PWR is "perfect withdrawal rate" or the constant rate that -- knowing in retrospect what the return series was or were, -- would result in (in this case) a zero fv at the end or as quoted in a previous post ""the maximum withdrawal rate possible over a fixed period of time if one had perfect foresight of investment returns." My other links/posts are in footnote [1]. A visualization of the math is here.
In one of the papers on PWRs that I used in my past posts, the authors mention that someone somewhere should in the future explore stochastic longevity and PWR rather than using a fixed number of periods (like 30). So I did. In doing 10,000 iterations (simulation) where I randomized a return of .05 with a standard deviation of .15, in place of using 30 periods for returns within the 10k iterations, I randomized the periods using Gompertz math for longevity using a mode of 90 and a dispersion of 9[2]. I assumed a 60 year old where I needed an age. For a comparison base-case I ran the same assumptions for a fixed 30 period term, which for a 60 year old would be to 90.
Here are some of the base case stats:
mean .069
median .067
min .012
max .177
5th percentile .034
And the base case distribution of PWRs looks like this:
Figure 1. dist. of PWRs for 30 years
with return = .07 and stdev = .15
Figure 2. mortality freq. dist. using Gompertz
M=90 and b=9 for a 60 year old. 10k sim
Then I re-ran the script with the variable longevity and setting "periods" = terminal-age - start-age so the modal expectation for periods is around 30 years but varies according to mortality distribution. At this point I was actually a little surprised. I was kinda expecting the distribution to merely be a little wider: a few more high PWRs and a few more low PWRs and maybe a little flatter kurtosis-wise. That's sort of what came out but it looked more like this (and that's where the disclaimer about checking for coding errors in the first sentence above comes in and also that is why this is an opening bid in an ongoing discussion rather than a tutorial):
Figure 3. distrb. of PWRs
using stochastic longevity
r=.07 sd=.15
But, I guess this makes sense. If one superannuates, the PWR is going to behave more like a type of limit while if one were to die in year 0 or 1 the PWR is going to be 1 or very very high. On the other hand, let's look at the stats since, from a planning perspective, we don't plan for 1-year retirements, we plan for either the worst case or maybe strategic scenarios and both of those should be conservative. I use, when looking at PWRs, a 95% level or the 5th percentile PWRs on the left. Let's take a look; SL means "stochastic longevity:"
| base | with SL | |
| mean | 0.069 | 0.088 |
| median | 0.067 | 0.074 |
| min | 0.012 | 0.013 |
| max | 0.177 | 1.000 |
| 5th percentile | 0.034 | 0.036 |
Conclusions?
I guess we could say: "spend 2/10ths of a percent more for tomorrow we may die!" or maybe it's the Jack Sparrow credo "take what yer can; give nothing back." In practice it doesn't look like this changes any planning considerations I might have when looking at the conservative planning cases. I suppose there is a legacy planning implication in there somewhere but that is not today's post.
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[1] some past PWR posts. These will have some references to other papers...
Trend Following Can Enhance Withdrawal Rates - Part 2 8/18/17
Trend Following Can Enhance Withdrawal Rates - Part 3
Trend Following Can Enhance Withdrawal Rates - Part 4
Some PWR acknowledgements and comments
Putting an adaptive PWR up against changing longevity estimates
Visualizing Sequence of Returns Risk
Here also are some posts I did subsequent to this one:
PWR v Kolmogorov v MC simulator
Simulation vs PDEs and other analytic methods
[2] kinda sorta matches the curve for a 60 year old using a SOA actuarial table data for annuitants.
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