Equilibrium investment strategy for a defined contribution pension plan under stochastic interest rate and stochastic volatility

https://doi.org/10.1016/j.cam.2019.112536Get rights and content

Highlights

  • Mean-variance DC pension problem with stochastic interest and volatility is modeled.

  • Extended HJB equation under the framework of Nash equilibrium game is solved.

  • Explicit equilibrium investment strategy and equilibrium value function are derived.

  • Numerical results are presented to demonstrate the impacts of the model parameters.

Abstract

This paper aims to find the equilibrium investment strategy for a defined contribution pension plan under the mean–variance criterion where both the interest rate and volatility are stochastic in the financial market. The financial market consists of a risk-free asset, a bond and a risky asset. Specifically, an affine model, which includes the Cox–Ingersoll–Ross model and the Vasicek model as special cases, is used to characterize the stochastic dynamics of the interest rate, and the price process of the risky asset is described by the Heston volatility model. Under the framework of Nash equilibrium, we first define the equilibrium strategy and the equilibrium value function. Then, by solving an extended Hamilton–Jacobi–Bellman equation, we obtain both the equilibrium investment strategy and the corresponding equilibrium value function explicitly. Furthermore, the effects of the stochastic interest rate and the stochastic volatility on the equilibrium investment strategy and the equilibrium efficient frontier are analyzed. Some numerical results and the economic meanings behind are also presented.

Introduction

The defined benefit (DB) pension plan and defined contribution (DC) pension plan are two mainly used types of pension plans all over the world. In a DB pension plan, the benefits are fixed in advance by the sponsor while the contributions are initially set and subsequently adjusted to maintain the fund balance. In contrast, a DC pension plan assumes that the contributions are fixed and the benefits depend on the pension fund investment return. Facing the aging population problem and the longevity risk, the past decades has witnessed the on the DC pension plan which has become the dominant form of pension plan in many countries. The reason for this phenomenon lies in the advantage of a DC pension plan over a DB pension plan by shifting the investment risk from the sponsor to the retiree. Moreover, the DC pension plan is a retirement plan that is typically tax-deferred, like a US 401(k) or a US 403(b), in which employees contribute a fixed amount or a percentage of their paychecks to an account that is intended to fund their retirements. From 1980 to 1998 in USA, the proportion of households covered by traditional DB type plan fell from 60% to 14%, and the proportion of households covered by the DC 401(k) plan rose from 18% to 56%, while the proportion of households participating in two plans rose from 22% to 30%. At of the end of 2009, the employer’s DB type pension plan was $2.1 trillion, while the corporate DC type plan was as large as $4.1 trillion. Currently, many countries adopt defined contribution pension funds, such as the United States, Japan, Australia, Chile, the Netherlands, and so on.

Since the investment management for DC pension plan usually lasts 20–40 years, the investor has to take wide categories of risks into consideration, such as the investment risk, the stochastic interest rate risk, the stochastic volatility risk, etc. Meanwhile, in order to keep the living standard after retirement, a minimum guarantee is usually required for a DC pension plan. Therefore, in the last two decades, the optimal investment strategies for DC pension plans have been extensively investigated. In discrete time model, Vigna and Haberman [1] considered both the investment risk and the annuitisation risk in the DC pension plan model, and derived the optimal investment strategy for the DC fund member by the stochastic dynamic programming method. Haberman and Vigna [2] formulated the model of DC pension fund in a market with n risk assets and studied the optimal investment and downside risk for the pension member. In continuous time model, Boulier et al. [3] considered the stochastic interest rate in the DC pension plan and obtained the optimal allocations to maximize the expected utility of terminal wealth over a guarantee which was an annuity after retirement. Deelstra et al. [4] extended the result to the case of stochastic contribution rate and general minimum guarantee. They considered a complete market, then martingale method was beautifully applied to solve the optimization problem. Later, Battochio and Menoncin [5], Zhang et al. [6] and Han and Hung [7] introduced inflation risk in the pension plan and derived optimal investment strategies to maximize the expected utility of terminal wealth. Giacinto et al. [8] used the stochastic programming method in DC pension plan when the wealth of the pension fund had to be higher than a solvency level all the time in the accumulation phase. However, most literature on DC pension plans are conducted in the framework of the expected utility maximization, while few researches focus on optimal investment problem of the DC pension plan under the mean–variance criterion.

Under the expected utility framework, it is difficult to determine the exact expression of the utility function for each investor. Meanwhile, little empirical studies presents the valuable guidance as to how to measure the investor’s risk aversion. First proposed by Markowitz [9], the mean–variance criterion has been widely used in the investment practice and the financial research due to its advantage in incorporating the information of financial asset return, the variance and the correlation, the decision-maker’s investment style, the investment goal and the investment constraint, etc. Besides, these two indexes, mean and variance, can be calculated based on the history information of financial asset. Especially since the work of Li and Ng [10] and Zhou and Li [11] on dynamic mean–variance portfolio optimization, there has been soaring interest in dealing with the optimal investment problems under the mean–variance criterion, such as [12], [13], [14], [15], [16], [17] and [18]. With the development of dynamic mean–variance portfolio optimization problems, there also appear several research papers on the DC pension plans under the mean–variance framework. In discrete time model, Yao et al. [19] studied the asset allocation for a DC pension fund with stochastic income and mortality risk. Zhang et al. [20] explored the multi-period investment strategy for a DC pension plan when the investor only obtained the partial information about the financial market. In continuous time setting, Højgaard and Vigna [21] studied a mean–variance portfolio selection problem in the accumulation phase of a DC pension scheme with multiple risky asset. He and Liang [22] considered the optimal investment problem of the DC pension with the return of premiums clauses. Vigna [23] used the embedding technique to solve an optimal investment problem of DC pension plan under mean–variance criterion, and showed that it was equivalent to a target-based optimization problem, consisting of minimization of a quadratic loss function. She also proved that when the investment horizon is relative long, e.g. for longer than 15 years, the optimal investment strategy for a DC pension fund under the Expected Utility (EU) is no longer mean–variance efficient which makes results far away from the investor’s appetite. Compared with the investigation of a DC pension plan under the framework of expected utility maximization, the study on the optimal investment management of a mean–variance DC pension plan is a relatively new area. The property of the optimal investment strategy for the mean–variance DC pension fund needs further investigation.

In the above-mentioned researches on mean–variance DC pension plan, the manager tries to seek the optimal investment strategy only based on the initial information, which is called the pre-commitment strategy. It means that the manager makes the optimal investment decision at the initial time point and will perform this optimal strategy at later time points even through this strategy will not be optimal any more. However, due to the non-separability of mean–variance criterion, the pre-commitment investment strategy is not time-consistent. So, at a later time, the manager of a DC pension plan may choose not to perform the optimal strategy made at the initial time since it is not optimal at current time. Hence, the pre-commitment investment strategy under mean–variance criterion has been criticized for a long time. Although the time-inconsistent optimization problem had been stated by Strotz [24], the study of time-consistent investment strategy for mean–variance optimization problem did not have a breakthrough until the Nash equilibrium decision framework was introduced by Björk and Murgoci [25]. Since then, some scholars have tried to find the time-consistent equilibrium strategy for mean–variance optimization problems, such as [15], [26], [27], [28] and [29].

During the long investment time-period of a DC pension plan, pension manager prefers to choose a strategy which is not only optimal at the initial time but also optimal at later time. If the investment strategy is short of time-consistency, the manager at a later time may not commit himself/herself to perform the pre-commitment strategy which is not optimal at the current time and state. So, some scholars pay more attention to the equilibrium investment strategy for a mean–variance DC pension plan instead of the pre-commitment strategy. For instance, Wu et al. [30] studied the equilibrium investment strategy with two background risks: the inflation risk and the stochastic salary risk in continuous time model. Wu and Zeng [31] introduced the mortality risk into the optimal investment problem of DC pension plan in discrete time model. Under the setting of jump-diffusion model, Sun et al. [32] compared the pre-commitment strategy and the equilibrium strategy for a mean–variance DC pension plan management problem.

Most of the existing literature on the equilibrium investment strategy for the mean–variance DC pension plan has not incorporated other factors that affect the actual investment yield of DC pension plans, such as the stochastic interest rate and the stochastic volatility. However, considerable empirical evidence suggests that interest rates change over time (cf. [33], [34]), and the stochastic interest rate models are more appropriate to incorporate the mean reversion, stationarity and increased volatility with the level of interest rate (cf. [35] and [36]). Similarly, there is substantial evidence of stochastic volatility, whose existence is sustained by the “smile curve” of volatilities (cf. [37] and [38]). Hence, in this paper, we assume that the interest rate and the volatility of the risky asset are stochastic rather than constants or the deterministic function of time (cf. [39] and [40]). The most widely used stochastic interest rate model are Cox–Ingersoll–Ross (CIR) model and Vasicek model, which are first introduced by Vasicek [35] and Cox et al. [36], respectively. By adopting these two models to describe the dynamic of interest rate, Boulier et al. [3], Deelstra et al. [4], Han and Hung [7] etc. considered the impact of the stochastic interest rate on the optimal investment strategy for DC pension plans. On the other hand, Heston stochastic volatility model [41] is classical and very popular for option pricing, and has been recognized as an important feature for asset price models. Meanwhile, the Heston stochastic volatility model can be seen as an explanation of many well-known empirical findings, such as the volatility smile, the volatility clustering and the heavy-tailed nature of return distributions. Currently, only [42] and [43] discussed the DC pension investment problem under the Heston model. However, all of the above-mentioned researches on DC pension plans, which consider the stochastic interest rate and stochastic volatility risks, are conducted to maximize the expected utility of the wealth at the terminal time. As far as we know, this paper is the first to introduce both the stochastic interest rate and stochastic volatility risks into the investment management problem of mean–variance DC pension plan. However, the pre-commitment strategies for the mean–variance DC pension plan investment management under stochastic volatility model are difficult to derive in the closed-form. Motivated by the ongoing progress on the time-consistent investment management strategy of DC pension plans, we investigate the equilibrium strategy for a mean–variance DC pension plan following the Nash equilibrium method proposed by Björk and Murgoci [25].

Therefore, in this paper, two kinds of background risks, stochastic interest rate risk and stochastic volatility risk, are introduced in the investment decision of mean–variance DC pension plan. The financial market consists of a risk-free asset, a zero-coupon bond and a risky asset. We apply an affine model, including the CIR model and the Vasicek model as special cases, to describe the dynamics of the stochastic interest rate. The dynamics evolution of the stock price is formulated by the Heston volatility model. Meanwhile, at each time point, the participant will contribute a stochastic amount of money into the account of DC pension plan. Therefore, the wealth of the DC pension plan account is affected by the stochastic interest rate, the stochastic volatility and the stochastic contribution of the plan participant. On the other hand, in order to keep the living standard of the participant, the terminal value of the DC pension plan should exceed a guarantee which serves as an annuity after retirement. The goal of the participant is to find the time-consistent investment strategy for this DC pension plan under the mean–variance criterion. The main contributions of this paper lie in the following aspects: (1) we model the time-consistent investment optimization problem of DC pension plan under the mean–variance criterion in the financial market with stochastic interest rate and stochastic volatility; (2) we derive the explicit expressions of the equilibrium investment strategy and the equilibrium efficient frontier for DC pension plan under the mean–variance criterion; (3) we analyze the action mechanism of the stochastic interest rate and the stochastic volatility on the equilibrium strategy.

This paper proceeds as follows. Section 2 describes the model setting and formulation of the time-consistent investment management problem for a mean–variance DC pension plan. An auxiliary problem and the corresponding verification theorem are given in Section 3. Closed-form expressions of the equilibrium investment strategy and the equilibrium value function for the mean–variance DC pension plan are derived in Section 4. Section 5 shows some illustrative numerical results. The paper is concluded in Section 6. The proofs for lemmas, theorems and propositions of this paper are given in the Appendix A, Appendix B.

Section snippets

Model formulation

In this section, we will present the model setting of the financial market, the admissible strategies set and the time-consistent optimization problem for a mean–variance DC pension plan.

Solution to the optimization problem

Inspired by Han and Hung [7] and Guan and Liang [42], we need to use an auxiliary problem to solve the original optimization problem (10). With the assumptions of complete market and no arbitrage, we construct a hypothetical financial instrument that pays the instantaneous contribution C(s) at time s. The value of this hypothetical asset at time t(ts) is denoted by D(t,s) and D(t,s) must satisfy the following partial differential equation with a boundary condition D(s,s)=C(s). Dt+DCCμ+Dr(abr)+

The equilibrium investment strategy and the equilibrium efficient frontier

In this section, we derive the explicit expressions of the equilibrium investment strategy and the equilibrium value function by using the method presented by Björk and Murgoci [47]. The results are summarized in Theorem 4.1.

Theorem 4.1

For the optimal investment problem of the DC pension plan with stochastic interest rate and stochastic volatility, the equilibrium investment strategy for problem (22) is specified as uB(t)=q(t)Yu+n̄3(t)+λRq(t)γn1(t)eq(t)r,uS(t)=σLρSLn̄4(t)+υγn1(t)eq(t)r,and the

Numerical simulation

Compared to the existing literature on the optimal investment strategy for DC pension plans, this paper focuses on the equilibrium investment strategy for a DC pension plan under the mean–variance criterion with stochastic interest rate and stochastic volatility. So, in this section, we will illustrate some numerical results about the impact of the stochastic interest rate and stochastic volatility on the equilibrium investment strategy as well as the equilibrium efficient frontier. For

Conclusion

Considering the long-term investment horizon of DC pension plan, the manager has to take the risk of stochastic interest rate of the risk-free asset and the stochastic volatility of the risky asset into account. In the financial market with stochastic interest rate and stochastic volatility, this paper investigates the equilibrium investment strategy for a mean–variance DC pension plan under the framework of Nash equilibrium. The generalized dynamics of the interest rate is characterized by an

Acknowledgments

The authors are very grateful to editors and two anonymous referees for their very helpful suggestions and comments.

This work was supported by National Natural Science Foundation of China (Nos. 71601055, 71971070, 11801179, 71701084), Chinese Postdoctoral Science Foundation (No. 2018M641594), the “Chenguang Program”, Shanghai, China (No. 18CG26), the Fundamental Research Funds for the Central Universities, PR China (No. 2019ECNU-HWFW028), the 111 Project, China (No. B14019), the Philosophy and

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