پس انداز و سرمایه گذاری برای بازنشستگی پیش از موعد: یک تحلیل نظری

We study optimal consumption and portfolio choice in a framework where investors adjust their labor supply through an irreversible choice of their retirement time. We show that investing for early retirement tends to increase savings and reduce an agent's effective relative risk aversion, thus increasing her stock market exposure. Contrary to common intuition, an investor might find it optimal to increase the proportion of financial wealth held in stocks as she ages and accumulates assets, even when her income and the investment opportunity set are constant. The model predicts a decrease in risk aversion following strong market gains like those observed in the nineties.

Two years ago, when the stock market was soaring, 401(k)'s were swelling and (…)(…) early retirement seemed an attainable goal. All you had to do was invest that big job-hopping pay increase in a market that produced double-digit gains like clockwork, and you could start taking leisurely strolls down easy street at the ripe old age of, say, 55. (Business Week December 31, 2001) The dramatic rise of the stock market between 1995 and 2000 significantly increased the proportion of workers opting for early retirement (Gustman and Steinmeier, 2002). The above quote from Business Week demonstrates the rationale behind the decision to retire early: a booming stock market raises the amount of funds available for retirement and allows a larger fraction of the population to exit the workforce prematurely. Indeed, for most individuals, increasing one's retirement savings seems to be one of the primary motivations behind investing in the stock market. Accordingly, there is an increased need to understand the interactions among optimal retirement, portfolio choice, and savings, especially in light of the growing popularity of 401(k) retirement plans. These plans give individuals a great amount of freedom when determining how to save for retirement. However, such increased flexibility also raises concerns about the extent to which agents’ portfolio and savings decisions are rational. Having a benchmark against which to determine the rationality of people's choices is crucial for both policy design and in order to form the basis of sound financial advice. In this paper we develop a theoretical model with which we address some of the interactions among savings, portfolio choice, and retirement in a utility maximizing framework. We assume that agents face a constant investment opportunity set and a constant wage rate while still in the workforce. Their utility exhibits constant relative risk aversion and is nonseparable in leisure and consumption. The major point of departure from preexisting literature is that we model the labor supply choice as an optimal stopping problem: an individual can work for a fixed (nonadjustable) amount of time and earn a constant wage but is free to exit the workforce (forever) at any time she chooses. In other words, we assume that workers can work either full time or retire. As such, individuals face three decision problems: (1) how much to consume, (2) how to invest their savings, and (3) when to retire. The incentive to quit work comes from a discrete jump in their utility due to an increase in leisure once retired. When retired, individuals cannot return to the workforce.1 We also consider two extensions of the basic framework. In the first extension we disallow the agent from choosing retirement past a pre-specified deadline. In a second extension we disallow her from borrowing against the net present value (NPV) of her human capital (i.e., we require that financial wealth be nonnegative). The major results that we obtain can be summarized as follows: First, we show that the agent will enter retirement when she reaches a certain wealth threshold, which we determine explicitly. In this sense, wealth plays a dual role in our model: not only does it determine the resources available for future consumption, but it also controls the “distance” to retirement. Second, the option to retire early strengthens the incentives to save compared to the case in which early retirement is not allowed. The reason is that saving not only increases consumption in the future but also brings retirement “closer.” Moreover, this incentive is wealth dependent. As the individual approaches the critical wealth threshold to enter retirement, the “option” value of retiring early becomes progressively more important and the saving motive becomes stronger. Third, the marginal propensity to consume (MPC) out of wealth declines as wealth increases and early retirement becomes more likely. The intuition is simple: an increase in wealth will bring retirement closer, therefore decreasing the length of time the individual remains in the workforce. Conversely, a decline in wealth will postpone retirement. Thus, variations in wealth are somewhat counterbalanced by the behavior of the remaining NPV of income and in turn the effect of a marginal change in wealth on consumption becomes attenuated. Once again this attenuation is strongest for rich individuals who are closer to their goal of early retirement. Fourth, the optimal portfolio is tilted more towards stocks compared to the case in which early retirement is not allowed. An adverse shock in the stock market will be absorbed by postponing retirement. Thus, the individual is more inclined to take risks as she can always postpone her retirement instead of cutting back her consumption in the event of a declining stock market. Moreover, in order to bring retirement closer, the most effective way is to invest the extra savings in the stock market instead of the bond market. Fifth, the choice of portfolio over time exhibits some new and interesting patterns. We show that there exist cases in which an agent might optimally increase the percentage of financial wealth that she invests in the stock market as she ages (in expectation), even though her income and the investment opportunity set are constant. This result obtains, because wealth increases over time and hence the option of early retirement becomes more relevant. Accordingly, the tilting of the optimal portfolio towards stocks becomes stronger. Indeed, as we show in a calibration exercise, the model predicts that, prior to retirement, portfolio holdings could increase, especially when the stock market exhibits extraordinary returns as it did in the late 1990s during which time many workers experienced rapid increases in wealth, that allowed them to opt for an earlier retirement date. In fact our model suggests a possible partial rationalization of the (apparently irrational) behavior of individuals who increased their portfolios as the stock market was rising and then liquidated stock as the market collapsed. 2 This paper is related to a number of strands in the literature that are surveyed in Ameriks and Zeldes (2001) and Jagannathan and Kocherlakota (1996). The paper closest to ours is that of Bodie et al. (1992) (henceforth BMS). The major difference between BMS and this paper is the different assumption we make about the ability of agents to adjust their labor supply. In BMS, labor can be adjusted in a continuous fashion. However, a significant amount of evidence suggests that labor supply is to a large extent indivisible. For example, in many jobs workers work either full time or they are retired. Moreover, it appears that most people do not return to work after they retire, or if they do, they return to less well-paying jobs or they work only part time. As BMS claim in the conclusion of their paper, Obviously, the opportunity to vary continuously one's labor without cost is a far cry from the workings of actual labor markets. A more realistic model would allow limited flexibility in varying labor and leisure. One current research objective is to analyze the retirement problem as an optimal stopping problem and to evaluate the accompanying portfolio effects. This is precisely the direction we take here. There are at least two major directions in which our results differ from BMS. First, we show that the optimal retirement decision introduces a nonlinear option-type element in the decision of the individual that is entirely absent if labor is adjusted continuously. Second, the horizon and wealth effects on portfolio and consumption choice in our paper are fundamentally different than those in BMS. For instance, stock holdings in BMS are a constant multiple of the sum of (financial) wealth and human capital. This multiple is not constant in our setup, but instead depends on wealth. 3 Third, the model we present here allows for a clear way to model retirement, which is difficult in the literature that allows for a continuous labor-leisure choice. An important implication is that in our setup, we can calibrate the parameters of the model to observed retirement decisions. In the BMS framework, on the other hand, calibration to microeconomic data is harder because individuals do not seem to adjust their labor supply continuously. 4 The model is also related to a strand of the literature that studies retirement decisions. A partial listing includes Stock and Wise (1990), Rust (1994), Lazear (1986), Rust and Phelan (1997), and Diamond and Hausman (1984). Most of these models are structural estimations that are solved numerically. Here our goal is different: rather than include all the realistic ramifications that are present in actual retirement systems, we isolate and very closely analyze the new issues introduced by the indivisibility and irreversibility of the labor supply–retirement decision on savings and portfolio choice. Naturally, there is a trade-off between adding realistic considerations and the level of theoretical analysis that we can accomplish with a more complicated model. Other studies in this literature include Sundaresan and Zapatero (1997), who study optimal retirement, but in a framework without disutility of labor, and Bodie et al. (2004), who investigate the effects of habit formation, but without optimal retirement timing. Some results of this paper share similarities with results that obtain in the literature on consumption and savings in incomplete markets. A highly partial listing includes Viceira (2001), Chan and Viceira (2000), Campbell et al. (2001), Kogan and Uppal (2001), Duffie et al. (1997), Duffie and Zariphopoulou (1993), Koo (1998), and Carroll and Kimball (1996) on the role of incomplete markets and He and Pages (1993) and El Karoui and Jeanblanc-Picque (1998) on issues related to the inability of individuals to borrow against the NPV of their future income. This literature provides insights on why consumption (as a function of wealth) should be concave, and also offers some implications on portfolio choice. However, while in the incomplete markets literature, the results are driven by the inability of agents to effectively smooth their consumption due to missing markets,5 in this paper the results are driven by an option component in an agent's choices that is related to the ability of agents to adjust their time of retirement. Throughout the paper we maintain the assumption that agents receive a constant wage. This is done not only for simplicity, but more importantly because it makes the results more surprising. It is well understood in the literature6 that allowing for a (positive) correlation between wages and the stock market can generate upward-sloping portfolio holdings over time. What we show is that optimal retirement choice can induce observationally similar effects even when labor income is perfectly riskless. Since the argument and the intuition for this outcome are orthogonal to those in existing models, we prefer to use the simplest possible setup in every other dimension, thereby isolating the effects of optimal early retirement. Technically, our model extends methods proposed by Karatzas and Wang (2000) (who do not allow for income) to solve optimal consumption problems with discretionary stopping. The extension that we consider in Section 3 uses ideas proposed by Barone-Adesi and Whaley (1987), and in Section 5, we extend the framework in He and Pages (1993) to allow for early retirement. Finally, three papers that present parallel and independent work on similar issues are Lachance (2003), Choi and Shim (2004), and Dybvig and Liu (2005). Lachance (2003) and Choi and Shim (2004) study a model with a utility function that is separable in leisure and consumption, but that abstracts from a deadline for retirement and/or borrowing constraints.7 The somewhat easier specification of separable utility does not allow consumption to fall upon retirement as we observe in the data. Technically, these papers solve the problem using dynamic programming rather than convex duality methods, which cannot be easily extended to models with deadlines, borrowing constraints, etc. Our approach overcomes these difficulties. Dybvig and Liu (2005) study a very similar model to that in Section 5 of this paper, with similar techniques. However, they do not consider retirement prior to a deadline as we do. A deadline makes the problem considerably harder (since the critical wealth thresholds become time dependent). Nonetheless, we are able to provide a fairly accurate approximate closed-form solution for this problem in Section 3. One can actually perform simple exercises that demonstrate that in the absence of a retirement deadline, the model-implied distribution of retirement times becomes implausible. Most importantly, compared to the papers above, the present paper goes into significantly greater detail in terms of the economic analysis and implications of the results. In particular, we provide applications (like the analysis of portfolios of agents saving for early retirement in the late 1990s) that demonstrate quite clearly the real-world implications of optimal portfolio choice in the presence of early retirement. The structure of the paper is as follows: Section 1 contains the model setup. In Section 2 we describe the analytical results if one places no retirement deadline. Section 3 contains an extension to the case in which retirement cannot take place past a deadline, Section 4 contains some calibration exercises, and Section 5 extends the model by imposing borrowing constraints. Section 6 concludes. We present technical details and all proofs in the appendix.

In this paper we propose a simple partial equilibrium model of consumer behavior that allows for the joint determination of a consumer's optimal consumption, portfolio, and time to retirement. The appendix provides essentially closed-form solutions for virtually all quantities of interest. The results can be summarized as follows. The ability to time one's retirement introduces an option-type character to the optimal retirement decision. This option is most relevant for individuals with a high likelihood of early retirement, that is, individuals with high wealth levels. This option in turn affects both an agent's incentive to consume out of current wealth and her investment decisions. In general, the possibility of early retirement will lead to portfolios that are more exposed to stock market risk. The marginal propensity to consume out of wealth will be lower as one approaches early retirement, reflecting the increased incentives to reinvest gains in the stock market in order to bring retirement “closer.” The model makes some intuitive predictions. Here we single out some of the predictions that seem to be particularly interesting. First, the model suggests that during stock market booms, there should be an increase in the number of people that opt for retirement as a larger percentage of the population hits the retirement threshold (some evidence for this may be found in Gustman and Steinmeier (2002) and references therein). Second, the models shows that it is possible that portfolios of aging individuals could exhibit increasing holdings of stock over time, even if there is no variation in the investment opportunity set and the income stream exhibits no correlation with the stock market (or any risk whatsoever). This is interesting in light of the evidence in Ameriks and Zeldes (2001) that portfolios tend to be increasing or hump-shaped with age for the data sets that they consider. Third, according to the model, there should be a discontinuity in the holdings of stock and in consumption upon entering retirement. Ample empirical evidence shows that indeed, this is the case for consumption. (See, e.g., Aguiar and Hurst, 2004 and references therein.) The discontinuity in stockholdings seems to have been less tested an hypothesis. Fourth, the model predicts that all else equal, switching to a more flexible retirement system that links portfolio choice with retirement timing should lead to increased stock market allocations. This is consistent with the empirical fact that stock market participation increased in the U.S. as 401(k)s were gaining popularity. Fifth, increasing levels of stockholdings during a stock market run-up and liquidations during a stock market fall might not be due to irrational herding; instead, both effects might be due to the behavior of the real option to retire that emerges during the run-up and becomes irrelevant after the fall. In this paper we try to outline the basic new insights that obtain by the timing of the retirement decision. By no means do we claim that we address all the issues that are likely to be relevant for actual retirement decisions (e.g., health shocks, unspanned income, etc.). Rather, we view the theory developed in this paper as a complement to our understanding of richer, typically numerically solved, models of retirement. Many interesting extensions to this model should be relatively tractable. A first important extension would be to include features that are realistically present in actual 401(k)-type plans such as tax deferral, employee matching contributions, and tax provisions related to withdrawals. The solutions developed in such a model could be used to determine the optimal saving, retirement, and portfolio decisions of consumers that are contemplating retirement and taking into account tax considerations. A second extension would be to allow the agent to reenter the workforce (at a lower income rate) once retired. We doubt this would alter the qualitative features of the model, but it is very likely that it would alter the quantitative predictions. It can be reasonably conjectured that the wealth thresholds would be significantly lower in that case, and the portfolios tilted even more towards stocks because of the added flexibility. A third extension of the model would be to introduce predictability and more elaborate preferences. If one were to introduce predictability, while keeping the market complete (like Wachter, 2002), the methods of this paper can be easily extended. It is also very likely that the model would not loose its tractability if one uses Epstein–Zin utilities in conjunction with the methods recently developed by Schroder and Skiadas (1999). A fourth extension of the model that we are currently pursuing is to study its general equilibrium implications.20 This is of particular interest as it would enable one to make some predictions about how the properties of returns are likely to change as worldwide retirement systems begin to offer more freedom to agents in making investment and retirement decisions.