Exploration of pension funding in case of exact vesting

A pension plan is said to be exactly vested if it provides in addition to the benefit available upon retirement, a benefit, upon termination for any cause prior to retirement, which is exactly equivalent to the actuarial accured liability for the terminating participant.The concept of exact vesting has simple application in defined contribution plans such as those of the Teachers Insurance and Annuity Association. It is also feasible to develop the exact vesting concept for a defined benefit plan which uses an individual type of actuarial cost method. An exactly vested plan would have more individual equity than is available under customary vesting and early retirement provisions of defined benefit plans.In this paper, theory is developed for an exactly vested model plan in parallel to the theory for a pure pension model plan discussed in previous papers on pension funding dynamics.     

Exponential smoothing methods in pension funding

‘Smoothed-market’ methods are used by actuaries,when they value pension plan assets, in order to dampen thevolatility in contribution rates recommended to plan sponsors.A method involving exponential smoothing is considered. Thedynamics of the pension funding process is investigated in thecontext of a simple model where asset gains and losses emergeas a result of random rates of investment return and where thegains and losses are spread. It is shown that smoothing marketvalues up to a point does improve the stability of contributionsbut excessive smoothing is inefficient. It is also shown thatconsideration should be given to the combined effect of theasset valuation and gain and loss adjustment methods. Practicaland efficient combinations of gain/loss spreading periods andasset value smoothing parameters are suggested.     

Pension funding problem with regime‐switching geometric Brownian motion assets and liabilities

This paper extends the pension funding model in (N. Am. Actuarial J. 2003; 7 :37–51) to a regime‐switching case. The market mode is modeled by a continuous‐time stationary Markov chain. The asset value process and liability value process are modeled by Markov‐modulated geometric Brownian motions. We consider a pension funding plan in which the asset value is to be within a band that is proportional to the liability value. The pension plan sponsor is asked to provide sufficient funds to guarantee the asset value stays above the lower barrier of the band. The amount by which the asset value exceeds the upper barrier will be paid back to the sponsor. By applying differential equation approach, this paper calculates the expected present value of the payments to be made by the sponsor as well as that of the refunds to the sponsor. In addition, we study the effects of different barriers and regime switching on the results using some numerical examples. The optimal dividend problem is studied in our examples as an application of our theory. Copyright © 2009 John Wiley & Sons, Ltd.     

Evaluating the UK and Dutch defined-benefit pension policies using the holistic balance sheet framework

This paper compares the UK and Dutch occupational defined-benefit pension policies using the holistic balance sheet (HBS) framework. The UK DB pension system differs from the Dutch one in terms of the steering tools and adjustment mechanisms. In addition to the sponsor guarantee, the UK system has the protection from the Pension Protection Fund (PPF) that guarantees DB pension schemes’ funding shortfalls if the sponsors of the schemes are insolvent. The paper first introduces a multi-period model called value-based ALM to value the embedded options implied by both UK and Dutch pension policies and build the HBS. The HBS framework allows us to have a holistic view on the real and contingent assets and liabilities of a pension scheme and evaluate the impact of introducing a new policy for the stakeholders of the pension scheme. Then, we compare the results of a typical UK policy with a typical Dutch one. The comparison suggests the UK policy is better for participants but worse for the sponsor compared to the Dutch policy. The UK policy is more generous in indexation and participants do not have the burden to contribute to the funding recovery of the pension scheme. The PPF provides protection of the benefits up to a certain level if the sponsor is insolvent, thus, participants in a scheme with a UK pension policy are exposed to limited downside risk. On the other hand, the sponsor of the pension scheme with the UK policy shoulders a heavier burden to contribute to the recovery of the pension funding shortfalls than that of the pension scheme with the Dutch policy.     

Mean–variance portfolio and contribution selection in stochastic pension funding

In this paper we study the problem of simultaneous minimization of risks, and maximization of the terminal value of expected funds assets in a stochastic defined benefit aggregated pension plan. The risks considered are the solvency risk, measured as the variance of the terminal fund’s level, and the contribution risk, in the form of a running cost associated to deviations from the evolution of the stochastic normal cost. The problem is formulated as a bi-objective stochastic problem of mean–variance and it is solved with dynamic programming techniques. We find the efficient frontier and we show that the optimal portfolio depends linearly on the supplementary cost of the fund, plus an additional term due to the random evolution of benefits.     

Longevity risk in portfolios of pension annuities

We analyze the importance of longevity risk for the solvency of portfolios of pension annuities. We distinguish two types of mortality risk. Micro-longevity risk quantifies the risk related to uncertainty in the time of death if survival probabilities are known with certainty, while macro-longevity risk is due to uncertain future survival probabilities. We use a generalized two-factor Lee-Carter mortality model to produce forecasts of future mortality rates, and to assess the relative importance of micro- and macro-longevity risk for funding ratio uncertainty. The results show that if financial market risk is fully hedged so that uncertainty in future lifetime is the only source of uncertainty, pension funds are exposed to a substantial amount of risk. Systematic and non-systematic deviations from expected survival imply that, depending on the size of the portfolio, buffers that reduce the probability of underfunding to 2.5% at a 5-year horizon have to be of the order of magnitude of 7% to 8% of the initial value of the liabilities.     

Time consistent pension funding in a defined benefit pension plan with non-constant discounting

We consider the time consistent management of a defined benefit stochastic pension plan where the participants have different rates of time preference and the fund manager collects this heterogeneity when discounting the future. The main objective is to select the amortization rate and the investment strategy minimizing both the contribution rate risk and the solvency risk. The problem is formulated as a stochastic control problem with non-constant rate of discount and is solved analytically by means of the dynamic programming approach and the technical interest rate is selected in order to keep stable the fund evolution within prescribed targets. A numerical illustration shows a comparative of the stability of the fund assets and the rate of contribution for a convex combination of exponential functions as discount function and for the constant discount case.     

Funding and investment decisions in a stochastic defined benefit pension plan with regime switching*

In this paper, we consider a continuous-time Markov regime-switching model for a pension plan with a collective defined benefit character. In particular, we focus on optimal funding and asset allocation problem for a fund manager who wants to maximize the expected utility of the difference ratio between the benefit and contribution rates to the total salary until ruin. Using the techniques and methods of stochastic control, we present a system of Hamilton–Jacobi–Bellman equations for this optimization problem and establish a verification theorem. In the special cases of logarithmic and power utility, we solve the problem explicitly and present some numerical examples to illustrate our results.     

Stochastic pension funding when the benefit and the risky asset follow jump diffusion processes

We study the asset allocation of defined benefit pension plans of the type designed and sponsored by firms with the aim of providing a lifetime pension to the employees at the age of retirement. Benefits are stochastic, combining Poisson jumps with Brownian uncertainty. The sponsor dynamically forms portfolios where the risky asset is also subjected to Poisson jumps and Brownian uncertainty, possibly correlated with the evolution of benefits. The objective is to assure future benefits, while controlling the contribution made to the fund reserves. The problem is solved analytically using dynamic programming techniques.     

Pension funds as institutions for intertemporal risk transfer

A continuous time overlapping generation model is used to analyse defined-contribution pension plans. Without intergenerational risk transfer between employees the optimal investment strategy results from the Merton model. Introducing intergenerational risk transfer leads to an increase in the risk tolerance of future employees and allows us to improve their anticipated expected utility resulting from accrued retirement benefits. Of course, this leads to a risk of temporary underfunding. But even for an underfunded pension plan one can guarantee that in the long run, the median of the funding ratio exceeds one.     

Valuation of intergenerational transfers in funded collective pension schemes

This paper applies contingent claim analysis to value pension contracts for real-life collective pension plans with intergenerational risk sharing and offering DB-like benefits. We rewrite the balance sheet of such a pension fund as an aggregate of embedded generational options. This implies that a pension fund is a zero-sum game in value terms, so any policy change inevitably leads to value transfers between generations. We explore intergenerational value transfers that may arise from a plan redesign or from changes in funding policy and risk sharing rules. We develop a stochastic framework which accounts for time-varying investment opportunities and computes the embedded generational options. Changes in the values of the generational options enable us to evaluate the impact of policy modifications in the pension contract with respect to intergenerational transfers and redistribution. We find that a switch to a less risky asset mix is beneficial to elderly members at the expense of younger members who lose value. A reallocation of risk bearing from a plan with flexible contributions and fixed benefits to a plan with fixed contributions and flexible benefits leads to value redistribution from older plan members to younger ones.     

Levelling the playing field: A VIX-linked structure for funded pension schemes

In this article, we study intergenerational risk and cost sharing for a variety of collective funded pension plans. Inspired by the literature on contingent claim analysis in pension insurance, we derive time-varying contribution and benefit levels. The latter specifically include the fund surplus, which accounts for (intergenerational) risk sharing, and the VIX, which is related to cost sharing among generations. We find that pension schemes with a well-structured volatility-risk-adjusted component can be welfare enhancing for the entry and future cohorts. In addition, these schemes are fair from an ex ante perspective, provide adequate consumption profiles and high levels of satisfaction.     

一类养老金精算成本模型

本文建立了一类养老金精算成本模型．该模型的基本思想是，当雇员加入养老金计划之后，与雇员的未来服务年限有关的养老金利益的精算现值由雇主（或雇主与雇员一起）用拨款的形式缴清，而与雇员在参加养老金计划之前的过去服务年限有关的养老金利益则另行处理．所以称该模型为应计利益精算成本模型．     

Allocating unfunded liability in pension valuation under uncertainty

This paper studied the cost allocation for the unfunded liability in a defined benefit pension scheme incorporating the stochastic phenomenon of its returns. In the recent literature represented by Cairns and Parker [Insurance: Mathematics and Economics 21 (1997) 43], Haberman [Insurance: Mathematics and Economics 11 (1992) 179; Insurance: Mathematics and Economics 13 (1993) 45; Insurance: Mathematics and Economics 14 (1994) 219; Insurance: Mathematics and Economics 14 (1997) 127], Owadally and Haberman [North American Actuarial Journal 3 (1999) 105], the fund level is modeled based on the plan dynamics and the returns are generated through several stochastic processes to reflect the current realistic economic perspective to see how the contribution changed as the cost allocation period increased. In this study, we generalize the previous constant value assumption in cost amortization by modeling the returns and valuation rates simultaneously. Taylor series expansion is employed to approximate the unconditional and conditional moments of the plan contribution and fund level. Hence the stability of the plan contribution and the fund size under different allocation periods could be estimated, which provide valuable information adding to the previous works.     

Optimal mix between pay-as-you-go and funding for DC pension schemes in an overlapping generations model

Public pension systems are usually pay-as-you-go financed, that is, current contributions cover the pension expenditures. However, some countries combine funding and pay-as-you-go within the first pillar. This article studies a mixed system where a part of the individual’s contribution accrues funded rights whereas the other part accrues pay-as-you-go rights. Diversification conditions between these two financing techniques are derived in a mean–variance framework for two distinct contexts: for a cohort entering the system (named ex-ante case) and for multiple cohorts coexisting at the same period of time (named ex-post case). The diversification benefits in presence of a liquidity constraint which ensures that the income from contributions is sufficient to cover the pension expenditures are also studied. We show that, on the one hand, diversification benefits individuals when the economy is dynamically efficient for the ex-ante case. On the other hand, diversification is unattractive when pay-as-ou-go and funding are positively correlated for the ex-post case.     

Robust decomposable Markov decision processes motivated by allocating school budgets

Motivated by an application to school funding, we introduce the notion of a robust decomposable Markov decision process (MDP). A robust decomposable MDP model applies to situations where several MDPs, with the transition probabilities in each only known through an uncertainty set, are coupled together by joint resource constraints. Robust decomposable MDPs are different than both decomposable MDPs, and robust MDPs and cannot be solved by a direct application of the solution methods from either of those areas. In fact, to the best of our knowledge, there is no known method to tractably compute optimal policies in robust, decomposable MDPs. We show how to tractably compute good policies for this model, and apply the derived method to a stylized school funding example.     

基于政府补贴的养老服务市场供给研究

随着我国老龄化速度加快,养老服务的有效供给问题是政府和学者关注的焦点。考虑政府财政补贴下,引入民办与公办养老服务的替代因子,构建民办养老机构与公办养老机构的服务动态供给模型,分析不同补贴方式和补贴力度对服务均衡供给量的影响,并进一步通过补贴乘数分析政府补贴对养老服务机构最优供给决策的影响程度。结果表明:政府对民办养老机构的财政补贴可以增加养老服务市场供给量,相较于运营补贴,政府建设补贴的政策效应更加明显;财政补贴降低了民办养老机构的建设成本和投资风险,刺激社会资本投入的积极性,民办养老服务供给增加幅度大于公办养老服务供给减少幅度,养老服务市场有效供给增加。在财政预算约束下,选择恰当的财政补贴方式,可以提高财政资源的配置效率,增加养老服务市场的有效供给,缓解养老服务财政压力。     

A risk-based model for the valuation of pension insurance

In the US, defined benefit plans are insured by the Pension Benefit Guaranty Corporation (PBGC). Taking account of the fact that the PBGC covers only the residual deficits of the pension fund the sponsoring company is unable to cover and that the plans can be prematurely terminated, we consider a model that accounts for the joint dynamics of the pension fund’s and sponsoring firm’s assets in order to effectively determine the risk-based pension premium for the insurance provided by the PBGC. We obtain a closed-form pricing formula for this risk-based premium. Its magnitude depends highly on the investment portfolio of the pension fund and of the sponsoring company as well as the correlation between these two portfolios.     

A comparison of two legislative approaches to the pay-as-you-go pension system in terms of adequacy. The Italian case

The aim of our work is to evaluate a new legislative proposal of the Italian pension system due to Giuliano Cazzola e Tiziano Treu and to compare it with the system in force due to former Minister Elsa Fornero. The evaluation is made in terms of adequacy. We make use of a mathematical model which, under the hypothesis of demographic equilibrium, formalizes the legislative changes of the pay-as-you-go pension system. The model is tested using Italian demographic and socio-economic data.The pay-as-you-go pension system in force is notional defined contribution and has the huge drawback that the replacement rate (the ratio between the monthly pension and the last wage perceived by the worker) is very low. We compare the two pension systems evaluating the dynamics of the replacement rate.     

Domain Decomposition of Optimal Control Problems for Dynamic Networks of Elastic Strings

We consider optimal control problems related to exact- and approximate controllability of dynamic networks of elastic strings. In this note we concentrate on problems with linear dynamics, no state and no control constraints. The emphasis is on approximating target states and velocities in part of the network using a dynamic domain decomposition method (d³m) for the optimality system on the network. The decomposition is established via a Uzawa-type saddle-point iteration associated with an augmented Lagrangian relaxation of the transmission conditions at multiple joints. We consider various cost functions and prove convergence of the infinite dimensional scheme for an exemplaric choice of the cost. We also give numerical evidence in the case of simple exemplaric networks.