One more obscure artifact of Portfolio Longevity

In the last post I was eyeballing the first derivative of the process that drives portfolios from finite longevity to infinite looking for a "tipping point" for the given portfolio return and vol assumptions.  Not even sure if that is legit but either way as long as I am on this road I might as well keep going.  In the last post I guessed that the max slope was around 3-4% spend.  This time I wanted to zero in on exactly where and get the min of dY/dw or where the second deriv goes to zero over an abbreviated interval.


Keep in mind as we do this that I have no formal training in calculus except a class or two in 1976 and a little bit of self study last year. If I embarrass myself, email me discretely (pun) and I'll fix it. Recall also that the data were hard to interpret. I'll try to come at meaning later.

If, for each withdrawal rate w(i), along the years over time for that w(i), there is a portfolio longevity distribution that is "defective," with a portion of the portfolios for w(i) that don't die and last to infinity --> then I am calculating here in this post the percent of each w(i) where L = infinity, or L(inf)/L(all) for w(i) [call it a success rate at infinity], and then I am looking at those %'s along all w(i), i = 2 to 12%.   Then I try to differentiate the curve. Can I do that? Am I supposed to? No idea. I did it anyway just for the hell of it.  A quant can beat me down on this later.  This is how I amuse myself on Fridays. I just wanted to see.

The data were discrete and jumpy so I fit a poly curve to the data and used that as a proxy. Differentiating that gives me this:

Figure 1. Looking for f''(Y) = 0

For this simple, quite fake set-up (though the portfolio assumptions are probably not all that different from mine if you squint your eyes and play with the "real return" a little bit) the minimum (thank god there is only one and not two or more or none) is a 3.4% spend rate.

While 3.4 happens to be very close to the middle of my current statistical control boundaries for spending, I'm thinking that this is coincidental or lucky or an artifact of the idea that from age 50 to 60 I have been playing a much longer planning horizon game than I will be in 20 years when I will have a higher control range well above 3 to 4.

But even if it is not coincidence, is this even interpretable? My guess is "no" in any practical sense though in the context of the model it might make sense.  I'll have to ask professor M some day. In this model with the r = ~N[.04,.10], a 3.4% spend rate is about where the portfolios along w(i=2:12) are tipping over to infinity at the highest rate. The closest I can come to meaning here is that below 3.4 consumption portfolios are probably robust (but self-denying in lifestyle?) and above 3.4 they are increasingly less so.

I have not connected any of this yet to the more common "fail rate" framework but at 3.4%, the cumulative fail rate over infinite time is approaching 35%.  That 35% is mostly meaningless since we have not scaled it to a human life.  But if we now go to the full scatter across all years and all w for the interval of interest, and look at a 30 year scale at the 3.4 spend rate, the fail at 30 (and not "last to infinity") has become a cumulative 4.4%. Still not sure how to interpret that but it at least sounds pretty good. Investigation for another post...