سرمایه گذاری وابسته به سن: سرمایه گذاری بهینه و استراتژی های سرمایه گذاری در تعریف برنامه سهم بازنشستگی هنگامی که اعضای چرخه عمر برنامه ریزان منطقی مالی هستند

A defined contribution pension plan allows consumption to be redistributed from the plan member’s working life to retirement in a manner that is consistent with the member’s personal preferences. The plan’s optimal funding and investment strategies therefore depend on the desired profile of consumption over the lifetime of the member. We investigate these strategies under the assumption that the member is a rational life cycle financial planner and has an Epstein-Zin utility function, which allows a separation between risk aversion and the elasticity of intertemporal substitution. We also take into account the member’s human capital during the accumulation phase of the plan and we allow the annuitisation decision to be endogenously determined during the decumulation phase. We show that the optimal funding strategy involves a contribution rate that is not constant over the life of the plan but is age-dependent and reflects the trade-off between the desire for current versus future consumption, the desire for stable consumption over time, the member’s attitude to risk, and changes in the level of human capital over the life cycle. We also show that the optimal investment strategy during the accumulation phase of the plan is ‘stochastic lifestyling’, with an initial high weight in equity-type investments and a gradual switch into bond-type investments as the retirement date approaches in a way that depends on the realised outcomes for the stochastic processes driving the state variables. The optimal investment strategy during the decumulation phase of the plan is to exchange the bonds held at retirement for life annuities and then to gradually sell the remaining equities and buy more annuities, i.e., a strategy known as ‘phased annuitisation’.

A typical individual’s life cycle consists of a period of employment followed by a period of retirement. Most individuals therefore need to reallocate consumption from their working life to retirement if they wish to avoid poverty in old age. A defined contribution (DC) pension plan can achieve this reallocation in a way that is consistent with the preferences of the individual plan member.1, 2 There are three key preferences to take into account. The first relates to the desire to smooth consumption across different possible states of nature within any given time period. The second relates to the desire to smooth consumption across different time periods. The third relates to the desire for current versus future consumption; saving for retirement involves the sacrifice of certain consumption today in exchange for uncertain consumption in the future. This uncertainty arises because both future labour income and the returns on the assets in which the retirement savings are invested are uncertain. The plan member therefore needs to form a view on both the trade-off between consumption in different states of nature in the same time period and the trade-off between consumption and consumption variability in different time periods. Attitudes to these trade-offs will influence the optimal funding and investment strategies of the pension plan. In a DC pension plan, the member allocates part of his labour income earned each year to the pension plan in the form of a plan contribution and, thus, builds up a pension fund prior to retirement. Then, at retirement, the member uses the accumulated pension fund to finance consumption in retirement by purchasing a life annuity, by keeping the fund invested and drawing an income from it, or some combination of these.3 The decisions regarding the level of the contribution rate in each year before retirement4 (i.e., the funding strategy) is driven by the member’s preference between current and future consumption. As a consequence, the optimal funding strategy might involve a contribution rate into the plan that is not, as in most extant plans, a fixed percentage of labour income, but is, instead, age-related. The investment strategy prior to retirement (i.e., the decision about how to invest the accumulating fund across the major asset categories, such as equities and bonds) will influence the volatility of the pension fund (and, hence, the amount available for consumption in future periods), and so will depend on the member’s attitude to that volatility, both across states of nature and across time. After retirement, hedging longevity risk becomes an important additional consideration, so the investment strategy will now include annuities as well as the traditional asset categories. In this paper, we investigate the optimal funding and investment strategies in a DC pension plan assuming the member is a rational life cycle financial planner. The model we use has three key features. The first key feature of the model is the assumption of Epstein-Zin (1989) recursive preferences by the plan member. This allows us to separate relative risk aversion (RRA) from the elasticity of intertemporal substitution (EIS). Risk aversion is related to the desire to stabilise consumption across different states of nature in a given time period5 and EIS measures the desire to smooth consumption over time.6 Thus, risk aversion and EIS are conceptually distinct and, ideally, should be parameterised separately. Within the commonly used power utility framework, the EIS is given by the reciprocal of the coefficient of relative risk aversion (e.g., see Campbell and Viceira (2002)). This restriction has been criticised because it does not appear to reflect empirical observations. For example, based on the consumption capital asset pricing model of Breeden (1979), Schwartz and Torous (1999) disentangle these two concepts using the term structure of asset returns. Using US data on discount Treasury bond returns, equity market returns and aggregate consumption for 1964-97, their best estimate for the coefficient of RRA is 5.65 (with a standard error of 0.22) and their best estimate of the EIS is 0.226 (with a standard error of 0.008). Thus, a high coefficient of RRA tends to be associated with a low level of EIS, but the estimated parameter values do not have the exact reciprocal relationship assumed in the power utility framework. Similarly, Blackburn (2006) rejects the reciprocal relationship on the basis of a time series of RRA and EIS parameters estimated from observed S&P 500 option prices for a range of different expiry dates between 1996 and 2003.7 The second key feature of the model is the recognition that the optimal investment strategy will depend not just on the properties of the available financial assets, but also on the plan member’s ‘human capital’, defined as the net present value of an individual’s future labour income.8,9 A commonly used investment strategy in DC pension plans is ‘deterministic lifestyling’.10 With this strategy, the pension fund is invested entirely in high risk assets, such as equities,11 when the member is young. Then, at a pre-set date (e.g., 5 to 10 years prior to retirement is quite common in practice), the assets are switched gradually (and often linearly) into lower risk assets such as bonds and cash. However, whilst intuitively appealing, there is no strong empirical evidence to date demonstrating that this is an optimal strategy. If equity returns are assumed to be mean reverting over time, then the lifestyling strategy of holding the entire fund in equities for an extended period prior to retirement might be justified, as the volatility of equity returns can be expected to decay over time (as a result of the ‘time diversification of risk’). However, there is mixed empirical evidence about whether equity returns are genuinely mean reverting: for example, Lo and Mackinley (1988), Poterba and Summers (1988) and Blake (1996) find supporting evidence in both US and UK markets, while Kim et al. (1991) and Howie and Davies (2002) find little support for the proposition in the same countries. We would therefore not wish an optimal investment strategy to rely on a debatable assumption of mean reversion holding true in practice. A more appealing justification for a lifestyling investment strategy comes from recognising the importance of human capital in individual financial planning. Human capital can be interpreted as a bond-like asset in which future labour income is fairly stable over time and can be interpreted as the ‘dividend’ on the individual’s implicit holding of human capital.12 Most young pension plan members are likely to have a significant holding of (bond-like) human capital, but a negligible holding of financial assets, especially equity. Their pension fund should initially compensate for this with a heavy weighting in equity-type assets.13 The ratio of human to financial wealth will therefore be a crucial determinant of the optimal lifecycle portfolio composition. At younger ages, as shown in Fig. 1, this ratio is large since the individual has had little time to accumulate financial wealth and expects to receive labour income for many years to come.14 Over time, as human capital decays and the value of financial assets in the pension fund grows, this ratio will fall and the pension fund should be rebalanced away from equities towards bonds. However to date, there has been no quantitative research exploring the human capital dimension in a DC pension framework. The third key feature of the model is the endogeneity of the annuitisation decision. In some jurisdictions, such as the UK, there is a mandatory requirement to purchase an annuity with the pension fund up to a specified limit. The limit in the UK, for example, is £20,000 per annum (as of 2011),15 and the annuity has to be purchased at the time of retirement. However, in many jurisdictions, including the US, Japan, Australia and most continental European countries, there is no requirement to purchase an annuity at all. In this study, we determine the optimal annuitisation strategy for the member.16

In this paper, we have examined optimal funding and investment strategies in a DC pension plan using a life cycle model that has been extended in three significant ways: • the assumption of Epstein-Zin recursive preferences by the plan member which enables a separation between relative risk aversion and the elasticity of intertemporal substitution, • the recognition of human capital as an asset class along with financial assets, such as equities and bonds, and • endogenising the decision about how much to annuitise in retirement. Our key findings with respect to funding are: • The optimal funding strategy involves an age-dependent annual contribution rate, increasing steadily from zero prior to age 35 to around 30-35% by age 55 (but then reducing slightly as labour income falls in the years immediately prior to retirement, so as to maintain pre-retirement consumption levels). • The effect of lower risk aversion is to reduce the level of pension contributions at all ages (in the expectation of achieving higher investment returns on those contributions during the accumulation phase). However, as would be expected, the downside of this is greater uncertainty in both the pension fund at retirement and the retirement consumption supported by this fund. • The effect of a lower EIS is greater consumption stability over the life cycle. The effect of a higher EIS is to increase the willingness of the plan member to accept consumption volatility over time. A high level of EIS in relation to RRA (such that EIS×RRA>1) implies that, after retirement, it is optimal for the member to spend less than the pension income received and use the resulting savings each year to purchase additional annuities (thereby benefiting from the mortality premium inherent in the return on life annuities). Since such behaviour is uncommon, a plausible restriction on the parameters in an Epstein-Zin utility function is EIS×RRA≤1. • A lower personal discount factor implies a preference for current (rather than future) consumption, leading to a lower contribution rate until the last 10 years or so before retirement when current consumption has to be reduced sharply each year, and thus retirement savings increased, to ensure a minimal level of pension wealth. Our key findings with respect to investment strategy are: • The optimal investment strategy is also age-dependent. Pre-retirement, the optimal strategy is stochastic lifestyling, rather than the more conventional deterministic lifestyling. While the optimal weighting in equities is initially very high and subsequently declines as the retirement date approaches, it does not do so in a predetermined manner as in the case of deterministic lifestyling. Instead, the optimal equity weighting over the life cycle depends on the realisations of the stochastic processes determining equity returns and labour income. Stochastic lifestyling is justified by recognising the importance of human capital and interpreting it as a bond-like asset which depreciates over the working life. An initial high weighting in equities is intended to counterbalance human capital in the combined ‘portfolio’ of human capital and financial wealth. In time, the weighting in equities falls stochastically, while that in bonds rises as human capital decays over time.47 • Another difference with deterministic lifestyling is that the portfolio is not completely switched into bonds by the retirement date. Depending on the member’s risk aversion, there could still be significant equity holdings in the pension fund on the retirement date. For the ranges of risk aversion that we considered in this study, the optimal equity weighing at retirement varied between 20% and 50%. • The optimal investment strategy at retirement is phased annuitisation. The first stage of this strategy is to exchange the bond fund for a life annuity, thereby securing lifelong income protection for the member as well as benefiting from the mortality premium in the return on the annuity. The optimal weight in the equity fund does not immediately change. However, each year that the member survives, the return from buying additional annuities increases and the equity weighting falls until a point is reached when the mortality premium exceeds the equity risk premium and it becomes optimal to switch the entire residual pension fund into annuities whatever the member’s attitude to risk. • The effects of lower risk aversion and a lower personal discount factor are to increase the length of time over which the pension fund is fully invested in equities and to reduce the length of the switchover period into bonds prior to retirement. Lower risk aversion leads to a higher post-retirement equity weighting, but does not affect the age at which it is optimal to switch the remaining pension fund assets into annuities (this decision depends purely on the relationship between the relative sizes of the mortality premium and the equity risk premium). The size of the personal discount factor has no effect on the optimal asset allocation after retirement. The size of the EIS has a marginal impact on the optimal asset allocation both before and after retirement. The results in this paper have some important implications for the optimal design of DC pension plans: • They provide some justification for age-related contribution rates in DC pension plans. Because individuals tend to prefer relatively smooth consumption growth, a plan design involving a zero contribution rate prior to around age 35 with an increasing age-dependent contribution rate thereafter (reaching, on average, around 30 to 35% per annum in the period immediately prior to retirement) offers higher expected lifetime utility than one with fixed age-independent contribution rates. Greater contribution rate flexibility would allow for the preferences of individual members (with regard to their desire for consumption smoothing, risk attitude and relative preference for current over future consumption) to be recognised. While high, heavily back-loaded, age-related contribution rates might be optimal for ‘econs’ (i.e., rational life cycle financial planners), they might not be optimal for ‘humans’ with their behavioural difficulties in starting and maintaining long-term savings programmes (see, e.g., Thaler and Bernartzi, 2004, Mitchell and Utkus, 2004 and Thaler and Sunstein, 2008). A compromise solution might be a compulsory minimum contribution rate at all ages (to ensure that all plan members have some minimal pension fund to support consumption in retirement) together with age-related additional voluntary contributions (AVCs) at higher ages. • It is important to get reliable measures of the member’s risk aversion and personal discount factor. This can be achieved using appropriately designed questionnaires (see, e.g., Coller and Williams, 1999, Holt and Laury, 2002, Andersen et al., 2008 and Laury et al., 2011). However, EIS seems to be less important according to our sensitivity analysis. This is helpful, since it is unlikely that we would be able to design a questionnaire that could elicit a member’s EIS even if the member understood what an EIS meant! The lack of sensitivity of both the contribution rate and investment strategy to the EIS suggests that we could fix the EIS at a level that happened to be convenient for us. A particularly convenient level would be to choose the EIS to equal the inverse of the RRA, i.e., at a level consistent with power utility. The study by Schwartz and Torous (1999) cited above showed that while there was an inverse relationship between EIS and RRA, the relationship is not exactly reciprocal. Nevertheless, it was fairly close for a typical individual, so using a reciprocal relationship in a practical application might be a reasonable approximation for most people and it would also help to speed up the numerical solution algorithm. • It is very important to incorporate the salary process in the optimal design of a DC pension plan. For most people, their human capital will be bond-like in nature and this will have a direct impact on the optimal contribution rate and asset allocation decisions, in particular, justifying a high weight for equities in the pension plan. However, for senior plan members whose salary levels (including bonus and dividends from their own stock holdings) may have a strong link with corporate profitability, their labour income growth rate might be much higher than the return on bonds and might also be more volatile. In this case, their human capital will be more equity-like in nature and so the optimal investment strategy will be more heavily geared towards bonds. The same would be true for unskilled workers who are unable to find regular employment. • An investment strategy involving a switch from equities to bonds as members approach retirement is appropriate for DC pension plans, even when equity returns are not mean reverting. However, the switch away from equities is stochastic rather than predetermined, and is dependent on past investment and salary growth experience. Nevertheless, the switch should typically be made earlier than in traditional lifestyle strategies (i.e., from age 45 or so rather than age 55, which is more common in practice). Also, unlike most traditional lifestyle investment strategies, the optimal equity weight in the portfolio immediately prior to retirement is not reduced to zero (rather it depends on the risk attitude of the individual). The practical implementation of such an investment strategy would not actually be that challenging if we had reliable measures of the member’s risk aversion and discount factor and could assume that the EIS was equal to the reciprocal of the RRA. Fig. 5 shows that for a given RRA, discount factor and EIS, the distribution of the optimal equity allocation is very narrow. An approximate solution for the optimal equity weighting could be that derived by Campbell and Viceira (2002), Equation (6.1) in the case where labour income is deterministic rather than stochastic (where Hxis the human capital at age x): Since the optimal investment strategy depends on the member’s RRA, discount factor and EIS, and since these differ across members, it is unlikely that a single default investment fund will be appropriate for all plan members. A life annuity is a critically important component of a well-designed pension plan. As a result of the mortality premium inherent in the return on a life annuity, the full amount of the pension fund should eventually be annuitised in old age (regardless of an individual’s RRA, EIS or personal discount factor).48 This is true despite the well-known aversion to annuitisation by ‘humans’ documented in Friedman and Warshawsky (1990) and Mitchell and Utkus (2004), and despite both their theoretical usefulness (Yaari, 1965 and Davidoff et al., 2005) and money’s worth (Mitchell et al., 1999 and Finkelstein and Poterba, 2002) and their recognised value by annuitants once purchased (Panis (2004)).