若干风险问题分析
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摘要
风险在人们生活中无处不在,概率统计是人们研究风险的一个很好的工具.本文主要给出概率统计方法在火灾风险分析中的应用,并研究投资组合中的最优化问题.
随着经济的发展,火灾风险越来越受到人们重视.我们将聚类和判别方法用在火灾探测上,对不同的材料的燃烧产物聚类.由于没有相应的数据,提出使用判别方法和logistic回归的方法来计算火灾发生概率.这为火灾探测提供了一个实用的方法.在对地区火灾损失的风险评估上,也使用了聚类方法.在火灾预测方面,发展了灰色GM(1,1)模型的适用性,证明了灰色GM(1,1)模型的数据要么单调增,要么单调减,单调增的时候,增幅越来越大,单调减的时候,减幅越来越小.另外,探讨火灾损失额重尾性和自相似性,并使用截断重尾分布(TPT模型)来拟合火灾损失额.
在风险组合方面,我们首先考虑了因子型相依违约风险模型.资产回报率之间的相依性体现在违约示性变量上.我们研究了施加于违约示性变量相依结构以及效用函数之上的几组充分条件,使得在此条件下可以得到不同资产最优投资份额的大小关系.对于保单限额和保单自留额,我们应用随机序关系的二元函数刻画性质,重新考虑文献中已有的模型,从保单持有人的角度,在扭曲风险度量最小的意义下,给出保单限额和保单自留额的最优配置序的一些关系.我们既考虑了风险的损失程度,也考虑了风险发生的时间效应.我们特别关注扭曲函数为凹(concave)函数或仅为单调递增的扭曲风险度量.文中大多数的结果都可以直接应用到一些具体的扭曲风险度量中去.所得结果丰富和发展了文献中已有的结果.
Risk exists in all our life. Statistics and probability is one of the best tools to study the risk. This paper gives applications of statistics in the analysis of fire risks and study the problem of optimal portfolio.
With the developing of the economy, fire risk is more and more important. In this paper, cluster and distinguish analyses are used in fire detection. The productions of different materials being combusted is clustered. Because we didn't have the actual fire detection data, distinguish analysis and logistic regression as a likely method to calculate the fire probability are used. For fire forecasting, the suitable criteria for the GM(1,1) is given. On the other hand, we investigate the heavy-tailed property and self-similarity of fire loss data, and use the truncated power-tail (TPT) distribution to model fire loss data.
In the theory of portfolio, we first consider the dependent default risk model of factor type. The dependence between the returns of assets is driven by default indicators. Sufficient conditions on the dependence structure of default indicators and on the utility function are investigated which enable one to order the optimal amount invested in each asset. For the optimal allocation of policy limits and deductibles, by applying the bivariate characterizations of stochastic ordering relations, we reconsider the same model and derive some new refined results on orderings of optimal allocations of policy limits and deductibles with respect to the family of distortion risk measures from the viewpoint of the policyholder. Both loss severities and loss frequencies are considered. Special attention is given to the optimization criteria of the family of distortion risk measures with concave distortions and with only increasing distortions. Most of results presented in this paper can be applied to some particular distortion risk measures. The main results extend and strengthen the known results in the literature.
随着经济的发展,火灾风险越来越受到人们重视.我们将聚类和判别方法用在火灾探测上,对不同的材料的燃烧产物聚类.由于没有相应的数据,提出使用判别方法和logistic回归的方法来计算火灾发生概率.这为火灾探测提供了一个实用的方法.在对地区火灾损失的风险评估上,也使用了聚类方法.在火灾预测方面,发展了灰色GM(1,1)模型的适用性,证明了灰色GM(1,1)模型的数据要么单调增,要么单调减,单调增的时候,增幅越来越大,单调减的时候,减幅越来越小.另外,探讨火灾损失额重尾性和自相似性,并使用截断重尾分布(TPT模型)来拟合火灾损失额.
在风险组合方面,我们首先考虑了因子型相依违约风险模型.资产回报率之间的相依性体现在违约示性变量上.我们研究了施加于违约示性变量相依结构以及效用函数之上的几组充分条件,使得在此条件下可以得到不同资产最优投资份额的大小关系.对于保单限额和保单自留额,我们应用随机序关系的二元函数刻画性质,重新考虑文献中已有的模型,从保单持有人的角度,在扭曲风险度量最小的意义下,给出保单限额和保单自留额的最优配置序的一些关系.我们既考虑了风险的损失程度,也考虑了风险发生的时间效应.我们特别关注扭曲函数为凹(concave)函数或仅为单调递增的扭曲风险度量.文中大多数的结果都可以直接应用到一些具体的扭曲风险度量中去.所得结果丰富和发展了文献中已有的结果.
Risk exists in all our life. Statistics and probability is one of the best tools to study the risk. This paper gives applications of statistics in the analysis of fire risks and study the problem of optimal portfolio.
With the developing of the economy, fire risk is more and more important. In this paper, cluster and distinguish analyses are used in fire detection. The productions of different materials being combusted is clustered. Because we didn't have the actual fire detection data, distinguish analysis and logistic regression as a likely method to calculate the fire probability are used. For fire forecasting, the suitable criteria for the GM(1,1) is given. On the other hand, we investigate the heavy-tailed property and self-similarity of fire loss data, and use the truncated power-tail (TPT) distribution to model fire loss data.
In the theory of portfolio, we first consider the dependent default risk model of factor type. The dependence between the returns of assets is driven by default indicators. Sufficient conditions on the dependence structure of default indicators and on the utility function are investigated which enable one to order the optimal amount invested in each asset. For the optimal allocation of policy limits and deductibles, by applying the bivariate characterizations of stochastic ordering relations, we reconsider the same model and derive some new refined results on orderings of optimal allocations of policy limits and deductibles with respect to the family of distortion risk measures from the viewpoint of the policyholder. Both loss severities and loss frequencies are considered. Special attention is given to the optimization criteria of the family of distortion risk measures with concave distortions and with only increasing distortions. Most of results presented in this paper can be applied to some particular distortion risk measures. The main results extend and strengthen the known results in the literature.
引文
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[2]Artzner,P.,Delbaen,F.,Eber,J.M.and Heath,D.(1999).Coherent measures of risk.Mathematical Finance,9,203-228.
[3]Bowers,N.L.,Gerber,H.U.,Hickman,J.C.,Jones,D.A.and Nesbitt,C.J.(1997).Actuarial Mathematics,2nd ed.The Society of Actuaries,Illinois.
[4]Cheung,K.C.(2006).Optimal portfolio problem with unknown dependency structure.Insurance:Mathematics and Economics,38,167-175.
[5]Chcung,K.C.(2007).Optimal allocation of policy limits and deductibles.Insurance:Mathematics and Economics,41,382-291.
[6]Chcung,K.C.and Yang,H.(2004).Ordering optimal proportions in the assct allocation problem with dcpendcnt default risks.Insurance:Mathematics and Economics 35,595-609.
[7]Chu,G.and Chen,T.(2007).Probabilistic risk assessment for evacuees in building fires.Building and Environment,42,1283-1290.
[8]Chu,G.and Sun,J.(2006).Pre-evacuation time for occupants and its effect on fire risk assessmont.Journal of Fire Sciences,24,237-259.
[9]Dclbaen,F.(2002).Coherent risk measure of risk on general probability spaces.In:Advances in Finance and Stochastics(cds.:K.Sandmann and J.P.Schonbucher),Springer,Berlin,pp.1-37.
[10]Dcnneberg,D.(1990).Premium calculation:why standard deviation should be replaced by absolute deviation.Astin Bulletion,20,181-190.
[11]Denuit,M.,Dhacne,J.,Goovaerts,M.J.and Kaas,R.(2005).Actuarial Theory for Dependent Risks:Measures,Orders and Models.John Wiley & Sons,Ltd.,West Sussex.
[12]Denuit,M.and Frostig,E.(2008).Comparison of dependence in factor models with application to credit risk portfolios.Probability in the Engineering and Informational Sciences 22,151-160.
[13]Dhaene,J.,Denuit,M.,Goovaerts,M.J.,Kaas,R.and Vyncke,D.(2002a).The concept of comonotonicity in actuarial science and finance:theory.Insurance:Mathematics and Economics,31,3-33.
[14]Dhaenc,J.,Denuit,M.,Goovaerts,M.J.,Kaas,R.and Vyncke,D.(2002b).The concept of comonotonicity in actuarial science and finance:application.Insurance:Mathematics and Economics,31,133-161.
[15]Dhaene,J.,Vanduffel,S.,Goovaerts,M.J.,Kaas,R.,Tang,Q.and Vyncke,D.(2006).Risk mcasurcs and comonotonicity:A review.Stochastic Models,22,573-606.
[16]Duffic,D.and Singleton,K.J.(2003).Credit Risk:Pricing,Measurement and Management.Princcton Univcrsity Press,Princeton,New Jersey.
[17]Duncanson,M.,Woodward.A.and Reid,P.(2002).Socioeconomic deprivation and fatal unintentional domcstic fire incidents in New Zealand 1993-1998.Fire Safety Journal,37,165-179.
[18]Embrechts,P.,Kl(u|¨)ppelberg,C.and Mikosch,T.(1997).Modelling Extreme Events for Insurance and Finance.Springer,New York.
[19]Fiorini,P.M.and Lipsky,L.(1996).A Markov renewal process with long-range autocorrelations.Technical Rcport.Department of Computer Science and Engineering,University of Connecticut,Storrs.
[20]F(o|¨)llmer,H.and Schicd,A.(2002).Convex measures of risk and trading constraints.Finance and Stochastics,6,429-447.
[21]Frittelli.M.and Gianin,E.R.(2005).Law invariant convex risk measures.Advances in Mathematical Economics,7,33-46.
[22]Goovaerts,M.J.,De Vijlder,F.E.and Haezendonck,J.(1984).Insurance Premiums:Theory and Applications.North-Holland,Amsterdam.
[23]Greiner,M.,Jobmann,M.and Lipsky,L.(1995).The importance of power-tail distributions for telecommunication traffic models.Technical Report TUM-I9521,Department of Informatics,Tcchnischc Univcrsit(a|¨)t Miinchen.
[24]Guiggin,J.(1982).A theory of anticipated utility.Journal of Economics Behavior and Organisation,3,323-343.
[25]Hadar,J.and Seo,T.K.(1988).Asset proportions in optimal portfolios.Review of Economic Studies 55,459-468.
[26]Hadar,J.and Seo,T.K.(1990).The effects of shifts in a return distribution on optimal portfolios.International Economic Review 31,721-736.
[27]Hatcm,J.E.(1997).Comparison of buffer Usage utilizing single and multiple servers in network systems with Power-tail distributions.Ph.D.Dissertation,Department of Computer Science and Engineering,Univcrsity of Connecticut,Storrs.
[28]Hennessy,D.A.and Lapan,H.E.(2002).The use of Archimedean copulas to model portfolio allocations.Mathematical Finance,12,143-154.
[29]Hu,T.and Wu,Z.(1999).On dependence of risks and stop-loss premiums.Insurance:Mathematics and Economics 24,323-332.
[30]Hua,L.and Cheung,K.C.(2008a).Stochastic orders of scalar products with applications.In-surance:Mathematics and Economics,42,865-872.
[31]Hua,L.and Cheung,K.C.(2008b).Worst allocations of policy limits and deductibles.Insurance:Mathematics and Economics,43,93-98.
[32]Grabner,K.W.,Dwyer,J.P.and Cutter,B.E.(2001).Fuel model selection for behave in mid-western oak savannas.Northern Journal of Applied Forestry,18,74-80.
[33]Kijima,M.and Ohnishi,M.(1996).Portfolio selection problems via the bivariate characterization of stochastic dominance relations.Mathematical Finance,6,237-277.
[34]Kijima,M.and Ohnishi,M.(1999).Stochastic orders and their applications in financial optimization.Mathematical Methods of Operations Research,50,351-372.
[35]Kusuoka S.(2001).On law-invariant coherent risk measures.Advances in Mathematical Economics,3,83-95.
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