Example: Vanguard Target Retirement 2045 Fund

We apply the tools developed so far to study asset allocation strategies of Vanguard target retirement fund, currently the largest lifecycle fund in AUM. In particular, we focus on the 2045 fund (VTIVX), which is designed for investors planning to leave the workforce in or within a few years of 2045. It would be interesting to find the forward performance process that best explains the fund’s strategy. Figure 3.7 displays the glide path adopted by the fund. The chart shows that the fund invests in five major asset classes. To simplify things, we consider the strategy as investing in only a stock with log-normal dynamics and a bond with zero interest rate. The proportion allocated to stocks, ˜π(t), as distinguished by the grey and blue area, starts at 90%, then gradually declines following a piecewise linear function, and eventually settles at 30% after year 55. Since ˜π(t) is deterministic, under log-normal market assumption, it is straightforward to calculate the expected total returns,

Ret(T ) = eµ

RT 0 π(t)dt˜ .

We apply the preference calibration tool to find the measure νx that

produces expected returns that best matches the returns implied by the Van- guard deterministic strategy. To increase accuracy, we sample the returns every quarter (∆T = 0.25) for 70 years, which generates a sequence of 281 returns, Ri = Ret(i∆T ), i = 0, 1, . . . , 280. We then solve the optimization

problem (3.10) for n0 = 1, 2, 3, 4. The outputs are reported in table 3.3.

Table 3.3: Estimation results for νx

n0 = 1 n0 = 2 n0 = 3 n0 = 4 y1 0.34 0.16 0.16 0.00 y2 - 0.00 0.00 0.16 y3 - - −2.88 −1.16 y4 - - - 0.00 p1 1.00 7.38 7.38 0.00 p2 - −6.38 −6.38 7.38 p3 - - 0.00 0.00 p4 - - - −6.38 L 4.00 × 103 _{371.49 317.49 317.49}

Surprisingly, after the large drop in L, the penalty function we try to minimize, when n0increases from 1 to 2, there seems to be no further improve-

ment by using a larger n0, which suggests that a two point measure might be

the best solution. Further confirming this are the distribution structures gen- erated under n0 = 2, 3, 4. All three of them show that νx is supported at 0.16

and 0 only, with weights 7.38 and −6.38. Therefore, we can firmly conclude the forward performance that best describes the allocation strategy of Vanguard

2045 retirement fund is generated by, the measure

νx = 7.38δ{0.16}− 6.38δ{0}. (3.11)

The forward optimal strategy derived from the above measure is actually quite simple. The investor with initial wealth $1 would borrow $6.38 from the bank, then invest the entire $7.38 with a CRRA manager with relative risk tolerance 0.16. Although, the forward optimal strategy is stochastic, hence different from the deterministic glide path of Vanguard fund, the expected returns they produce are reasonably close (see panel A of figure 3.8).

0 10 20 30 40 50 60 70 Horizon (years) 0 5 10 15 20 25 30 35 40 Expected Return Panel A VTIVX Forward Approximation 0 10 20 30 40 50 60 70 time (years) 0 0.5 1 1.5 2 2.5 Stock Proportion Panel B

VTIVX Glide path Forward Implied Determinstic Strategy

Figure 3.8: A: expected returns. B: stock proportion

A second, more illustrative approach is to compare the strategies them- selves. Since ˜π for the forward performance process is stochastic, we have to introduce some kind of averaging before comparing to the deterministic glide path. Here we introduce the forward implied deterministic strategy ˜πfImp_(t)

0 2 4 6 8 10 Wealth after a decade

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cumulative Probability T=10 VTIVX Forward 0 50 100 150 200 Wealth at Retirement 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cumulative Probability T=55 VTIVX Forward

Figure 3.9: cdf for Xv_{(T ) and X}f_{(T ).}

as the forward optimal strategy, i.e.

˜ πfImp(t) = 1 µ ∂ ∂t ln(E[ X∗(t) x ])
. where X∗(t) is the optimal wealth process derived from νx.

Panel B of figure 3.8 shows that ˜πfImp_{(t) is a much steeper “glide path”. With}

an initial leverage higher than 200%, it rapidly declines and comes down to the same level as the VTIVX glide path around year 10, then it slowly converges to the post-retirement level, tracking the fund glide path more closely. Judging from the high leverage in the first decade, one might suspect that the dynamic forward strategy is too risky. To find out if this is case, we need to compute the risk-return profiles for both strategies. Assume for simplicity that the ini- tial wealth is one dollar. Let Xv(T ) and Xf(T ) denote the wealth processes by following the Vanguard glide path and the forward optimal strategy. We plot the cumulative distribution functions for both wealth variables at horizons T = 10 and T = 55. As shown in figure 3.9, at year 10 the forward strategy is indeed riskier since it poses a greater downside risk, as there is a 6% chance

of losing 50% or more, while for the Vanguard glide path the probability of such loss is only slightly above 1%. However, apart from the worst cases, the forward strategy does dominate in most of the other scenarios. For example, it offers a 48% chance of at least tripling the initial wealth, while under the glide path, the probability goes down to 17%. In fact, under the notion of “almost stochastic dominance”, introduced by Leshno and Levy (2002), the forward strategy has “almost first order stochastic dominance” over the Van- guard glide path, with violation parameter  = 0.031 ( ≤ 0.059 is commonly considered as acceptable).

Surprisingly, the situation reverses at the time of retirement (T = 55). The forward strategy is actually more conservative in that it offers a higher con- siderably probability of getting a decent return while forgoes some chances of exceptional returns. This can also be observed from the summary statistics in table 3.4. While the mean returns are about the same, standard deviation for the forward strategy is much lower. Among the recorded quantiles, the Vanguard glide path only outperforms at the highest decile.

Table 3.4: Retirement wealth summary statistics

D1 Q1 Median Q3 D10 Mean Stdev forward strategy 8.1 12.7 19.6 29.1 40.5 22.5 13.9 Vanguard glide path 3.2 6.1 12.8 27 52.8 23.6 24.3