Income Drawdown Schemes for a Defined-Contribution Pension Plan

In retirement a pensioner must often decide how much money to withdraw from a pension fund, how to invest the remaining funds, and whether to purchase an annuity. These decisions are addressed here by introducing a number of income drawdown schemes, which are relevant to a defined-contribution personal pension plan. The optimal asset allocation is defined so that it minimizes the expected loss of the pensioner as measured by the performance of the pension fund against a benchmark. Two benchmarks are considered: a risk-free investment and the price of an annuity. The fair-value income drawdown rate is defined so that the fund performance is a martingale under the objective measure. Annuitization is recommended if the expected fair-value drawdown rate falls below the annuity rate available at retirement. As an illustration, the annuitization age is calculated for a Gompertz mortality distribution function and a power law loss function.     

Deterministic Modeling of Defined-Contribution Pension Funds

Abstract

Formulas are derived to model the response of a money purchase, defined-contribution pension fund to the following types of investment shock: (1) an instantaneous fall in the value of the fund assets and (2) a permanent, uniform reduction in the rate of earned interest. An alternative defined-contribution fund, incorporating a variable defined-benefit scale, is then proposed, and its responses to the same investment shocks are compared with those of the money purchase fund. The comparison shows that the variable defined-benefit approach offers a useful alternative to the money purchase principle for defined-contribution pension funds.     

The Value and Risk of Defined Contribution Pension Schemes: International Evidence

We use historical data on investment returns and labor income from 16 countries to quantify the value and risk of defined contribution pension plans, building frequency distributions of pension fund and pension replacement ratios for each country. We show that pension risk is substantial and find that pension fund ratios are lower and less variable than when the correlation between wage growth and investment returns is ignored, typically halving the median pension fund ratio. We also show that an all‐equity fund is the dominant investment strategy across all countries, although sometimes a life‐cycle strategy insures against downside risk.     

Dynamic preferences for popular investment strategies in pension funds

In this paper, we characterize dynamic investment strategies that are consistent with the expected utility setting and more generally with the forward utility setting. Two popular dynamic strategies in the pension funds industry are used to illustrate our results: a constant proportion portfolio insurance (CPPI) strategy and a life-cycle strategy. For the CPPI strategy, we are able to infer preferences of the pension fund’s manager from her investment strategy, and to exhibit the specific expected utility maximization that makes this strategy optimal at any given time horizon. In the Black–Scholes market with deterministic parameters, we are able to show that traditional life-cycle funds are not optimal to any expected utility maximizers. We also prove that a CPPI strategy is optimal for a fund manager with HARA utility function, while an investor with a SAHARA utility function will choose a time-decreasing allocation to risky assets in the same spirit as the life-cycle funds strategy. Finally, we suggest how to modify these strategies if the financial market follows a more general diffusion process than in the Black–Scholes market.     

金融危机对拉美国家私营养老金的影响及经验启示

2008年爆发的金融危机对全球经济产生了重要影响,也对拉美国家私营养老金制度的发展产生了巨大冲击。文章介绍了拉美国家私营养老金制度的改革路径和养老基金概况,从养老金资产价值、投资收益率、投资组合、雇员、制度覆盖面和缴费密度方面切入,分析了金融危机对拉美私营养老金制度的影响。在现状分析的基础上,得出了要保障私营养老金制度可持续运行的一些经验启示:构建多元化的混合型养老金体系,增强退休和养老金支付的灵活性,积极运用生命周期投资策略,实施多元化和分散化投资,以养老金长期收益作为投资目标,建立社会风险应急储备基金和政府担保机制,以及加强员工退休教育。     

Around the Life Cycle: Deterministic Consumption-Investment Strategies

We study a classical continuous-time consumption-investment problem of a power utility investor with deterministic labor income with the important feature that the consumption-investment process is constrained to be deterministic. This is motivated by the design of modern pension schemes of defined contribution type where, typically, the savings rate is constant and the proportional investment in growth stocks is a function of age or time-to-retirement, a so-called life-cycle investment strategy. We derive and study the optimal behavior corresponding to the optimal product design within this realistic family of products with deterministic decision profiles. We also propose a couple of suboptimal deterministic strategies inspired from the optimal stochastic strategy and compare the optimal stochastic control, the optimal deterministic control, and these suboptimal deterministic controls. The conclusion is that only little is lost by constraining to deterministic strategies and only little is lost by implementing the suboptimal simple explicit strategies rather than the optimal one we derive.     

企业年金缩小企业与机关事业单位职工养老金差距的政策仿真研究——工资异质性的视角

引入工资差异对企业职工企业年金缴费意愿的影响,采用仿真方法测量企业年金对企业与机关事业单位职工养老金差距的影响。首先,根据我国养老制度安排,构建了企业年金基金、基本养老金、退休金预测模型和养老金差距测量指标;其次,选取工资水平从社平工资的0．5倍～5．0倍的14类代表性职工,模拟了基准情境和提高企业年金税收优惠的四种情境——提高雇主缴费的企业所得税免税上限、降低雇主缴费计入个人账户部分的个人所得税税率、降低领取阶段的个人所得税税率、免征个人缴费的个人所得税,并进行了敏感性分析。仿真结果表明,企业年金及其税收优惠政策能够有效缩小企业与机关事业单位职工养老金差距,提高雇主缴费的企业所得税免税上限的作用最显著。企业年金投资收益率提高、企业年金管理费率降低、余寿减少、社平工资增长率提高和工作年限增加都会放大这种作用,反之亦然。工资越高企业职工从企业年金中获益越大,可能扩大企业职工内部养老金差距。     

Pricing Participating Inflation Retirement Funds Through Option Modeling and Copulas

Abstract

Pension plans and life insurances offering minimum performance guarantees are very common worldwide. In the Brazilian market, the customers of a common type of defined contribution plan have the right to receive, over their savings, the positive difference between the return of a specified investment fund, usually a fixed income fund, and the minimum guaranteed rate, commonly defined as the composition of a fixed interest rate and a floating inflation rate. This instrument can be characterized as an option to exchange one asset, the minimum guaranteed rate, for another, the return of the specified investment fund. In this paper we provide a closed formula to evaluate this liability that depends on two stochastic rates assuming bivariate normality. We also explore the use of copulas for the modeling of the dependence structure and price the options using Monte Carlo simulation to compare the effects of the copula specification in their values. An application with real data is provided. The model makes use of a one-factor Vasicek framework for the term structures of interest rate and inflation rate.     

“Author’s Reply: Critique of Mexico’s New Social Security Act”, Octavio Maupomé-Carvantes,July, 1999

Abstract

This paper considers the pension plan as part of the capital structure of the sponsoring employer. This enables lessons from financial theory concerning capital structure to be used to answer the question, “What assets should a pension fund hold?” The standard Modigliani-Miller framework is expanded on to consider the implications of corporate tax. This leads to the conclusion that bond investment for pension plans has tangible advantages over holding risky assets (e.g., equities). The paper considers a case study of the pension plan of the Boots Company, a U.K. pharmacy retailer with a pension fund of around £2.3 billion ($3.5 billion), where these ideas were put into practice. Finally, the paper discusses the value released to shareholders and the extra security members of the pension fund have derived from putting theory into practice.     

The Impact of DC Pension Systems on Population Dynamics

Abstract

This study investigates the risk inherent in defined contribution (DC) pension plans on an individual and aggregate basis, based on U.S. data. Our aim is to gain insight into the consequences of a DC pension scheme becoming the predominant pillar of retirement income for an entire society. Using the stochastic simulated output of a DC flexible age-of-retirement model, we first determine the optimal investment strategies. We then examine the demographic retirement dynamics of an entire population of DC pension plan participants.

We observe that even for the most risk-averse plan members there is a high level of uncertainty in an individual’s age at retirement. At the aggregate population level, we find that this uncertainty does not get dampened to any great extent by a diversification effect. Instead, the central role played by the market in determining retirement dates results in significant variation in the dependency ratio (the ratio of retirees to workers) over time. In addition, an attempt to ameliorate the outcome by introducing additional realistic features in the DC population modeling did little to dampen this volatility, which suggests that countries dominated by DC schemes of this type may, over time, be exposed to significant risk in the size of its labor force.     

Optimal Control of DC Pension Plan Management under Two Incentive Schemes

Since the late 1990s, a performance fee arrangement has been approved as a managerial incentive in direction contribution (DC) pension plan management to motivate managers. However, the fact that managers may take undue risk for the larger performance fees and thus reduce members’ utility has been a subject of debate. As such, this study investigates the optimal risk-taking policies of DC pension fund managers under both the single management fee scheme and a mixed scheme with a lower management fee, as well as an additional performance fee. The analytical solutions are derived by using the duality method and concavification techniques in a singular optimization problem. The results show the complex risk-taking structures of fund managers and recognize the win-win situation of implementing performance-based incentives in DC pension plan management. Under the setting of geometric Brownian motion asset price dynamics and constant relative risk aversion utility, the optimal risk investment proportion shows a peak-valley pattern under the mixed scheme. Further, the manager gambles for gain when fund wealth is low and time to maturity is short. As opposed to the existing literature, this study found that the risk-taking policy is more conservative when fund wealth is relatively large. Furthermore, the utilities of the manager and members could both be improved by appropriately choosing the performance fee rate.     

Asset–liability modelling and pension schemes: the application of robust optimization to USS

This paper uses a novel numerical optimization technique – robust optimization – that is well suited to solving the asset–liability management (ALM) problem for pension schemes. It requires the estimation of fewer stochastic parameters, reduces estimation risk and adopts a prudent approach to asset allocation. This study is the first to apply it to a real-world pension scheme, and the first ALM model of a pension scheme to maximize the Sharpe ratio. We disaggregate pension liabilities into three components – active members, deferred members and pensioners, and transform the optimal asset allocation into the scheme's projected contribution rate. The robust optimization model is extended to include liabilities and used to derive optimal investment policies for the Universities Superannuation Scheme (USS), benchmarked against the Sharpe and Tint, Bayes–Stein and Black–Litterman models as well as the actual USS investment decisions. Over a 144-month out-of-sample period, robust optimization is superior to the four benchmarks across 20 performance criteria and has a remarkably stable asset allocation – essentially fix-mix. These conclusions are supported by six robustness checks.     

Optimal and Simple,Nearly Optimal Rules for Minimizing the Probability Of Financial Ruin in Retirement

Abstract

The increasing risk of poverty in retirement has been well documented; it is projected that current and future retirees’ living expenses will significantly exceed their savings and income. In this paper, we consider a retiree who does not have sufficient wealth and income to fund her future expenses, and we seek the asset allocation that minimizes the probability of financial ruin during her lifetime. Building on the work of Young (2004) and Milevsky, Moore, and Young (2006), under general mortality assumptions, we derive a variational inequality that governs the ruin probability and optimal asset allocation. We explore the qualitative properties of the ruin robability and optimal strategy, present a numerical method for their estimation, and examine their sensitivity to changes in model parameters for specific examples. We then present an easy-to-implement allocation rule and demonstrate via simulation that it yields nearly optimal ruin probability, even under discrete portfolio rebalancing.     

How much does tax erode fund excess returns?

We model the tax drag from active fund management based on reported monthly holdings of active equity funds. Tax drag erodes 65 percent of the 0.74 percent excess return in Broad Market funds, but only 21 percent of the 1.80 percent excess return in Small-Cap funds for Australian superannuation (pension) fund investors. Tax drag varies with investment style; market state, which is most detrimental during bull markets; and fund turnover. For high-income individual investors, tax drag is exacerbated to the extent that active management only generates meaningful after-tax excess return for Small-Cap funds of certain styles.     

Polynomial goal programming and the implicit higher moment preferences of US institutional investors in hedge funds

Polynomial goal programming (PGP) is a flexible method that allows investor preferences for different moments of the return distribution of financial assets to be included in the portfolio optimization. The method is intuitive and particularly suitable for incorporating investor preferences in higher moments of the return distribution. However, until now, PGP has not been able to meet its full potential because it requires quantification of “real” preference parameters towards those moments. To date, the chosen preference parameters have been selected somewhat “arbitrarily”. Our goal is to calculate implied sets of preference parameters using investors’ choices of and the importance they attribute to risk and performance measures. We use three groups of institutional investors—pension funds, insurance companies, and endowments—and derive implied sets of preference parameters in the context of a hedge fund portfolio optimization. To determine “real” preferences for the higher moments of the portfolio return distribution, we first fit implied preference parameters so that the PGP optimal portfolio is identical to the desired hedge fund portfolio. With the obtained economically justified sets of preference parameters, the well-established PGP framework can be employed more efficiently to derive allocations that satisfy institutional investor expectations for hedge fund investments. Furthermore, the implied preference parameters enable fund of hedge fund managers and other investment managers to derive optimal portfolio allocations based on specific investor expectations. Moreover, the importance of individual moments, as well as their marginal rates of substitution, can be assessed.     

Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure

Abstract

In this paper, we consider the optimal proportional reinsurance problem in a risk model with the thinning-dependence structure, and the criterion is to minimize the probability that the value of the surplus process drops below some fixed proportion of its maximum value to date which is known as the probability of drawdown. The thinning dependence assumes that stochastic sources related to claim occurrence are classified into different groups, and that each group may cause a claim in each insurance class with a certain probability. By the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman equation, the optimal reinsurance strategy and the corresponding minimum probability of drawdown are derived not only for the expected value principle but also for the variance premium principle. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.     

Dynamic asset–liability management in a Markov market with stochastic cash flows

This paper provides a general model to investigate an asset–liability management (ALM) problem in a Markov regime-switching market in a multi-period mean–variance (M–V) framework. Emphasis is placed on the stochastic cash flows in both wealth and liability dynamic processes, and the optimal investment and liquidity management strategies in achieving the M–V bi-objective of terminal surplus are evaluated. In this model, not only the asset returns and liability returns, but also the cash flows depend on the stochastic market states, which are assumed to follow a discrete-time Markov chain. Adopting the dynamic programming approach, the matrix theory and the Lagrange dual principle, we obtain closed-form expressions for the efficient investment strategy. Our proposed model is examined through empirical studies of a defined contribution pension fund. In-sample results show that, given the same risk level, an ALM investor (a) starting in a bear market can expect a higher return compared to beginning in a bull market and (b) has a lower expected return when there are major cash flow problems. The effects of the investment horizon and state-switching probability on the efficient frontier are also discussed. Out-of-sample analyses show the dynamic optimal liquidity management process. An ALM investor using our model can achieve his or her surplus objective in advance and with a minimum variance close to zero.     

An Investigation of the Pay-As-You-Go Financing Method Using a Contingency Fund and Optimal Control Techniques

Abstract

In many countries, aging populations are expected to lead to substantial rises in the cost of public pension systems financed by the pay-as-you-go (PAYGO) method. These systems will need to be adapted to cope with these changes. This paper considers one approach to reform, described in the literature as “parametric” (see, e.g., Disney 2000), and develops a model for adapting the PAYGO method using a contingency fund and optimal control techniques. The solution of the original model is investigated within two different frameworks: a deterministic-continuous one and a stochastic-discrete one. Finally, a case study applied to Greece is discussed, leading to a potentially acceptable proposal of a smooth path for contribution rates and the age of eligibility for the normal retirement pension.     

The optimal asset allocation of the main types of pension funds: a unified framework

The existing literature deals with the optimal investment strategy of defined benefit (DB) or defined contribution (DC) pension plans. This article’s objective is to compare the optimal policies of different types of pension plans. This is done by first defining an original framework, which is based on the distinction between the nature of the guarantee—which can be internal or external—offered by or to a pension fund. This framework allows to establish links between optimization programs of DC, DB and targeted money purchase schemes. The case of an internal guarantee appears as a standard portfolio insurer’s problem. The second kind of guarantee, not analyzed in the literature yet with regard to the resulting optimal policy, is characterized by the existence of an option in the final wealth definition. Four funds are present in the internal guarantee optimal allocation: the speculative component, the preference independent guarantee- and contribution-hedge terms and the preference dependent state variable-hedge fund. The external guarantee program, solved with an original method using the principles of standard options theory, yields an optimal policy incorporating the delta of the option embodied in the final wealth definition. The conclusion is that the resulting optimal portfolio policy becomes riskier.

Optimal Dividends In An Ornstein-Uhlenbeck Type Model With Credit And Debit Interest

Abstract

In the absence of investment and dividend payments, the surplus is modeled by a Brownian motion. But now assume that the surplus earns investment income at a constant rate of credit interest. Dividends are paid to the shareholders according to a barrier strategy. It is shown how the expected discounted value of the dividends and the optimal dividend barrier can be calculated; Kummer’s confluent hypergeometric differential equation plays a key role in this context. An alternative assumption is that business can go on after ruin, as long as it is profitable. When the surplus is negative, a higher rate of debit interest is applied. Several numerical examples document the influence of the parameters on the optimal dividend strategy.