پس انداز بازنشستگی با پرداخت سهم و درآمد نیروی کار به عنوان یک معیار برای سرمایه گذاری

In this paper we study the retirement saving problem from the point of view of a plan sponsor, who makes contribution payments for the future retirement of an employee. The plan sponsor considers the employee's labor income as investment-benchmark in order to ensure the continuation of consumption habits after retirement. We demonstrate that the demand for risky assets increases at low wealth levels due to the contribution payments. We quantify the demand for hedging against changes in wage growth and find that it is relatively small. We show that downside-risk measures increase risk-taking at both low and high levels of wealth.

In this paper we study the problem of saving for retirement with contribution payments and labor income as a benchmark for investments. We consider the retirement saving problem from the point of view of a plan sponsor. The plan sponsor makes contribution payments to an investment fund in order to save for the future retirement of an employee. The goal is to ensure that the employee can continue his consumption pattern after retirement. As the consumption habits of the employee are related to his wages, the plan sponsor considers the labor income of the employee as a benchmark for investments. Clearly, the plan sponsor is not only concerned about the welfare of the employee and also wants to minimize his contribution payments. We formalize this in a multi-period retirement saving model, where the plan sponsor makes a trade-off between the utility of low contribution payments and the utility of high fund values at retirement, relative to the labor income of the employee. The solution of the model reveals the optimal dynamic investment strategy and the optimal funding policy of the plan sponsor. The retirement saving model presented in this paper includes both a defined contributions pension plan and a ‘final pay’ defined benefits plan as special cases. The generality of the model allows us to circumvent the large difference in pension schemes throughout the world: we focus on the core of the retirement saving problem. Clearly, not every plan sponsor or pension fund considers labor income as a benchmark for investments. However, we believe that it is in the interest of the employees to do so, without any adverse consequences. An important assumption throughout the paper is that the labor income of the employee cannot be replicated with the available assets: consequently, the financial market is incomplete. Moreover, we assume that the wage growth rate is partly predictable. Given the basic model setup, we derive optimal decision rules by applying dynamic programming. The optimal decision rules specify the asset weights and the contribution payment as a function of the state variables (the wealth-to-income ratio and the wage growth rate) and provide direct insight into the underlying problem. We are particularly interested in the following issues, which are relevant for retirement saving and have not been studied adequately in the literature yet 1. What is the magnitude of the demand for the hedge portfolio against random changes of the wage growth rate? 2. What is the impact of contribution payments on the optimal asset allocation? 3. What is the optimal multi-period investment strategy for investment objectives based on downside-risk measures, which are very popular in practice? 4. What is the effect of an objective that assigns utility to a high fund value before the retirement date (on top of utility at retirement)? With our implementation of the dynamic programming algorithm we were able to address these three main questions about optimal investment and funding in the retirement saving model. The conclusions to be drawn from our analysis and computational experiments are as follows: 1. Regardless of his utility function, the plan sponsor invests in a hedge portfolio against random fluctuations of the employee's labor income. The hedge portfolio depends on the covariance of the asset returns with the wage growth rate. The numerical results demonstrate that the correlation between asset returns and wage growth has a substantial influence on portfolio composition. Dynamic adjustments of the hedging strategy due to changes of the wage growth rate are relatively small. 2. Contribution payments change the optimal investment strategy considerably, even for a plan sponsor with constant relative risk aversion over fund value. The portfolio weights are no longer constant and there is a strong tendency to gamble at low levels of wealth. We also find that contribution payments lead to a strong investment horizon effect. 3. If a plan sponsor maximizes the expected fund value subject to a penalty on downside-risk, then the optimal investment policy is to increase the weight of risky assets at low levels of wealth. This gambling policy can be attributed to the property of increasing relative risk aversion of the downside-risk objective. 4. Intertemporal measurement of utility over wealth reduces the gambling effects due to contribution payments considerably. The insights gained from the basic retirement saving model studied in this paper may help plan sponsors to formulate dynamic investment policies and choose reasonable objectives. Furthermore, the optimal decision-rules derived here can be implemented in simulation-based systems for ALM, where additional market imperfections such as transaction costs could be added. In order to place this paper in the literature, we could interpret the retirement saving problem with labor income as an asset–liability management (ALM) problem. In the literature many single-period ALM models have been studied (Sharpe and Tint, 1990; Leibowitz et al., 1994). These models indicate that investors should take the correlation between assets and liabilities into account, while deciding about the investment strategy. However, saving for retirement typically involves a long-term investment goal, and one-period models are, therefore, inappropriate due to stochastic opportunity sets and non-myopic preferences. There is a large stream of literature about the application of stochastic programming methods for multi-period ALM (see Mulvey and Ziemba (1998) for an overview). Stochastic programming models formulate an accurate answer to the question: how to invest today, given optimal recourse in the future? However, the optimal policies do not explicitly reveal the relation between the decisions and the state-variables. Our main objective is to gain insight in optimal decision rules for portfolio choice and funding in a simple micro-economic model without market imperfections. Dynamic retirement saving problems have also been studied in the actuarial literature: we refer to O'Brien (1987) for an early continuous-time formulation and Haberman and Sung (1994) for a more recent discrete-time model. Haberman and Sung (1994) provide a closed-form solution for a discrete-time pension model where the objective function is to minimize the squared deviations from the target contribution rate, and the squared deviations from the target funding ratio. The mathematical results of Haberman and Sung (1994) are impressive but the approach can not be easily extended to objective functions that are non-quadratic, or to problems with constraints on the policies, as studied in this paper. Due to the focus on dynamic programming and micro-economic analysis this paper is also related to the individual consumption–investment literature (Mossin, 1968; Hakansson, 1969; Samuelson, 1969). Our retirement saving model can be classified as a model with a stochastic opportunity set (predictable wage growth), with both negative and positive consumption (the net contribution payment) and without a riskless asset (due to market incompleteness). Koo (1999) analyzes a simple discrete-time model with labor income for power utility and has to apply numerical techniques to solve it. As our model is more elaborate and as we additionally want to study more general objectives than power utility, we clearly need a numerical solution method. For the computation of the optimal investment policy we apply a dynamic programming algorithm. We add several numerical improvements to the basic algorithm in order to increase efficiency, including transformation and interpolation of the value function, which facilitate the solution of investment models with power and HARA-utility. As we apply dynamic programming, the optimal policies are derived in feedback form and we acquire direct insight into the structure of the strategies. This paper can be placed in a growing literature that solves consumption–investment problems numerically in order to study the impact of relevant, but analytically complicated, issues such as predictability, transaction costs and parameter uncertainty (Brennan et al., 1997; Balduzzi and Lynch, 1999; Barberis, 2000). This paper is organized as follows. In Section 2, we introduce the retirement saving model and the main assumptions. Moreover, we review the literature and introduce our implementation of the dynamic programming algorithm. In Section 3, we study the optimal investment strategies in detail for different investment objectives. We show that the widely used class of downside-risk measures leads to peculiar investment policies in a multi-period setting. In Section 4, we study the additional effect of funding payments on the investment strategy, while Section 5 considers horizon effects and the impact of utility over intertemporal fund value. Section 6 concludes and summarizes the paper.

In this paper we studied a retirement saving model with labor income as a benchmark for investments and contribution payments by the plan sponsor. Due to market incompleteness it is very hard to derive closed-form solutions for the optimal investment and contribution policies of the plan sponsor. We used an efficient implementation of the dynamic programming algorithm in order to solve the problem numerically. The main conclusions of our analysis and computational experiments are as follows: 1. Regardless of his utility function, the plan sponsor invests in a hedge portfolio against random fluctuations of the employee's labor income. The hedge portfolio depends on the covariance of the asset returns with the wage growth rate. The numerical results demonstrate that the correlation between asset returns and wage growth has a substantial influence on portfolio composition. Dynamic adjustments of the hedging strategy due to changes of the wage growth rate are relatively small. 2. Contribution payments change the optimal investment strategy considerably, even for a plan sponsor with constant relative risk aversion over fund value. The portfolio weights are no longer constant and there is a strong tendency to gamble at low levels of wealth. We also find that contribution payments lead to a investment horizon effect. Moreover, additional constraints on the funding policy can have a large impact on the optimal investment strategy. 3. If a plan sponsor maximizes the expected fund value subject to a penalty on downside-risk, then the optimal weight of risky assets in the portfolio increases at low levels of wealth. The gambling policy can be attributed to the increasing relative risk aversion property of the downside-risk measure. 4. Intertemporal measurement of utility over wealth can reduce gambling effects. The insights gained through the numerical computations in this paper may aid fund-managers to formulate dynamic policies and choose reasonable investment objectives. Furthermore, the decision rules derived here could be applied in simulation-based models for asset–liability management, where additional market imperfections such as transaction costs and position limits can be taken into account.