具有Regime Switching 模型的资本分配问题
|
摘要
这篇论文致力于研究保险和金融领域的资本分配和投资组合问题。资本分配就是将资本在多种资产(包括债券、股票、无风险资产等)中进行分配,使得投资者最终获得预期范围内效用最大的收益。现代投资组合理论起源于Markowitz (1952)提出的均值一方差模型。Markowitz均值-方差模型的基本思想是：用期望表示投资者的预期投资收益,用方差(或者标准差)来度量投资风险；该模型的目标是：在投资收益水平给定的前提下,最小化投资风险,或者在投资风险水平给定的条件下,最大化投资收益。众所周知,高收益一般伴随着高风险,于是,平衡均值和方差两项指标进行资本分配是投资者需要解决的问题。继Markowitz (1952)一文问世之后,其作者Markowitz被公认为“现代投资组合理论之父”,由于此文章中的模型是单期情形,随后许多研究者都进行了扩展。例如,Harrison and Kreps (1979), Cheung and Yang (2007)研究离散时间的多期情形投资组合问题,Merton (1969), Chiu and Li (2006)研究连续时间的多期情形投资组合问题,Bajeux-Besnainou and Portait (1998), Li and Ng (2000), Basak and Chabakauri (2010)分别对动态投资组合问题进行了研究,并取得了较丰硕的成果。本篇论文由五个部分组成,我们将在第一部分详细介绍资本分配和投资组合理论的发展历史和实际应用。
本论文的第二部分由第二章和第三章组成,研究了保险学中保单限额和免赔额的最优分配问题。我们假设投保人面临着n种风险,并且他需要买n份保险。在总保单限额和总免赔额给定的情况下,我们从投保人角度研究了保单限额、免赔额的最优分配问题,该思想源于Cheung (2007)一文中的模型。首先我们在原有模型的基础上作了扩展：假设n种风险都受离散的随机环境影响,故每一种风险都可以表示成一些基础随机变量(fundamental r.v.)的复合形式。之后,我们又进一步延伸了模型：一方面,假设n种风险都受离散的随机环境影响；另一方面,损失发生的频率也被考虑到其中,并且这些频率都是随机的。借助于两种随机序一Likelihood ratio order, Hazard rate order,第二章和第三章分别在不同意义上得到了保单限额和免赔额的最优分配量的排序结果,并且后者是前者的推广
从风险管理的角度看,财富的分配方法有许多种,不同的方法通常诱导出不同的分配策略。Cummins (2000)给出了关于保险业中资本分配的多种方法综述,在资本分配和决策工具之间建立了桥梁纽带。Denault(2001)研究了基于博弈论的资本分配,其中采用风险度量作为成本函数。本论文的第三部分讨论了两种分配法的应用问题：其一是一类公理化分配法,由Kalkbrener (2005)提出；另一种是广义加权分配法,由Furman and Zitikis (2008b)的模型推广得到。
目前,马尔可夫调制的体制转换模型(regime switching模型)在保险和金融领域中的应用已经引起了大量研究学者的兴趣。模型中连续时间马尔可夫链的状态被看作市场经济状态,经济状态的转移是由于经济和商业周期的结构变化而引起的。如：Elliott and Van der Hoek (1997), Cheung and Yang (2004)给出了regime switching模型在资本分配方面的应用,Guo (2001), Elliott et al. (2005)讨论了regime switching模型在期权定价方而的应用。
投资组合选择问题意味着在多种债券的投资方法中选择最优的分配法。有时候,消费、债务、通货膨胀、交易费等因素也被考虑在其中。需要指出的是,最大化利润不是投资组合的唯一目标,限制和控制投资过程中的风险也是极其重要的。有些学者已经在风险控制方面获得了一些成果,具有风险限制的投资组合问题也成为研究者讨论的一大热点。Cuoco,He and Issaenko (2001)采用鞅方法研究了VaR风险限制下的最优动态投资策略；Gabih, Sass, Wunderlich (2005)研究了具有shortfall风险限制的模型,其中,股票收益由一个连续时间、有限状态的隐形的马尔科夫链调制。在论文的第四部分中,我们考察了风险限制下的投资-消费问题。这里,用一个gime switching模型来刻画经济状态。对于每一个状态,我们对短时间内的投资组合限制一个VaR风险值。MVaR表示所有经济状态下的VaR风险值的最大值,第四部分模型中的风险限制采用了MVaR风险限制。此外,还假设所有的市场参数,比如,银行利率、风险资产和债务的收益率以及离差率,都受regime switching模型影响。本部分的目标是最大化消费的折现效用,在获得一组Hamilton-Jacobi-Bellman方程(简记为HJB方程)之后,采用拉格朗日乘数法得到最优投资和最优消费。最后,我们给出了一个具体数值实例,刻画了几对参数间的相互影响关系。本部分的模型以Yiu et al. (2010)的模型为基础,其中,原文作者只考察了总财富的分配问题,没有讨论盈余和债务分开的情形。我们将盈余和债务分开讨论,假设风险资产和债务的价值动态过程均由马尔科夫调制的几何布朗运动控制。
现今,资产-债务管理问题在风险管理中具有重要的意义。事实上,资产-债务管理问题就是盈余管理问题,其中,通常采用不可控制的债务来研究问题。Sharpe and Tint (1990), Leippold et al. (2004)认为债务的动态变化过程应该依赖于资产交易策略,即债务是不可控制的。在Markowitz均值-方差准则下,本文的第五部分研究了一个连续时间的、具有regime switching模型的资产-债务管理问题。将Xie(2009)一文中一种风险债券的情形推广至多种风险债券情形,并且还考虑了风险资产和债务的相关性。假设市场有m+1(m>1)种债券和一种债务,并且它们的价值动态过程都由布朗运动控制。此外,还假设regime switching模型和布朗运动是独立的。本部分模型的目标是：当给定最终盈余的期望值后,最小化最终盈余的风险。借助于线性二次控制技巧,我们获得了最优分配策略、有效边界、最小方差投资组合以及二基金定理。
This dissertation is devoted to study of asset allocation and optimal portfolio in in-surance and finance field. Asset allocation means allocating total wealth among several kinds of assets (including securities, stocks, riskless asset, and so on), and makes the in-vestors get the most profit within an expectant range. Modern Portfolio Theory derives from mean-variance analysis method proposed in Markowitz (1952) [Markowitz, H.,1952. Portfolio selection. Journal of Finance 7,77-91]. The basic thought of Markowitz portfo-lio is:expectation is considered as investors'return, and variance (or standard deviation) is considered as risk. Its objective is to minimize risk of investment when expectation is given, or to maximize return of investment when risk is given. It is well known, the more return is, the riskier the investment is. Therefore, obtaining balance between mean and variance becomes the investors'objective. Due to Markowitz (1952) where the model is single-period, the author Markowitz is regarded as the pioneer of modern portfolio theory, afterwards, many researchers develop Markowitz's model to a more general case, respectively. For example, Harrison and Kreps (1979) [Harrison, M., Kreps D.,1979. Martingales and arbitrage in multi-period securities market. Journal of Economic Theory 2,381-408], Cheung and Yang (2007) [Cheung, K.C., Yang, H.L.,2004. Asset allocation with regime-switching: discrete-time case. ASTIN BULLETIN 34(1),99-111] considered a portfolio problem in discrete time and multi-period case, Merton (1969) [Merton, R.C., 1969. Lifetime portfolio selection under uncertainty:the continuous time case. The Re-view of Economics and Statistics 51(3),247-257], Chiu and Li (2006) [Chiu, M.C., Li, D., 2006. Asset and liability management under a continuous-time mean-variance optimiza-tion framework. Insurance:Mathematics and Economics 39,330-355] studied a portfolio problem in continuous time and multi-period case, Bajeux-Besnainou and Portait (1998) [Bajeux-Besnainou I., Portait R.,1998. Dynamic Asset Allocation in a Mean-variance Framework. Management Science 44,79-95], Li and Ng (2000) [Li, D., Ng, W.L.,2000. Optimal dynamic portfolio selection:Multi-period mean-variance formulation. Mathe-matical Finance 10(3),387-406], Basak and Chabakauri (2010) [Basak, S., Chabakauri, G.,2010. Dynamic Mean-Variance Asset Allocation. Review of Financial Studies 23(8), 2970-3016] all investigated dynamic portfolio problems, and gained abundant fruits. This thesis consists of five parts, we make a detailed introduction about the history and appli-cations of asset allocation and optimal portfolio theory.
The second part of this thesis is composed by Chapter 2 and Chapter 3, it considers the optimal allocation problem of policy limits and deductibles in insurance field. We suppose that a policyholder is exposed to n kinds of risks, and he has n policies. When the total policy limit or the total deductible is granted, we study the optimal allocation problem of policy limits and deductibles from the viewpoint of a policyholder. The model stems from that of Cheung (2007) [Cheung, K.C.,2007. Optimal allocation of policy limits and deductibles. Insurance:Mathematics and Economics 41,382-391]. Firstly, we extend his model as follows:we assume that n risks are influenced by a discrete random environment, so each risk is a mixture of some fundamental random variables. Secondly, we extend the model once again:on one hand, n risks are influenced by a discrete random environment; On the other hand, loss frequencies which are stochastic are also considered. In the help of two kinds of stochastic orders-likelihood ratio order and Hazard rate order, Chapter 2 and Chapter 3 get the orderings of the optimal allocations amounts of policy limits and deductibles in different senses, respectively. The latter is an extension of the former.
There are lots of methods about wealth allocation in risk management area, and different methods usually leads to different decision strategies. Cummins(2000) [Cum-mins, J.D.,2000. Allocation of capital in the insurance industry. Risk Management and Insurance Review 3(1),7-27] provided an overview of several methods usually lead to dif-ferent strategies. Denault(2001) [Denault, M.,2001. Coherent allocation of risk capital. Journal of Risk 4(1),1-34] discussed capital allocation based on game theory, where a risk measure was used as cost function. Part 3 of this thesis considers applications of two kinds of capital allocation principles, one is axiomatic allocation proposed by Kalkbrener (2005) [Kalkbrener, M.,2005. An axiomatic approach to capital allocation. Mathemat-ical Finance 15(3),425-437], and the other is generalized weighted allocation which is extended from the model of Furman and Zitikis (2008b) [Furman, E., Ztikis, R.,2008b. Weighted risk capital allocations. Insurance:Mathematics and Economics 43,263-269]. After obtaining theoretic conclusions and discussing their properties, we give some specific numerical examples for applications of the two allocation methods.
Recently, lots of researchers have been interested in applications of a regime switch-ing model in insurance and finance field, and the regime switching model is modulated by a continuous time Markov chain. The states of the continuous time Markov chain can be interpreted as the states of the economy. The transitions of the states of the economy may be attributed to structural changes of the economy and business cycles. For example, Elliott and Van der Hoek (1997) [Elliott, R.J., Van der Hoek, J.,1997. An application of hidden Markov models to asset allocation problems. Finance and Stochastics 3,229-238], Cheung and Yang (2004) [Cheung, K.C., Yang, H.L.,2004. Asset allocation with regime-switching:discrete-time case. ASTIN BULLETIN 34(1),99-111] represent the applications of regime switching on asset allocation, Guo (2001) [Guo, X.,2001. Infor-mation and option pricings. Quantitative Finance 1,38-44], Elliott et al. (2005) [Elliott, R.J., Chan, L. L., Siu, T.K.,2005. Option pricing and Esscher transform under regime switching. Annals of Finance 1(4),423-432] mean the applications of regime switching on option pricing.
Portfolio selection problem is to search the best allocation of wealth among some kinds of securities. Sometimes, consumption, liability, currency inflation, transaction costs are considered. It's necessary to point out:maximizing profits is not the unique goal, constraining and controlling risk of investment is also very important. Some researchers have obtained achievements in risk controlling, and risk controlling problem has become hot spots. Cuoco, He and Issaenko (2001) [Cuoco, D., He, H., Issaenko, S.,2001, Optimal dynamic trading strategies with risk limits. Reprint. School-Yale School of Management-The Whartou School] adopted martingale method to study optimal dynamic investment strategies with VaR constraint; Gabih, Sass, Wunderlich (2005) considered a related model with shortfall constraint, where return of stocks is modulated by a continuous-time, finite-states markov chain. In the fourth part of this thesis, we investigate a optimal investment-consumption problem with liability risk constraints. Here, we use regime switching model to represent economy states. For each state of regime switching, we constrain a VaR value for the portfolio in a short time duration. MVaR means the maximum value of all VaR values in all economy states. The model of the fourth part adopts MVaR risk constraint. Moreover, we assume that all market parameters, such as interest rate of a bank account, the appreciation rates of the risky asset and the liability, the volatilities of the risky asset and the liability, switch according to regime switching model. The objective of this part is to maximize the discounted utility of consumption. After obtaining a system of Hamilton-Jacobi-Bellman (denoted by HJB) equations and utilizing Lagrange multiplier method, we derive the optimal investment and the optimal consumption. Finally, we investigate a numerical example, and characterize the effects between several pair of parameters. The idea of Part 4 is based on Yiu et al. (2010) [Yin, K.F.C., Liu, J.Z., Siu, T.K., Ching, W.K.,2010. Optimal portfolios with regime switching and value-at-risk constraint. Automatica 46,979-989], where the authors only studied the total wealth. Here, we consider the liability and surplus separately, and we suppose that the price dynamics of both risky asset and liability value are governed by markov modulated geometric Brown motions.
Nowadays, asset-liability management problem is very significant. Indeed, asset-liability management problem is surplus management problem, where the liability is un-controllable. Sharpe and Tint (1990) [Sharpe, W.F., Tint, L.G.,1990. Liabilities-a new approach. Journal of Portfolio Management 16(2),5-10], Leippild et al. (2004) [Leip-pold, M., Trojani, F., Vanini, P.,2004. A geometric approach to multi-period mean variance optimization of assets and liabilities. Journal of Economic Dynamics and Con-trol 28,1079-1113] suggested that the dynamics of liability should not be affected by the asset trading strategy, i.e., the liabilities are not controllable. Under Markowitz's mean-variance criteria, the fifth part of this thesis considers a continuous time asset-liability management problem with regime switching model. Our idea is based on Xie (2009) [Xie, S.X.,2009. Continuous-time mean variance portfolio selection with liability and regime switching. Insurance:Mathematics and Economics 45,148-155], where only considered one risky security. We assume that there are m+1(m>1) securities and one liability in the market, and the price of each security and the liability value are governed by Brown motions. Furthermore, we also investigate the correlation between the risky asset and the liability, and we assume that the markov chain and the underlying Brown motion are independent. When the investor is given a fixed expected terminal surplus level in advance, the objective is to minimize the risk of terminal wealth surplus. Under the help of linear quadratic control technique, we investigate the feasibility, derive the optimal strategy, and obtain the efficient frontier, gain minimum variance portfolio and mutual fund theorem.
本论文的第二部分由第二章和第三章组成,研究了保险学中保单限额和免赔额的最优分配问题。我们假设投保人面临着n种风险,并且他需要买n份保险。在总保单限额和总免赔额给定的情况下,我们从投保人角度研究了保单限额、免赔额的最优分配问题,该思想源于Cheung (2007)一文中的模型。首先我们在原有模型的基础上作了扩展：假设n种风险都受离散的随机环境影响,故每一种风险都可以表示成一些基础随机变量(fundamental r.v.)的复合形式。之后,我们又进一步延伸了模型：一方面,假设n种风险都受离散的随机环境影响；另一方面,损失发生的频率也被考虑到其中,并且这些频率都是随机的。借助于两种随机序一Likelihood ratio order, Hazard rate order,第二章和第三章分别在不同意义上得到了保单限额和免赔额的最优分配量的排序结果,并且后者是前者的推广
从风险管理的角度看,财富的分配方法有许多种,不同的方法通常诱导出不同的分配策略。Cummins (2000)给出了关于保险业中资本分配的多种方法综述,在资本分配和决策工具之间建立了桥梁纽带。Denault(2001)研究了基于博弈论的资本分配,其中采用风险度量作为成本函数。本论文的第三部分讨论了两种分配法的应用问题：其一是一类公理化分配法,由Kalkbrener (2005)提出；另一种是广义加权分配法,由Furman and Zitikis (2008b)的模型推广得到。
目前,马尔可夫调制的体制转换模型(regime switching模型)在保险和金融领域中的应用已经引起了大量研究学者的兴趣。模型中连续时间马尔可夫链的状态被看作市场经济状态,经济状态的转移是由于经济和商业周期的结构变化而引起的。如：Elliott and Van der Hoek (1997), Cheung and Yang (2004)给出了regime switching模型在资本分配方面的应用,Guo (2001), Elliott et al. (2005)讨论了regime switching模型在期权定价方而的应用。
投资组合选择问题意味着在多种债券的投资方法中选择最优的分配法。有时候,消费、债务、通货膨胀、交易费等因素也被考虑在其中。需要指出的是,最大化利润不是投资组合的唯一目标,限制和控制投资过程中的风险也是极其重要的。有些学者已经在风险控制方面获得了一些成果,具有风险限制的投资组合问题也成为研究者讨论的一大热点。Cuoco,He and Issaenko (2001)采用鞅方法研究了VaR风险限制下的最优动态投资策略；Gabih, Sass, Wunderlich (2005)研究了具有shortfall风险限制的模型,其中,股票收益由一个连续时间、有限状态的隐形的马尔科夫链调制。在论文的第四部分中,我们考察了风险限制下的投资-消费问题。这里,用一个gime switching模型来刻画经济状态。对于每一个状态,我们对短时间内的投资组合限制一个VaR风险值。MVaR表示所有经济状态下的VaR风险值的最大值,第四部分模型中的风险限制采用了MVaR风险限制。此外,还假设所有的市场参数,比如,银行利率、风险资产和债务的收益率以及离差率,都受regime switching模型影响。本部分的目标是最大化消费的折现效用,在获得一组Hamilton-Jacobi-Bellman方程(简记为HJB方程)之后,采用拉格朗日乘数法得到最优投资和最优消费。最后,我们给出了一个具体数值实例,刻画了几对参数间的相互影响关系。本部分的模型以Yiu et al. (2010)的模型为基础,其中,原文作者只考察了总财富的分配问题,没有讨论盈余和债务分开的情形。我们将盈余和债务分开讨论,假设风险资产和债务的价值动态过程均由马尔科夫调制的几何布朗运动控制。
现今,资产-债务管理问题在风险管理中具有重要的意义。事实上,资产-债务管理问题就是盈余管理问题,其中,通常采用不可控制的债务来研究问题。Sharpe and Tint (1990), Leippold et al. (2004)认为债务的动态变化过程应该依赖于资产交易策略,即债务是不可控制的。在Markowitz均值-方差准则下,本文的第五部分研究了一个连续时间的、具有regime switching模型的资产-债务管理问题。将Xie(2009)一文中一种风险债券的情形推广至多种风险债券情形,并且还考虑了风险资产和债务的相关性。假设市场有m+1(m>1)种债券和一种债务,并且它们的价值动态过程都由布朗运动控制。此外,还假设regime switching模型和布朗运动是独立的。本部分模型的目标是：当给定最终盈余的期望值后,最小化最终盈余的风险。借助于线性二次控制技巧,我们获得了最优分配策略、有效边界、最小方差投资组合以及二基金定理。
This dissertation is devoted to study of asset allocation and optimal portfolio in in-surance and finance field. Asset allocation means allocating total wealth among several kinds of assets (including securities, stocks, riskless asset, and so on), and makes the in-vestors get the most profit within an expectant range. Modern Portfolio Theory derives from mean-variance analysis method proposed in Markowitz (1952) [Markowitz, H.,1952. Portfolio selection. Journal of Finance 7,77-91]. The basic thought of Markowitz portfo-lio is:expectation is considered as investors'return, and variance (or standard deviation) is considered as risk. Its objective is to minimize risk of investment when expectation is given, or to maximize return of investment when risk is given. It is well known, the more return is, the riskier the investment is. Therefore, obtaining balance between mean and variance becomes the investors'objective. Due to Markowitz (1952) where the model is single-period, the author Markowitz is regarded as the pioneer of modern portfolio theory, afterwards, many researchers develop Markowitz's model to a more general case, respectively. For example, Harrison and Kreps (1979) [Harrison, M., Kreps D.,1979. Martingales and arbitrage in multi-period securities market. Journal of Economic Theory 2,381-408], Cheung and Yang (2007) [Cheung, K.C., Yang, H.L.,2004. Asset allocation with regime-switching: discrete-time case. ASTIN BULLETIN 34(1),99-111] considered a portfolio problem in discrete time and multi-period case, Merton (1969) [Merton, R.C., 1969. Lifetime portfolio selection under uncertainty:the continuous time case. The Re-view of Economics and Statistics 51(3),247-257], Chiu and Li (2006) [Chiu, M.C., Li, D., 2006. Asset and liability management under a continuous-time mean-variance optimiza-tion framework. Insurance:Mathematics and Economics 39,330-355] studied a portfolio problem in continuous time and multi-period case, Bajeux-Besnainou and Portait (1998) [Bajeux-Besnainou I., Portait R.,1998. Dynamic Asset Allocation in a Mean-variance Framework. Management Science 44,79-95], Li and Ng (2000) [Li, D., Ng, W.L.,2000. Optimal dynamic portfolio selection:Multi-period mean-variance formulation. Mathe-matical Finance 10(3),387-406], Basak and Chabakauri (2010) [Basak, S., Chabakauri, G.,2010. Dynamic Mean-Variance Asset Allocation. Review of Financial Studies 23(8), 2970-3016] all investigated dynamic portfolio problems, and gained abundant fruits. This thesis consists of five parts, we make a detailed introduction about the history and appli-cations of asset allocation and optimal portfolio theory.
The second part of this thesis is composed by Chapter 2 and Chapter 3, it considers the optimal allocation problem of policy limits and deductibles in insurance field. We suppose that a policyholder is exposed to n kinds of risks, and he has n policies. When the total policy limit or the total deductible is granted, we study the optimal allocation problem of policy limits and deductibles from the viewpoint of a policyholder. The model stems from that of Cheung (2007) [Cheung, K.C.,2007. Optimal allocation of policy limits and deductibles. Insurance:Mathematics and Economics 41,382-391]. Firstly, we extend his model as follows:we assume that n risks are influenced by a discrete random environment, so each risk is a mixture of some fundamental random variables. Secondly, we extend the model once again:on one hand, n risks are influenced by a discrete random environment; On the other hand, loss frequencies which are stochastic are also considered. In the help of two kinds of stochastic orders-likelihood ratio order and Hazard rate order, Chapter 2 and Chapter 3 get the orderings of the optimal allocations amounts of policy limits and deductibles in different senses, respectively. The latter is an extension of the former.
There are lots of methods about wealth allocation in risk management area, and different methods usually leads to different decision strategies. Cummins(2000) [Cum-mins, J.D.,2000. Allocation of capital in the insurance industry. Risk Management and Insurance Review 3(1),7-27] provided an overview of several methods usually lead to dif-ferent strategies. Denault(2001) [Denault, M.,2001. Coherent allocation of risk capital. Journal of Risk 4(1),1-34] discussed capital allocation based on game theory, where a risk measure was used as cost function. Part 3 of this thesis considers applications of two kinds of capital allocation principles, one is axiomatic allocation proposed by Kalkbrener (2005) [Kalkbrener, M.,2005. An axiomatic approach to capital allocation. Mathemat-ical Finance 15(3),425-437], and the other is generalized weighted allocation which is extended from the model of Furman and Zitikis (2008b) [Furman, E., Ztikis, R.,2008b. Weighted risk capital allocations. Insurance:Mathematics and Economics 43,263-269]. After obtaining theoretic conclusions and discussing their properties, we give some specific numerical examples for applications of the two allocation methods.
Recently, lots of researchers have been interested in applications of a regime switch-ing model in insurance and finance field, and the regime switching model is modulated by a continuous time Markov chain. The states of the continuous time Markov chain can be interpreted as the states of the economy. The transitions of the states of the economy may be attributed to structural changes of the economy and business cycles. For example, Elliott and Van der Hoek (1997) [Elliott, R.J., Van der Hoek, J.,1997. An application of hidden Markov models to asset allocation problems. Finance and Stochastics 3,229-238], Cheung and Yang (2004) [Cheung, K.C., Yang, H.L.,2004. Asset allocation with regime-switching:discrete-time case. ASTIN BULLETIN 34(1),99-111] represent the applications of regime switching on asset allocation, Guo (2001) [Guo, X.,2001. Infor-mation and option pricings. Quantitative Finance 1,38-44], Elliott et al. (2005) [Elliott, R.J., Chan, L. L., Siu, T.K.,2005. Option pricing and Esscher transform under regime switching. Annals of Finance 1(4),423-432] mean the applications of regime switching on option pricing.
Portfolio selection problem is to search the best allocation of wealth among some kinds of securities. Sometimes, consumption, liability, currency inflation, transaction costs are considered. It's necessary to point out:maximizing profits is not the unique goal, constraining and controlling risk of investment is also very important. Some researchers have obtained achievements in risk controlling, and risk controlling problem has become hot spots. Cuoco, He and Issaenko (2001) [Cuoco, D., He, H., Issaenko, S.,2001, Optimal dynamic trading strategies with risk limits. Reprint. School-Yale School of Management-The Whartou School] adopted martingale method to study optimal dynamic investment strategies with VaR constraint; Gabih, Sass, Wunderlich (2005) considered a related model with shortfall constraint, where return of stocks is modulated by a continuous-time, finite-states markov chain. In the fourth part of this thesis, we investigate a optimal investment-consumption problem with liability risk constraints. Here, we use regime switching model to represent economy states. For each state of regime switching, we constrain a VaR value for the portfolio in a short time duration. MVaR means the maximum value of all VaR values in all economy states. The model of the fourth part adopts MVaR risk constraint. Moreover, we assume that all market parameters, such as interest rate of a bank account, the appreciation rates of the risky asset and the liability, the volatilities of the risky asset and the liability, switch according to regime switching model. The objective of this part is to maximize the discounted utility of consumption. After obtaining a system of Hamilton-Jacobi-Bellman (denoted by HJB) equations and utilizing Lagrange multiplier method, we derive the optimal investment and the optimal consumption. Finally, we investigate a numerical example, and characterize the effects between several pair of parameters. The idea of Part 4 is based on Yiu et al. (2010) [Yin, K.F.C., Liu, J.Z., Siu, T.K., Ching, W.K.,2010. Optimal portfolios with regime switching and value-at-risk constraint. Automatica 46,979-989], where the authors only studied the total wealth. Here, we consider the liability and surplus separately, and we suppose that the price dynamics of both risky asset and liability value are governed by markov modulated geometric Brown motions.
Nowadays, asset-liability management problem is very significant. Indeed, asset-liability management problem is surplus management problem, where the liability is un-controllable. Sharpe and Tint (1990) [Sharpe, W.F., Tint, L.G.,1990. Liabilities-a new approach. Journal of Portfolio Management 16(2),5-10], Leippild et al. (2004) [Leip-pold, M., Trojani, F., Vanini, P.,2004. A geometric approach to multi-period mean variance optimization of assets and liabilities. Journal of Economic Dynamics and Con-trol 28,1079-1113] suggested that the dynamics of liability should not be affected by the asset trading strategy, i.e., the liabilities are not controllable. Under Markowitz's mean-variance criteria, the fifth part of this thesis considers a continuous time asset-liability management problem with regime switching model. Our idea is based on Xie (2009) [Xie, S.X.,2009. Continuous-time mean variance portfolio selection with liability and regime switching. Insurance:Mathematics and Economics 45,148-155], where only considered one risky security. We assume that there are m+1(m>1) securities and one liability in the market, and the price of each security and the liability value are governed by Brown motions. Furthermore, we also investigate the correlation between the risky asset and the liability, and we assume that the markov chain and the underlying Brown motion are independent. When the investor is given a fixed expected terminal surplus level in advance, the objective is to minimize the risk of terminal wealth surplus. Under the help of linear quadratic control technique, we investigate the feasibility, derive the optimal strategy, and obtain the efficient frontier, gain minimum variance portfolio and mutual fund theorem.
引文
[1]Artzner P., Delbaen F., Eber J.M., Heath D.,1999. Coherent measures of risk. Mathe-matical Finance 9(3),203-228.
[2]Athayde, G., Flores, R.,2002. Portfolio Frontier with Higher Moments:The Undiscovered Country. Computing in Economics and Finance 209,1-42.
[3]Bai, Z.D., Liu, H.X., Wong, W.K.,2009. Enhancement of the applicability of markowitz's portfolio optimization by utilizing random matrix theory. Mathematical Finance 19(4), 639-667.
[4]Bajeux-Besnainou I., Portait R.,1998. Dynamic Asset Allocation in a Mean-variance Framework. Management Science 44,79-95.
[5]Basak, S., Chabakauri, G.,2010. Dynamic Mean-Variance Asset Allocation. Review of Financial Studies 23(8),2970-3016.
[6]Basak, S., Shapiro, A.,2001. Value-at-risk-based risk management:optimal policies and asset prices. The Review of Financial Studies 14(2),371-405.
[7]Best, M.J., Grauer R.R.,1990. The efficient set mathematics when mean-variance prob-lems are subject to general linear constraints. Journal of Economics and Business 42, 105-120.
[8]Best, M.J., Grauer R.R.,1991. On the sensitivity of mean-variance efficient portfolio to changes in asset means:some analytical and computational results. Review of Financial Studies 4,315-342.
[9]Best, M.J., Jaroslava H.,2000. The efficient frontier for bounded assets. Mathematical methods of operations research 52,195-212.
[10]Black, F.,1972. Capital Market Equilibrium with Restricted Borrowing. Journal of Business 45(3),444-455.
[11]Black, F., Scholes, M.,1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3),637-654.
[12]Boyle, P.P., Yang, H.L.,1997. Asset allocation with time variation in expected returns. Insurance:Mathematics and Economics 21,201-218.
[13]Brennan, M.J.,1970. Taxes, market valuation and corporate financial policy. National Tax Journal ⅩⅩⅢ(4),417-427.
[14]Brinson, G.P., Hood, L.R., Beebower, G.L.,1986. Determinants of Portfolio Performance. Financial Analysts Journal 42(4),39-44.
[15]Brinson. G.P.. Singer, B.D., Beebower. G.L.,1991. Determinants of Portfolio Performance Ⅱ:An Update. Financial Analysts Journal 47(3).40-48.
[16]Browne. S.1997. Survival and growth with a liability:optimal portfolio strategies in continuous time. Mathematics of Operations Research 22.468-493.
[17]Cai, X.Q., Teo, K.L., Yang, X.Q.,and Zhou, X.Y.,2000. Portfolio Optimization Under Minimax Rule. Management Science 46(7),957-972.
[18]Chen, P.. Yang. H.L., Yin G.,2008. Markowitz's mean-variance asset-liability manage-ment with regime switching: A continuous-time model. Insurance:Mathematics and Economics 43.456-465.
[19]Chen, P.. Yang. H.L.. Zhao, L.,2010. Continuous-time optimal investment-consumption strategies with uncontrollable liabilities and regime-switching: the CRRA case. Submitted.
[20]Cheung, K.C.,2006. Optimal portfolio problem with unknown dependency structure. Insurance: Mathematics and Economics 38,167-175.
[21]Cheung. K.C..2007. Optimal allocation of policy limits and deductibles. Insurance: Mathematics and Economics 41.382-391.
[22]Cheung, K.C.. Yang. H.L..2004. Asset allocation with regime-switching:discrete-time case. ASTIN BULLETIN 34(1).99-111.
[23]Cheung, K.C.. Yang, H.L..2007. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete and Continuous Dynamical Systems, Series B 8(2).315-332.
[24]Cheung, K.C.. Yang. H.L.,2008. Ordering of optimal portfolio allocation in a model with mixture of fundamental risks. Journal of Applied Probability 45.55-66.
[25]Chiu. M.C.. Li, D.,2006. Asset and liability management under a continuous-time mean-variance optimization framework. Insurance:Mathematics and. Economics 39. 330-355.
[26]Cox. J.C., Huang C.F.,1989. Optimal consumption and portfolio policies when asset prices follow a diffusion process. Journal of Economic Theory 49,33-83.
[27]Cox. J.C., Ross, S.A.,1976. The valuation of options for alternative stochastic processes Journal of Financial Economics 3.145-166.
[28]Cummins. J.D.,2000. Allocation of capital in the insurance industry. Risk Management and Insurance Review 3(1),7-27.
[29]Cuoco, D., He, H., Issaenko, S.,2001. Optimal dynamic trading strategies with risk limits. Reprint. School-Yale School of Management-The Wharton School.
[30]Dai, M., Jiang, L.S., Li, P.F., Yi, F.H.,2009. Finite horizon optimal investment and consumption with transaction costs. SIAM Journal on Control and Optimization 48(2), 1134-1154.
[31]Decamps, M., Schepper, A.D., Goovaerts, M.,2006. A path integral approach to asset-liability management. Physica A:Statistical Mechanics and its Applications 363(2),404-416.
[32]Denault, M.,2001. Coherent allocation of risk capital. Journal of Risk 4(1).1-34.
[33]Dhaene, J., Denuit, M.,1999. The safest dependency structure among risks. Insurance: Mathematics and Economics 25,11-21.
[34]Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D.,2002. The concept of comonotonicity in actuarial science and finance:theory. Insurance:Mathematics and Economics 31,3-33.
[35]Dhaene, J., Goovaerts, M.J.,1997. On the dependency of risks in the individual life model. Insurance:Mathematics and Economics 19,243-253.
[36]Dhaene, J., Goovaerts, M.J., Kaas, R.,2003. Economic capital allocation derived from risk measures. North American Actuarial Journal 7(2),44-59.
[37]Dhaene J., Henrard L., Landsman Z., Vandendorpe A., Vanduffel S.,2008. Some results on the CTE-based capital allocation rule. Insurance:Mathematics and Economics 42. 855-863.
[38]Dhaene, J., Tsanakas, A., Valdez, E., Vanduffel, S.,2010. Optimal capital allocation principles. www.kuleuven.be/insurance, publications.
[39]Dumas B., Luciano E.,1991. An exact solution to a dynamic portfolio choice problem under transactions costs. Journal of Finance 46,577-595.
[40]Elliott, R.J., Aggoun, L., Moore, J.B.,1995. Hidden markov models:estimation and control. Berlin, Heidelberg, New York:Springer.
[41]Elliott, R.J., Chan, L. L., Siu, T.K.,2005. Option pricing and Esscher transform under regime switching. Annals of Finance 1(4),423-432.
[42]Elliott, R.J., Siu, T.K.,2009. Robust Optimal Portfolio Choice Under Markovian Regime-switching Model. Methodology and Computing in Applied Probability 11,145-157.
[43]Elliott, R.J., Van der Hoek, J.,1997. An application of hidden Markov models to asset allocation problems. Finance and Stochastics 3,229-238.
[44]Feldstein, M.S.,1969. Mean-variance analysis in the theory of liquidity, preference and portfolio selection. Review of Economic Studies 36,5-12.
[45]Fischer T.,2003. Risk capital allocation by coherent risk measures based on one-sided moments. Insurance:Mathematics and, Economics 32.135-146.
[46]Furman E., Zitikis R.,2008a. Weighted premium calculation principles. Insurance: Math-ematics and Economics 42,459-465.
[47]Furman, E., Ztikis, R.,2008b. Weighted risk capital allocations. Insurance:Mathematics and Economics 43.263-269.
[48]Gabih. A., Sass. J., Wunderlich. R.2005. Utility maximization with bounded shortfall risk in an HMM for the stock returns. In:N.Kolev. P. Morettin (eds.):Proceedings of the Second Brazilian Conference:on Statistical Modelling in Insurance and Finance. Maresias August 28-September 3.2005, Institute of Mathematics and Statistics. University of Sao Paulo.116-121.
[49]Gerber. H.U.. Shiu. E.S.W.,2004. Geometric Brownian motion models for assets and liabilities:From pension funding to optimal dividends. North American Actuarial Journal 7(3),37-56.
[50]Gonedes. N.J.,1976. Capital Market Equilibrium for a Class of Heterogeneous Expecta-tions in a Two-Parameter World. Journal of Finance 31(1).1-15.
[51]Goovaerts M. J., Laeven R. J. A.,2008. Actuarial risk measures for financial derivative pricing. Insurance: Mathematics and Economics 42.540-547.
[52]Guo. X..2001. Information and option pricings. Quantitative Finance.1,38-44.
[53]Hakansson N.H..1970. Optimal investment and consumption strategies under risk for a class of utility functions. Econometrica 38(5),587-607.
[54]Hamilton, J.D..1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2).357-384.
[55]Harlow, W.V.,1991. Asset Allocation in a Downside Risk Framework. Financial Analysts Journal 47(5):28-40.
[56]Harrison. M., Kreps D.,1979. Martingales and arbitrage in multiperiod securities market. Journal of Economic Theory 2,381-408.
[57]Harrison. M., Pliska S.,1981. Martingales and stochastic intergrals in theory of continuous trading. Stochastic Processes and their Applications 11(3).215-260.
[58]Heilmann W.R..1989. Decision theoretic foundations of credibility theory. Insurance: Mathematics and, Economics 8,77-95.
[59]Hicks. J.R.,1935. A Suggestion for Simplifying the Theory of Money. Econornica 2(5). 1-19.
[60]Hicks. J.R..1939. Value and Capital.2nd. London:Oxford University Press.
[61]Hicks, J.R.,1962. Liquidity. Economic Journal 72(288),787-802.
[62]Hogan, W.W., Warren, J.M.,1974. Toward the development of an equilibrium capital-market model based on semivariance. Journal of Financial and Quantitative Analysis 9(1),1-11.
[63]Hua, L., Cheung, K.C.,2008. Stochastic orders of scalar products with applications. Insurance:Mathematics and Economics 42,865-872.
[64]Huang, X.X.,2007. Two new models for portfolio selection with stochastic returns taking fuzzy information. Eurppean Journal of Operational Research 180(1),396-405.
[65]Jensen, M.C.,1969. Risk, The Pricing of Capital Assets, and The Evaluation of Invest-ment Portfolios. The Journal of Business 42(2),167-247.
[66]Kalkbrener, M.,2005. An axiomatic approach to capital allocation. Mathematical Finance 15(3),425-437.
[67]Kamps U.,1998. On a class of premium principles including the Esscher premium. Scan-dinavian Actuarial Journal 1,75-80.
[68]Kell, A., Muller, H.,1995. Efficient portfolios in the asset liability content. ASTIN Bulletin 25,33-48.
[69]Konno, H., Suzuki, K.,1995. A mean-variance-skewness portfolio optimization model. Journal of the Operations Research Society of Japan 38(2),173-187.
[70]Konno, H., Yamazaki, H.,1991. Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Management Science 37(5),519-531.
[71]Koo, H.K.,1998. consumption and portfolio selection with labor income: A continuous time approach. Mathematical Finance 8(1),49-65.
[72]Laeven, R.J.A., Goovaerts, M.J.,2004. An optimization approach to the dynamic alloca-tion of economic capital. Insurance:Mathematics and Economics 35,299-319.
[73]Leippold, M., Trojani, F., Vanini, P.,2004. A geometric approach to multi-period mean variance optimization of assets and liabilities. Journal of Economic Dynamics and Control 28,1079-1113.
[74]Li, D., Ng, W.L.,2000. Optimal dynamic portfolio selection: Multi-period mean-variance formulation. Mathematical Finance 10(3),387-406.
[75]Lim, A.E.B., Zhou, X.Y.,2002. Mean-variance portfolio selection with random coefficients in a complete market. Mathematical and Operation Research 27,101-120.
[76]Lindenberg, E.B.,1979. Capital Market Equilibrium with Price Affecting Institutional Investors.25 Years After-Essays in honor of Harry Markowitz, Management Science.
[77]Lintner. J.,1965. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics 47(1). 13-37.
[78]Liptser. R.S., Shiryayev, A.N.,1977. Statistics of Random Process:General Theory. Vol.1. Springer-Verlag, Berlin.
[79]Luenberger, D.G.,1968. Optimization by Vector Space Methods. John Wiley, New York.
[80]Markowitz, H.,1952. Portfolio selection. Journal of Finance 7,77-91.
[81]Markowitz, H.,1959. Portfolio selection:Efficient Diversification of Investment. New York:John Wiley and Sons.
[82]Marschak. J.,1938. Money and the Theory of Assets. Econometrica 6(4),311-325.
[83]Mayers. D.,1972. Nonmarketable Assets and Capital Market Equilibrium under Uncer-tainty. In Studies in the Theory of Capital Markets, edited by Michael C. Jensen. New York:Praeger.
[84]Merton, R.C.,1969. Lifetime portfolio selection under uncertainty:the continuous time case. The Review of Economics and Statistics 51(3),247-257.
[85]Merton. R.C.,1971. Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3,373-413.
[86]Merton, R.C.,1972. An analytic derivation of the efficient frontier. Journal of Finance and Quantitative Ananlysis 9.1851-1872.
[87]Merton, R.C.,1973. An Intertemporal Capital Asset Pricing Model. Econometrica 41(5). 867-887.
[88]Merton. R.C.,1976. Option Pricing When Underlying Stock Returns are Discontinuous Journal of Financial Economics 3,125-144.
[89]Mossin. J.,1966. Equilibrium in a Capital Asset Market. Econometrica 34(4).768-783.
[90]Mossin. J.,1969. Optimal multiperiod portfolio selection policies. Journal of Busine 41.215-229.
[91]Muthuraman. K. KUMAR, S..2006. Multi-dimensional Portfolio Optimization with Proportional Transaction Costs. Mathematical Finance 16(2),301-335.
[92]Myers, S.C., Read, J.A.,2001. Capital allocation for insurance companies. Journal of Risk and Insurance 68(4),545-580.
[93]Patil. G.P., Rao,C.R., Ratnaparkhi, M.V.,1986. On discrete weighted distributions and their use in model choice for observed data. Communications in Statistics. A:Theory and Methods 15.907-918.
[94]Ross, S.A.,1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory 13(3),341-360.
[95]Roy, A.D.,1952. Safety-first and the holding of assets. Econometrics 20,431-449.
[96]Schechtman, E., Shelef, A., Yitzhaki, S., Zitikis, R.,2008. Testing hypotheses about ab-solute concentration curves and marginal conditional stochastic dominance. Econometric Theory 24,1044-1062.
[97]Sethi, S.P.,1997. Optimal consumption and investment with bankruptcy. Boston:Kluwer Academic Publishers.
[98]Shaked, M., Shanthikumar, J.G.,1994. Stochastic Orders and Their Applications. Aca-demic Press, New York.
[99]Shaked, M., Shanthikumar, J.G.,2007. Stochastic Orders. Springer.
[100]Sharpe, W.F.,1963. A Simplified Model for Portfolio Analysis. Management Science 9(2),277-293.
[101]Sharpe, W.F.,1964. Capital Asset Prices:A Theory of Market Equilibrium under Con-ditions of Risk. The Journal of Finance 19(3),425-442.
[102]Sharpe, W.F.,1970. Portfolio Theory and Capital Markets. New York: McGraw-Hill.
[103]Sharpe, W.F., Tint, L.G.,1990. Liabilities-a new approach. Journal of Portfolio Man-agement 16(2),5-10.
[104]Shreve, S.E., Soner, H.M., Xu, G.L.,1991. Optimal investment and consumption with two bonds and transaction costs. Mathematical Finance 1(3),53-84.
[105]Sotomayor, L.R., Cadenillas, A.,2009. Explicit solutions of consumption-investment prob-lems in financial markets with regime switching. Mathematical Finance 19(2),251-279.
[106]Tobin, J.,1958. Liquidity preference as behavior toward risk. Review of Economic Srudies 25(2),25-86.
[107]Tsanakas, A.,2004. Dynamic capital allocation with distortion risk measures. Insurance: Mathematics and Economics 35,223-243.
[108]Tsanakas, A.,2008. Risk measurement in the presence of background risk. Insurance: Mathematics and Economics 42,520-528.
[109]Tsanakas, A.,2009. To split or not to split:capital allocation with convex risk measures. Insurance:Mathematics and Economics 44,268-277.
[110]Van Heerwaarden A. E., Kaas R.,1992. The Dutch premium principle. Insurance: Math-ematics and Economics 11,129-133.
[111]Vasicek. O.,1971. Capital Market Equilibrium with No Riskless Borrowing. Wells Fargo Bank memorandum. San Francisco. California.
[112]Von Neumann.J., Morgrnstern. O.,1947. Thoery of games and economic behavior. Princeton University Press.
[113]Williams. J.,1938. The Theory of Investment Value. Cambridge, Mass.:Harvard Uni-versity Press.
[114]Xie. S.X..2009. Continuous-time mean variance portfolio selection with liability and regime switching. Insurance:Mathematics and Economics 45.148-155.
[115]Yin, K.F.C.. Liu. J.Z.. Siu, T.K.. Ching, W.K..2010. Optimal portfolios with regime switching and value-at-risk constraint. Automatica 46.979-989.
[116]Young, M.R.,1998. A Minimax Portfolio Selection Rule with Linear Programming Solu-tion. Management Science 44(5).673-683.
[117]Zhou,X.Y., Yin. G..2003. Markowitz mean-variance portfolio selection with regime switching: a continuous-time model. SIAM Journal on Control and Optimization 42(4). 1466-1482.
[2]Athayde, G., Flores, R.,2002. Portfolio Frontier with Higher Moments:The Undiscovered Country. Computing in Economics and Finance 209,1-42.
[3]Bai, Z.D., Liu, H.X., Wong, W.K.,2009. Enhancement of the applicability of markowitz's portfolio optimization by utilizing random matrix theory. Mathematical Finance 19(4), 639-667.
[4]Bajeux-Besnainou I., Portait R.,1998. Dynamic Asset Allocation in a Mean-variance Framework. Management Science 44,79-95.
[5]Basak, S., Chabakauri, G.,2010. Dynamic Mean-Variance Asset Allocation. Review of Financial Studies 23(8),2970-3016.
[6]Basak, S., Shapiro, A.,2001. Value-at-risk-based risk management:optimal policies and asset prices. The Review of Financial Studies 14(2),371-405.
[7]Best, M.J., Grauer R.R.,1990. The efficient set mathematics when mean-variance prob-lems are subject to general linear constraints. Journal of Economics and Business 42, 105-120.
[8]Best, M.J., Grauer R.R.,1991. On the sensitivity of mean-variance efficient portfolio to changes in asset means:some analytical and computational results. Review of Financial Studies 4,315-342.
[9]Best, M.J., Jaroslava H.,2000. The efficient frontier for bounded assets. Mathematical methods of operations research 52,195-212.
[10]Black, F.,1972. Capital Market Equilibrium with Restricted Borrowing. Journal of Business 45(3),444-455.
[11]Black, F., Scholes, M.,1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3),637-654.
[12]Boyle, P.P., Yang, H.L.,1997. Asset allocation with time variation in expected returns. Insurance:Mathematics and Economics 21,201-218.
[13]Brennan, M.J.,1970. Taxes, market valuation and corporate financial policy. National Tax Journal ⅩⅩⅢ(4),417-427.
[14]Brinson, G.P., Hood, L.R., Beebower, G.L.,1986. Determinants of Portfolio Performance. Financial Analysts Journal 42(4),39-44.
[15]Brinson. G.P.. Singer, B.D., Beebower. G.L.,1991. Determinants of Portfolio Performance Ⅱ:An Update. Financial Analysts Journal 47(3).40-48.
[16]Browne. S.1997. Survival and growth with a liability:optimal portfolio strategies in continuous time. Mathematics of Operations Research 22.468-493.
[17]Cai, X.Q., Teo, K.L., Yang, X.Q.,and Zhou, X.Y.,2000. Portfolio Optimization Under Minimax Rule. Management Science 46(7),957-972.
[18]Chen, P.. Yang. H.L., Yin G.,2008. Markowitz's mean-variance asset-liability manage-ment with regime switching: A continuous-time model. Insurance:Mathematics and Economics 43.456-465.
[19]Chen, P.. Yang. H.L.. Zhao, L.,2010. Continuous-time optimal investment-consumption strategies with uncontrollable liabilities and regime-switching: the CRRA case. Submitted.
[20]Cheung, K.C.,2006. Optimal portfolio problem with unknown dependency structure. Insurance: Mathematics and Economics 38,167-175.
[21]Cheung. K.C..2007. Optimal allocation of policy limits and deductibles. Insurance: Mathematics and Economics 41.382-391.
[22]Cheung, K.C.. Yang. H.L..2004. Asset allocation with regime-switching:discrete-time case. ASTIN BULLETIN 34(1).99-111.
[23]Cheung, K.C.. Yang, H.L..2007. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete and Continuous Dynamical Systems, Series B 8(2).315-332.
[24]Cheung, K.C.. Yang. H.L.,2008. Ordering of optimal portfolio allocation in a model with mixture of fundamental risks. Journal of Applied Probability 45.55-66.
[25]Chiu. M.C.. Li, D.,2006. Asset and liability management under a continuous-time mean-variance optimization framework. Insurance:Mathematics and. Economics 39. 330-355.
[26]Cox. J.C., Huang C.F.,1989. Optimal consumption and portfolio policies when asset prices follow a diffusion process. Journal of Economic Theory 49,33-83.
[27]Cox. J.C., Ross, S.A.,1976. The valuation of options for alternative stochastic processes Journal of Financial Economics 3.145-166.
[28]Cummins. J.D.,2000. Allocation of capital in the insurance industry. Risk Management and Insurance Review 3(1),7-27.
[29]Cuoco, D., He, H., Issaenko, S.,2001. Optimal dynamic trading strategies with risk limits. Reprint. School-Yale School of Management-The Wharton School.
[30]Dai, M., Jiang, L.S., Li, P.F., Yi, F.H.,2009. Finite horizon optimal investment and consumption with transaction costs. SIAM Journal on Control and Optimization 48(2), 1134-1154.
[31]Decamps, M., Schepper, A.D., Goovaerts, M.,2006. A path integral approach to asset-liability management. Physica A:Statistical Mechanics and its Applications 363(2),404-416.
[32]Denault, M.,2001. Coherent allocation of risk capital. Journal of Risk 4(1).1-34.
[33]Dhaene, J., Denuit, M.,1999. The safest dependency structure among risks. Insurance: Mathematics and Economics 25,11-21.
[34]Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D.,2002. The concept of comonotonicity in actuarial science and finance:theory. Insurance:Mathematics and Economics 31,3-33.
[35]Dhaene, J., Goovaerts, M.J.,1997. On the dependency of risks in the individual life model. Insurance:Mathematics and Economics 19,243-253.
[36]Dhaene, J., Goovaerts, M.J., Kaas, R.,2003. Economic capital allocation derived from risk measures. North American Actuarial Journal 7(2),44-59.
[37]Dhaene J., Henrard L., Landsman Z., Vandendorpe A., Vanduffel S.,2008. Some results on the CTE-based capital allocation rule. Insurance:Mathematics and Economics 42. 855-863.
[38]Dhaene, J., Tsanakas, A., Valdez, E., Vanduffel, S.,2010. Optimal capital allocation principles. www.kuleuven.be/insurance, publications.
[39]Dumas B., Luciano E.,1991. An exact solution to a dynamic portfolio choice problem under transactions costs. Journal of Finance 46,577-595.
[40]Elliott, R.J., Aggoun, L., Moore, J.B.,1995. Hidden markov models:estimation and control. Berlin, Heidelberg, New York:Springer.
[41]Elliott, R.J., Chan, L. L., Siu, T.K.,2005. Option pricing and Esscher transform under regime switching. Annals of Finance 1(4),423-432.
[42]Elliott, R.J., Siu, T.K.,2009. Robust Optimal Portfolio Choice Under Markovian Regime-switching Model. Methodology and Computing in Applied Probability 11,145-157.
[43]Elliott, R.J., Van der Hoek, J.,1997. An application of hidden Markov models to asset allocation problems. Finance and Stochastics 3,229-238.
[44]Feldstein, M.S.,1969. Mean-variance analysis in the theory of liquidity, preference and portfolio selection. Review of Economic Studies 36,5-12.
[45]Fischer T.,2003. Risk capital allocation by coherent risk measures based on one-sided moments. Insurance:Mathematics and, Economics 32.135-146.
[46]Furman E., Zitikis R.,2008a. Weighted premium calculation principles. Insurance: Math-ematics and Economics 42,459-465.
[47]Furman, E., Ztikis, R.,2008b. Weighted risk capital allocations. Insurance:Mathematics and Economics 43.263-269.
[48]Gabih. A., Sass. J., Wunderlich. R.2005. Utility maximization with bounded shortfall risk in an HMM for the stock returns. In:N.Kolev. P. Morettin (eds.):Proceedings of the Second Brazilian Conference:on Statistical Modelling in Insurance and Finance. Maresias August 28-September 3.2005, Institute of Mathematics and Statistics. University of Sao Paulo.116-121.
[49]Gerber. H.U.. Shiu. E.S.W.,2004. Geometric Brownian motion models for assets and liabilities:From pension funding to optimal dividends. North American Actuarial Journal 7(3),37-56.
[50]Gonedes. N.J.,1976. Capital Market Equilibrium for a Class of Heterogeneous Expecta-tions in a Two-Parameter World. Journal of Finance 31(1).1-15.
[51]Goovaerts M. J., Laeven R. J. A.,2008. Actuarial risk measures for financial derivative pricing. Insurance: Mathematics and Economics 42.540-547.
[52]Guo. X..2001. Information and option pricings. Quantitative Finance.1,38-44.
[53]Hakansson N.H..1970. Optimal investment and consumption strategies under risk for a class of utility functions. Econometrica 38(5),587-607.
[54]Hamilton, J.D..1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2).357-384.
[55]Harlow, W.V.,1991. Asset Allocation in a Downside Risk Framework. Financial Analysts Journal 47(5):28-40.
[56]Harrison. M., Kreps D.,1979. Martingales and arbitrage in multiperiod securities market. Journal of Economic Theory 2,381-408.
[57]Harrison. M., Pliska S.,1981. Martingales and stochastic intergrals in theory of continuous trading. Stochastic Processes and their Applications 11(3).215-260.
[58]Heilmann W.R..1989. Decision theoretic foundations of credibility theory. Insurance: Mathematics and, Economics 8,77-95.
[59]Hicks. J.R.,1935. A Suggestion for Simplifying the Theory of Money. Econornica 2(5). 1-19.
[60]Hicks. J.R..1939. Value and Capital.2nd. London:Oxford University Press.
[61]Hicks, J.R.,1962. Liquidity. Economic Journal 72(288),787-802.
[62]Hogan, W.W., Warren, J.M.,1974. Toward the development of an equilibrium capital-market model based on semivariance. Journal of Financial and Quantitative Analysis 9(1),1-11.
[63]Hua, L., Cheung, K.C.,2008. Stochastic orders of scalar products with applications. Insurance:Mathematics and Economics 42,865-872.
[64]Huang, X.X.,2007. Two new models for portfolio selection with stochastic returns taking fuzzy information. Eurppean Journal of Operational Research 180(1),396-405.
[65]Jensen, M.C.,1969. Risk, The Pricing of Capital Assets, and The Evaluation of Invest-ment Portfolios. The Journal of Business 42(2),167-247.
[66]Kalkbrener, M.,2005. An axiomatic approach to capital allocation. Mathematical Finance 15(3),425-437.
[67]Kamps U.,1998. On a class of premium principles including the Esscher premium. Scan-dinavian Actuarial Journal 1,75-80.
[68]Kell, A., Muller, H.,1995. Efficient portfolios in the asset liability content. ASTIN Bulletin 25,33-48.
[69]Konno, H., Suzuki, K.,1995. A mean-variance-skewness portfolio optimization model. Journal of the Operations Research Society of Japan 38(2),173-187.
[70]Konno, H., Yamazaki, H.,1991. Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Management Science 37(5),519-531.
[71]Koo, H.K.,1998. consumption and portfolio selection with labor income: A continuous time approach. Mathematical Finance 8(1),49-65.
[72]Laeven, R.J.A., Goovaerts, M.J.,2004. An optimization approach to the dynamic alloca-tion of economic capital. Insurance:Mathematics and Economics 35,299-319.
[73]Leippold, M., Trojani, F., Vanini, P.,2004. A geometric approach to multi-period mean variance optimization of assets and liabilities. Journal of Economic Dynamics and Control 28,1079-1113.
[74]Li, D., Ng, W.L.,2000. Optimal dynamic portfolio selection: Multi-period mean-variance formulation. Mathematical Finance 10(3),387-406.
[75]Lim, A.E.B., Zhou, X.Y.,2002. Mean-variance portfolio selection with random coefficients in a complete market. Mathematical and Operation Research 27,101-120.
[76]Lindenberg, E.B.,1979. Capital Market Equilibrium with Price Affecting Institutional Investors.25 Years After-Essays in honor of Harry Markowitz, Management Science.
[77]Lintner. J.,1965. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics 47(1). 13-37.
[78]Liptser. R.S., Shiryayev, A.N.,1977. Statistics of Random Process:General Theory. Vol.1. Springer-Verlag, Berlin.
[79]Luenberger, D.G.,1968. Optimization by Vector Space Methods. John Wiley, New York.
[80]Markowitz, H.,1952. Portfolio selection. Journal of Finance 7,77-91.
[81]Markowitz, H.,1959. Portfolio selection:Efficient Diversification of Investment. New York:John Wiley and Sons.
[82]Marschak. J.,1938. Money and the Theory of Assets. Econometrica 6(4),311-325.
[83]Mayers. D.,1972. Nonmarketable Assets and Capital Market Equilibrium under Uncer-tainty. In Studies in the Theory of Capital Markets, edited by Michael C. Jensen. New York:Praeger.
[84]Merton, R.C.,1969. Lifetime portfolio selection under uncertainty:the continuous time case. The Review of Economics and Statistics 51(3),247-257.
[85]Merton. R.C.,1971. Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3,373-413.
[86]Merton, R.C.,1972. An analytic derivation of the efficient frontier. Journal of Finance and Quantitative Ananlysis 9.1851-1872.
[87]Merton, R.C.,1973. An Intertemporal Capital Asset Pricing Model. Econometrica 41(5). 867-887.
[88]Merton. R.C.,1976. Option Pricing When Underlying Stock Returns are Discontinuous Journal of Financial Economics 3,125-144.
[89]Mossin. J.,1966. Equilibrium in a Capital Asset Market. Econometrica 34(4).768-783.
[90]Mossin. J.,1969. Optimal multiperiod portfolio selection policies. Journal of Busine 41.215-229.
[91]Muthuraman. K. KUMAR, S..2006. Multi-dimensional Portfolio Optimization with Proportional Transaction Costs. Mathematical Finance 16(2),301-335.
[92]Myers, S.C., Read, J.A.,2001. Capital allocation for insurance companies. Journal of Risk and Insurance 68(4),545-580.
[93]Patil. G.P., Rao,C.R., Ratnaparkhi, M.V.,1986. On discrete weighted distributions and their use in model choice for observed data. Communications in Statistics. A:Theory and Methods 15.907-918.
[94]Ross, S.A.,1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory 13(3),341-360.
[95]Roy, A.D.,1952. Safety-first and the holding of assets. Econometrics 20,431-449.
[96]Schechtman, E., Shelef, A., Yitzhaki, S., Zitikis, R.,2008. Testing hypotheses about ab-solute concentration curves and marginal conditional stochastic dominance. Econometric Theory 24,1044-1062.
[97]Sethi, S.P.,1997. Optimal consumption and investment with bankruptcy. Boston:Kluwer Academic Publishers.
[98]Shaked, M., Shanthikumar, J.G.,1994. Stochastic Orders and Their Applications. Aca-demic Press, New York.
[99]Shaked, M., Shanthikumar, J.G.,2007. Stochastic Orders. Springer.
[100]Sharpe, W.F.,1963. A Simplified Model for Portfolio Analysis. Management Science 9(2),277-293.
[101]Sharpe, W.F.,1964. Capital Asset Prices:A Theory of Market Equilibrium under Con-ditions of Risk. The Journal of Finance 19(3),425-442.
[102]Sharpe, W.F.,1970. Portfolio Theory and Capital Markets. New York: McGraw-Hill.
[103]Sharpe, W.F., Tint, L.G.,1990. Liabilities-a new approach. Journal of Portfolio Man-agement 16(2),5-10.
[104]Shreve, S.E., Soner, H.M., Xu, G.L.,1991. Optimal investment and consumption with two bonds and transaction costs. Mathematical Finance 1(3),53-84.
[105]Sotomayor, L.R., Cadenillas, A.,2009. Explicit solutions of consumption-investment prob-lems in financial markets with regime switching. Mathematical Finance 19(2),251-279.
[106]Tobin, J.,1958. Liquidity preference as behavior toward risk. Review of Economic Srudies 25(2),25-86.
[107]Tsanakas, A.,2004. Dynamic capital allocation with distortion risk measures. Insurance: Mathematics and Economics 35,223-243.
[108]Tsanakas, A.,2008. Risk measurement in the presence of background risk. Insurance: Mathematics and Economics 42,520-528.
[109]Tsanakas, A.,2009. To split or not to split:capital allocation with convex risk measures. Insurance:Mathematics and Economics 44,268-277.
[110]Van Heerwaarden A. E., Kaas R.,1992. The Dutch premium principle. Insurance: Math-ematics and Economics 11,129-133.
[111]Vasicek. O.,1971. Capital Market Equilibrium with No Riskless Borrowing. Wells Fargo Bank memorandum. San Francisco. California.
[112]Von Neumann.J., Morgrnstern. O.,1947. Thoery of games and economic behavior. Princeton University Press.
[113]Williams. J.,1938. The Theory of Investment Value. Cambridge, Mass.:Harvard Uni-versity Press.
[114]Xie. S.X..2009. Continuous-time mean variance portfolio selection with liability and regime switching. Insurance:Mathematics and Economics 45.148-155.
[115]Yin, K.F.C.. Liu. J.Z.. Siu, T.K.. Ching, W.K..2010. Optimal portfolios with regime switching and value-at-risk constraint. Automatica 46.979-989.
[116]Young, M.R.,1998. A Minimax Portfolio Selection Rule with Linear Programming Solu-tion. Management Science 44(5).673-683.
[117]Zhou,X.Y., Yin. G..2003. Markowitz mean-variance portfolio selection with regime switching: a continuous-time model. SIAM Journal on Control and Optimization 42(4). 1466-1482.