Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump–diffusion model

In this paper, we study an optimal investment problem under the mean–variance criterion for defined contribution pension plans during the accumulation phase. To protect the rights of a plan member who dies before retirement, a clause on the return of premiums for the plan member is adopted. We assume that the manager of the pension plan is allowed to invest the premiums in a financial market, which consists of one risk-free asset and one risky asset whose price process is modeled by a jump–diffusion process. The precommitment strategy and the corresponding value function are obtained using the stochastic dynamic programming approach. Under the framework of game theory and the assumption that the manager’s risk aversion coefficient depends on the current wealth, the equilibrium strategy and the corresponding equilibrium value function are also derived. Our results show that with the same level of variance in the terminal wealth, the expected optimal terminal wealth under the precommitment strategy is greater than that under the equilibrium strategy with a constant risk aversion coefficient; the equilibrium strategy with a constant risk aversion coefficient is revealed to be different from that with a state-dependent risk aversion coefficient; and our results can also be degenerated to the results of He and Liang (2013b) and Björk et al. (2014). Finally, some numerical simulations are provided to illustrate our derived results.     

Nash equilibrium strategies for a defined contribution pension management

This paper studies the time-consistent investment strategy for a defined contribution (DC) pension plan under the mean–variance criterion. Since the time horizon of a pension fund management problem is relatively long, two background risks are taken into account: the inflation risk and the salary risk. Meanwhile, there are a risk-free asset, a stock and an inflation-indexed bond available in the financial market. The extended Hamilton–Jacobi–Bellman (HJB for short) equation of the equilibrium value function and the verification theorem corresponding to our problem are presented. The closed-form time-consistent investment strategy and the equilibrium efficient frontier are obtained by stochastic control technique. The effects of the inflation and stochastic income on the equilibrium strategy and the equilibrium efficient frontier are illustrated by mathematical and numerical analysis. Finally, we compare in detail the time-consistent results in our paper with the pre-commitment one and find the distinct properties of these two results.     

Time-consistent investment strategy under partial information

This paper considers a mean–variance portfolio selection problem under partial information, that is, the investor can observe the risky asset price with random drift which is not directly observable in financial markets. Since the dynamic mean–variance portfolio selection problem is time inconsistent, to seek the time-consistent investment strategy, the optimization problem is formulated and tackled in a game theoretic framework. Closed-form expressions of the equilibrium investment strategy and the corresponding equilibrium value function under partial information are derived by solving an extended Hamilton–Jacobi–Bellman system of equations. In addition, the results are also given under complete information, which are need for the partial information case. Furthermore, some numerical examples are presented to illustrate the derived equilibrium investment strategies and numerical sensitivity analysis is provided.     

基于加权效用和VaR-PI约束下DC型养老金计划的最优资产配置

本文从养老金计划参与人和基金经理的双重视角出发,以最大化双方加权的期望效用为目标,研究了在最低保障和VaR约束下,DC养老金计划的最优资产配置问题。假设养老金计划参与人和基金经理均是损失厌恶的,分别用两个S型的效用函数来刻画双方的损失厌恶行为。VaR约束和加权的效用函数使得本文所研究的优化问题成为一个复杂的非凹效用最大化问题。利用拉格朗日对偶理论和凹化方法求得了最优财富和最优投资组合的封闭解。数值结论表明当更为看重养老金计划参与人的利益时,基金经理会采取更为激进的投资策略,VaR约束可以改进对DC养老金计划的风险管理。     

Asset allocation for a DC pension fund with stochastic income and mortality risk: A multi-period mean–variance framework

This paper investigates an asset allocation problem for defined contribution pension funds with stochastic income and mortality risk under a multi-period mean–variance framework. Different from most studies in the literature where the expected utility is maximized or the risk measured by the quadratic mean deviation is minimized, we consider synthetically both to enhance the return and to control the risk by the mean–variance criterion. First, we obtain the analytical expressions for the efficient investment strategy and the efficient frontier by adopting the Lagrange dual theory, the state variable transformation technique and the stochastic optimal control method. Then, we discuss some special cases under our model. Finally, a numerical example is presented to illustrate the results obtained in this paper.     

Defined Contribution Pension Planning with the Return of Premiums Clauses and HARA Preference in Stochastic Environments

This paper studies a defined contribution(DC) pension fund investment problem with return of premiums clauses in a stochastic interest rate and stochastic volatility environment. In practice, most of pension plans were subject to the return of premiums clauses to protect the rights of pension members who died before retirement. In the mathematical modeling, we assume that a part of pension members could withdraw their premiums if they died before retirement and surviving members could equally sh...     

Mean–variance efficiency of DC pension plan under stochastic interest rate and mean-reverting returns

This paper studies the optimization problem of DC pension plan under mean–variance criterion. The financial market consists of cash, bond and stock. Similar to Guan and Liang (2014), we assume that the instantaneous interest rate is an affine process including the Cox–Ingersoll–Ross (CIR) model and Vasicek model. However, we assume that the expected return of the stock follows a completely different mean-reverting process, which can well display the bear and bull features of the market, and the market price of the stock index is the Ornstein–Uhlenbeck process. The pension manager thus has to undertake the risks of interest rate and market price of stock index. Besides, a special stochastic contribution rate is formulated. The goal of the pension manager is to maximize the expected terminal value and minimize the variance of terminal value. We will use the technique developed by Guan and Liang (2014) to tackle this problem and derive the closed-forms of efficient frontier and strategies. Numerical analysis is given in the end of this paper to show the economic behavior of the efficient frontier and strategies.     

Optimal assets allocation and benefit outgo policies of DC pension plan with compulsory conversion claims

In this paper, we study optimal asset allocation and benefit outgo policies of DC (defined contribution) pension plan. We extend He and Liang model (2013a,b) to describe dynamics of individual fund scale during distribution period. The fund scale is affected by investment return, benefit outgo and mortality credit. The management of the pension plan controls the asset allocation and benefit outgo policies to achieve the objective of pension members. The goal of the management is to minimize accumulated deviations between the actual benefit outgo and a pre-set target during the whole distribution period. The performance function (criterion) is the weighted average of the square and linear deviations to express more penalty on negative deviation than positive deviation. Using HJB (Hamilton–Jacobi–Bellman) equations and variational inequality methods, the closed-forms of the optimal policies are derived. The counterintuitive effect of the optimal proportion allocated in the risky asset with respect to the fund scale is also derived, and the optimal benefit outgo has the form of the spread method. Moreover, we use Monte Carlo Methods (MCM) to analyze economic behaviors of the optimal asset allocation and benefit outgo policies.     

Mean–variance target-based optimisation for defined contribution pension schemes in a stochastic framework

We solve a mean–variance optimisation problem in the accumulation phase of a defined contribution pension scheme. In a general multi-asset financial market with stochastic investment opportunities and stochastic contributions, we provide the general forms for the efficient frontier, the optimal investment strategy, and the ruin probability. We show that the mean–variance approach is equivalent to a “user-friendly” target-based optimisation problem which minimises a quadratic loss function, and provide implementation guidelines for the selection of the target. We show that the ruin probability can be kept under control through the choice of the target level. We find closed-form solutions for the special case of stochastic interest rate following the Vasiček (1977) dynamics, contributions following a geometric Brownian motion, and market consisting of cash, one bond and one stock. Numerical applications report the behaviour over time of optimal strategies and non-negative constrained strategies.     

Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information

In this paper, based on equilibrium control law proposed by Björk and Murgoci (2010), we study an optimal investment and reinsurance problem under partial information for insurer with mean–variance utility, where insurer’s risk aversion varies over time. Instead of treating this time-inconsistent problem as pre-committed, we aim to find time-consistent equilibrium strategy within a game theoretic framework. In particular, proportional reinsurance, acquiring new business, investing in financial market are available in the market. The surplus process of insurer is depicted by classical Lundberg model, and the financial market consists of one risk free asset and one risky asset with unobservable Markov-modulated regime switching drift process. By using reduction technique and solving a generalized extended HJB equation, we derive closed-form time-consistent investment–reinsurance strategy and corresponding value function. Moreover, we compare results under partial information with optimal investment–reinsurance strategy when Markov chain is observable. Finally, some numerical illustrations and sensitivity analysis are provided.     

On minimizing drawdown risks of lifetime investments

Drawdown measures the decline of portfolio value from its historic high-water mark. In this paper, we study a lifetime investment problem aiming at minimizing the risk of drawdown occurrences. Under the Black–Scholes framework, we examine two financial market models: a market with two risky assets, and a market with a risk-free asset and a risky asset. Closed-form optimal trading strategies are derived under both models by utilizing a decomposition technique on the associated Hamilton–Jacobi–Bellman (HJB) equation. We show that it is optimal to minimize the portfolio variance when the fund value is at its historic high-water mark. Moreover, when the fund value drops, the proportion of wealth invested in the asset with a higher instantaneous rate of return should be increased. We find that the instantaneous return rate of the minimum lifetime drawdown probability (MLDP) portfolio is never less than the return rate of the minimum variance (MV) portfolio. This supports the practical use of drawdown-based performance measures in which the role of volatility is replaced by drawdown.     

Robust equilibrium excess-of-loss reinsurance and CDS investment strategies for a mean–variance insurer with ambiguity aversion

This paper considers the robust equilibrium reinsurance and investment strategies for an ambiguity-averse insurer under a dynamic mean–variance criterion. The insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market consisting of a risk-free asset and a credit default swap (CDS). Following a game theoretic approach, robust equilibrium strategies and equilibrium value functions for the pre-default case and the post-default case are derived, respectively. For the ambiguity-averse insurer, in general the equilibrium strategies can be characterized by unique solutions to some algebraic equations. For the degenerate case with an ambiguity-neutral insurer, closed-form expressions of equilibrium strategies and equilibrium value functions are obtained. Numerical examples demonstrate that the consideration of model uncertainty and CDS investment improves the insurer’s utility. In this regard, our paper establishes theoretical and numerical support for the importance of ambiguity aversion, credit risk and their interplay in insurance business.     

Optimal pension fund management under multi-period risk minimization

In this paper, a multi-period stochastic optimization model for solving a problem of optimal selection of a pension fund by a pension plan member is presented. In our model, members of the pension plan are given a possibility to switch periodically between J types of funds with different risk profiles and so actively manage their risk exposure and expected return. Minimization of a multi-period average value-at-risk deviation measure under expected return constraint leads to a large-scale linear program. A theoretical framework and a solution for the case of the pension system of Slovak Republic are presented.     

Can asset allocation limits determine portfolio risk–return profiles in DC pension schemes?

In defined contribution (DC) pension schemes, the regulator usually imposes asset allocation constraints (minimum and maximum limits by asset class) in order to create funds with different risk–return profiles. In this article, we challenge this approach and show that such funds can exhibit erratic risk–return profiles that deviate significantly from the intended design. We propose to replace all minimum and maximum asset allocation constraints by a single risk metric (or measure) that controls risk directly. Thus, funds with different risk–return profiles can be immediately created by adjusting the risk tolerance parameter accordingly. Using data from the Chilean DC pension system, we show that our approach generates funds whose risk–return profiles are consistently ordered according to the intended design, and outperforms funds created by means of asset allocation limits.     

Minimization of risks in defined benefit pension plan with time‐inconsistent preferences

In this paper, we investigate the defined benefit pension plan, where the object of the manager is to minimise the contribution rate risk and the solvency risk by considering a quadratic performance criterion. To incorporate some well‐documented behavioural features of human beings, we consider the situation where the discounting is non‐exponential. It leads to a time‐inconsistent control problem in the sense that the Bellman optimality principle does no longer hold. In our model, we assume that the benefit outgo is constant, and the pension fund can be invested in a risk‐free asset and a risky asset whose return follows a geometric Brownian motion. We characterise the time‐consistent strategies and value function in terms of the solution of a system of integral equations. The existence and uniqueness of the solution is verified, and the approximation of the solution is obtained. Some numerical results of the equilibrium contribution rate and equilibrium investment policy are presented for three types of discount functions. Copyright © 2015 John Wiley & Sons, Ltd.     

基于Stein-Stein波动率和动态VaR约束下DC型养老基金的最优投资策略

研究了确定缴费型养老基金在退休前累积阶段的最优资产配置问题.假设养老基金管理者将养老基金投资于由一个无风险资产和一个价格过程满足Stein-Stein随机波动率模型的风险资产所构成的金融市场.利用随机最优控制方法,以最大化退休时刻养老基金账户相对财富的期望效用为目标,分别获得了无约束情形和受动态VaR (Value at Risk)约束情形下该养老基金的最优投资策略,并获得相应最优值函数的解析表达形式.最后通过数值算例对相关理论结果进行数值验证并考察了最优投资策略关于相关参数的敏感性.     

Equilibrium in an ambiguity-averse mean–variance investors market

In a financial market composed of n risky assets and a riskless asset, where short sales are allowed and mean–variance investors can be ambiguity averse, i.e., diffident about mean return estimates where confidence is represented using ellipsoidal uncertainty sets, we derive a closed form portfolio rule based on a worst case max–min criterion. Then, in a market where all investors are ambiguity-averse mean–variance investors with access to given mean return and variance–covariance estimates, we investigate conditions regarding the existence of an equilibrium price system and give an explicit formula for the equilibrium prices. In addition to the usual equilibrium properties that continue to hold in our case, we show that the diffidence of investors in a homogeneously diffident (with bounded diffidence) mean–variance investors’ market has a deflationary effect on equilibrium prices with respect to a pure mean–variance investors’ market in equilibrium. Deflationary pressure on prices may also occur if one of the investors (in an ambiguity-neutral market) with no initial short position decides to adopt an ambiguity-averse attitude. We also establish a CAPM-like property that reduces to the classical CAPM in case all investors are ambiguity-neutral.     

Optimal investment for a pension fund under inflation risk

This paper investigates an optimal investment problem faced by a defined contribution (DC) pension fund manager under inflationary risk. It is assumed that a representative member of a DC pension plan contributes a fixed share of his salary to the pension fund during the finite time horizon [0, T]. The pension contributions are invested continuously in a risk-free bond, an index bond and a stock. The objective is to maximize the expected utility of terminal value of the pension fund. By solving this investment problem we present a way to deal with the optimization problem, in case there is a (positive) endowment (or contribution), using the martingale method.     

Time-consistent reinsurance–investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk

In this paper, we consider the time-consistent reinsurance–investment strategy under the mean–variance criterion for an insurer whose surplus process is described by a Brownian motion with drift. The insurer can transfer part of the risk to a reinsurer via proportional reinsurance or acquire new business. Moreover, stochastic interest rate and inflation risks are taken into account. To reduce the two kinds of risks, not only a risk-free asset and a risky asset, but also a zero-coupon bond and Treasury Inflation Protected Securities (TIPS) are available to invest in for the insurer. Applying stochastic control theory, we provide and prove a verification theorem and establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) equation. By solving the extended HJB equation, we derive the time-consistent reinsurance–investment strategy as well as the corresponding value function for the mean–variance problem, explicitly. Furthermore, we formulate a precommitment mean–variance problem and obtain the corresponding time-inconsistent strategy to compare with the time-consistent strategy. Finally, numerical simulations are presented to illustrate the effects of model parameters on the time-consistent strategy.     

Optimal investment for the defined-contribution pension with stochastic salary under a CEV model

In this paper, we study the optimal investment strategy of defined-contribution pension with the stochastic salary. The investor is allowed to invest in a risk-free asset and a risky asset whose price process follows a constant elasticity of variance model. The stochastic salary follows a stochastic differential equation, whose instantaneous volatility changes with the risky asset price all the time. The HJB equation associated with the optimal investment problem is established, and the explicit solution of the corresponding optimization problem for the CARA utility function is obtained by applying power transform and variable change technique. Finally, we present a numerical analysis.