First derivative of deterministic portfolio longevity

Using a deterministic formula (see figure 1) that I borrowed from Milevsky's book 7 Equations as well as from his lecture notes, provides some insight into the effect of withdrawal rates on portfolio longevity (L): High w = shorter, low w = longer. That is trivially easy and intuitively accessible without the math.

Figure 1. Portfolio Longevity

A little bit more obscure but perhaps of more interest is the derivative of L, also lifted from the lecture notes and seen in figure 2, here presented over different w . 

Figure 2. Derivative of L wrt w

which could have also been gathered from the chart in figure 1.  Figure 3 varies v the growth rate for different w where w/M > v. ...less than v tends to be a little perpetual. 

Figure 3. Derivative of L wrt w for different v 

This post is not ready-set for conclusions but if I were to make conclusions they might go like this:

• Spend more = shorter lived portfolio, obviously 

• Withdrawal has a non-linear impact on L with a little counter-intuition that at high w the sensitivity is lower.  

• Also counter-intuitively, the impact of v (growth rate) is higher for lower withdrawal rates  

• It may not be obvious but I'll say that this supports the contention I made in my 5-process series that first and second derivatives, speed and acceleration, of a process, and retirement is a process, are often of more interest than "position." They reveal aspects of the "personality" of a process that single numbers and point estimates cannot. That's one of the reasons why that ad series "what's your number" was always kind of a joke.