Economic Capital Allocation Derived from Risk Measures

Abstract

We examine properties of risk measures that can be considered to be in line with some “best practice” rules in insurance, based on solvency margins. We give ample motivation that all economic aspects related to an insurance portfolio should be considered in the definition of a risk measure. As a consequence, conditions arise for comparison as well as for addition of risk measures. We demonstrate that imposing properties that are generally valid for risk measures, in all possible dependency structures, based on the difference of the risk and the solvency margin, though providing opportunities to derive nice mathematical results, violates best practice rules. We show that so-called coherent risk measures lead to problems. In particular we consider an exponential risk measure related to a discrete ruin model, depending on the initial surplus, the desired ruin probability, and the risk distribution.     

Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment

Abstract

This article considers the compound Poisson insurance risk model perturbed by diffusion with investment. We assume that the insurance company can invest its surplus in both a risky asset and the risk-free asset according to a fixed proportion. If the surplus is negative, a constant debit interest rate is applied. The absolute ruin probability function satisfies a certain integro-differential equation. In various special cases, closed-form solutions are obtained, and numerical illustrations are provided.     

Ruin Problems for Phase-Type(2) Risk Processes

In this paper we consider a risk process in which claim inter-arrival times have a phase-type(2) distribution, a distribution with a density satisfying a second order linear differential equation. We consider some ruin related problems. In particular, we consider the compound geometric representation of the infinite time survival probability, as well as the (defective) distributions of the surplus immediately prior to ruin and of the deficit at ruin. We also consider explicit solutions for the infinite time ruin probability in the case where the individual claim amount distribution is phase-type.     

The Expected Discounted Penalty at Ruin for a Markov-Modulated Risk Process Perturbed by Diffusion

Abstract

A Markov-modulated risk process perturbed by diffusion is considered in this paper. In the model the frequencies and distributions of the claims and the variances of the Wiener process are influenced by an external Markovian environment process with a finite number of states. This model is motivated by the flexibility in modeling the claim arrival process, allowing that periods with very frequent arrivals and ones with very few arrivals may alternate. Given the initial surplus and the initial environment state, systems of integro-differential equations for the expected discounted penalty functions at ruin caused by a claim and oscillation are established, respectively; a generalized Lundberg’s equation is also obtained. In the two-state model, the expected discounted penalty functions at ruin due to a claim and oscillation are derived when both claim amount distributions are from the rational family. As an illustration, the explicit results are obtained for the ruin probability when claim sizes are exponentially distributed. A numerical example also is given for the case that two classes of claims are Erlang(2) distributed and of a mixture of two exponentials.     

On the Joint Distributions of the Time to Ruin,the Surplus Prior to Ruin,and the Deficit at Ruin in the Classical Risk Model

Abstract

The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated.     

The Moments of the Time of Ruin in Markovian Risk Models

Abstract

We present an approach based on matrix-analytic methods to find moments of the time of ruin in Markovian risk models. The approach is applicable when claims occur according to a Markovian arrival process (MAP) and claim sizes are phase distributed with parameters that depend on the state of the MAP. The method involves the construction of a sample-path-equivalent Markov-modulated fluid flow for the risk model. We develop an algorithm for moments of the time of ruin and prove the algorithm is convergent. Examples show that the proposed approach is computationally stable.     

流动性剩余、流动性风险与银行业系统性风险测量

全球性金融危机过后,世界范围内的政策制定者、经济学家以及时评人士对加强金融市场监管、防范系统性风险基本达成共识。本文从流动性角度出发,对我国商业银行部门系统性风险进行了度量,同时衡量了我国上市银行的系统重要性程度。通过对商业银行个体的资产负债表数据分析,本文提出一种基于系统相对流动性剩余的概率分布来度量流动性风险,以及基于绝对流动性剩余的方差贡献度来衡量银行系统重要性程度的方法,并对我国上市银行系统性风险进行了实证分析。本文的研究结果表明,从流动性风险视角来看,部分银行对我国商业银行体系的系统风险贡献程度较高。     

Can a Coherent Risk Measure Be Too Subadditive?

We consider the problem of determining appropriate solvency capital requirements for an insurance company or a financial institution. We demonstrate that the subadditivity condition that is often imposed on solvency capital principles can lead to the undesirable situation where the shortfall risk increases by a merger. We propose to complement the subadditivity condition by a regulator's condition. We find that for an explicitly specified confidence level, the Value‐at‐Risk satisfies the regulator's condition and is the “most efficient” capital requirement in the sense that it minimizes some reasonable cost function. Within the class of concave distortion risk measures, of which the elements, in contrast to the Value‐at‐Risk, exhibit the subadditivity property, we find that, again for an explicitly specified confidence level, the Tail‐Value‐at‐Risk is the optimal capital requirement satisfying the regulator's condition.     

The Ruin Probability of a Discrete Time Risk Model under Constant Interest Rate with Heavy Tails

This paper investigates the ultimate ruin probability of a discrete time risk model with a positive constant interest rate. Under the assumption that the gross loss of the company within one year is subexponentially distributed, a simple asymptotic relation for the ruin probability is derived and compared to existing results.     

基于信用风险管理视角的信用衍生产品创新

20世纪90年代以来,信用衍生品市场蓬勃发展,已经成为商业银行进行信用风险管理的重要手段,研究和引进这一新兴的金融工具,对于提高我国商业银行信用风险管理水平具有积极的意义。本文对银行业信用风险的产生和管理以及信用衍生产品作了介绍,对我国引进这一金融产品进行了分析并提出了建议。     

Extreme Value Theory as a Risk Management Tool

The financial industry, including banking and insurance, is undergoing major changes. The (re)insurance industry is increasingly exposed to catastrophic losses for which the requested cover is only just available. An increasing complexity of financial instruments calls for sophisticated risk management tools. The securitization of risk and alternative risk transfer highlight the convergence of finance and insurance at the product level. Extreme value theory plays an important methodological role within risk management for insurance, reinsurance, and finance.     

The Effect of Shortfall as a Risk Measure for Portfolios with Hedge Funds

Abstract:  Current research suggests that the large downside risk in hedge fund returns disqualifies the variance as an appropriate risk measure. For example, one can easily construct portfolios with nonlinear pay-offs that have both a high Sharpe ratio and a high downside risk. This paper examines the consequences of shortfall-based risk measures in the context of portfolio optimization. In contrast to popular belief, we show that negative skewness for optimal mean-shortfall portfolios can be much greater than for mean-variance portfolios. Using empirical hedge fund return data we show that the optimal mean-shortfall portfolio substantially reduces the probability of small shortfalls at the expense of an increased extreme crash probability. We explain this by proving analytically under what conditions short-put payoffs are optimal for a mean-shortfall investor. Finally, we show that quadratic shortfall or semivariance is less prone to these problems. This suggests that the precise choice of the downside risk measure is highly relevant for optimal portfolio construction under loss averse preferences.     

金融发展的测度理论研究

文章探讨金融发展的测度理论问题。     

流量覆盖风险——网络小额信贷风险控制新思路

本文总结网络金融在促进小微金融机理方面的作用,提出一种新的风险控制原则和盈利模式——流量覆盖风险:网络金融以大数据和技术帮助更好实现“大数法则”,网络贷款应恪守小额、产品简单、贷款分散的原则,以较少的风险控制指标和精简的信贷流程提升客户贷款体验,以低风险、低成本金融服务普惠更广人群。商业银行的小额贷款数据提示网络金融从业者去抵押担保化、把握好借款人、低杠杆率、本地化开展个人贷款,有助于在降低小额贷款的风险前提下,加快业务健康发展。     

Detecting Linear and Nonlinear Dependence in Stock Returns: New Methods Derived from Chaos Theory

Interest in the relevance of nonlinear dynamics to fields such as finance and economics has spurred the development of new methods of analysis for time series data. Early tests for chaos led to problems when applied to financial and economic data. This motivated development of the BDS family of statistics to test for nonlinearity generally. More recently, another method of analysis has been introduced into the scientific literature. It uses a test for chaos which is relatively simple and appropriate for financial data. A quantitative version of this test is developed here and is used to analyze stock return data.     

Asymptotic Theory for a Risk Process with a High Dividend Barrier

The only way to avoid ruin in the classical model of the collective risk theory is that the surplus increases to infinity. We consider a modified model with a dividend barrier that prevents this behavior. It is shown that there is a simple approximation formula for the time of ruin when the level of the dividend barrier is high and the Cramér-Lundberg condition is satisfied. A numerical example is presented in the case when the claims are exponentially distributed. The relation to queuing theory is used to derive the proportion of time the surplus is below some given level.     

Direct Derivation of Finite-Time Ruin Probabilities in the Discrete Risk Model with Exponential or Geometric Claims

Abstract

Growing research interest has been shown in finite-time ruin probabilities for discrete risk processes, even though the literature is not as extensive as for continuous-time models. The general approach is through the so-called Gerber-Shiu discounted penalty function, obtained for large families of claim severities and discrete risk models. This paper proposes another approach to deriving recursive and explicit formulas for finite-time ruin probabilities with exponential or geometric claim severities. The proposed method, as compared to the general Gerber-Shiu approach, is able to provide simpler derivation and straightforward expressions for these two special families of claims.     

贷款风险度量在农村信用社信贷管理中的应用

防范信贷风险是农村信用社风险管理的主要任务。贷款风险度量是贷款风险管理的重点和难点,对防范和控制贷款风险起着至关重要的作用。本文描述了贷款对象、形式、期限和形态对于贷款风险度影响的差异性及其风险权数的确定,重点探讨了贷款风险度量在农村信用社贷款决策审批、信贷检查、风险监测及控制管理方面的应用方法。     

Implied Equity Duration: A New Measure of Equity Risk

Duration is an important and well-established risk characteristic for fixed income securities. We use recent developments in financial statement analysis research to construct a measure of duration for equity securities. We find that the standard empirical predictions and results for fixed income securities extend to equity securities. We show that stock price volatility and stock beta are both positively correlated with equity duration. Moreover, estimates of common shocks to expected equity returns extracted using our measure of equity duration capture a strong common factor in stock returns. Additional analysis shows that the book-to-market ratio provides a crude measure of equity duration and that our more refined measure of equity duration subsumes the Fama and French (1993) book-to-market factor in stock returns. Our research shows how structured financial statement analysis can be used to construct superior measures of equity security risk.