I was contemplating, the other day, a project that would use backward induction and stochastic dynamic programming in order to visualize approximate optimal spend rates at different ages. I wanted to try something like this because the form of analysis is powerful and efficient, the method is economically rigorous, consumption is a big deal in a multi-period retirement decumulation setting, and I had done it once before, on asset allocation at that time, in a post from January 2017, though that now seems like a million years ago. I paused a long while on this idea for several reasons:
1. This, surprisingly to me, is becoming less and less trivial but my aging, decaying eyes are starting to limit what I can and want to do in terms of chasing down the financial economics of retirement. My ability to sit and stare at a screen of code for hours or days or weeks -- which is what it would take for this idea -- is much more constrained than it was even 5 years ago. These are the type of wages paid for entering late middle age.
2. I could not figure out how to make the proper focus on spend rates. The lowest spend rate will always have the highest probability of success which will therefore always make self-denial the dominant strategy for success, something that is both common sense and ridiculous at the same time, and, not unrelatedly,
3. If using utility to evaluate spend rates or lifetime consumption -- so that they do not tend towards self denial -- makes more sense, then I have no idea how to chain things backward. In my prior attempt it was relatively trivial (not really, technically, but conceptually) to chain probabilities backward. Chaining utility makes no sense.
Stepping into the Project
But I started down the path anyway. To begin I re-read an article by Gordon Irlam that lays out the method[1] of backward induction for asset allocation choices (and levels of wealth) in order to refresh my memory on how he did it and how I tried to do it last year. I banged my head against this article at least 5 times over the last month until it dawned on me (ahhh, now I get it) that, like wearing ruby slippers, he had already given me the answer. All I had to do was see it.
What I realized was that Mr. Irlam had, conveniently for me, done his analysis in RPS (relative portfolio size) or what I've called wealth units. RPS is basically the idea that a 4% spend rate consumes 1 of 25 wealth units per unit of time so from an RPS of 25 we can impute a 4% spend rate. Since he did his backward induction in RPS terms we can impute spend rates on his optimization map, or at least the one for a fixed lifetime. It's not perfect for what I wanted to do but it solves at least issue #1 above.
A Mini Primer on Backward Induction
This has been explained elsewhere better by others but let's do a short refresher to set this up. Backward induction is basically the way we decide when we should leave for the airport. We work backwards from when we want to get there and then factor in time, distance and some contingencies. The same here. We start at the last year of allocation and spend choices (in a multi-period world) and, for each level of wealth (RPS and hence spend rates) and for each allocation we do a single time-step simulation to choose the best allocation at that level. Then we go back one year and repeat the time-step sim, use the end state RPS to look up what we did in the previous step for each RPS, combine the probabilities and choose the allocation with the highest joint probability. This is repeated back to time zero and provides an economically rigorous way to view portfolio choice over time. And, as Mr Irlam points out, it is easier than working forwards where 101 allocation choices over 70 years (30, say, if we are just looking at decumulation) creates 101^70 combinations...which is hard, or at least time consuming.
Extracting Spend Rates From Someone Else's Backward Induction Project
I wanted to do a similar backward induction thing for spending but given the constraints I mentioned above I realized I would have a hard time if it were even to be possible. But I also realized, via my epiphany, that I could probably at least extract something useful from the BI work already completed by Mr. Irlam. For example, the following figure is Figure 1 from the paper. It is not in color because I did not get a color copy printed and I no longer have access to the paper online (so I am not sure I am allowed to do this but we'll proceed with caution here):
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| Figure 1 from Gordon Irlam's "Portfolio Size Matters" 2014 |
This figure is a representation of success rates at different ages given the optimal asset allocation coming out of the BI, fixed spend rates, and fixed longevity. Those last two are not particularly happy assumptions but we work with what we have. The "given optimal asset allocation" statement is a really big "given" and people that know this kind of stuff should be suspiciously wary of me trying to use something like that at this point in the post. But it's the key point here since it is the nexus to the induction process in the paper. That's as close as I can get.
To extract the highest probability "optimal" spend rates from this, we'll focus on age 65 and later so that we are in decumulation mode given Irlam's assumptions and constraints. Then we'll look at the region of the map, at each age level, where the success rates stabilize in RPS terms (to the north of the pyramid where the color stops changing). At that point we'll eyeball (this is pretty sketchy don't you think) the RPS on the Y axis and then invert it to get the maximum spend rate (min RPS) at which (or below) we have the highest success rates in Irlam's backward induction project. Don't forget that this is "given" optimal asset allocations that vary by age and RPS, a given on which this post depends and that is not defined or explained. When we do this -- again, this is by eye not by math or programming -- we get something roughly like the following (rough but close enough for me today) over the interval 65-80 which is my interval of interest:
65 - 3.7% or lower
70 - 4.2% or lower
75 - 4.8% or lower
80 - 5.6% or lower
Age 60 interests me personally, too, but I can't get it off the map without extrapolating. The result of extrapolating is consistent, though, if I were to do it, with other work I've done that is described in the next paragraph.
Comparing the Results Here to Other Work Done Using Utility Math and Forward Simulation
In a recent post I used forward simulation and a value function for the expected discounted utility of lifetime consumption to evaluate optimal spend rates for different ages across different allocations to risk. In that post, for a risk aversion level or coefficient of 3, this is what I came up with for optimal spend rates by age[2]. These are in ranges to reflect the different outcomes for different allocations with the high values of the ranges being the optima:
65: 3.5 - 4.00%
70: 4.0 - 4.50%
75: 4.5 - 5.00 %
80: 5.0 - 5.75%
In general, it looks, to my untrained and age-decayed eye, that these two approaches are speaking the same language if not playing the same game or at least they are if you believe in utility and risk aversion.
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[1] Irlam, G. (2014). Portfolio Size Matters. The Journal of Personal Finance. 13(2), 9-16. Note that his single time step simulations use historical data rather than computer generated distributions.
[2] this forward utility sim was based on unrealistic and illustrative-only assumptions and parameters, and there was no attempt to optimize using lifetime income, but it'll have to stand in here today as a placeholder until some other day when I try a little harder. Also, I am cherry-picking the risk aversion coefficient of 3 since it makes the comparison work. That choice might perhaps be called marketing or self-promotion rather than data science. Still think it's interesting, though.
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