Optimal retirement with borrowing constraints and forced unemployment risk

In this paper, we study optimal retirement in a two-dimensional incomplete market caused by borrowing constraints and forced unemployment risk. We show that the two aspects jointly affect an individual’s optimal consumption, investment, and retirement strategies. In contrast to the complete market case, the endogenously determined wealth threshold for retirement is significantly affected by the two-dimensional market incompleteness, resulting in a lower wealth threshold. We also discuss a possible unemployment insurance scheme for the borrowing-constrained individual to respond to the shocks of forced unemployment.     

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.     

带有通胀风险的退休后最优投资-年金化时刻决策

研究了带通货膨胀的确定缴费养老计划退休后最优投资-年金化决策。假设通货膨胀过程是一个随机过程,建立了真实财富的波动过程。先相对固定年金化时刻,采取目标定位型模型,预设未来各时期的投资目标,利用贝尔曼优化原理,得到从退休时刻到相对固定年金化时刻之间的最优投资策略。接着建立了最优年金化时刻的评估标准,最优的年金化时刻使得年金化前后的累加消费折现均值得到最大。证明了在随机通货膨胀的假设下,传统的自然投资目标不存在;当随机通胀过程退化到确定过程时,求出了自然投资目标的显式表达式,并且在这两种情况下,分析了通胀情况对最优投资策略的影响。最后利用数值分析手段, 研究了通货膨胀、风险偏好、折现率对最优年金化时刻的影响。     

考虑决策者心理行为的证券投资组合决策方法研究

在现实的证券投资组合决策中,决策者的心理行为是不可忽视的重要因素。本文针对考虑决策者心理行为的证券投资组合问题,给出了一种基于累积前景理论和心理账户的决策分析方法。首先,依据累积前景理论,将决策者对不同市场状态下的预期收益率作为参考点,计算各备选证券收益率相对于参照点的收益和损失,并计算不同市场状态下针对所有备选证券的综合前景价值;然后,依据决策者的心理账户,即以证券投资组合的收益总体综合前景价值最大为目标、以投资期末总财富阈值以及满足财富约束的概率不小于决策者设定的概率阈值为约束,构建了具有概率约束条件的证券投资组合优化模型,通过将概率约束转化为线性约束并求解优化模型,可得到最优的证券投资组合方案。最后,通过一个算例对本文提出方法的可行性和有效性进行了验证。研究结果表明,本文提出的方法能够较好地解决考虑决策者心理行为的证券投资组合问题。     

Evaluating the Advanced Life Deferred Annuity — An annuity people might actually buy

Although annuities provide longevity insurance that should be attractive to households facing an uncertain lifespan, rates of voluntary annuitization remain extremely low. We evaluate the Advanced Life Deferred Annuity, an annuity purchased at retirement, providing an income commencing in advanced old age. Using numerical optimization, we show that it would provide a substantial proportion of the longevity insurance provided by an immediate annuity, at much lower cost. At plausible levels of actuarial unfairness, households should prefer it to both immediate and postponed annuitization and an optimal decumulation of unannuitized wealth. Few households would suffer significant losses were it used as a 401(k) plan default.     

To borrow or insure? Long term care costs and the impact of housing

We assess the impact of housing, the availability of reverse mortgages and long-term care (LTC) insurance on a retiree’s optimal portfolio choice and consumption decisions using a multi-period life cycle model that takes into consideration individual longevity risk, health shocks and house price risk. We determine how much an individual should borrow against their home equity and how much to insure health care costs with LTC insurance. We introduce an endogenous grid method, along with a regression based approach, to improve computational efficiency and avoid the curse of dimensionality. Our results confirm that borrowing against home equity provides higher consumption in earlier years and longevity insurance. LTC insurance transfers wealth from healthy states to disabled states, but reduces early consumption because of the payment of insurance premiums. Housing is an illiquid asset that is important in meeting bequest motives, and it reduces the demand for LTC insurance for the wealthy. We show that the highest welfare benefits come from combining a reverse mortgage with LTC insurance because of strong complementary effects between them. This result highlights the benefits of innovative products that bundle these two products together.     

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.     

Optimal investment–consumption problem: Post-retirement with minimum guarantee

We study the optimal investment–consumption problem for a member of defined contribution plan during the decumulation phase. For a fixed annuitization time, to achieve higher final annuity, we consider a variable consumption rate. Moreover, to have a minimum guarantee for the final annuity, a safety level for the wealth process is considered. To solve the stochastic optimal control problem via dynamic programming, we obtain a Hamilton–Jacobi–Bellman (HJB) equation on a bounded domain. The existence and uniqueness of classical solutions are proved through the dual transformation. We apply the finite difference method to find numerical approximations of the solution of the HJB equation. Finally, the simulation results for the optimal investment–consumption strategies, optimal wealth process and the final annuity for different admissible ranges of consumption are given. Furthermore, by taking into account the market present value of the cash flows before and after the annuitization, we compare the outcomes of different scenarios.     

Relative deprivation as a cause of risky behaviors

ABSTRACT

Combining a standard measure of concern about low relative wealth and a standard measure of relative risk aversion leads to a novel explanation of variation in risk-taking behavior identified and documented by social psychologists and economists. We obtain two results: (1) Holding individual i’s wealth and his rank in the wealth distribution constant, the individual’s relative risk aversion decreases when he becomes more relatively deprived as a result of an increase in the average wealth of the individuals who are wealthier than he is. (2) If relative deprivation enters the individual’s utility function approximately linearly then, holding constant individual i’s wealth and the average wealth of the individuals who are wealthier than he is, the individual’s relative risk aversion decreases when he becomes more relatively deprived as a result of a decline in his rank. Our findings provide a theoretical support for evidence about the propensity of relatively deprived individuals to gamble and resort to other risky behaviors.     

Optimal portfolio strategies with a liability and random risk: The case of different lending and borrowing rates

This paper deals with two problems of optimal portfolio strategies in continuous time. The first one studies the optimal behavior of a firm who is forced to withdraw funds continuously at a fixed rate per unit time. The second one considers a firm that is faced with an uncontrollable stochastic cash flow, or random risk process. We assume the firm’s income can be obtained only from the investment in two assets: a risky asset (e.g., stock) and a riskless asset (e.g., bond). Therefore, the firm’s wealth follows a stochastic process. When the wealth is lower than certain legal level, the firm goes bankrupt. Thus how to invest in the fundamental problem of the firm in order to avoid bankruptcy. Under the case of different lending and borrowing rates, we obtain the optimal portfolio strategies for some reasonable objective functions that are the piecewise linear functions of the firm’s current wealth and present some interesting proofs for the conclusions. The optimal policies are easy to be operated for any relevant investor.     

随机市场中均值-方差模型最优投资策略的时间不相容性及其修正

由于方差算子在动态规划意义下不可分,导致随机市场中多期均值一方差模型的最优投资策略不满足时间相容性,即Bellman最优性原理．为此,首先提出了随机市场中比Bellman最优性原理更弱的时间相容性,并证明在投资区间的任意中间时刻,当投资者的财富不超过某一给定的财富阈值时,最优投资策略满足弱时间相容性;当投资者的财富超过该阈值时,最优投资策略将不再是弱时间相容的,且导致投资者变为非理性,即他会同时极小化终期财富的均值和方差．在这种情形下,通过放松自融资约束,对最优投资策略进行了修正,使得其满足：修正策略可使投资者回归理性;相对于终期财富,修正策略可以获得与最优投资策略相同的均值和方差．在策略修正过程中,投资者可以从市场中获得一个严格正的现金流．这些结果表明修正策略要优于原最优投资策略,拓展了现有关于确定市场下多期均值．方差模型的求解以及策略时间相容性的结论．     

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.     

Minimizing the probability of lifetime ruin under borrowing constraints

We determine the optimal investment strategy of an individual who targets a given rate of consumption and who seeks to minimize the probability of going bankrupt before she dies, also known as lifetime ruin. We impose two types of borrowing constraints: First, we do not allow the individual to borrow money to invest in the risky asset nor to sell the risky asset short. However, the latter is not a real restriction because in the unconstrained case, the individual does not sell the risky asset short. Second, we allow the individual to borrow money but only at a rate that is higher than the rate earned on the riskless asset.We consider two forms of the consumption function: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of her wealth. The first is arguably more realistic, but the second is closely connected with Merton’s model of optimal consumption and investment under power utility. We demonstrate that connection in this paper, as well as include a numerical example to illustrate our results.     

Optimal consumption and portfolio selection with quadratic utility and a subsistence consumption constraint

In this article, we analyze the optimal consumption and investment policy of an agent who has a quadratic felicity function and faces a subsistence consumption constraint. The agent's optimal investment in the risky asset increases linearly for low wealth levels. Risk taking continues to increase at a decreasing rate for wealth levels higher than subsistence wealth until it hits a maximum at a certain wealth level, and declines for wealth levels above this threshold. Further, the agent has a bliss level of consumption, since if an agent consumes more than this level she will suffer utility loss. Eventually her risk taking becomes zero at a wealth level which supports her bliss consumption.     

Purchasing life insurance to reach a bequest goal

We determine how an individual can use life insurance to meet a bequest goal. We assume that the individual’s consumption is met by an income from a job, pension, life annuity, or Social Security. Then, we consider the wealth that the individual wants to devote towards heirs (separate from any wealth related to the afore-mentioned income) and find the optimal strategy for buying life insurance to maximize the probability of reaching a given bequest goal. We consider life insurance purchased by a single premium, with and without cash value available. We also consider irreversible and reversible life insurance purchased by a continuously paid premium; one can view the latter as (instantaneous) term life insurance.     

Optimal insurance under the insurer’s risk constraint

In this paper, we impose the insurer’s risk constraint on Arrow’s optimal insurance model. The insured aims to maximize his/her expected utility of terminal wealth, under the constraint that the insurer wishes to control the expected loss of his/her terminal wealth below some prespecified level. We solve the problem, and it is shown that when the insurer’s risk constraint is binding, the solution to the problem is not linear, but piecewise linear deductible. Moreover, it can be shown that the insured’s optimal expected utility will increase if the insurer increases his/her risk tolerance.     

The design of equity-indexed annuities

There is a rich variety of tailored investment products available to the retail investor in every developed economy. These contracts combine upside participation in bull markets with downside protection in bear markets. Examples include equity-linked contracts and other types of structured products. This paper analyzes these contracts from the investor’s perspective rather than the issuer’s using concepts and tools from financial economics. We analyze and critique their current design and examine their valuation from the investor’s perspective. We propose a generalization of the conventional design that has some interesting features. The generalized contract specifications are obtained by assuming that the investor wishes to maximize end of period expected utility of wealth subject to certain constraints. The first constraint is a guaranteed minimum rate of return which is a common feature of conventional contracts. The second constraint is new. It provides the investor with the opportunity to outperform a benchmark portfolio with some probability. We present the explicit form of the optimal contract assuming both constraints apply and we illustrate the nature of the solution using specific examples. The paper focusses on equity-indexed annuities as a representative type of such contracts but our approach is applicable to other types of equity-linked contracts and structured products.     

基于CCAPM框架的财富不平等与房价关系研究

运用含有房屋的CCAPM模型,在同时考虑房屋的消费属性与投资属性基础上分析了财富(或收入)分配对房价的影响;并通过数值模拟解释了我国房价高企的根源.结论表明,我国房价高企是财富分配不均所致,财富不均等程度加剧致使房屋从消费属性向投资属性转变进而推高了房价.文章也分析了财富分配对居民消费结构的影响,财富分配不均等程度加剧使得贫穷阶层的房屋消费比重下降而富裕阶层的投资性购房比重增加,进而拉高房价.文章最后指出,要调控我国当前房地产市场价格,必须从财富(或收入)分配入手,明晰产权.     

Optimal investment,consumption and timing of annuity purchase under a preference change

In this paper, we study the optimal investment and consumption strategies for a retired individual who has the opportunity of choosing a discretionary stopping time to purchase an annuity. We assume that the individual receives a fixed annuity income and changes his/her preference after paying a fixed cost for annuitization. By using the martingale method and the variational inequality method, we tackle this problem and obtain the optimal strategies and the value function explicitly for the case of constant force of mortality and constant relative risk aversion (CRRA) utility function.