May 28, 2019

Obscure artifacts, now with a constrained horizon

I've about beat this subject to death but let's do one more.  This time I'll take the same portfolio longevity simulation I did before that was unconstrained, or run to infinity, and here create an arbitrary horizon. I'll set the horizon H to 40 periods to perhaps reflect an early retiree or a 60 or 65 year old with long survival expectations. I'll also, as before, look for artifacts in the fake model that suggest what I was calling a "tipping point" while being reluctant to conclude whether that point has any meaning or interpretability in real life.


Here is the original 3d scatter-frequency diagram with H = 40 indicated by the red line.




which is being driven by this recursive net wealth process:

net wealth process
or alternatively seen like this in past posts

µW – 1 

where 1 is one unit of consumption and W is wealth in units (4% spend would mean 25 units). This happens, not so uncoincidentally, to be the coefficient of the second term of the Kolmogorov equation for lifetime probability of ruin. 1, by the way, or w, is net consumption. The presence of lifetime income changes things for the better even if it is endogenously allocated at some point over time via (fair) income annuities inside the world of the model.

Now I'll re-render frequency as a simulation-probability-mass (one down each w) and then look at the "percent succeed" as I travel down the red line from withdrawal = 2 to 12. I'll chart that scatter, draw a poly trend line and estimate the first derivative of the change in success along a slightly shorter interval. Doing that (while also marking the old 3.4 critical point we had when we had PL unconstrained to infinity) we get the following:




Discussion

1 - I'm still not sure if the critical point at ~4.8% is perfectly meaningful. It is, to me, the point at which the success rate is increasing the fastest which signifies, to me (again), the beginning (or end, I guess) of a "robustness" or "fecundity" range.  4.8%, by the way, is around a 43% "fail rate" at H=40 using the common lingo. The meaning on that 43 is somewhere out there, too, but not discussed here.

2 - The 3.4 spend, given the infinite horizon that was used to come up with that number, happens to land around a 90% success rate which is a common but arbitrary threshold I see in planning and academic papers.  Given the portfolio parameters[1], that seems to be an ok lower bound due to: a) horizons can't get longer than infinity, b) more than a 90% success rate or the 3.4 level seems to be self-denial at some personal level absent a bequest motive, and c) annutization would create a dynamic where spend rates could be higher across the board.

3 - The last two points give me, in this entirely false model-world, a range to work with.  It looks, to my untrained eye, like spending wants to be some where between 3.4% and 4.8% for the endowment to be fecund while spending is not completely self-denying. That range happens to have a mid point just above 4% which might give some context to the 4% rule given the presence of a horizon constraint. I personally lean a little lower than that spend level at 60 because I am young and because my forward expectations for real return have been constrained by my viewpoint from the heights of 2019.

4 - My last post on my "error" in doing a wealth depletion utility sim now looks a little more reasonable. There, when using a utility value function, we: a) used similar portfolio assumptions, b) had a lifetime constraint using an actuarial life table, and c) varied (arbitrarily) the risk aversion coefficient between 1 and 2. In that run the optimal spend rate wrt the expected discounted utility of lifetime consumption looked like this which feels like it has (probably coincidentally?) a similar range of spend rates.  Higher risk aversion would of course create possible non-rational behaviors so very low spend rates might make sense given the presence of fear:

Add caption

These are being driven by an animated value function(V):
Expected (E) discounted (k) utility (g) of lifetime (T*) consumption (c)

the terms of which are described here and where consumption snaps to income or some baseline when wealth depletes:  A WDT Model

5 - Shorter horizons would clearly change all this and open the door to much higher spend rates. If I am 90 and can still drink and dance, I will be throwing a lot of parties.


Notes
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[1] The portfolio parameters may look unreasonable at first glance but my estimate of returns, using 1928-2016 data for a 50/50 US large cap/10year mix, was somewhere around 8% nominal return with 10-11% sd.  Knock of 3% inflation and 1% fee and/or tax and 4% r starts to look optimistic in real life but maybe ok for this simulation.


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