Stochastic Optimization Problems Of Asset Allocation For Different Groups

Asset allocation is a process to select asset classes and determine their proportions in the portfolio.Based on the investor 's specific requirement,it predicts the future trend of each asset from its past performance and mathematical models.Then it determines the asset classes and their proportions through mathematical optimization methods,and dynamically rebalance the portfolio.Compared to investing a single asset,asset allocation portfolio reduces the volatility of the portfolio by decentralized investments.Thus it is more flexible,return-risk balanced,and has lower volatility and less proba-bility of extreme case.It is applicable to investors with specific investment objectives,such as professional institutions,high net worth clients,etc.On the one hand,as the asset allocation has the above advantages,people gradually realize its importance.This makes that asset allocation plays an increasingly important role in modern investment,and its vigorous development in the industry also requires and promotes the theoretical follow-up in academia.However,on the other hand,the diversity of asset classes leads to the complexity and difficulty of theoretical research model,and thus asset allocation becomes an important and focused issue in modern financial mathematics.During the development of theoretical research on asset allocation,many scholars have conducted fruitful research from different perspectives,and have derived many profound results.Nevertheless,these studies are often based on individual investors,and there are few studies on other types of investors.But in real life,asset allocation has different meanings for different types of investors.Therefore,it is necessary to study and compare the asset allocation of different investor groups.Due to the defi-ciencies of the above theoretical research and the need for investment decisions in real life,this thesis studies the problem of asset allocation for individuals,families,and in-stitutions from the multiple dimensions of modern portfolio theory,behavioral finance,risk control,and fund management.This thesis has three parts according to the three different groups.The first part considers the asset allocation of individual investors under the coefficient of relative risk aversion(CRRA)utility function from the perspective of behavioral finance.Life insurance is included in the framework of asset allocation,and it is assumed that there is a cointegration relationship between income and interest rate,and the stock follows the stochastic volatility model.We firstly study the stochastic optimization control problem without labor income,then under the assumption of complete financial mar-kets,replicate and value investor 's human capital through tradable assets.This converts the stochastic optimization control problem with income into the situation without in-come,and derives the explicit solution through the dynamic programming approach.The results show that the level of risk aversion can hardly affect the proportion of consumption and life insurance.In addition,investors with high risk aversion level will hold more bond positions and less stock positions.The second part deals with some family problems caused by late marriages,late childbirth and aging.We con-sider the asset allocation problem under mean-variance utility from the perspective of risk management,and also take into account the inflation risk since its a long-term fi-nancial investment process.Besides,life insurance and unemployment insurance are included in the investment system.Now the Hamilton-Jacobi-Bellman(HJB)equation becomes nonlinear,and the traditional linear quadratic(LQ)method and dynamic pro-gramming method become more difficult.In order to overcome these difficulties,the Lagrange dual method is applied to handle the constraints and the problem turns out to be an unconstrained problem.Then with the assumption of the complete market and no arbitrage,we construct some hypothetical financial instruments to replicate the labor income,and finally employ the dynamic programming approach to derive the nonlinear HJB equation from which we get the optimal feedback control.The results show that a higher inflation level makes the family bear more risk at the same level of expectation,and a higher interest rate or early retirement also reduces the family 's risk tolerance.Moreover,the family can still get a positive feedback from working despite the negative effect of its consumption.In the third part,we take mutual fund as an example to study the influence of fund managers ' personal interests on their asset al-location behavior,and reveal the possible tunneling behaviors between different funds managed by the same fund manager.From the perspective of the fund manager,tunnel-ing behaviors can maximize benefits,but on the other hand,these behaviors may cause losses to investors ' interests.This thesis first study the value function of the mutual fund manager under the one-dimensional framework as a benchmark,then the two-dimensional stochastic control problem without transaction costs is established,and the semi-analytical solution is obtained by the heuristic method.After that we consid-er the transaction cost,and obtain the numerical solution by the penalty method and the finite difference method.The result shows high management fee rate could neither make the fund manager work efficiently,nor induce the beneficial tunneling behaviors.In addition,the scale effect of fund size is more important than the high management fee rates.All three problems studied in the thesis are based on reality and driven by practical problems,then make theoretical innovation.Therefore,on the one hand this thesis theoretically fills the blank in the study of asset allocation for different groups in academia.On the other hand,it also has practical significance and application value for decision making of individual,family or institutional investor 's in real life.