**1. Introduction**

Defined benefit (DB) pension plans can be high risk liabilities for both private and public sector organizations. To de-risk balance sheets, many employers have been gradually converting to defined contribution (DC) plans. This trend has been particularly strong in Australia and the US: as of 2017, 87% of pension assets in Australia were in DC plans, while the corresponding figure for the US was 60%. In contrast, in countries such as the UK, Canada, and Japan, pension plan assets were still predominantly in DB plans.<sup>1</sup> Of course, considering just the percentage of assets in DC plans at a point in time can hide the underlying trend. For example, about 95% of Canadian pension assets were in DB plans as of 2017. However, Gougeon (2009) points out that the number of DC plan participants in Canada almost doubled between 1991 and 2006, while the number of DB plan participants shrank by about 4% over the same period.

<sup>1</sup> Statistics from the Thinking Ahead Institute's "Global Pension Assets Study 2018". Available online: www. thinkingaheadinstitute.org/-/media/Pdf/TAI/Research-Ideas/GPAS-2018.pdf.

Under a DC plan, the employee and employer contribute to a retirement savings account, which is usually tax-advantaged. In some cases, the employee is able to select from a list of investments (e.g., stock and bond index funds) which have been approved by the employer. However, in terms of the important decisions about asset allocation, the employee is typically left up to her own devices.

Although DC plans are desirable from the employers' point of view, the retirement savings risk has simply been transferred to those who are perhaps least able to manage it. Many studies have confirmed that the average retail investor is very poor at managing investment portfolios (e.g., Barber and Odean (2013)). A typical DC plan investor will accumulate savings for thirty or more years, followed by a decumulation phase of perhaps twenty years. As a result, inefficiencies in asset allocation, even if small in annual terms, can have very large cumulative effects over a typical long-term retirement savings investment horizon.

Target date funds (TDFs) have become a popular means of providing an *autopilot* asset allocation strategy. The prototypical TDF uses a high allocation to equities during the early years of DC savings. The stock allocation is decreased, and the portfolio becomes more weighted towards bonds, as the retirement date nears. The transition from high to low equity allocations is pre-determined by means of a *glide path* strategy. In the US, TDFs are *Qualified Default Investment Alternatives*, implying that the assets of employees enrolled in an employer-managed DC plan can be invested in a TDF by default.<sup>2</sup> Holt (2018) indicates that total assets in US TDFs at the end of September 2018 were almost \$1.19 trillion, and notes that "given their role as the default investment in many defined contribution retirement plans, investors can expect to see assets in target-date funds continue to rise over the long run".<sup>3</sup> This reflects the tendency of many DC plan participants to stick with the default option that is presented to them. This has been noted in the US context by Madrian and Shea (2001) and Choi et al. (2002) amongs<sup>t</sup> several others, and by other authors in various other countries as well.<sup>4</sup> An implication of investors' choosing default options is that default asset allocations will have a strong effect on the accumulation of savings over time (Dobrescu et al., 2018). Choi et al. (2002) note that

Sophisticated employers should choose their plan defaults carefully, since these defaults will strongly influence the retirement preparation of their employees. Policymakers should also recognize the role of defaults, since policymakers can facilitate, with laws and regulations, the socially optimal use of defaults (Choi et al. 2002, p. 104).

However, several studies have reported that TDFs do not appear to outperform simpler constant proportion strategies (Arnott et al., 2013; Basu et al., 2011; Esch and Michaud, 2014; Estrada, 2014; Forsyth and Vetzal, 2019). Clearly, constant proportion strategies which have a significant allocation to equities could result in a wide range of terminal wealth at the end of the accumulation period. The fact that TDFs are not superior to such strategies indicates that TDFs are not a panacea for DC plan investors.

Our objective in this article is to describe realistic outcomes for a typical DC plan investor who contributes a fixed (real) amount per year to an account containing two possible assets, a low risk bond fund and an equity index fund, over a lengthy (30 year) accumulation period, and who follows one of three different asset allocation strategies. The first two of these strategies are often available as default options in US DC plans, these being a constant proportion strategy and a linear glide path, which approximately mimics a prototypical TDF. These strategies are both deterministic since the asset allocation between the bond and equity funds is pre-specified and does not change as a result of realized investment returns. The third strategy is an optimal dynamic (multi-period) quadratic

<sup>2</sup> In other words, the employee must explicitly decide to opt out of a TDF if she desires a different asset allocation strategy.

<sup>3</sup> Although our focus in this article is on the US setting, we note that TDFs are now being marketed in various parts of Europe, in part due to regulatory developments (Pielichata, 2018). Major US vendors such as Vanguard and Fidelity have launched TDFs in Canada in recent years. In addition, some life-cycle products such as Time Pension which has been popular in Denmark are similar in some respects to TDFs.

<sup>4</sup> See, for example, Hedesström et al. (2004) (Sweden), O'Connell (2009) (New Zealand), and Dobrescu et al. (2018) (Australia).

shortfall (QS) strategy. While such a strategy is not (to our knowledge) currently available for DC plan investors, it is an interesting extension to consider since it is adaptive, i.e., the asset allocation is a function of the prevailing state of the investment portfolio and the time remaining until anticipated retirement. Our investigation of this strategy allows us to draw conclusions about the extent to which adaptive strategies could potentially improve savings outcomes compared to the types of strategies that are currently popular default options. Unlike the constant proportion and glide path strategies, the QS strategy relies on a parametric model for the equity index. Our implementation models the real (inflation-adjusted) stock index as following a double exponential jump diffusion model (Kou and Wang, 2004), which incorporates the risk of sudden market crashes.<sup>5</sup> The parameters of the jump diffusion model are estimated by fitting to nine decades of US market data.

The QS strategy is computed by numerically solving a Hamilton–Jacobi–Bellman (HJB) equation (Dang and Forsyth, 2014). It is interesting to note that the QS strategy, which is naturally time consistent, is equivalent to the pre-commitment multi-period mean-variance (MV) strategy, in the sense that both objective functions lead to the same optimal controls. This can be proven based on the embedding technique (Li and Ng, 2000; Zhou and Li, 2000). Our numerical approach allows us to impose realistic no-leverage constraints. This avoids the impractical situation of a highly leveraged portfolio, which often results for unconstrained pre-commitment MV optimal strategies (Lioui and Poncet, 2016) 6.

As a benchmark case, we consider a constant proportion strategy involving the two assets listed above: a risk-free bond index and a risky stock index. The portfolio is annually rebalanced. The weights on the two assets are chosen to achieve a target expected level of real wealth at the end of the savings period. We then compare this strategy with a glide path and an optimal QS strategy, both of which are constrained to have the same expected value of final wealth as the constant proportion strategy. The glide path and constant proportion asset allocation strategies can be considered to be typical of many DC plan holders. The optimal QS strategy is effectively a *best case* scenario, in terms of reducing the probability of shortfall (over a wide range of the terminal wealth distribution) compared to glide path or constant proportion strategies (Forsyth and Vetzal, 2017b); Forsyth and Vetzal (2019).

We compare these strategies in a *synthetic* market, in which the equity price process is assumed to follow a jump diffusion with constant parameters (i.e., the average historical parameters). We also carry out tests using bootstrap resampling with actual historical data (Cogneau and Zakalmouline, 2013; Dichtl et al., 2016).

We make several assumptions which allow us to convert a desired final salary replacement ratio into a target real portfolio value after a 30-year contribution period. Our base case example assumes a target real final salary replacement ratio of 50%, a combined annual employee-employer contribution of 20% of real salary, and investments in short-term risk-free governmen<sup>t</sup> bonds and a value-weighted equity index.

Our two main findings are as follows:


<sup>5</sup> An obvious extension would be a GARCH/stochastic volatility model. However, Ma and Forsyth (2016) document that mean-reverting stochastic volatility effects for Heston-type stochastic volatility models are negligible for long-term investors. Since multivariate GARCH/stochastic volatility models are typically mean-reverting, this suggests that stochastic volatility may be unimportant for long-term investors under these models as well, but this has not been proven.

<sup>6</sup> Recall the time consistent QS strategy has the same controls as the pre-commitment MV strategy.

but this may not work in practice due to higher costs associated with equal-weighted indexes, which are not recognized in our model.

Our results sugges<sup>t</sup> that most DC plan holders are unlikely to be able to maintain their desired standard of living upon retirement. In our opinion, the true risks of DC plans have not been communicated in realistic terms to plan participants. It seems that this situation can be improved only by simultaneously lowering expectations, increasing savings rates, and improving asset allocation strategies. We emphasize that while, in principle, an optimal QS strategy could be offered as an option to DC plan members by a financial intermediary, and this would give those who selected it improved odds of achieving sufficient levels of retirement savings to maintain their desired living standards in retirement, by itself this is unlikely to be enough. It will also be necessary to raise savings rates and reduce expectations about living standards during retirement.

A possible criticism of our findings is based on the fact that our computations use the past 90 years of US market data. Some would argue that real market returns going forward will be lower than historically observed values. However, this would only lead to worse results, so it could be argued that our conclusions should be even more pessimistic.
