The Optimal Investment And Risk Management Problem For Open-end Fund With Dynamic Fund Flows

With over 200 years development,mutual fund has become one of the most impor-tant financial production in the international financial market.Many investors join the financial market through trading mutual funds.For the flexible trading mechanism and professional management,open-end mutual fund has been the most important component of international mutual industry.Due to the investors’purchase and redemptions,the fund manager operates a fund pool with dynamic fund flows,which may influence the invest-ment and management strategies of the fund.This thesis aims at investigating the optimal investment and management problem for an open-end mutual fund with the dynamic fund flows.There are only a few of publications concentrating on this topic.This thesis studies this problem in the following aspects.The first part of this thesis considers the optimal investment problem of open-end fund with dynamic fund flows in the view of the fund manager.The dynamic fund inflow and outflow processes are modeled by two independent compound Poisson processes,and the risky asset price process is modeled by a geometric Brownian motion.By the diffusion approximation to the fund inflow process,we take the correlation between the fund inflow process and the risky asset price process into consideration.Referring to the results of Berk and Van Binsbergen（2015）^[1],the investor’s expected return when purchasing the fund shares is taken into account in this model.We first introduce the accumulated value added process to measure the fund performance,which overcomes the shortages of lacking persistence and missing fund size information when measure the fund performance by the measurements based on fund returns.The fund manager chooses the optimal investment strategy for the fund to maximize the expected utility of the accumulated value added of the fund.By the dynamic programming approach,we derive the Hamilton-Jacobi-Bellman（HJB）equation for the manager,and solve the optimal investment strategy and the corresponding value function explicitly.Our result shows that the fund manager considers the investor’s expected returns when formulating the investment strategies,she/he will make efforts to satisfy the investor’s expected return and attract more fund inflows.On the other hand,the correlation between the fund flows and the risky asset price also influences the manager’s optimal strategies.The second part of this thesis investigates a Stackelberg game between an open-end fund manager and an individual investor.In this game,the fund manager is the leader,while the investor is the follower.The investor can balance her/his investment among a risk-free asset,a passive index fund and an active managed mutual fund.On the other hand,with the fund manager’s professional stock picking ability,the mutual fund is only invested in some of the profitable risky assets,which constitute the passive index fund,and a risk-free asset.A fixed proportion of the asset under management is charged as the management fee.The investor tries to maximize the expected utility of her/his terminal wealth,and the fund manager can adjust the fund portfolio to maximize the expected util-ity of the cumulative management fees.Different from the martingale method and back-ward stochastic differential equation（BSDE）approach adopted by the existing related research,in this chapter,we solve this problem by the dynamic programming approach.After solving the corresponding HJB equations of the investor and the manager,we derive the Stackelberg equilibrium strategies explicitly.Our result shows that the passive index fund serves as a stochastic benchmark for the investor and the manager,which has a deep influence on their optimal investment strategies.Moreover,a higher management fee rate would encourage the manager to take aggressive strategies,and lower down the investor’s investment in the mutual fund.The third part considers the optimal investment and management fee problem of open-end fund under the jump diffusion model.The risky asset price process follows a jump diffusion process,and both the fund inflow and outflow processes are correlated with the risky asset price process in this model.The fund manager balances the investment between a risky asset and a risk-free asset,and she/he can also adjust the proportional management fee rate of the fund.In this model,we consider the fund manager’s short-term and long-term goals,i.e.,making profits to fund company and the manager’s career concern,at the same time.The manager chooses the optimal investment strategy and management fee rate to maximize the sum of the expected utility of the accumulated management fees and the expected utility of the terminal wealth of the fund.By the maximum principle,we present the Hamiltonian function and the BSDE which the adjoint process satisfies.By solving the BSDE,we get the equations that the optimal strategy pair satisfies.Moreover,the existence and uniqueness of the optimal strategy pair are verified.The result shows that the management fee is an important manner for the fund manager to control the risk of the dynamic fund flows,no mater if they are related with the risky asset price process or not.However,this kind of correlation affects the optimal investment strategy of the fund until it vanishes.This thesis formulates models to investigate the optimal investment and management problem of an open-end fund,which reveals the effects of the fund flows and market fac-tors on the optimal management strategy of the fund.Some economic phenomenons are well explained in this work,and this thesis also provides references and theoretical sup-ports to the factual management of an open-end fund.In order to analyze the investment strategy intuitively,some numerical examples are provided to illustrate the influences of some model parameters on the optimal strategies.