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The Chart
Benchmark (black line) - The black line is my RH40 spend rule which is an adaptation of a rule by Evan Inglis called Divide-by-20. He said you can maybe divide age by 20 at 65 and by 10 at 90 so I regressed this and got a formula for RH40-spend rate = age/(40-age/3). This coincidentally matches the line -- over the interval 60-80 -- that one would get solving the the Kolmogorov equation for a lifetime ruin probability = .05 where the mode of the longevity distribution is set to 88 and the dispersion to 9. So, RH40 is a proxy for age-related spend policy that I am comfortable is conservative but not too far off the mark. For now. Plus it is easy to calculate and deploy. In the last post you can see how it aligns with the more complex model.
Data (black dots) - The dots are the machine-recommended spend rates at each age for $1M endowment at that age. The variance comes from a variety of sources like: a) inadequate amount of training, b) outliers from too-short of iterations in the inner mini-sims (sampling errors), c) model not caring about splitting hairs when close to the optimal solution, d) other TBD. The early runs give some sense of the tendency, though.
Regression Line (blue line) - This is a set of R functions that fit and predict a poly line given the data. I am using it to visualize what the data might be like if more smooth.
Discussion from Last Post
In the last post, I had roughly the same chart and commented on the following:
a) it trained itself up again at later ages which is good,
b) it has vaguely similar convexity to the benchmark, again good,
c) it is playing a similar game in response to higher risk aversion, and
d) there is a big divergence at earlier ages from the benchmark
That last item was bugging me.
Discussion from this post
This is still a relatively un-evolved training set-up, but a couple things jump out as I think:
1. Risk aversion of "2" has, in the past, been relatively consistent with how I seem to behave myself but that is subjective,
2. I use the RH40 guideline (plus a half point or so) myself in early retirement,
3. My personal spend rate target right now is ~ 3.5 which makes this chart more interesting,
4. In general, at later ages, I'm now comfortable that the machine seems to know what it's doing,
5. I still am wondering about the divergence at the earlier ages...
Some Thoughts on the Divergence in #5 above
First of all, the benchmarks, whether formal Kolmogorov or my more casual RH40, or others, do not really have any idea of things like spending floors or long term entities like perpetuities (or consumption utility). My model assumes, for example, because I trust Ed Thorpe, that ~2% spend is roughly the bottom spend rate for perpetuities, something I confirmed in my own simulations (depending on some assumptions). So I bake that in as a floor-spend in my model. That's different from the benchmark methods.
Then, because I assume that no one really consumes at zero - institutions of some kind like family, government or associations, or even friends step in - I also assume that there is a hard bottom floor consumption of (arbitrary) 1K when wealth depletes. The utility math sucks if I don't do that. The other models don't care.
So, that means that my software is a bit at odds with the benchmarks for early retirements where one might expect lower levels of consumption (or lower levels of consumption, alternatively, when wealth runs out at later ages). Let me put it this way. Let's say that at 5 years old one were to embark on a career at lemonade stands, and by age 10 one has amassed enough capital to retire ;-) The RH40 rule says spend .27% which seems a little off ... which is because the math is only tuned to a 60-90 interval and not to "age ten to 1000" or other long retirements. Likewise, the Kolmogorov model says spend 1.56% for whatever reasons it has. That seems off, too.
BUT, common sense says that those hyper low spends are likely wrong. Ed Thorpe, in his last book, says 2% spend is a theoretical "forever" lower bound. I agree, except for some quibbles about residual risk. Also, in an early retirement before 60 -- let's say age 10, just to be glib -- we are looking at something that looks more like a long duration trust or an endowment, which it is. At that point the benchmarks I've used don't apply as well as they do for traditional retirement. Rather, as in the case of endowments, we are talking now about maintaining the fecundity of a portfolio over the really, really long haul -- as opposed to retirement models where mortality and shorter horizons come into play.
At this "fecundity point" desideratum, desired because we want our money last to infinity, if we were to make a bunch of limiting assumptions[1], the "threshold spend" is going to be, if it's not at least the theoretical 2%, a lot closer to the geometric mean return that one might expect for the portfolio over long-haul multi-period time[1]. If, for example, we look at a 4% real return and 12% std dev, then the long term "expectation" for those (stationary) returns might be, at infinity: 3.28%. For a 90 year old with a 10 year horizon it's perhaps closer to 3.35%.[2] In other words, we could spend for a very long time at more than 2%: something closer to low to mid threes, but probably not much more than that if we are trying to spend forever or if we doubt stationarity of returns, vol, spending, etc... and not much less if we value consumption utility over, well, over nothing.
This is my guess for the reason for the divergence of the blue and black lines. A lower bound for spending, given some limiting assumptions, is not going to be zero, it's going to be 2 or 3% or so (while wealth remains). And so that means that a wise model will bend like the blue line above as spending recommendations get lower and lower but also approach a limit somewhere in the low-to-mid 3% range. They get higher, of course, when longevity expectations come in at those later ages. Or at least they should in the absence of annuitized income. But we knew that. Smart machine, I guess.
The graph no longer bugs me.
But all this is just a thought.
------Notes-----------------------------------
[1] Assumptions like no taxes or fees, plus the idea that stochastic returns are stationary, as is vol. Spend variance doesn't exist here either even though we know it does.
[2] I'm thinking of R Michaud's formula for N-period (less than infinity) geo return approximator:
E[G] = mu - [(1-1/N)*sigma^2]/2
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