What is the spend rate that maximizes median terminal wealth at a horizon of 500 years (why 500? idk, just messin' around and anyway that's the game we played in the last post)? Obviously the proper spend rate for doing that is zero. That's why it is sometimes hard to talk about portfolio growth optimality in a retirement spending context, granting that the 500 years here is absurd. Better to discuss portfolio longevity + mortality, and/or (gasp) life consumption utility.
Ok, what is the threshold spend rate that minimizes median terminal wealth at a 500 year horizon? idk, let's look.
The game is: N[.04,.12]* real; 500 years*; spend rate = .0250->.0308. Net wealth is W(t) = W(t-1)*r - spend; W(0)=1; if W(t) <=0 set W(t) to Min(W)* and spend(t) to Min(spend)*. Calculate ln(W[500]) for the diff spend rates and chart it. Looks like this:
* All of these are arbitrary and illustrative. Min(spend) and Min(W) are close to but not exactly zero. That's a setup for a future game where 0 spend is undefined in utility terms. fwiw I had to run 200k iterations due to instability in the aggregated median with too few cycles. Slow.
Answer in this post only: ln(W) goes undefined (downward to "bad") at around a rate of .0308. Me? I had figured the undefined would be rather at the long horizon expected geometric mean return of .0328. So, why not there? No idea yet. I mean, E(gm) is about the place in a portfolio longevity context that the longevity flips to infinite the fastest. My guesses so far on the current question:
- Mis-coding and stupidity. I mean, I have miscoded quite a bit before. Maybe not stupid, just incautious and a bad code tester.
- Being too loose with assumptions about begin- and end- of period timing of spend and returns. Moshe Milevsky was crystal clear on how this needs to be done in his recent book on R Recipes. I need to look at this more carefully.
- The way I set up the game rules with zeroing out wealth and spend in a wealth fail state and not understanding the impact.
Feel free to chime in on where I am on this....
Misc Background Notes
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1. What the wealth paths look like when I stop the game for paths that hit zero. Blue is median Wealth. Interval constrained to 80 years. Spend is .0306 here.
2. The same as #1 except with the full 500 years....
3. The geo return time-average for #1 and #2 for only the first two of 200k paths. Just showing the convergence of time average of individual paths over time, sort of. Don't really need to know this it's just that we were talking about time averages in a recent post and this seems like a decent illustration.
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