Optimal retirement strategy with a negative wealth constraint

This paper investigates an optimal consumption, portfolio, and retirement time choice problem of an individual with a negative wealth constraint. We obtain analytical results of the optimal consumption, investment, and retirement behaviors and discuss the effect of the negative wealth constraint on the optimal behaviors. We find that, as an individual can borrow more with better credit, she is more likely to retire at a higher wealth level, to consume more, and to invest more in risky assets.     

Annuitization and asset allocation with borrowing constraint

We generalize the result of Yaari (1965) on annuitization with borrowing constraint. We show that inability to borrow against future labor income has a significant influence on an individual’s consumption and asset allocation strategies. We also show that there exists a certain threshold of wealth for annuitization. We find that the wealth threshold is lower in the presence of borrowing constraint than in its absence, implying the individual’s earlier retirement.     

Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints

In this paper we consider a general optimal consumption-portfolio selection problem of an infinitely-lived agent whose consumption rate process is subject to subsistence constraints before retirement. That is, her consumption rate should be greater than or equal to some positive constant before retirement. We integrate three optimal decisions which are the optimal consumption, the optimal investment choice and the optimal stopping problem in which the agent chooses her retirement time in one model. We obtain the explicit forms of optimal policies using a martingale method and a variational inequality arising from the dual function of the optimal stopping problem. We treat the optimal retirement time as the first hitting time when her wealth exceeds a certain wealth level which will be determined by a free boundary value problem and duality approaches. We also derive closed forms of the optimal wealth processes before and after retirement. Some numerical examples are presented for the case of constant relative risk aversion (CRRA) utility class.     

The effect of objective formulation on retirement decision making

For a retiree who must maintain both investment and longevity risks, we consider the impact on decision making of focusing on an objective relating to the terminal wealth at retirement, instead of a more correct objective relating to a retirement income. Both a shortfall and a utility objective are considered; we argue that shortfall objectives may be inappropriate due to distortion in results with non-monotonically correlated economic factors. The modelling undertaken uses a dynamic programming approach in conjunction with Monte-Carlo simulations of future experience of an individual to make optimal choices. We find that the type of objective targetted can have a significant impact on the optimal choices made, with optimal equity allocations being up to 30% higher and contribution amounts also being significantly higher under a retirement income objective as compared to a terminal wealth objective. The result of these differences can have a significant impact on retirement outcomes.     

Optimal Asset Allocation for Retirement Saving: Deterministic Vs. Time Consistent Adaptive Strategies

We consider optimal asset allocation for an investor saving for retirement. The portfolio contains a bond index and a stock index. We use multi-period criteria and explore two types of strategies: deterministic strategies are based only on the time remaining until the anticipated retirement date, while adaptive strategies also consider the investor’s accumulated wealth. The vast majority of financial products designed for retirement saving use deterministic strategies (e.g., target date funds). In the deterministic case, we determine an optimal open loop control using mean-variance criteria. In the adaptive case, we use time consistent mean-variance and quadratic shortfall objectives. Tests based on both a synthetic market where the stock index is modelled by a jump-diffusion process and also on bootstrap resampling of long-term historical data show that the optimal adaptive strategies significantly outperform the optimal deterministic strategy. This suggests that investors are not being well served by the strategies currently dominating the marketplace.     

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.     

Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk

In this paper we investigate an optimal investment strategy for a defined-contribution (DC) pension plan member who is loss averse, pays close attention to inflation and longevity risks and requires a minimum performance at retirement. The member aims to maximize the expected S-shaped utility from the terminal wealth exceeding the minimum performance by investing her wealth in a financial market consisting of an indexed bond, a stock and a risk-free asset. We derive the optimal investment strategy in closed-form using the martingale approach. Our theoretical and numerical results reveal that the wealth proportion invested in each risky asset has a V-shaped pattern in the reference point level, while it always increases in the rising lifespan; with a positive correlation between salary and inflation risks, the presence of salary decreases the member’s investment in risky assets; the minimum performance helps to hedge the longevity risk by increasing her investment in risky assets.     

Less is more: Increasing retirement gains by using an upside terminal wealth constraint

We solve a portfolio selection problem of an investor with a deterministic savings plan who aims to have a target wealth value at retirement. The investor is an expected power utility-maximizer. The target wealth value is the maximum wealth that the investor can have at retirement.By constraining the investor to have no more than the target wealth at retirement, we find that the lower quantiles of the terminal wealth distribution increase, so the risk of poor financial outcomes is reduced. The drawback of the optimal strategy is that the possibility of gains above the target wealth is eliminated.     

Optimal portfolio, consumption and retirement decision under a preference change

We investigate an optimal portfolio, consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time with constant labor wage and disutility. We allow the preference of the agent to be changed before and after retirement. It is assumed that the agent's coefficient of relative risk aversion becomes higher after retirement. Under a constant relative risk aversion (CRRA) utility function, we obtain the optimal policies in closed-forms using martingale methods and variational inequality methods. We give some numerical results of the optimal policies. We also consider the relation between the level of disutility and the labor wage with the optimal retirement wealth level.     

On the sub-optimality cost of immediate annuitization in DC pension funds

We consider the position of a member of a defined contribution (DC) pension scheme having the possibility of taking programmed withdrawals at retirement. According to this option, she can defer annuitization of her fund to a propitious future time, that can be found to be optimal according to some criteria. This option, that adds remarkable flexibility in the choice of pension benefits, is not available in many countries, where immediate annuitization is compulsory at retirement. In this paper, we address and try to answer the questions: “Is immediate annuitization optimal? If it is not, what is the cost to be paid by the retiree obliged to annuitize at retirement?”. In order to do this, we consider the model by Gerrard et?al. in Quant Finance, (2011) and extend it in two different ways. In the first extension, we prove a theorem that provides necessary and sufficient conditions for immediate annuitization being always optimal. The not surprising result is that compulsory immediate annuitization turns out to be sub-optimal. We then quantify the extent of sub-optimality, by defining the sub-optimality cost as the loss of expected present value of consumption from retirement to death and measuring it in many typical situations. We find that it varies in relative terms between 6 and 40%, depending on the risk aversion of the member. In the second extension, we make extensive numerical investigations of the model and seek the optimal annuitization time. We find that the optimal annuitization time depends on personal factors such as the retiree’s risk aversion and her subjective perception of remaining lifetime. It also depends on the financial market, via the Sharpe ratio of the risky asset. Optimal annuitization should occur a few years after retirement with high risk aversion, low Sharpe ratio and/or short remaining lifetime, and many years after retirement with low risk aversion, high Sharpe ratio and/or long remaining lifetime. This paper supports the availability of programmed withdrawals as an option to retirees of DC pension schemes, by giving insight into the extent of loss in wealth suffered by a retiree who cannot choose programmed withdrawals, but is obliged to annuitize immediately on retirement.     

Optimal consumption-leisure, portfolio and retirement selection based on α-maxmin expected CES utility with ambiguity

This article studies optimal consumption-leisure, portfolio and retirement selection of an infinitely lived investor whose preference is formulated by ??-maxmin expected CES utility which is to differentiate ambiguity and ambiguity attitude. Adopting the recursive multiplepriors utility and the technique of backward stochastic differential equations (BSDEs), we transform the ??-maxmin expected CES utility into a classical expected CES utility under a new probability measure related to the degree of an investor??s uncertainty. Our model investigates the optimal consumption-leisure-work selection, the optimal portfolio selection, and the optimal stopping problem. In this model, the investor is able to adjust her supply of labor flexibly above a certain minimum work-hour along with a retirement option. The problem can be analytically solved by using a variational inequality. And the optimal retirement time is given as the first time when her wealth exceeds a certain critical level. The optimal consumption-leisure and portfolio strategies before and after retirement are provided in closed forms. Finally, the distinctions of optimal consumption-leisure, portfolio and critical wealth level under ambiguity from those with no vagueness are discussed.     

On “optimal pension management in a stochastic framework” with exponential utility

This paper reconsiders the optimal asset allocation problem in a stochastic framework for defined-contribution pension plans with exponential utility, which has been investigated by Battocchio and Menoncin [Battocchio, P., Menoncin, F., 2004. Optimal pension management in a stochastic framework. Insurance: Math. Econ. 34, 79-95]. When there are three types of asset, cash, bond and stock, and a non-hedgeable wage risk, the optimal pension portfolio composition is horizon dependent for pension plan members whose terminal utility is an exponential function of real wealth (nominal wealth-to-price index ratio). With market parameters usually assumed, wealth invested in bond and stock increases as retirement approaches, and wealth invested in cash asset decreases. The present study also shows that there are errors in the formulation of the wealth process and control variables in solving the optimization problem in the study of Battocchio and Menoncin, which render their solution erroneous and lead to wrong results in their numerical simulation.     

HARA frontiers of optimal portfolios in stochastic markets

In this paper, we consider the optimal portfolio selection problem in continuous-time settings where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has the structure of the HARA family and the market states change according to a Markov process. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. This includes the distributions of the random asset returns as well as the utility function. We analyzed Black–Scholes type continuous-time models where the market parameters are driven by Markov processes. The Markov process that affects the state of the market is independent of the underlying Brownian motion that drives the stock prices. The problem of maximizing the expected utility of the terminal wealth is investigated and solved by stochastic optimal control methods for exponential, logarithmic and power utility functions. We found explicit solutions for optimal policy and the associated value functions. We also constructed the optimal wealth process explicitly and discussed some of its properties. In particular, it is shown that the optimal policy provides linear frontiers.     

Optimal mean–variance efficiency of a family with life insurance under inflation risk

We study an optimization problem of a family under mean–variance efficiency. The market consists of cash, a zero-coupon bond, an inflation-indexed zero-coupon bond, a stock, life insurance and income-replacement insurance. The instantaneous interest rate is modeled as the Cox–Ingersoll–Ross (CIR) model, and we use a generalized Black–Scholes model to characterize the stock and labor income. We also take into account the inflation risk and consider our problem in the real market. The goal of the family is to maximize the mean of the surplus wealth at the retirement or death of the breadwinner and minimize its variance by finding a portfolio selection. The efficient frontier and optimal strategies are derived through the dynamic programming method and the technique of solving associated nonlinear HJB equations. We also present a numerical illustration to explore the impact of economical parameters on the efficient frontier.     

基于通货膨胀和最低保障的DC养老金最优投资

主要研究了通货膨胀和最低保障下的DC养老金的最优投资问题。 首先, 应用伊藤公式得到通胀折现后真实股票价格的微分方程。 然后, 在DC养老金终端财富外部保障约束下, 引入欧式看涨期权, 考虑随机通胀环境下的退休时刻终端财富期望效用最大化问题, 应用鞅方法推导退休时刻以及退休前任意时刻DC养老金最优投资策略的显式解。 最后, 应用蒙特卡洛方法对结果进行数值分析, 分析最低保障对DC养老金最优投资策略的影响。     

含不动产项目的保险公司再保险-投资策略

站在保险公司管理者的角度, 考虑存在不动产项目投资机会时保险公司的再保险--投资策略问题. 假定保险公司可以投资于不动产项目、风险证券和无风险证券, 并通过比例再保险控制风险, 目标是最小化保险公司破产概率并求得相应最佳策略, 包括: 不动产项目投资时机、 再保险比例以及投资于风险证券的金额. 运用混合随机控制-最优停时方法, 得到最优值函数及最佳策略的显式解. 结果表明, 当且仅当其盈余资金多于某一水平(称为投资阈值)时保险公司投资于不动产项目. 进一步的数值算例分析表明: (a)~不动产项目投资的阈值主要受项目收益率影响而与投资金额无明显关系, 收益率越高则投资阈值越低; (b)~市场环境较好(牛市)时项目的投资阈值降低; 反之, 当市场环境较差(熊市)时投资阈值提高.     

Optimal investment strategy to minimize occupation time

We find the optimal investment strategy to minimize the expected time that an individual’s wealth stays below zero, the so-called occupation time. The individual consumes at a constant rate and invests in a Black-Scholes financial market consisting of one riskless and one risky asset, with the risky asset’s price process following a geometric Brownian motion. We also consider an extension of this problem by penalizing the occupation time for the degree to which wealth is negative.     

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.     

Optimal multi-period mean–variance policy under no-shorting constraint

We consider in this paper the mean–variance formulation in multi-period portfolio selection under no-shorting constraint. Recognizing the structure of a piecewise quadratic value function, we prove that the optimal portfolio policy is piecewise linear with respect to the current wealth level, and derive the semi-analytical expression of the piecewise quadratic value function. One prominent feature of our findings is the identification of a deterministic time-varying threshold for the wealth process and its implications for market settings. We also generalize our results in the mean–variance formulation to utility maximization with no-shorting constraint.     

Portfolio selection for individual passive investing

This paper considers passive fund selection from an individual investor's perspective. The growth of the passive fund market over the past decade is staggering. Individual investors who wish to buy these funds for their retirement and brokerage accounts have many options and are faced with a difficult selection problem. Which funds do they invest in, and in what proportions? We develop a novel statistical methodology to address this problem by adapting recent advances in posterior summarization. A Bayesian decision‐theoretic approach is presented to construct optimal sparse portfolios for individual investors over time.