Continuous-Time Optimal Management For The Pension Funds

By the development of the world economy and the society, the pensioninsurance system plays a more and more important role for the basic livelihoodof the elderly. Under the background of the whole world old age, the propor-tion and number of the elderly are both much higher. The pension insuranceensures the basic needs in the life of the elderly, which means guarantee thebasic life of the quite partial population. Therefore, the pension insurance hasbecome a focus of global attention, and it has a very important role for thesocial stability and development. As the population evolution and economicfluctuation, the pension insurance faces many uncertain factors, such as thesalary structure, the worker flow, the stock price fluctuation, inflation and in-terest rate change and so on. As is known to all, these demographic variablesand economic variables are not certain, but are random. Thus, the randomsimulation and random control theory have already been used in the optimalmanagement of the pension insurance. By the mathematical method, this pa-per researches the optimal management of defined-contribution pension fundsunder the continuous time in order to enrich the theory and practice.Because the pension investment is a long-term investment, generally20to40years, in such a long time, the interest rates are random fluctuation; thepension investment must consider the interest rate risk. Chapter3studies theoptimal investment of defined-contribution pension under the afne interestrate model (including CIR model and Vasicek model). In the model, thepension funds are allowed to invest in a risk-free asset, a zero coupon bondand a risk asset. By the application of HJB equation, Legendre conversion anddual theory, we find the explicit solution of the utility functions of constantrelative risk aversion and absolute risk aversion. The research conclusionshave: in the power utility function, along with the retirement is near, thehigh-risk stock investment began gradually to turn to the low-risk cash or bond investments, but the cash and bond investment trend are uncertain; inthe exponential utility function, with the approach of the retired moment, thefund manager will invest more in high-risk stock, and less in low-risk cashand bonds, but can’t also determine the low-risk assets allocation proportion.Finally, make the numerical analysis.Based on the deficiencies of the utility maximization, Chapter4adoptsthe mean-variance model to know the decision-making process of the weight ofthe risks and benefits better. Because the constant variance elasticity (CEV)model is often the natural extension of geometric Brownian motion (GBM),the risky assets price is more practical which obeys the CEV model. By therandom control theory, establishes the optimal investment problem of the pen-sion funds under the mean-variance model. By the Legendre transformationand the dual theory, obtains the optimal investment strategies before andafter retirement, and devises the efective frontier under the mean-variancemodel. Although Chapter4assumes that the risky assets price obeys theCEV model, the financial market only contains a risky asset and doesn’t con-sider the stochastic salary. These assumptions are stricter, so Section5.1assumes that the n kinds of risky assets prices all obeys the GBM model, andthere is a correlation coefcient qi∈[1,1] between the risky-assets stochasticsources and the salary stochastic sources. Moreover, the mean-variance modelis still the optimizing target. By constructing a Riccati equation as the HJBequation’s solution, ultimately obtain the explicit solution and the efectivefrontier.The above studies don’t all consider combining the CEV model and thestochastic salary, so Section5.2establishes the mathematical models under thelogarithmic utility and utility, respectively, and obtains the explicit solutionsby the application of stochastic dynamic programming techniques. Assumingthat the volatility of the salary completely comes from the financial market,there is the multiple relationship (η), which measuring how the risk sources ofstock afect the salary. In the logarithmic utility function, studies the optimal investment problem of the pension funds with the salary as a benchmark, andobtains the optimal investment proportion by the stochastic control, Legendreconversion and duality theory. In the exponential utility function, throughthe power transformation and variable replacement, converts the nonlinearequations into linear ones, and ultimately obtains the explicit solution byconstructing the solution.Finally, the CEV model is the price dependent volatility model, whosevolatility isn’t completely stochastic and is completely related to the stockprice, so it is not ideal to overcome the “volatility smile efect”. Thus, Chapter6extends the risky-asset prices to Heston stochastic volatility model, studiesthe optimal investment problem of the defined-contribution pension undermaximizing the expected power utility, and obtains the explicit solution bythe HJB equation, power transformation and variable replacement techniques.