سرمایه گذاری هدف محور: استراتژی های سرمایه گذاری بهینه در تعریف طرح های سهم بازنشستگی تحت عامل زیان گریزی

Assuming the loss aversion framework of Tversky and Kahneman (1992), stochastic investment and labour income processes, and a path-dependent fund target, we show that the optimal investment strategy for defined contribution pension plan members is a target-driven ‘threshold’ strategy, whereby the equity allocation is increased if the accumulating fund is below target and is decreased if it is above. However, if the fund is sufficiently above target, the optimal investment strategy switches to ‘portfolio insurance’. We show that the risk of failing to attain the target replacement ratio is significantly lower with target-driven strategies than with those associated with the maximisation of expected utility.

The purpose of this paper is to determine the optimal dynamic investment strategies for defined contribution (DC) pension plans when plan members experience loss aversion. The concept of ‘loss aversion’ was first proposed by Kahneman and Tversky (1979) within the framework of prospect theory (PT), the foundation stone of behavioural finance. The recent literature on behavioural finance has provided powerful evidence that the standard optimisation paradigm, expected utility maximisation within a framework of risk-averse economic agents, does not correspond well with how economic agents actually behave in real world risk situations.1 Real world investors are prone, among other things, to overconfidence in their investment abilities, regret and, especially, loss aversion. They also tend to monitor the performance of their portfolios (particularly their long-term portfolios) ‘too frequently’. As a result, they tend to become risk averse when winning and sell winning investments too quickly, and avoid cutting losses and even take extra risks when they have made losses.2 Loss aversion (LA) is defined in terms of gains and losses in wealth relative to a pre-defined reference or endowment point, rather than in terms of changes in the absolute level of total wealth, as with expected utility theory (EUT). Rabin and Thaler (2001) have argued that EUT is manifestly not a suitable explanation for most observed risk attitudes: ‘we have also often been surprised by economists’ reluctance to acknowledge the descriptive inadequacies of [the] theory’. They suggest that LA and the tendency to isolate each risky choice and analyse it separately should replace EUT as the foremost descriptive theory of risk attitudes. Given the behavioural traits exhibited by many investors, it is important to investigate the consequences of using a PT utility function to determine the optimal investment strategy in a DC plan and to compare the results with those implied by the traditional expected utility model. In a DC plan, members contribute part of their income each year to building a pension fund for retirement.3 The accumulated fund is then used to buy a life annuity to provide a pension income after retirement. Members are assumed to have a target replacement ratio4 at retirement age 65. This translates into a target pension fund at retirement which will depend, in part, on their longevity prospects during retirement. Members are assumed to be loss averse with respect to the target retirement pension fund and to a series of annual interim target fund levels prior to retirement. The interim targets reflect the discounted value of the final target retirement fund level and, for convenience, we will treat the interim targets as being age-related.5 Members are also assumed to have an investment strategy (i.e., make asset allocation decisions) which aims to maximise the expected total discounted value of PT utility over the period until retirement. To do this, we use a two-asset, dynamic-programming-based numerical solution method with stochastic labour income and both borrowing and short selling constraints. Within this proposed framework, it will be shown that the optimal dynamic asset allocation strategy is consistent with a target-driven strategy known as ‘threshold’, ‘funded status’ or ‘return banking’, an example of which was discussed in Blake et al. (2001). With this strategy, the weight in risky assets such as equities is increased if the accumulating fund is below the relevant interim target and is decreased if the fund is above target. This is because, under loss aversion, the member is risk seeking in the domain of losses and risk averse in the domain of gains. Close to each target (whether above or below), the plan member has the lowest equity weighting (for that target) in order to minimise the risk of a significant loss relative to the target. However, if the fund is sufficiently above the target, there is a discrete change in the investment strategy and the equity weighting is increased (subject to the member's degree of risk aversion in the domain of gains), since the risk of the fund falling below the target is now considered to be acceptably low. This strategy of increasing the equity weight as the fund value continues to rise above the target is consistent with the investment strategy known as ‘portfolio insurance’ and its role in portfolio choice under loss aversion has been noted by other researchers (e.g., Berkelaar et al., 2004 and Gomes, 2005). If the target-driven strategy is successful in the sense that the series of interim targets has been met, the overall equity weight will tend to fall with age, since the fund is in line to meet the final target fund level at retirement. Although this is similar to what happens in conventional (deterministic) ‘lifestyle’ strategies,6 the target-driven strategy considered here is very different. In particular, whilst conventional lifestyle strategies typically involve switching mechanically from 100% equities only in the last 5–10 years before retirement and often end up holding 100% of the fund in bond-type assets at retirement, the optimal strategy under loss aversion involves a much more gradual reduction in the equity holding if the fund remains close to the sequence of targets. If, however, the fund is either well below or well above a particular target, even one near to the retirement date, the optimal equity holding will be high for reasons given in the previous paragraph. We also show that under loss aversion, the risk of failing to attain the desired replacement ratio at retirement is significantly lower with target-driven strategies than those arising out of a traditional risk aversion framework aimed at maximising a power utility function on retirement. where F is the actual value of the pension fund when the plan member is a given age, f is the pre-defined target value of the pension fund at the same age, v1 and v2 are the curvature parameters for gains and losses, respectively, and λ is the loss aversion ratio. As shown in Fig. 1, the two key properties of the PT utility function are: the PT utility function is ‘S’-shaped (i.e., convex below the reference point and concave above it) when 0<v1<1 and 0<v2<1, implying that individuals are risk seeking in the domain of losses and risk averse in the domain of gains (this contrasts with the concave shape in standard utility functions, where individuals are assumed to be risk averse for all levels of wealth and have diminishing marginal utility of wealth); and the PT utility function is steeper below the reference point than above when λ>1, implying that individuals are λ times more sensitive to a unit loss than to a corresponding unit gain. Conventional lifestyle investment strategies are currently widely used by many such pension plans as the default investment option. However, as will be shown below, there can be substantial uncertainty over the size of the fund at retirement when a lifestyle investment strategy is used and this makes it difficult for the plan member to be confident about the level of retirement income.7 Thus, for DC plan members seeking greater certainty in their retirement planning, the plan's investment strategy needs to be far more focused on achieving the target pension. We will assume that plan members evaluate the plan's investment performance on an annual basis. Members have a final target replacement ratio at retirement and a series of corresponding interim targets before retirement. They are assumed to be ‘loss averse’ with respect to these targets (which define the reference points in the PT framework outlined above) and to make asset allocation decisions to maximise the total discounted value of the PT utility over the period until retirement. They are also, as mentioned, assumed to be ‘risk-averse’ above the target (since v1<1) and ‘risk-seeking’ below the target (since v2<1). Having target fund levels when formulating the optimal investment strategy for a DC pension plan is not a new idea. Vigna and Haberman (2001) and Haberman and Vigna (2002) derive a dynamic-programming-based formula for the optimal investment allocation in DC plans. In their model, members are assumed to face a quadratic cost (or disutility) function each year based on actual and targeted fund levels and to make investment decisions that minimise the cost of deviations of the fund from these corresponding targets. Their analysis suggests that a lifestyle investment strategy remains optimal for a risk-averse member and that the age at which the member begins to switch from equities to bonds depends on both the member's risk aversion and age when the plan started: the more risk averse the member or the longer the accumulation period prior to retirement, the earlier the switch to bonds. However, as the authors acknowledge, one obvious limitation of this approach is that the quadratic cost function penalises equally both under- and over-performance relative to the specified targets. In summary, our proposed model differs from the existing literature in three significant respects: •Loss aversion: Most existing studies (e.g., Haberman and Vigna, 2002 and Gerrard et al., 2004) assume that the individual plan member has a quadratic cost function with respect to deviations in the actual fund from the appropriate targets. However, given the behavioural traits exhibited by many investors, we believe it is more appropriate to reflect this by using a PT function to determine the optimal asset allocation model in a DC pension plan. •Stochastic labour income: Previous studies (e.g., Vigna and Haberman, 2001) used a simple deterministic model, whereas our study uses a more realistically-calibrated stochastic model for labour income. •Choice of investment targets: The choice of an appropriate investment target is crucial in a target-driven model. Previous studies oversimplified the problem by assuming fixed targets (derived by assuming a fixed investment return over time). In our model, the final fund target and, hence, the series of corresponding interim targets are path-dependent (and, thus, vary over time in accordance with the evolution of the member's income). The rest of the paper is organised as follows. Section 2 formulates the target-driven asset allocation problem for a DC pension plan under loss aversion and outlines our model, including the optimisation method used. Section 3 calibrates the model's parameters. Section 4 presents the output from the optimisation exercise and conducts a sensitivity analysis. Conclusions are presented in Section 5.

We have applied some of the most powerful lessons of behavioural finance to solve the optimal investment strategy in a defined contribution pension plan. The key lesson is that loss aversion with respect to a series of reference wealth levels in the form of increasing target fund values over the life of the plan might well provide a better representation of a typical plan member's attitude to risk taking than risk aversion in respect of the terminal fund value at retirement, the most commonly used approach from expected utility theory. The most popular investment strategy by practitioners, conventional lifestyling, does not even take the plan members’ attitude to risk or loss into account. Nor does it take into account the stochastic nature of asset returns and the plan member's labour income.22 Loss aversion (as originally outlined in Tversky and Kahneman (1992)) leads to a new type of target-driven approach to deriving the dynamic optimal asset allocation. The key findings from using this approach are as follows: •The optimal investment strategy under loss aversion is consistent with a ‘threshold’ strategy. With this strategy, the weight in equities is increased if the accumulating fund is below the interim target (since plan members are risk seeking in the domain of losses) and is decreased if the fund is above target (since plan members are risk averse in the domain of gains), unless the fund is very much above target, in which case plan members are comfortable with assuming more equity risk again and accordingly adopt a ‘portfolio insurance’ investment strategy. However, as the retirement date approaches and assuming the fund is on target, the overall equity weight begins to fall and the value of the fund is ‘banked’ by switching to lower risk investments, such as bonds. The strategy is highly focused on achieving a target replacement ratio at retirement. •The switch to a more conservative asset allocation strategy is implemented at lower current fund values (relative to target) and at a lower age, the higher is the loss aversion ratio, λ. Although the mean replacement ratio falls as a consequence, the expected shortfall from the target decreases. • The effect of higher risk aversion in the domain of gains (measured by a lower v1 in the PT utility function) leads, unsurprisingly, to an earlier switch out of equities and a lower mean replacement ratio, but also to a lower expected shortfall. The effect of greater risk seeking in the domain of losses (measured by a lower v2 in the PT utility function) leads to a later switch away from equities, a higher mean replacement ratio, a higher probability of achieving the target but also a higher expected shortfall. •Compared with the standard framework of risk aversion, loss-averse plan members are committed to achieving interim and final target fund levels and, accordingly, adopt a more conservative asset allocation strategy. Although this leads to a lower mean replacement ratio at retirement, there is a greater likelihood of achieving the desired target replacement ratio and a lower expected shortfall. • Compared with a conventional lifestyle investment approach, the optimal target-driven investment strategy significantly increases the likelihood of achieving the chosen target, thereby providing a much greater degree of certainty in retirement planning. The risks inherent in the conventional lifestyle strategy appear to be much higher than generally understood. Thus, for DC plan members who seek greater certainty of outcome in retirement planning, the investment strategy adopted over time needs to be far more focused on achieving the specified target replacement ratio. The target-driven investment strategy arising from within a framework of loss aversion is an example of a highly focused investment strategy. However, the framework is not easy to implement since it requires the solution of a nonlinear dynamic programming problem whenever there is new information about the key state variables. Nevertheless, in practice, it should be possible to design an approximately optimal investment strategy, given member profile characteristics (such as age and occupation) and values of the key state variables (e.g., interim fund level and current labour income). Financial advisers would then be able to advise on the appropriate investment strategy for the coming year depending on how far the actual fund level was away from the interim target fund level.