The private value of public pensions

As individual retirement savings accounts replace public pensions and defined benefit schemes, more retirees will decumulate using commercial income streams rather than public or corporate annuities. Here we use an approximation to the retirement income problem [Huang, H., Milevsky, M.A., Wang, J., 2004. Ruined moments in your life: How good are the approximations? Insurance: Math. Econom. 34, 421–447] to compute the cost of replicating a public real life annuity (the Australian Age Pension) using commercial decumulation products. We treat the public pension as a phased withdrawal plan, matching insurance and payment features, and back out the stochastic present value of the plan under an arbitrarily small ruin probability. To reproduce the pension payment with 99% certainty, a male retiree needs 3.6 times the current average retirement savings account balance, and a female retiree needs more than 10 times the average female account balance. At 95% certainty, required wealth falls by around 25%. We measure separately the impact of gender, investment strategy, retirement age and management fees on this valuation.     

Options on tontines: An innovative way of combining tontines and annuities

Increases in the life expectancy, the low interest rate environment and the tightening solvency regulation have led to the rebirth of tontines. Compared to annuities, where insurers bear all the longevity risk, policyholders bear most of the longevity risk in a tontine. Following Donnelly and Young (2017), we come up with an innovative retirement product which contains the annuity and the tontine as special cases: a tontine with a minimum guaranteed payment. The payoff of this product consists of a guaranteed payoff and a call option written on a tontine. Extending Donnelly and Young (2017), we consider the tontine design described in Milevsky and Salisbury (2015) for designing the new product and find that it is able to achieve a better risk sharing between policyholders and insurers than annuities and tontines. For the majority of risk-averse policyholders, the new product can generate a higher expected lifetime utility than annuities and tontines. For the insurer, the new product is able to reduce the (conditional) expected loss drastically compared to an annuity, while the loss probability remains fairly the same. In addition, by varying the guaranteed payments, the insurer is able to provide a variety of products to policyholders with different degrees of risk aversion and liquidity needs.     

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.     

Stochastic utilities with subsistence and satiation: Optimal life insurance purchase,consumption and investment

We introduce stochastic utilities such that utility of any fixed amount of interest is a stochastic process or random variable. Also, there exist stochastic (or random) subsistence and satiation levels associated with stochastic utilities. Then, we consider optimal consumption, life insurance purchase and investment strategies to maximize the expected utility of consumption, bequest and pension with respect to stochastic utilities. We use the martingale approach to solve the optimization problem in two steps. First, we solve the optimization problem with an equality constraint which requires that the present value of consumption, bequest and pension is equal to the present value of initial wealth and income stream. Second, if the optimization problem is feasible, we obtain the explicit representations of the replicating life insurance purchase and portfolio strategies. As an application of our general results, we consider a family of stochastic utilities which have hyperbolic absolute risk aversion (HARA).     

Financial planning for young households

We analyze the financial planning problems of young households whose main decisions are how to finance the purchase of a house (liabilities) and how to allocate investments in pension savings schemes (assets). The problems are solved using a multi-stage stochastic programming model where the uncertainty is described by a scenario tree generated from a vector auto-regressive process for equity returns and interest rate evolution. We find strong evidence of the importance of taking into account the multi-stage nature of the problem, as well as the need to consider the asset and liability sides jointly.     

Following the rules: Integrating asset allocation and annuitization in retirement portfolios

Financial advisers have developed standardized payout strategies to help Baby Boomers manage their money in their golden years. Prominent among these are phased withdrawal plans offered by mutual funds including the “self-annuitization” or default rules encouraged under US tax law, and fixed payout annuities offered by insurers. Using a utility-based framework, and taking account of stochastic capital markets and uncertain lifetimes, we first evaluate these rules on a stand-alone basis for a wide range of risk aversion. Next, we permit the consumer to integrate these standardized payout strategies at retirement and compare the results. We show that integrated strategies can enhance retirees’ well-being by 25%-50% for low/moderate levels of risk aversion when compared to full annuitization at retirement. Finally, we examine how welfare changes if the consumer is permitted to switch to a fixed annuity at an optimal point after retirement. This affords the retiree the chance to benefit from the equity premium when younger, and exploit the mortality credit in later life. For moderately risk-averse retirees, the optimal switching age lies between 80 and 85.     

具有遣赠动机的生命周期模型

本文研究遗赠动机对个人消费与储蓄的影响.利用解优化问题的一阶条件,得出在给定的条件下个人存在最优的消费流和遗产使得个人的终身效用达到最大化,当个人获得的收入或遗产增加时,个人将消费更多并留下更多的遗产.进一步,文中在四种具体的遗赠动机函数下研究了遗赠动机对个人消费与储蓄行为的影响,证明在后三种遗赠动机下,储蓄与遗产随遗赠动机的强度增加而赠加;在门槛动机下,随着门槛的提高,个人的消费上升,储蓄下降.     

Unhedgeable inflation risk within pension schemes

Pension schemes generally aim to protect the purchasing power of their participants, but cannot completely do this when due to market incompleteness inflation risk cannot be fully hedged. Without a market price for inflation risk the value of a pension contract depends on the investor’s risk appetite and inflation risk exposure. We develop a valuation framework to deal with two sources of unhedgeable inflation risk: the absence of instruments to hedge general consumer price inflation risk and differences in group-specific consumption bundles from the economy-wide bundle. We find that the absence of financial instruments to hedge inflation risks may reduce lifetime welfare by up to 6% of certainty-equivalent consumption for commonly assumed degrees of risk aversion. Regulators face a dilemma as young (workers) and old participants (retirees) have different capacities to absorb losses from unhedgeable inflation risks and as a consequence have a different risk appetite.     

Longevity risk and retirement income tax efficiency: A location spending rate puzzle

In this paper we model and solve a retirement consumption problem with differentially taxed accounts, parameterized by longevity risk aversion. The work is motivated by some observations on how Canadians de-accumulate financial wealth during retirement — which seem rather puzzling. While the Modigliani lifecycle model can justify a variety of (pre-tax) de-accumulation or draw down rates depending on risk preferences, the existence of asymmetric taxes implies that certain financial accounts should be depleted faster than others. Our analysis of data from the Survey of Financial Security indicates that Canadian retirees maintain approximately two-thirds of their financial wealth in tax-sheltered accounts and a third in taxable accounts regardless of age. The ratio of taxable to tax-sheltered wealth increases slightly or remains relatively constant depending on household income which is not what one would expect from the lifecycle model. Indeed, using our model we cannot locate a plausible tax function that justifies a constant “account ratio” regardless of age. For example under flat rates taxable accounts should be depleted well before tax-sheltered accounts are ever touched. The account ratio should go to zero quite rapidly in the absence of government mandated withdrawals. We also demonstrate that under progressive income taxes withdrawals are made from both accounts but at different rates depending on account size, pension income and longevity risk preferences. Again, the “account ratio” should eventually decline. We postulate that this sort of behavior is likely due to irrational considerations linked to mental accounting, etc. It remains to be seen whether this will persist over time and under a more careful analysis of Canadian cohorts or if retirees in other countries exhibit the same behavior.     

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.     

Valuation of risk-based premium of DB pension plan with terminations

This paper concentrates on the premium valuation of pension insurance provided by the Pension Benefit Guaranty Corporation (PBGC). The PBGC provides a defined benefit pension sponsor with coverage in case that the pension fund fails to make pension payments as promised or that the plan sponsor does not stay in business any more. In practice, both the pension fund and the sponsor assets play a critical role in fulfilling the commitment of pension payments, and thereby it is not reasonable to isolate the risk of distress termination of the sponsor assets from that of the premature termination of the pension fund. Different from previous works in which the premature termination of the pension fund and the distress termination of the sponsor assets are analyzed separately, our model examines the situation in which retirees suffer the risk of two types of terminations at the same time. We evaluate the risk-based fair premium under the framework that the pension fund and the sponsor assets are correlated and subject to the risk of the involuntary termination (i.e., premature termination) and the distress termination, respectively. In this framework, we manage to obtain closed-form pricing formulas. Our model is more practical because of the realistic design of termination schemes. Numerical simulations are also carried out to demonstrate our findings. Our numerical experiments validate that a variable rate premium is more appropriate for the PBGC to implement.     

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.     

Dynamic hybrid products with guarantees—An optimal portfolio framework

Dynamic hybrid products are pension products that consist of a dynamic combination of classical with profits participating life insurance contracts (or a bank account) and fund savings plans. To put such products in an optimal utility framework, we derive an optimal combination of a money market account, a CPPI-style fund and a free fund in continuous trading via transforming the original investment problem into a conventional portfolio problem in the presence of a guarantee condition. By this, we obtain (semi-) explicit forms of the dynamic weights for the different ingredients of a dynamic hybrid product.     

Optimal investment strategies and risk-sharing arrangements for a hybrid pension plan

A continuous time stochastic model is used to study a hybrid pension plan, where both the contribution and benefit levels are adjusted depending on the performance of the plan, with risk sharing between different generations. The pension fund is invested in a risk-free asset and multiple risky assets. The objective is to seek an optimal investment strategy and optimal risk-sharing arrangements for plan trustees and participants so that this proposed hybrid pension system provides adequate and stable income to retirees while adjusting contributions effectively, as well as keeping its sustainability in the long run. These goals are achieved by minimizing the expected discount disutility of intermediate adjustment for both benefits and contributions and that of terminal wealth in finite time horizon. Using the stochastic optimal control approach, closed-form solutions are derived under quadratic loss function and exponential loss function. Numerical analysis is presented to illustrate the sensitivity of the optimal strategies to parameters of the financial market and how the optimal benefit changes with respect to different risk aversions. Through numerical analysis, we find that the optimal strategies do adjust the contributions and retirement benefits according to fund performance and model objectives so the intergenerational risk sharing seem effectively achieved for this collective hybrid pension plan.     

Return smoothing mechanisms in life and pension insurance: Path-dependent contingent claims

Traditional with-profits pension saving schemes have been criticized for their opacity, plagued by embedded options and guarantees, and have recently created enormous problems for the solvency of the life insurance and pension industry. This has fueled creativity in the industry’s product development departments, and this paper analyzes a representative member of a family of new pension schemes that have been introduced in the new millennium to alleviate these problems. The complete transparency of the new scheme’s smoothing mechanism means that it can be analyzed using contingent claims pricing theory. We explore the properties of this pension scheme in detail and find that in terms of market value, smoothing is an illusion, but also that the return smoothing mechanism implies a dynamic asset allocation strategy which corresponds with traditional pension saving advice and the recommendations of state-of-the-art dynamic portfolio choice models.     

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 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.     

On allocation of upper limits and deductibles with dependent frequencies and comonotonic severities

With the assumption of Archimedean copula for the occurrence frequencies of the risks covered by an insurance policy, this note further investigates the allocation problem of upper limits and deductibles addressed in Hua and Cheung (2008a). Sufficient conditions for a risk averse policyholder to well allocate the upper limits and the deductibles are built, respectively.     

Optimal investment strategies and risk measures in defined contribution pension schemes

In this paper, we derive a formula for the optimal investment allocation (derived from a dynamic programming approach) in a defined contribution (DC) pension scheme whose fund is invested in n assets. We then analyse the particular case of n=2 (where we consider the presence in the market of a high-risk and a low-risk asset whose returns are correlated) and study the investment allocation and the downside risk faced by the retiring member of the DC scheme, where optimal investment strategies have been adopted. The behaviour of the optimal investment strategy is analysed when changing the disutility function and the correlation between the assets. Three different risk measures are considered in analysing the final net replacement ratios achieved by the member: the probability of failing the target, the mean shortfall and a value at risk (VaR) measure. The replacement ratios encompass the financial and annuitisation risks faced by the retiree. We consider the relationship between the risk aversion of the member and these different risk measures in order to understand better the choices confronting different categories of scheme member. We also consider the sensitivity of the results to the level of the correlation coefficient.     

How to Spend and Invest Retirement Savings

The question of post-retirement optimal consumption and investment of retirement savings is addressed. This problem has received considerably less attention than that of how to invest for retirement. With the increase in life span and an increase in private pension funds, a retiree has considerable flexibility in both how to consume and how to continue to invest their retirement funds. Our interest is in developing a platform that allows a wide variety of behavioural aspects to be modeled and also enables explicit constraints to be imposed. To enable the flexibility we seek it is necessary to model the problem as a large-scale nonlinearly constrained optimization problem, which is solved using a sequential quadratic programming algorithm. Fortunately, modern optimization methods are now sufficiently powerful as to enable solving such problems. A key point is that, though the problems are large, they have a rich structure. Problems in this class have been addressed assuming that an investor is rational in the sense that when making financial decisions the preference relation of the investor satisfies all the axioms of choice. Research in behavioural science indicates that not all financial decisions of an average person satisfy the axioms of choice. The algorithm we propose enables the problem to be solved for a user-specified utility function that does not satisfy all the axioms of choice.