Visualizing the impact of spend choice

As in the last post, nothing really new to me or the literature here. 

This is how it works at RH: if I go back to old code I wrote -- even if it is well commented -- long ago, I often have no idea what I did then and I usually can't make it work well without a bunch of work and we here at RH are a little lazy. So when I have new code I will sometimes pile on: "hmmm, I wonder what x would look like as long as I have this code up?" That was certainly the case with the last post where a reader, reasonably, asked "what am I missing, this is simple?" yep, just goofin around. 

The same kind of thing can be said for this post. I mean, how many MC sim charts have we seen? A ton. So, I am adding nothing here, I just wanted to look at what spending -- which I, and we, already know pretty well -- does to a portfolio but only in terms that the code "I was just working on" speaks. 

1. Spending in a Standard Simulation Overlay

Here is an N[.04,.12] portfolio over 30 periods with and without spend. I am not displaying the geometric return here as a separate chart because the internal return engine is unaffected by spending (unless we were to re-allocate in response to behavioral stimuli, which we might). 

- grey/blue(median) lines are "no spend"

- red/black(median) lines are "with spending" 

- Y axis is net wealth with initial wealth = 1, and X is time period 

- spend rate = (0% grey, 4% red) in this example

Figure1. Sim paths for different spending assumptions (0 v 4%)
with median wealth in blue and black for (0 and 4%)

Ok, so this is a bit banal. What, if anything, can we say?

• net wealth in accumulation mode can approach zero but not punch through
• the upside outliers without spending can get quite large
• the median wealth without spending at each step rises convexly 
• spending can force wealth to and through zero (ignore negative wealth for now)
• the median wealth under these "with spending" parameters looks like it is in a process of slowly wasting the fecundity of the portfolio. 
• The outlier "with spend" paths breach the zero bound as early as 17-18 years
• I probably should have put in "percentile of sim distribution" lines....but didn't
• The mass of red (with spend) is dispersed much lower than grey but that's always a "duh."
• It might be possible to meaningfully compare 2 well defined spend-portfolio combos using this method but looking at broad array of choices would be a little ad hoc and hit-and-miss I think

2. Spend Choice Viewed via "Portfolio Longevity in Years" 

This is not something new to me. I did this effort in a series back in 2019. But, contrasted with the above, I do think this was kinda additive. I have seen exactly zero papers or books that look at the problem like this. I even got a thumbs up emoji from Prof Milevsky on Twitter once for something related to this which is like getting a gold medal, maybe silver. Heh.  

The idea here, in this section, is to roll a sim not in terms of net wealth paths or fail rates or "magnitudes" but rather in terms of a distribution (probability mass from empirical sim data, or in frequency terms as in Figure 2) of years that a portfolio lasts (horizon = infinity or 100y in this example) under a set of assumptions that include a pegged but stochastic return assumption as well as then rolling through different spend rates. In this case the peg was a portfolio r = N[.04,.10] so a little apples and oranges vis-a-vis #1 above, but close enough.

Figure 2. Spending effects on Portfolio Longevity

Here, 2% withdrawal is kind of like the no spend scenario, while the 100 year horizon stands in, imperfectly, for time=infinity. What, if anything can we say about this?

• There is quite a bit more, a ton more, info content packed in here compared to Figure 1
• As the spend rate goes down the rate of, or % of portfolios lasting to, infinity goes up faster then slower (ie non linear)
• High spend rates have a very small, to the left, distribution of PL. Good for someone with a near term death sentence perhaps
• The rate at which portfolios tip over to infinity -- the d(PL)/ds at inf (?) -- is maxed at about the geometric return assumption for the horizon. This is not proven but seems like it visually and makes intellectual sense for now. Correct me on this if you have a better view. 
• None of this includes any chaos from outside the model, a nuke strike which would maybe vote for lower spend rates...which is also a way of saying "redundant capital" but no one I know speaks like that, so: lower spend is a reserve for the unknown in longer retirements. 
• If I wanted to visualize spend impact, "many" spends, at a meta-level like figure 2 might be useful. On the other hand, if I were to do it side by side, comparing only 2 strategies, some variation of Figure 1 and/or simpler models might be more useful. In this case I am thinking of the Fibonacci formula for PL in Milevsky's book but it might be something else. The only thing you want to make sure you capture is the dispersion/uncertainty and there are some different ways to do it. It's just that, as far as I can remember, when we add spending we mostly leave deterministic var models behind. Might be wrong. TBD