This is my valley. My house is just about right in the middle of that rain storm maybe 10-15 miles away (haven't looked the distance up). It is not an unpleasant place to be. There is certainly more to do here than quant retirement finance. On this day I was taking advantage of that fact and hiking up a hill on the west side of the Bridger range. If we are all lucky there will be more posts like this and fewer like the last one ;-)
I thought I'd try this kind of thing again just for "fun." Fun! ;-) Heh.
I've used a number of different evaluative frameworks over the years including fail rates/Monte Carlo, Lifetime Probability of Ruin, Perfect Withdrawal Rates (PWR), consumption utility, formulas, etc. A while back I took PWR (Suarez 2015) and feathered in random life time by way of Gompertz math that I used to take a random draw on terminal age consistent with a distribution tuned to actuarial tables. That was interesting and while I don't have my link handy I recall that I got some counter-intuitive results (misc measures of spend rates went up) due to the inclusion of the possibility of very very short lifetimes vs the typical "30 years" combined with the fact that long lifetimes will still be circumspect with respect to spending.
This time I wanted to shake up the "PWR + random life" thing a little more. This was spurred by reading Huang et al 2011 which presented a way of randomizing force of mortality via diffusion process. Just out of curiosity I wanted to see what happened in a PWR context when doing that. (recall that PWR is the consumption rate that with perfect foreknowledge of returns would allow one to spend to zero over a given horizon.). After messing around with this a little too much for some very minor new info I had these various ways of shaking things up:
1. No shakeup, just PWR and 30 year horizon
2. PWR with a simple parameterized Gompertz derived distribution.
3. PWR with parameter uncertainty
4. PWR with parameter uncertainty - bias to the dispersion param
5. PWR with parameter uncertainty - bias to the mode param
6. PWR with a faked pseudo-chaotic nudge
7. PWR with stochastic force of mortality - lower sigma
8. PWR with stochastic force of mortality - higher sigma
The important role of uncertainty around expected lifetime in making plans for retirement consumption has been around for a while and has seemed, based on my n=1 read of the literature, to be something found more often in economics papers than in finance though I think that is changing. The pivotal role of this complicator became most manifest in Yaari (Uncertain Lifetime, Life Insurance and the Theory of the Consumer, 1965) which focused on optimal consumption and annuitization given random life among other things.
"Wealth / Life Expectancy" as it Relates to Consumption Choice
Subsequent papers over the years (e.g., Merton 1969, Samuelson 1969, Milevsky & Huang 2010, Irlam & Tomlinson 2014 among others) have made various heuristic conclusions that, due to the role of longevity uncertainty, a proxy for optimal consumption can sometimes devolve to "wealth divided by a measure of remaining life expectancy."
"Their [Merton and Samuelson] work also addressed consumption over the life cycle, and they were able to show that, under one particular set of reasonable assumptions[note 1 ... pay attention to the particularity], optimal spending each year could be approximately determined as current wealth divided by remaining life expectancy. Milevsky and Huang [2010] later demonstrated a similar result." [Irlam and Tomlinson, italics added]
My fin-bro David chastises me for sandbagging in my blog but in this post you must take everything after this sentence with a big fat grain of salt. I have no idea if what I am doing here is either correct or legit and the fact that it looks like it works in the end is pretty much dispositive of zip. Plus my notation is sketchy.
Two years ago I tried to do this thing where I inferred optimal spend rates by working backwards from the end of a 30 year interval and used the results of a value function in the last period to help figure out what to spend in the period just prior and then on to the beginning. This is called backward induction and stochastic dynamic programming and is often described as Bellman equations if I have it right.
Sorta worked then. I went back a week ago to look at my code and I noticed two things: 1) I had no idea what the code was doing, and 2) there were some coding errors. Those two things, combined with incipient dementia (kidding), caused me to decide to exercise my brain and try this again. Silly me. There is no real purpose other than do it and what I noticed in the re-try is that it's a lot of work for results one can get from easier and simpler methods: the juice is probably not worth the squeeze.
I've been through this paper -- Uncertain Lifetime, Life Insurance, and the Theory of the Consumer, Menahem E. Yaari 1965 -- maybe 10 times now. I've done this because it was a seminal paper and and it frames the lifecycle dynamics of the retirement problem well in quantitative terms and it is a good read with some sly comments here and there. Most decent finance papers that go beyond simple finance and into econ and actuarial topics either do or should reference it. Here are some stray thoughts on my 10th (or so) read.
< with some corrections 2/8 >