Are target date funds dinosaurs? Failure to adapt can lead to extinction
aa r X i v : . [ q -f i n . P M ] M a y Are target date funds dinosaurs?Failure to adapt can lead to extinction.
Peter A. Forsyth ∗ Yuying Li † Kenneth R. Vetzal ‡ May 1, 2017
Abstract
Investors in Target Date Funds are automatically switched from high risk to low risk assetsas their retirements approach. Such funds have become very popular, but our analysis bringsinto question the rationale for them. Based on both a model with parameters fitted to historicalreturns and on bootstrap resampling, we find that adaptive investment strategies significantlyoutperform typical Target Date Fund strategies. This suggests that the vast majority of TargetDate Funds are serving investors poorly.
Conventional defined benefit (DB) plans are becoming a thing of the past. Most organizations donot want to take on the risk of providing a DB plan. More employees are participating in definedcontribution (DC) plans, and the trend is likely to continue.In a typical DC plan, the employee contributes a fraction of his/her salary into a tax-advantagedsavings account. The employer may also contribute to the DC account. In some cases, the employermanages the DC plan, in the sense that the employee picks from a list of approved investmentvehicles, usually bond and stock mutual funds. Upon retirement, the employee has to decide whatto do with the accumulated amount in the portfolio. Typical options include buying an annuityor continuing to manage the portfolio to generate a stream of income. There is, of course, noguarantee of the level of income that will be produced in a DC plan.It would not be unusual for a DC plan member to accumulate for thirty years (possibly withdifferent employers), and then to decumulate for another twenty years. This implies a fifty yearinvestment cycle, making DC plan holders truly long term investors.
Before getting into some technical details, let’s consider two common investment strategies, and wewill examine how a DC investor would have fared during the 30 year period from 1985 − ∗ David R. Cheriton School of Computer Science, University of Waterloo, Waterloo ON, Canada N2L 3G1, [email protected] , +1 519 888 4567 ext. 34415. † David R. Cheriton School of Computer Science, University of Waterloo, Waterloo ON, Canada N2L 3G1, [email protected] , +1 519 888 4567 ext. 7825. ‡ School of Accounting and Finance, University of Waterloo, Waterloo ON, Canada N2L 3G1, [email protected] , +1 519 888 4567 ext. 36518. ime R ea l W ea l t h (t hou sa nd s ) Optimal Adaptive 60% stocks40% bonds (a)
Comparison of optimal adaptive vs.constant proportion. time R ea l W ea l t h (t hou sa nd s ) Optimal AdaptiveLinear Glide Path (b)
Comparison of optimal adaptive vs.linear glide path.
Figure 2.1:
Historical data: real (inflation adjusted) US total stock and short term bond returns,1985:1 to 2015:12. Linear glide path: 80% stocks, 20% bonds in 1985, dropping linearly over time to20% stocks, 80% bonds in 2015. Investor starts with $10 , in dollars, and invests $10 , in dollars each year. Optimal adaptive strategy described in Section 3.3. Yearly rebalancing for allstrategies. will assume that the investor had two possible assets in her DC fund: a short term bond index fundand a market capitalization weighted stock index fund. The investor had $10 ,
000 in 1985 dollarsto start with, and contributed $10 ,
000 (in 1985 dollars) each year to the DC fund.The simplest strategy is based on rebalancing to a constant proportion stock-bond mix. Atypical weight would be 60% stocks and 40% bonds. This rebalancing to a constant mix wasrecommended by Graham (2003) for defensive investors.However, in order to avoid sudden drops in a DC portfolio just before a retirement date, it isoften suggested that investors should use a glide path strategy. In this case, we start off with ahigh allocation to stocks, and then decrease the stock fraction as time goes on. We will consider astrategy where the initial mix in 1985 is 80% stocks and 20% bonds adjusting linearly over time to20% stocks and 80% bonds in 2015.We will compare these strategies with an optimal adaptive strategy. We will describe how wecome up with this strategy in Section 3.3. For now, we’ll just note that this adaptive strategydepends depends only on the total real wealth accumulated so far, and on the time remainingbefore retirement.Figure 2.1(a) compares the adaptive strategy with a constant proportion strategy. We can seethat both strategies ended up with roughly the same total real wealth, but the 60 −
40 strategyhad a very rough ride during the dot-com bubble (2002) and the financial crisis (2008).Alternatively, Figure 2.1(b) compares the adaptive strategy with the linear glide path. Asadvertised, the glide path strategy is much smoother than the 60 −
40 strategy, but it ends up withconsiderably smaller (i.e. about 30% lower) real wealth in 2015 compared to the optimal adaptivestrategy.The optimal adaptive strategy appears to offer some advantages over both the constant propor-tion and the glide path strategies. Perhaps you are intrigued. How does this strategy work? We’llsee in the next Section. 2
Three possible solutions
Many studies have shown that individual investors generally do a poor job of investing. They tendto buy at market peaks, sell at bottoms, and are not well-diversified (see, e.g. Barber and Odean,2013). Target Date Funds (TDFs) (also known as Lifecycle Funds) are an attempt by the investmentindustry to provide a solution for retail clients, specifically those enrolled in a DC plan. The mostbasic TDF has only two possible investments: a bond index and an equity index. Given a specifiedtarget date (which would be the anticipated retirement date of the plan member), we considerhere three possible methods to specify the bond/stock allocation in the DC investment portfolio:deterministic glide path, constant proportion, and an adaptive strategy.
In this case, the allocation of stocks and bonds is determined by a glide path . This is currently apopular method used by many TDFs. Denoting time by t , a simple example of a glide path isFraction invested in equities = p ( t ) = 110 − your age at t . The logic behind this idea is that you should take on more risk when you are young (with manyyears to retirement) and then take on less risk when you are older, with less time to recover frommarket shocks. This seems quite sensible. The investment portfolio is typically rebalanced atquarterly or yearly intervals, so that the equity fraction is reset back to the glide path value p ( t ).This idea is so attractive that TDFs are Qualified Default Investment Alternatives (QDIAs) in theUS. If an employee has enrolled in an employer-managed DC plan, the assets may be placed in aQDIA as a default option, in the absence of any instructions from the employee.It is important to note that the glide path p ( t ) in the age-based example above is only a functionof time t . We call this type of strategy a deterministic glide path, i.e. this strategy does not adaptto market conditions or the investment goals of the DC plan member. A much simpler method is a constant proportion policy, which is also a common asset allocationmethod. In this strategy, we rebalance to a constant equity fraction p const at all rebalancing times.Of course, a constant proportion allocation is a special case of a glide path, where p ( t ) = p const . In deterministic glide paths, rebalancing strategies are only a function of time. Let’s consider astrategy which allows the fraction invested in the risky asset to be a function of both time t andaccumulated wealth in the DC portfolio at t , denoted by W t . Then p = p ( W t , t ), so that this an adaptive strategy.We consider a target-based strategy, where we choose p ( W t ,t ) to minimize E h ( W T − W ∗ ) i , (3.1)where W T denotes terminal wealth at time T , W ∗ is target final wealth, and E [ · ] indicates expected(or mean) value. In other words, we seek the asset allocation strategy which minimizes the expectedquadratic shortfall with respect to the target wealth W ∗ . According to Morningstar, there was over USD 750 billion invested in TDFs in the US at the end of 2015. We assume that the initial wealth W < W ∗ . Dang and Forsyth (2016) show that the optimal strategy has W $10,000Real investment each year $10,000Rebalancing interval 1 year Table 4.1:
Long term investment scenario. After the initial investment, cash is injected and rebal-ancing occurs at t = 1 , . . . , years. To provide a realistic comparison, we consider a plausible investment scenario and evaluate thethree strategies under both (i) a parametric model which captures the broad statistical propertiesof the historical market, and (ii) bootstrap resamples of the historical market.
We consider the prototypical DC investor example shown in Table 4.1. We assume that the investormakes an initial investment in the portfolio of $10,000 at time zero (i.e. W = $10 , T = 30 years, with the last investment of $10,000 (real) beingmade at t = 29 years. We construct two real indexes: a real total return equity index and a real short term bond index.Our data was obtained from the Center for Research in Security Prices (CRSP) through WhartonResearch Data Services. We use the CRSP value-weighted total return index (“vwretd”), whichincludes all distributions for all domestic stocks trading on major US exchanges. We also use the30-day Treasury bill return index from CRSP. Both this index and the equity index are in nominalterms, so we adjust them for inflation by using the US CPI index (also supplied by CRSP). Weuse real indexes since long term retirement saving should be trying to achieve real (not nominal)wealth goals.Figure 4.1(a) shows a histogram of the monthly returns from the real total return equity index,scaled to unit standard deviation and zero mean. We superimpose a standard normal (Gaussian)density onto this histogram. The plot shows that the empirical data has a higher peak and fattertails than a normal distribution, consistent with previous empirical findings for virtually all financialtime series.The fat left tails of the historical density function can be attributed to large downward equityprice movements which are not well modelled assuming normally distributed returns. From a longterm investment perspective, it is advisable to take into account these sudden downward pricemovements. W t ≤ W ∗ , so that W T ≤ W ∗ . This means that only shortfall (not excess) is penalized in (3.1). More specifically, results presented here were calculated based on data from Historical Indexes, c (cid:13) Return scaled to zero mean, unit standard deviation
Probability Density
HistoricalObservationsStandard NormalJump Diffusion (a)
Probability densities. -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
Return scaled to zero mean, unit standard deviation
Probability Density
Jump DiffusionFat Left TailStandard Normal (b)
Zoom of Figure 4.1(a), showing the fat left tail.
Figure 4.1:
Probability density of monthly returns for real CRSP VWD index. Monthly data, 1926:1- 2015:12, scaled to unit standard deviation and zero mean. Standard normal density and fitted jumpdiffusion model also shown.
We fit the data over the entire historical period using a jump diffusion model (Kou and Wang,2004). This provides a more accurate fit to the data, as illustrated in Figure 4.1(a). To show thefat left tail of the jump diffusion model, we have zoomed in on a portion of the fitted distributionsin Figure 4.1(b).
We now compare the constant proportion, (optimal) deterministic glide path and optimal adaptivestrategies under the estimated jump diffusion model with parameters estimated from the entire1926:1 - 2015:12 data set (subsequently referred to as the synthetic market ). Note that the jumpdiffusion model is applied only to the equity returns. Bond returns are simply determined fromthe sample average monthly change in the real bond index. We will use this synthetic market todetermine optimal strategies and carry out Monte Carlo simulations.In the following comparison, we consider a 60 −
40 equity-bond split as the constant proportionstrategy. In other words, at each annual rebalancing date we rebalance so that 60% of the portfoliois invested in equities and 40% in bonds. This is a special case of a glide path strategy, with p ( t ) = .
60 for all times t . The equity fractions at rebalancing times for the other two strategies aredetermined as follows: • Deterministic glide path strategy: we calculate the equity fraction at each rebalancing datesuch that the standard deviation of terminal wealth std [ W T ] is as small as possible underthe restriction that the expected value of terminal wealth E [ W T ] matches the correspondingexpected value for the constant proportion strategy. • Adaptive strategy: we calculate the equity fraction at each rebalancing date such that themean quadratic target error E h ( W T − W ∗ ) i is as small as possible with the target W ∗ set sothat the expected value of terminal wealth E [ W T ] is the same as for the constant proportionstrategy.We determine the optimal rebalancing fractions by using a computational optimization method(glide path) and solving a Hamilton Jacobi Bellman equation (adaptive strategy). In each case, weconstrained the equity fraction p so that 0 ≤ p ≤ ime (years) E qu i t y f r ac t i on ( p ) Optimal Deterministic Glide Path (a)
Optimal deterministic glide path.
Time (years) p = fraction in risky assetMean of pStandard deviation of p (b)
Optimal adaptive path.
Figure 4.2:
Optimal deterministic and adaptive paths. For the adaptive case, we show the expectedvalue of the fraction invested in the risky asset ( p ) and its standard deviation. The adaptive case isshown based on Monte Carlo simulations. fraction invested in equities for the adaptive strategy, and its standard deviation. On average, theadaptive strategy maintains a high allocation to stocks for longer than the deterministic strategy,but de-risks faster as we approach retirement. The constraint p ≤ p = . .
97 of the constant proportion strategy, avery small improvement. The standard deviations of the glide path and constant proportion aremore than twice as large as that of the adaptive strategy. Table 4.2 also shows some informationabout shortfall probabilities. For example, the probability of achieving a final real wealth less than$650,000 is about 43% for both the constant proportion the and optimal deterministic strategies.For the adaptive strategy, the probability of achieving a final wealth less than $650,000 is only 22%.Based on these synthetic market results for this case with regular contributions, it is possibleto come up with a deterministic glide path which beats a constant proportion strategy, but not byvery much. The adaptive strategy, on the other hand, significantly outperforms both the optimaldeterministic glide path and the constant proportion strategies. We have repeated these tests withmany different synthetic market parameters. As long as the investment horizon is longer than 20years, the differences in performance between the optimal deterministic glide path and the constantproportion strategy with equivalent terminal wealth are very small, and each of these are clearlydominated by the adaptive strategy.
The synthetic market results are based on fitting the historical returns to a jump diffusion model.This model assumes that monthly equity returns are statistically independent, which is debatable.In order to get around artifacts introduced by our modelling assumptions, we test the three strate-gies using a bootstrap resampling method. This can be viewed as a more realistic test, in the sense Since the adaptive strategy depends on the accumulated wealth so far, it is non-deterministic. As a result, weplot both the mean and the standard deviation of the equity fraction. robability of ShortfallStrategy E [ W T ] std [ W T ] W T < $500 , W T < $650 , W T < $800 , Table 4.2:
Scenario data in Table 4.1. W T is accumulated real portfolio wealth at T = 30 years.Synthetic market results for a constant proportion strategy with p = . , an optimal deterministicglide path strategy, and an optimal adaptive strategy. Probability of ShortfallStrategy E [ W T ] std [ W T ] W T < $500 , W T < $650 , W T < $800 , Table 4.3:
Example in Table 4.1. W T is accumulated real portfolio wealth at T = 30 years. Bootstrapresampling results based on historical data from Jan. 1926 to Dec. 2015 for a constant proportionstrategy with p = . , an optimal deterministic glide path strategy, and an optimal adaptive strategy.10,000 bootstrap resamples were used, with a blocksize of 2 years. that we can observe how the strategies would have performed on actual historical data.Our investment horizon is T = 30 years. Each bootstrap path is determined by dividing T into k blocks of size b years, so that T = kb . We then select k blocks at random (with replacement)from the historical data set. Each block starts at a random month. We then concatenate theseblocks to form a single path. We repeat this procedure 10 ,
000 times and generate statistics basedon this resampling method.The idea here is that if the real data shows some serial dependence, then this will show up inthe bootstrap resampling. Based on some econometric criteria, we use a blocksize of 2 . . − . .5 More on the optimal adaptive strategy The target W ∗ for the adaptive strategy was selected so that expected terminal wealth in thesynthetic market matched that achieved by the constant proportion strategy. In practice, howshould we pick W ∗ ? A reasonable approach is to enforce the constraint E [ W T ] = W goal = investment goal . (4.1)The investment goal in the case of retirement saving would be the amount required to fund areasonable replacement level of income for a retiree. Of course in general, W goal < W ∗ , i.e. in orderto have W goal wealth on average, we have to aim at a higher target.It is also interesting to note that the adaptive strategy turns out to be dynamically meanvariance optimal (Li and Ng, 2000; Dang and Forsyth, 2016). This means that for a specified meanterminal wealth E [ W T ], no other strategy has smaller variance. In addition, the adaptive strategyoffers the opportunity in some cases to withdraw cash without compromising the probability ofreaching the target (Dang and Forsyth, 2016; Forsyth and Vetzal, 2016). Of course, we have looked at only one possible adaptive strategy. There are many other possibil-ities. We argue that the adaptive strategy we have considered is especially appealing because itsimultaneously minimizes two measures of risk: quadratic shortfall and variance (standard devia-tion).However, our main point here is that restricting attention to deterministic glide paths is sub-optimal. Investors can do a lot better by considering adaptive strategies. It is worthwhile to notethat the vast majority of target date funds use deterministic strategies. Do target date funds needto adapt? The answer is clear: you need to adapt to your target.
References
Barber, B. M. and T. Odean (2013). The behavior of individual investors. In G. Constantinides,M. Harris, and R. Stulz (Eds.),
Handbook of Economics and Finance , Chapter 22, pp. 1533–1569.Elsevier.Dang, D.-M. and P. Forsyth (2016). Better than pre-commitment mean-variance portfolio allocationstrategies: a semi-self-financing Hamilton-Jacobi-Bellman equation approach.
European Journalof Operational Research 250 , 827–841.Forsyth, P. and K. Vetzal (2016). Robust asset allocation for long-term target-based investing.submitted to the International Journal of Theoretical and Applied Finance, working paper, Uni-versity of Waterloo.Graham, B. (2003).
The Intelligent Investor . New York: HarperCollins. Revised edition, forwardby J. Zweig.Kou, S. and H. Wang (2004). Option pricing under a double exponential jump diffusion model.
Management Science 50 , 1178–1192.Li, D. and W.-L. Ng (2000). Optimal dynamic portfolio selection: Multiperiod mean-varianceformulation.