*5.5. Equal-Weighted Equity Index*

As shown above in Figure 2, an equal-weighted equity index has historically outperformed the value-weighted equity index that has been used for all results to this point. We now consider the impact of replacing the value-weighted index by its equal-weighted counterpart. Note that the bond index used here is the 3-month T-bill index. Table 9 provides the results. Whether in the idealized synthetic market or the backtest historical market, the QS optimal strategy clearly outperforms the constant proportion and glide path alternatives by the criteria given in the table. In the synthetic market, the QS optimal strategy achieves the same *E*[*WT*] with dramatically lower standard deviation of *WT* and shortfall probability, along with the possibility of a modest amount of surplus cash. The same general conclusions apply in the historical market, although it is worth noting that the average real terminal wealth for the other strategies is somewhat lower for all expected blocksizes considered. Comparing the results in Table 9 for the equal-weighted equity index with those reported above in Table 3 for the value-weighted index, it can be seen that the shortfall probabilities are now considerably lower for the QS optimal strategy, but almost unchanged for the other strategies. The standard deviation of *WT*, however, is substantially lower for all of the strategies.



Figure 8 plots the cumulative distributions of real terminal wealth in both the synthetic and the historical markets. In both cases, the distributions for the glide path and constant proportion strategies are virtually indistinguishable. Figure 8a indicates that the QS optimal strategy outperforms over a wide range of terminal wealth values in the synthetic market, although it does perform worse in the tails of the distribution. As mentioned earlier, this is due to two features of the strategy: (i) it automatically de-risks once the quadratic wealth target is achievable by investing only in the bond index (so it does not take advantage of continued strong equity market performance afterwards on paths where that happens); and (ii) it continually tries to reach the quadratic wealth target by using maximum equity market exposure (and this gamble for resurrection fails on paths where the equity market has persistently poor performance). The same comments apply to the historical market shown in Figure 8b, but it is worth noting that the underperformance of the QS optimal strategy in the tails is considerably reduced here compared to the synthetic market.

### *5.6. Effect of Replacement Ratio*

All of the results provided to here assume a replacement ratio *R* = 50% of final real salary, in accordance with Table 2. We now explore the effects of lowering this to 40% and increasing it to 60%. For each case, we determine the desired expected real wealth target by using Equation (11). Table 10 shows the results. Decreasing *R* to 40% reduces the expected wealth target *Wd* to \$732,000 from \$915,000, while raising *R* to 60% increases *Wd* to \$1,098,000. The remaining wealth values in the table are normalized by *Wd*. Whether we consider the synthetic or the historical market, it is clear that increasing *R* requires taking on more risk, as measured by either the standard deviation or the shortfall probabilities. This is borne out in the cumulative distribution plots of normalized terminal real wealth provided in Figure 9. The synthetic market results in Figure 9a clearly indicate that the QS optimal strategy performs better for the lowest value of *R*. Recall that the strategy attempts to come as close as possible to *Wd* (normalized to 1 for this plot), and the cumulative distribution when *R* = 40% shows relatively low probability of normalized real wealth being much above or below 1. For the highest replacement ratio ( *R* = 60%), there is a substantial chance of being either significantly below or above 1. This is because the strategy must take on more risk in order to attain the higher expected wealth target. Of course, the base case with *R* = 50% lies in between these other two cases. The results for the historical market shown in Figure 9b are generally similar, though the differences across the range of values of *R* are somewhat less pronounced.

**Figure 8.** Cumulative distribution of real terminal wealth when the equal-weighted equity index is used. Wealth units: thousands of dollars. Input data provided in Tables 1 and 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12 with expected blocksize ˆ *b* = 2 years; surplus cash flow included for the QS optimal strategy. (**a**) synthetic market; (**b**) historical market.

**Table 10.** QS optimal results with varying salary replacement ratios *R*. Units for *Wd*: thousands of dollars. Remaining wealth values are normalized by *Wd* for each case. Input data provided in Tables 1 and 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12.


**Figure 9.** Cumulative distributions of real terminal wealth with different salary replacement ratios *R* for the QS optimal strategy. Input data provided in Tables 1 and 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using historical data from 1926:1 to 2015:12 with expected blocksize ˆ *b* = 2 years; surplus cash included. (**a**) synthetic market; **b**) historical market.

### *5.7. Summary Regarding Alternative Assumptions*

Sections 5.1–5.6 above provided detailed results concerning the effects of


Table 11 summarizes the results of various strategies in terms of probability of shortfall with respect to the desired wealth goal *Wd*. These results were all obtained using bootstrap resampling (i.e., for the historical market) with an expected blocksize of two years.

It is interesting to note that the results for the Base Case, constant proportion strategy (contribution rate 20%) are worse than the results for the QS optimal strategy, *Fc* = 0.15 (contribution rate 15%), at least in terms of the two points of the cumulative distribution function listed in the table. In other words, the shortfall increase with a constant proportion strategy compared to the quadratic shortfall strategy can be interpreted as losing 5% of lifetime salary, which is very significant. However, this comparison does not take into account the entire cumulative distribution function. In general, constant proportion strategies are superior to quadratic shortfall policies in the extreme left tail of the distribution. However, the improvement over quadratic shortfall is very small, with a very low probability.

As a filter to determine an acceptable combination of DC plan parameters and investment strategies, suppose we specify that there should be at least a 90% probability of achieving at least 80% of the desired expected wealth goal *Wd*. Based on attempting to achieve the final target expected real wealth for the base case (see Table 2) and applying this filter, we can see that the shortfall probabilities using standard strategies (constant proportion or glide path) are unacceptably high. Using the QS optimal strategy leads to a substantial reduction in these shortfall probabilities, but still not to the desirable range of less than 10%.

From Equation (11), it is clear that the case with *R* = 0.6 and *wr* = 4% leads to the same expected wealth target as specifying *R* = 0.5 and *wr* = 3.3%. Table 11 therefore indicates that, if we assume that the safe (real) withdrawal rate is 3.3% and the replacement ratio is 50%, the probability of shortfall is quite high even if the QS optimal strategy is followed.

**Table 11.** Comparison of shortfall probabilities. Results are normalized by *Wd* for each case. Input data provided in Tables 1 and 2, except as noted. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12 with expected blocksize ˆ *b* = 2 years.

Assuming we use the QS optimal allocation strategy, we are then forced to take other actions to attempt to reduce the shortfall probability. Increasing the contribution rate to 25% of annual salary meets our criterion, but this might be difficult to implement in terms of agreemen<sup>t</sup> from employees and employers. Decreasing the replacement ratio (40% of final salary) also achieves the shortfall objective. We note that many institutions effectively do this by targeting a final career average salary replacement ratio (instead of a final salary replacement ratio).

Finally, the use of the alternative equal-weighted equity index also achieves the shortfall probability target. As noted earlier, this type of index has historically outperformed its value-weighted counterpart owing to higher exposure to value, size, and market factors (Plyakha et al., 2014). In effect, the equal-weighted portfolio is a *smart beta* portfolio, with a long track record. However, equal-weighted portfolios have higher costs, which have not been factored in to our analysis. This suggests that there may be a market opportunity for a low cost synthetic ETF which tracks the equal-weighted index.
