**5. Alternative Assumptions**

The base case results in Section 4 relied on the input parameters given in Tables 1 and 2, for the value-weighted equity and 3-month T-bill indexes. We now explore the effects of altering our assumptions about factors such as the contribution fraction, the salary escalation rate, the maximum amount of leverage permitted, the underlying indexes to be used, and the salary replacement ratio. We consider each of these in turn. For the most part, we only use the QS optimal strategy since it has been shown above to be generally superior to the constant proportion and glide path strategies.

### *5.1. Effect of Contribution Fraction*

Our base case described by Table 2 assumed a total combined contribution by the employee and employer of *Fc* = 20% of salary. Table 5 reports the effects of dropping this to 15% or increasing it to 25% for the QS optimal strategy. As is to be expected, the table shows that risk (measured either in terms of standard deviation or the reported shortfall probabilities) decreases significantly as *Fc* rises. However, even in the case where 25% of the employee's salary is contributed to the retirement savings plan, there is still almost a 15% chance that real terminal wealth is less than \$800,000, considerably lower than the target of \$915,000. A broader comparison of these cases is provided in Figure 4, which depicts the cumulative distributions of real terminal wealth. The cases where *Fc* = 20% and *Fc* = 25% appear quite comparable for values of *WT* ≥ \$915,000 over the plotted range, but the higher contribution fraction case appears to be much safer over a wide range below the target. The high savings rate leads to a notably increased amount of expected surplus cash. Along paths with strong equity market returns, the target can be reached relatively early, but the model assumes that savings continue each year (in this case at a high rate), so surplus cash can build up. The case where *Fc* = 15% exhibits poor performance on the downside, but is somewhat better on the upside. With a low amount of money saved, more risk must be taken on in order to reach the target. Doing so works out very well if realized returns are strong, and quite poorly if they are not.

**Table 5.** Effect of varying contribution fraction *Fc* 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 4.** Cumulative distributions of real terminal wealth for various contribution fractions *Fc* for the QS optimal strategy. Wealth units: thousands of dollars. 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.

### *5.2. Effect of Salary Escalation Rate*

We now examine the effect of changing the salary escalation rate *μI* from the base case value of 1.27% given in Table 2. Note that this will result in a different expected real wealth target *Wd*, based on Equation (11). Table 6 summarizes the results for both the synthetic market and the historical market, using the QS optimal control. For comparability, the table expresses the results in terms of *WT*/*Wd*, i.e., real terminal wealth as a fraction of the expected wealth target. Obviously, a higher escalation rate leads to a higher final salary. Given a fixed salary replacement ratio, this translates into a higher expected value of terminal wealth *Wd*. For example, with *μI* = 1.75%, we have *Wd* = \$1,056,000 instead of \$915,000 as in the base case. The results in Table 6 are quite similar in the synthetic and historical markets, on a case by case basis. For each set of tests, the standard deviation and shortfall probabilities show an increase with *μ<sup>I</sup>*. With a higher salary and a fixed contribution fraction, there will obviously be a higher amount of saving. Despite this, the associated higher real terminal wealth target results in a higher level of risk. This is borne out in Figure 5, which plots the cumulative distribution functions of normalized real terminal wealth *WT*/*Wd* for the various values of *μ<sup>I</sup>*. The highest salary escalation rate has the worst performance for low *WT*/*Wd*, and the best performance for high *WT*/*Wd*. Taking on more risk to reach the higher wealth target works out well if investment returns are favourable, and poorly if they are not. Conversely, the lowest value of *μI* results in the best performance if investment returns are weak, and the worst performance if they are not.

**Table 6.** Effect of varying salary escalation rate *μI* for the QS optimal strategy. 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.

