Asset Allocation Problems With Regime Switching Model

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.