*5.4. Long-Term Bond Index*

All of the results presented thus far were based on the value-weighted equity index and the 3-month T-bill index. In this section, we explore the effects of substituting the 10-year T-bond index for the T-bill index. Recall from Table 1 that the long-term bond index had average annual real returns of 2.16%, about 1.3% higher than the T-bill index. Of course, these higher average returns are accompanied by higher volatility (Figure 1), but that is ignored when we formulate the QS optimal strategy since we assume constant interest rates. Table 8 shows results for the constant proportion, linear glide path, and QS optimal strategies when the 10-year T-bond index is used in lieu of the 3-month T-bill index. The general pattern in Table 8 is similar to that seen earlier in Table 3. The QS optimal approach achieves roughly the same expected real terminal wealth, but takes on considerably lower risk compared to the other two strategies, as measured by standard deviation of *WT* or either of the two shortfall probabilities considered. Comparing the results from Tables 3 and 8, we observe that the standard deviation and shortfall probabilities for each strategy are a little lower in the synthetic market when the 10-year T-bond index is used. This also holds in the historical market for the constant proportion and glide path strategies, but is not the case for the QS optimal strategy in terms of the shortfall probabilities. For example, the probability of real terminal wealth being lower than \$800,000 is 23% if the 3-month T-bill index is used, but 26% when the 10-year T-bond index is used. The overall conclusion, however, is that replacing the 3-month T-bill index by the 10-year T-bond index does not make much difference. This picture is reinforced by comparing the cumulative distributions of real terminal wealth shown in Figure 7 based on the 10-year T-bond index with those shown previously in Figure 3a for the 3-month T-bill index.

**Table 8.** Results for the optimal QS strategy when the 10-year T-bond index is used instead of the 3-month T-bill index. 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.


**Figure 7.** Cumulative distribution of real terminal wealth for QS optimal, glide path, and constant proportion strategies when the 10-year T-bond index is used. Wealth units: thousands of dollars. Input data provided in Tables 1 and 2, except as noted; surplus cash included for the QS optimal strategy. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12. Expected blocksize ˆ *b* = 2 years.
