**Methodology**

The base case was designed with simple but reasonably plausible assumptions in order to observe the pure effect on retirement wealth of several assumptions and strategies, most notably the glide paths. All of the examples are far simpler than any typical real-world retirement investment plan, but sufficiently complex to illustrate several important concepts. We assume a simple portfolio with two assets, intended to stand for a stock fund and a bond fund. At first we assume an independent and identical distribution for each time period. This distribution was selected to approximately match long-term total returns of typical stock and fixed-income funds in the U.S.: expected annual returns of 10 and 3 percent, and standard deviations of 17 and 5 percent for stocks and bonds, respectively, and a correlation of 0.1 between the two assets.^{7} Our base case is matched to a static portfolio of the two assets through the entire investment period, choosing a portfolio of 40 percent stocks and 60 percent bonds as a base case.^{8} These simulation parameters allow for a wide range of outcomes, including negative returns for both stocks and bonds, occasional extreme downside returns, and both bull and bear markets. Although the expected returns match the historical nominal and not the real returns for these assets, the conclusion is exactly the same for either choice—there is a wide range of glide paths with the same risk level, all with roughly equivalent retirement wealth distributions. The terminal wealth for the real returns case would be less than for the nominal returns, but the conclusions are unaffected.

For the purpose of maintaining a comparable scale across experiments, we arbitrarily set the initial investment amount at 10, corresponding to an initial $10,000 contribution to the plan, growing by 2 percent for each yearly contribution. Of course, a different starting amount would simply change the scales on the graphs, but not the conclusions.

We solve for the matching glide paths using numerical grid search and bisection algorithms. We then can compare the various glide paths with the expected or median target-date wealth, or any other statistic, to investigate the impact of changing the glide path, with the retirement risk amount held constant. Among the glide paths, there is a winner in terms of expected or median wealth for all the cases we studied.

After examining the results with these baseline assumptions in the next section, we systematically vary the assumptions and examine how glide-path results are affected.

**Results**

Figure 1 is a plot for the base case simulation, overlaying the family of matched glide paths for the 40/60 static investment plan with summaries of their target wealth distributions for each glide path. The simulations were performed with the base assumptions mentioned in the Methodology section. The glide paths were matched to a standard deviation of target-date wealth of 953.68—the same as the 40 percent stocks, 60 percent bonds static investment plan. For example, the left-most linear glide path begins with 98.04 percent stocks, ends with 0 percent stocks and results in the same standard deviation of target-date wealth as the 40/60 static plan. Starting with a higher stock percentage will yield a retirement wealth with higher standard deviation for any linear glide path. The right-most glide path begins with 0 percent stocks, ends with 59.48 percent stocks and also results in the same standard deviation of retirement wealth. Schematically, every such linear glide path is shown by an arrow in the plot, with its tail and head corresponding to starting and ending stock allocations given in the legend on the right-hand side of the plot. The heavy blue line shows the expected retirement wealth under each glide path, and the other roughly parallel curved lines show the quantiles of retirement wealth as annotated in the legend below the plot. Corresponding wealth values are shown on the left vertical axis. The static 40/60 investment over the period is marked with a red vertical rule, and corresponds to an arrow of zero length. The mean target-date wealth attains its maximum within the matched family with a gently descending glide path, starting at 53.24 percent stocks and ending at 32.38 percent stocks at retirement, although its mean is only slightly better than those of the other glide paths. The median target wealth is maximized for an even-more-gently descending glide path that starts at 48.29 percent stocks and ends at 35.31 percent stocks. Typical industry glide paths that start with a large allocation to equities and end with a small one fare somewhat worse than either the optimal glide path or the static 40/60 investment.