July 6, 2010
The ranking of the curves is shown in the matrix in Figure 6. The rankings are not always obvious. Consider which curve should be higher: that of good luck with returns and observable inflation (3,2) or skillful returns and minimal inflation (2,1)? The calculations reveal the former is superior to the latter.
A SWR of 4% can be viable for time horizons ranging from 16 to over 50 years, depending on which path is realized.
I can now simplify these results by taking expectations over the probabilities we assign to the occurrence of the paths themselves.
Figure 6: Mean SWR for each path
Figure 7 smooths the nine outcomes according to our beliefs about each path’s probability of occurrence. If the beliefs change, so will the graph. The middle curve is the expected value of SWR, and the two other curves show the mean and the standard deviation around the mean. Keep in mind that the mean and standard deviation SWR curves are generated by the particular probability distributions chosen by the investor. Different distributions produce different results.
For these beliefs, a horizon of 20 years has a SWR of 4.2% 
1% and 30 years requires a SWR of 2.6% 1%.
These results illustrate the sensitivity of a SWR to the assumptions. Investors should use this approach to understand the required returns for a desired SWR and the constraints on a SWR that his beliefs induce.
Figure 7: SWR as function of investor beliefs
Lloyd Nirenberg, Ph.D., provides investment services through Rocket Science Capital Advisors, LLC. He comes to the financial world after a long career in the electronics industry, in which he was an engineer, executive, deal maker and consultant. He may be contacted at: www.rocketcap.com.
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