Abstract: Pension fund management is an important public policy issue. In most countries, pension funds account for a large share of government spending, a share that will continue to grow as the “baby boom” generation retires. In addition, private pension funds are an important part of modern financial markets and have a significant impact on savings, investment and economic growth. Defined-benefit (DB) pension plans and defined-contribution (DC) pension plans are the two most common types of pension fund plans. The core of the DB pension plan is to pay the pension with a certain income, and the payment level of the pension fund after retirement is determined in advance, and has no relationship with the investment income in the accumulation stage and the inflation in the later stage, and the risk is borne by the fund management. This brings a lot of inconveniences to the investment management, preservation and appreciation of pension funds, and may even lead to a deficit. In the DC pension plan, the contribution rate is determined in advance, the payment level of pension fund is determined by the contribution rate in the accumulation stage and the corresponding investment income, and the risk is borne by the members of pension fund. It can be seen that the DC pension fund plan is a pension management model widely adopted by social security systems in the world. It is of great theoretical and practical significance to study the investment strategies of DC pension funds under different financial market environments.
Interest rate is one of the most important and direct factors affecting the level of pension fund payment after retirement. This paper assumes that the instantaneous interest rate meets the CIR interest rate model. In order to maintain and increase the value of pension funds, fund managers invest their funds in the financial market, hoping to find an optimal strategy that can maximize investment returns and minimize investment risks. Based on this situation, we build a continuous-time mean-variance model for DC pension investment. For the mean-variance model under the stochastic interest rate market model, the traditional linear quadratic control theory and the backward stochastic differential equation theory are all difficult to solve this problem. In this paper, the mean-variance model is transformed into an unconstrained optimization problem by Lagrange multiplier method, and it satisfies the application conditions of stochastic optimal control theory. The analytical solutions of Precommitment strategies and efficient frontier are obtained by using Hamilton-Jacobi-Bellman (HJB) equation method and Lagrange duality theorem. Numerical examples explain the influence of model parameters on Precommitment strategies and efficient frontiers, and explain some economic implications.
The theoretical and empirical analysis results are summarized as follows. (i)The efficient strategy depends on the instantaneous interest rate level, and the risk level does not depend on the instantaneous interest rate level, but on the initial interest rate level. (ii)The capital market line in the mean-standard deviation plane remains a straight line. When the investment risk is zero, the expected return is equivalent to the asset value derived from investing all of the accumulated discounted initial wealth and contribution rate in the zero coupon bond at the initial moment. (iii)When the mean-reverting speed of interest rate increases, fund managers should reduce the amount of investment in risk-free assets, and should invest more funds in stocks and zero-coupon bonds. At the same time, the risks faced by investors will increase. (iv)When the volatility level of interest rates increases, fund managers should reduce the amount of funds invested in stocks and zero-coupon bonds, and should invest more funds in risk-free assets, and the investment risk will decrease accordingly.

Key words: CIR interest rate model, DC pension investment, mean-variance model, effective strategy, efficient frontiers

摘要： 假定金融市场包括一种无风险资产、多种股票和一种零息票债券,其中利率满足Cox-Ingersoll-Ross 利率模型。基金管理人旨在寻找一种最优投资策略使投资收益最大化的同时最小化投资风险。首先,运用拉格朗日乘子法和拉格朗日对偶定理将原均值-方差模型转化为二次效用函数下的等价问题。然后,通过求解Hamilton-Jacobi-Bellman方程得到了预先承诺策略和有效前沿。最后,给出数值算例对所得结果进行了解释。

关键词: CIR利率模型, DC型养老基金投资, 均值-方差准则, 有效策略, 有效前沿