Long Horizon Spending (con't.)

This post, still about "Long Horizon Spending," follows the last post on the same topic:

• On the Behavior of Long Horizon Adaptive Spending

where I was playing around with what a percent-of-portfolio approach does to spending and portfolios at a 50 year mark. 50 is pretty arbitrary but one of the cited papers used 50 years for some kind of reasonable endowment policy cycle. 50 years is pretty long and not a typical assumption in the retirement finance I read but it is not terribly unreasonable were we to be given both early retirements and extended longevity.[1]  

In that last post I foreswore, for a moment, dealing with consumption volatility, consumption utility evaluation and considerations of human longevity in favor of just looking at median spending capacity at year 50 given the distribution of spending generated by the simulation at that year. This time I'll throw in consumption utility and longevity over the full lifecycle.  I could say that we are dealing as well with spending vol because the convexity of utility math will punish that variation (I'll skip the refresher on certainty equivalents) but that is not my main interest and we'll just call it implicit.    

The big point is maybe more that we are now moving from apples to oranges and these two posts are really evaluating very different things. My goal in both posts, though, was mostly to look at long horizon spending using whatever tools I have and then look for general shapes or boundaries or something, anything, of interest. In the former post we saw, I hope, that:1)  spending under 2% doesn't make much sense unless you don't have to spend at all, and 2) spending at about the expected geometric mean looks like it is the maximum level that preserves some "intergenerational equity" between the present and the mid-future, pretty much like the referenced papers suggested(Coiner, Dybvig). Now on to utility evaluation...

"So, RH, haven't you done this consumption utility thing about 4 million times before?" "Uh, yeah, but I think it is different this time."  What is different?  Not much but maybe like this:

• I am using a pretty young start age. Here I am using 50. FIRE retirements can sometimes be earlier but me? I retired at 50 and then realized at 52 I should have been more concerned than I was. Simple logic tells me that 50 starts to open up a long horizon. If we were to plan to 95 or 100 or 105, there's your 50 years. 50 years old is early. 
  
  
• I am grotesquely distorting longevity in this post using a Gompertz model that does not match any reasonable actuarial table. This is mostly just for fun. I took an SOA table, RP2014, and tried to curve-fit it using Gompertz and then I blew out the mode and dispersion to create something vaguely human like but weirdly long lived but still within the "unlikely to live much past 120" range. See Appendix 1. 
  
  
• The spending is rendered as a % of portfolio not constant. This is a relatively uncommon assumption in the retirement lit I read. This because, I think, economics favors, and has a decent rationale for, consumption smoothing and constant spend. %P is more endowment-like though most endowments will typically engage in smoothing/rules for practical reasons. Here in this post %P is just a stand-in for "adaptive rule" with all the consequences of that choice that were mentioned in the last post. I personally do not use a rule like this. Spending is what it is and is also hard to change especially to the downside except maybe in dire circumstances, which for me was just-post-divorce. 
  
  
• As mentioned, I am using consumption utility here vs. just looking at the median of the spend distribution at year 50. To do this I am evaluating additive M & vN utility over a whole life where a life is an iteration and the mean of the iterations life-summed and discounted utility is the target value function in Eq1. 
  
  
• I have a floor for spending/utility in the sim now. This is set, again very very arbitrarily, to 1000 real/yr. My rationale is primarily convenience in that it avoids the disutility of zero consumption (though that would not really be possible in a %P model anyway). Also, no one really consumes zero (before death). Even if one is on SS or welfare or getting help from family/church or cadging coins at intersections, there is always some minimal level of income and spend. 
  
  
• The consumption utility is discounted. Not time-value discounted since I am working in real terms but strictly through a conditional survival probability that can be visualized in the right panel of Fig A.1. This point is probably redundant with bullet 2, I'm just emphasizing it. The Yaari model and others will also allow for a time-bias discount factor, still separate from time-value discounting, that I have not included for simplicity. 
  
  

To be clear on points 1 and 2, this graphic is basically what I am trying to do in this post with the planning horizon. See Appendix 1.

Fig 1. Long retirement

The Value Function

We've seen this before but the value function for these runs can be characterized in non-continuous form like this for each spend rate and parameter set in the sim: 

Eq1. simulation utility value function

The Assumptions

• Four portfolio scenarios in real terms: 4/12, 4/20, 6/12, 6/20. I want to say this is arbitrary but maybe it isn't. I don't know what to assume for inflation anymore. What is it now? 20%? for longer term maybe we can say 3%. If so, then 4/12 or 6/12 is about the right range for 60/40 funds and etfs from some past work. I've used either Vanguard 60/40 or AOM. There are some others. 
  
  
• Utility is evaluated via constant relative risk aversion. Here I am using this form:
  U(c) = (c^(1-ra)-1)/(1-ra) | ra < > 1 and U(c) = ln(c) | ra = 1
  
  
• RA is set to 1, 2 or 3 in yet another dimension of scenarios
  
  
• Gompertz mode and dispersion set weirdly high to 95/9.5 - see appendix 1
  
  
• Spending is strictly % of portfolio and filtered on the state of the portfolio in each period. There is a consumption floor equivalent to 1000/yr.
  
  
• Iterations were generally somewhere around 50k which created some simulation instability. Or maybe I just had coding errors. 
  
  
  

The Process

1. Run sim for four scenarios and for 3 levels of risk aversion = 12 scenarios total
   
   
2. For each scenario, run for ~30 spend rates between 2.5 and 5.5 or so in .1 increments 
   
   
3. Look for a local max E[V(c)] and if there is any simulation ambiguity, which there often is, articulate a range of spend rates that might maximize the expected discounted utility of lifetime consumption.

The Output

I'm skipping the charts this time. In table form and including the results of the last post, I get something like this:  

Table 1

Discussion

I'm not sure what to say. For lower risk aversion here in this post the spend rates are higher than before but not terribly unconscionable. For higher risk aversion they look familiar to me and not radically out of line with the last post but that is pretty subjective. That spending is lower at 6/12 than in the prior post perplexes me. That, say, 4/12 is higher in this post for lower risk aversion makes sense to me. a) because it is lower aversion, and b) I think there is a very strong bias introduced by the conditional survival probability. At year 50 its at around .19 given our 50 year old with super longevity - closer to zero in a real model -- but that still starts to moot both spend volatility and the degradation in the portfolio that'd happen at those rates over that horizon. Maybe that is the point.  

My guess is that any sane person would also evaluate the state of the portfolio and the environment in each and every period and then re-adapt things as the situation demands. This is why the older retires who find out what I do sometimes laugh at me. They always know what's going on irl and don't typically spend themselves into holes or bankruptcy willy nilly. But that is another post or at least a new sim-within-a-sim thing that David C always talks about. My PC isn't fast enough for that, though. 

Appendix 1. Goosing Longevity a Bit

I used SOA RP2014 tables to get the (male annuitant) hazard and inferred from that the survival probabilities conditioned on age 50 if I got it right. I then curve fit that using Gompertz without the Makeham param. Like this

I then, very very arbitrarily, pushed that out even further to make a long-lived kind of thing. In charts it'd look like this using m=95 and b=9.5 for the push. The red parameterization is what I used in the simulation runs. Blue is the RP2014 data. Grey was my manual attempt to fit a curve which is not used in the simulation. 

Fig A.1

------------ Notes ----------------------------------------

[1] This extension of longevity can easily be dealt with using simple formulas and pushing out the time frame to feel out what might happen. That makes posts like this overkill which, of course, it is. But like a train-hobbyist playing with trains in the basement, one must make the little towns and paint all those tiny trees green or it wouldn't be as much fun. Also, If I wear my Milevsky hat, I would ditch human longevity considerations -- when being simple -- entirely and just look at portfolio longevity. This is easily done with what he referred to as the Fibonacci formula, in his 7-equations book, which looks like this. This solves a lot of retirement questions and has the virtue of transparency.  M is nest egg, w is withdrawal, and g is the annual real growth rate of the portfolio as a % and probably net of fees and taxes, too.