*5.3. Effect of Leverage*

All of the results presented thus far have specified *L*max = 1, as indicated in Table 2. We now consider increasing this to *L*max = 1.5, thereby allowing the use of 50% leverage. Table 7 documents the results for the QS optimal strategy. It is interesting to observe that the risk measures shown (standard deviation of *WT* and the two shortfall probabilities) each indicate somewhat lower risk when the use of leverage is permitted. In one sense, this is obvious: relaxing a constraint cannot lead to worse performance. On the other hand, leverage is often not allowed since it is perceived to be "risky". The issue here is what is meant by risk. We have defined it on the basis of the value of terminal wealth. Leverage constraints, however, are typically motivated by concerns that portfolio values during the accumulation period (not at the end of the period) will fluctuate excessively.

**Figure 5.** Cumulative distributions of normalized real terminal wealth for various salary escalation rates *μI* for the QS optimal strategy. 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; surplus cash included.

**Table 7.** Effect of varying maximum leverage indicator *L*max for the QS optimal strategy. 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 6a shows the cumulative distributions of real terminal wealth when leverage is allowed and when it is not. The cumulative distributions are fairly similar for high levels of terminal wealth. Over a wide range of wealth levels below the target of \$915,000, the strategy which permits leverage performs better. However, in the extreme left tail of the distribution, it turns out to be worse to allow leverage. These very low wealth levels occur as a result of very poor equity market returns over most of the investment horizon. The QS optimal strategy continues to try to reach the quadratic wealth target, so it invests completely in the equity market to the extent possible. With continued poor returns, leverage in this case amounts to doubling down on a losing bet.

**Figure 6.** Results for cases allowing and excluding leverage. 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. (**a**) cumulative distribution function of real terminal wealth for cases allowing leverage (*L*max = 1.5) and excluding it (*L*max = 1). Wealth units: thousands of dollars; surplus cash included; (**b**) mean and standard deviation of the fraction allocated to the equity market for the case where leverage (*L*max = 1.5) is allowed.
