July 6, 2010
I assume independent probability distributions for returns and inflation over the time horizon. This assumption has some empirical support in Dexia Asset Management’s Special Report, “Inflation and equity returns: what’s the link?” (March 2010), which says:
"Our general conclusion is that it is extremely hard to define hard-and-fast rules with respect to the link between inflation and equity returns."
The investor also must specify his subjective probabilities that each scenario will be realized. Thus, he produces two other distributions like those in Figure 5. The matrix in Figure 5 shows the joint probabilities for the nine paths, assuming returns and inflation are independent.
Figure 5: Scenario Probabilities
Sensitivity of SWR to beliefs
Armed with this set of beliefs and assumptions, I can draw some conclusions about the mean SWR. Figure 6 shows the mean SWR versus horizon-years curve for each of the nine paths. These curves reveal the strong sensitivity to different paths (recall that a path is a pair of return rate and inflation rate probability distributions that generate the future evolution of the account balance).
The matrix below the curves shows the assumed probabilities of the return and inflation scenarios, thereby determining the probability of each path. The cells in the matrix correspond to the labels on the graph. For example, the highest curve in the graph is for path (3,1), which is return scenario 3 “Good Luck”, inflation scenario 1 “Minimal”. For any horizon N, this path among the nine gives the largest mean SWR. The lowest mean SWR curve results from path (1,3), which is the path (Bad Luck Returns, Runaway Inflation).
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