of the benefits of mean reversion. We have projected wealth adjusted for taxes and inflation. We have expressed risk in terms of the volatility of ending results. There is a great deal of useful and perhaps surprising information in this chart. II It demonstrates that it is difficult to grow wealth net of taxes and inflation. Twenty years is a long time by most individuals' standards. This data covered some very robust markets, but two of the combinations, on average, were unable to achieve even a doubling of real wealth. This was based on the assumption that none of the portfolio was used to meet spending requirements. As described later, even modest spending requirements will have a big impact on wealth accumulation. II The minimum results were almost identical for each combination, and the riskier combinations actually had higher terminal minimum results. When looking at asset allocation issues, many individuals attempt to identify their ideal risk levels based on estimates of worst-case outcomes. The assumption is that the riskier strategies have inferior worst-case outcomes. That is true only for short holding periods. The interim low points for the riskier strategies are likely to be meaningfully lower than their worst-case terminal result. When we extend the holding period to 20 years, results tend to converge toward long-term averages. The higher expected return of riskier strategies compounds and raises even the worst-case outcomes. Many individuals use their investment portfolios to fund their ongoing expenditures. We next want to see how a spending requirement will alter these results. We will model the impact of spending based on the following assumption. We will continue to use 20-year holding periods. We will set spending as a percentage of the portfolio's initial value. Thereafter, spending will rise with the rate of inflation and will not respond to changing portfolio values. Spending will increase with inflation but not increase in response to very good markets nor will it be cut back in response to poor markets. This is probably a good approximation of investors' desires but may not be a good approximation of investors' actual behavior. Tables 30.9, 30.10, and 30.11 demonstrate the impact of spending. A comparison to Table 30.8 shows that even modest spending requirements would have had meaningful impact. Conservative strategies could not, on average, have supported more than a 2 percent initial spending requirement without the expectation of a decline in real wealth. One of the key differences among Tables 30.8 through 30.11 is the deterioration of minimum results. Our model assumed that spending would continue to grow with the rate of inflation regardless of market conditions. In other words, the investor would not cut back on spending if investment results were poor. Under this assumption, worst-case results become much worse. Even modest spending requirements can meaningfully increase the decline of principal value in weak markets. If assets are sold off at depressed prices, those assets cannot contribute to a recovery in principal value in any subsequent market rebound. This point is best illustrated in Table 30.11, where the interim worst-case results were about the same as the terminal worst-case results. In these simulations, the worst cases were the result of market conditions reducing the