which makes it very complicated to deal with it in mathematical approaches. Nevertheless, the results of this paper will allow us to transform computation and optimization problems involving V@R into continuous and differentiable problems.     

Asymptotics of multivariate conditional risk measures for Gaussian risks

This paper investigates accurate approximations of marginal moment excess, marginal conditional tail moment and marginal moment shortfall for multivariate Gaussian system risks. Based on the dimension reduction property via the quadratic programming problem, the super-exponential and polynomial convergence speeds are specified. Two interesting questions involved in risk management are well addressed, namely the minimal additional risk capital injection to avoid infinite risk contagion and a sufficient and necessary condition to alternate the convergence speeds. Numerical study and typical examples are given to illustrate the efficiency of our findings. Due to the flexible moment order, additional applications may involve in risk management, including tail mean–variance portfolio and multivariate conditional risk measures of tail covariance, tail skewness with dependence and extremal risk contagion under consideration.