In the last post on allocation optimization I was trying to home in on a reasonable asset allocation that might be "best" given my age, endowment, risk sensibility and, importantly, the objectives against which I would judge any optimality or near optimality, objectives that are quite a bit different for a retiree than they would be for a young person or an institution where risk-return (mean variance) is the gold standard and long time frames are the norm. Using a combination of a simulator, with all the implied flaws of simulators that comes with it, and the hopefully strong economic insight provided by using backward induction (BI) and dynamic programming to suggest optimal allocations, I figured out that something around a 40% risk allocation using some combination of a static or dynamic allocation (using 40% as a base in the dynamic case) probably made a lot of sense for the reasons I described before.
Now I want to see if I can push it a little further. In writing up an addendum to the prior post it dawned on me that when I happened to simplify the BI map of suggested allocations -- I was looking at spending shocks in the addendum at that point -- into a simple binary of 100% stocks when the map said 100 and 70% stocks when the map said 70 (year of plan and portfolio size are the keys to the map) I happened to end up with a strategy that dominated anything else I was looking at (for that shock-scenario only).
So, in this post I want to see, for my non-spending-shock example, if there is a very simple binary pair of BI dynamic allocation values that will either: a) change the optimal zone I was interested in last time or b) enhance the retirement outcomes against which I am optimizing. Basically the intuition I had was that if in the shock example there was a pair that dominated other strategies, then maybe there was as well for the non-shock example. I thought there might be two portfolios between which I could toggle while still using the insights from BI: one portfolio would be a "risk-on" version that was more heavy on stocks than bonds, say 70/30, when the BI table says that I can take or expand risk. The other would be a "risk-off " or safety portfolio and it would be bond "heavy," say 40/60 stock/bond or more. The now-simplified BI table would tell the simulator which allocation pair would be used for any given sim-year. Whether by doing this simplification I have strayed too far from economic theory or not is not a question I can answer or probably even want to answer. That's the nature of games and jokes: kills the fun to ask too many questions.
To do this mini-project, I did what I did before and I took the raw BI table recommendations and simplified the allocations into "bands." In the last post I had three bands; now I'll have two. To clarify what I'm doing here, let me illustrate with a simple table. The table below is what comes out of the dynamic programming code (hours of waiting!) and is specific to the plan horizon under consideration (30 years in this case), expected future wealth range (depends on starting portfolio and where it might go), and spending plan projected and inflated into the 30 year future (my veru idiosyncratic plan here). This made up data is an example of the BI output:
The raw optimal allocation recommendations are in column 3 above. Column 4 is created by deciding which value in column three goes to what "band" in 4. Right now there is no science to it; it's a little arbitrary except maybe in the process of testing out of what works. In this example, everything .70 and above goes to .70 otherwise to .30. The same is done for 40 and so on.
The simulator will then use the revised table and if the year and wealth level in the sim match, in any sim year, the wealth and year in the table then that is the allocation to the risk asset used for that sim year. My job here at this point was to try to systematically test and select combinations of two bands (and the thresholds used for the bands) that might plausibly move the needle on my optimization attempt.
First, before we recap the results from last time let's clarify what is happening here because it may not be totally obvious. For a "fixed allocation" simulation I do trial and error on different allocations one by one to find the one that: a) has the best success rate, and b) also minimizes the length of the simulated fail (the years between portfolio depletion and death). I then put those results on the y-axis for that allocation level on the x-axis. For the sims where I use the BI table and attempt to dynamically allocate, it's very different. I use a table like the one above and if the bands I have in the table are 70(30)/40(60) [1] for example, then I put the results of that run on the y-axis for the 40% allocation because we are testing 70/40 but in reality the simulator in any given sim year could be using either 70/30 or 40/60 based on the BI map and where we are in a sim-life.[2] Then I move on to 70/50 and do the same with the results for "50" put to the y axis for that 50 on the x-axis. It's just a way of doing a comparative chart with dynamic data that's hard to pin down. Ugly but necessary. Perhaps you have a better way...
From last time:
- For a fixed 40-60% allocation the fail rate was 7.5% and the median duration of fail was 8 years.
- Using the raw BI table without any modifications and allocate dynamically, the fail rate was 19.7% and the med fail duration was 5 years across its many possible allocations.
- Using my "wing-it" three band simplification of the prior point, and when using 40% for the most conservative band, the fail rate was 7.4% and median fail duration was 5 years.
- I recall that we decided that that last strategy, doing a dynamic, banded allocation, with a 40% conservative band, was probably best of all, all else being equal.
Now here in this post we are going to look at simple binary bands or simple pairs of allocations. I'll spare you the grind necessary to go through this stuff in a reasonably methodical way. If you'll trust me I'll just say that an "upper" band of 70(30) and a lower band of 40(60) looked like the center of the universe. (If you want to snigger at the methods, feel free and remember that we can mock the non-science if not burn it to the ground, I am still just playing a game here). Most all of the combos I looked at were worse in some way, many were similar. The closer one gets to 70/40, though, the more there is a visible difference in the spread between strategies. This is what it looks like after zooming in on the 30-70% allocation range for 4 different strategies:
Black lines - original fixed-allocation simulation w/out using BI table
Red - uses dynamic BI table with three bands: 100, 70, and xx
Green - uses BI table with two bands: 100 and xx and a threshold of .7
Dotted - uses BI table with two bands: 70 and xx and a threshold of .7
Fail rates are on left Y axis
Median duration of fail in years is on the right axis
Observations:
- 40% still looks like the the zone where we can look for some optimal results.
- There are what looks like some decent material improvements to fail rates by using a binary-but-dynamic approach. If you believe the programming (I'm still thinking about it...) and ignore real statistics (I sometimes do) that is about a 11% reduction in fail rates or abt 80bps.
- The median fail duration improves, too vs. the base case but same qualifier as before means that I might not read too much into the differences in the median fail duration lines...except: a) the dynamic allocation seems effective in general, and b) watch out for green. Green means there is the potential for a 100% equity allocation. That 100% can mean getting out of some scrapes every once in a while when it works in your favor, like a lottery ticket, but it can also sometimes mean digging a bigger hole when it doesn't work.
- I'd be a little circumspect, if I were you, about making any conclusions based on this data or analysis -- not least that, even if it were to be correct, it only applies to me.
- It all makes sense to me in a way, though. Risk assets are a two edge sword especially if you are in or are headed into risk. See the previous comment about digging holes. If risk really comes home to roost, equities sometimes help but can also sometimes do you no favors especially if you spend heavily into the risk. You'll probably fail with either bonds or stocks at that point...it's may just a question of how bad and for how long.
- The weirdness of the pure BI results (my version anyway) was that it implied that I should go straight to all bonds now and all equities in 15-20 years or so if I'm in the right wealth zone. That is a big change in plan. The binary 70(30)/40(60), on the other hand, makes sense to me since that plan was already on the table even without this analysis. It means to me: stay moderately conservative now early in retirement when the concept of portfolio erosion either really is risky or just feels more risky as compared to later. It also says: adapt and expand risk (moderately) later as wealth warrants and/or time gets shorter. Then, if wealth heads down again into a "BI caution zone" then it's back to risk-off until things change. I can do that; it makes sense. And it's implementable now because I don't have to change or do anything.
- Pure BI was pretty much a non-starter for me anyway. The fail rates were high and the median duration of fail is too abstract when I know that I can course-correct with enough advance warning. The execution would have been complex and sometimes dramatic. Allocations could be unstable and be jumping around continuously on the way up or down. It might be a jarring and bumpy ride.
- All of this, frankly, sounds a little like a utility function I know like DRRA or declining relative risk aversion. That's the one where risk aversion goes down with increasing wealth and/or shortening time. Seems like an awful lot of retirees might fit that bill. Cringe now and be tight fisted early in retirement; loosen up and spend like mad (e.g., bequests) the older you get (while actually probably spending less on yourself. At least the aversion goes down...)
- The binary-dynamic-allocation-thing I am doing, whether or not it happens to be economically legit which it might not be, looks, on paper at least, like it might help optimize towards the outcomes I need which was the whole point in the first place. I mean who doesn't like their fail rate estimates to go down without having to lift a finger (today).
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[1] it doesn't need to add to 100, btw. each number is the risk asset allocation in the risk-on and risk-off portfolios. So if I am testing 70/40 then when times are great (according to the BI map) then I use a 70/30 allocation and when times are not so great I use a 40/60 allocation...if that makes sense.
[2] Here is the original BI map I produced but now recast as a 70-30/40-60 map rather than anything between 100-0 and 0-100. Its binary in that for any pair of year+wealth-level there are only two allocation choices: risk-on portfolio and risk-off, yellow and blue. Reductive but it seems to work.
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References:
- Portfolio Size Matters, Gordan Irlam, Journal of Personal Finance Vol 13, iss. 2.
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